Financial Econometrics Lecture Notes
Professor Doron Avramov The Hebrew University of Jerusalem & Chinese University of Hong Kong Introduction: Why do we need a course in Financial Econometrics?
2 Professor Doron Avramov, Financial Econometrics Syllabus: Motivation
The past few decades have been characterized by an extraordinary growth in the use of quantitative methods in the analysis of various asset classes; be it equities, fixed income securities, commodities, and derivatives.
In addition, financial economists have routinely been using advanced mathematical, statistical, and econometric techniques in a host of applications including investment decisions, risk management, volatility modeling, interest rate modeling, and the list goes on.
3 Professor Doron Avramov, Financial Econometrics Syllabus: Objectives
This course attempts to provide a fairly deep understanding of such techniques.
The purpose is twofold, to provide research tools in financial economics and comprehend investment designs employed by practitioners.
The course is intended for advanced master and PhD level students in finance and economics.
4 Professor Doron Avramov, Financial Econometrics Syllabus: Prerequisite
I will assume prior exposure to matrix algebra, distribution theory, Ordinary Least Squares, Maximum Likelihood Estimation, Method of Moments, and the Delta Method.
I will also assume you have some skills in computer programing beyond Excel. MATLAB and R are the most recommended for this course. OCTAVE could be used as well, as it is a free software, and is practically identical to MATLAB when considering the scope of the course. If you desire to use STATA, SAS, or other comparable tools, please consult with the TA.
5 Professor Doron Avramov, Financial Econometrics Syllabus: Grade Components
Assignments (36%): there will be two problem sets during the term. You can form study groups to prepare the assignments - up to three students per group. The assignments aim to implement key concepts studied in class.
Class Participation (14%) - Attending all sessions is mandatory for getting credit for this course.
Final Exam (50%): based on class material, handouts, assignments, and readings.
6 Professor Doron Avramov, Financial Econometrics Syllabus: Topics to be Covered - #1 Overview: Matrix algebra Regression analysis Law of iterated expectations Variance decomposition Taylor approximation Distribution theory Hypothesis testing OLS MLE
7 Professor Doron Avramov, Financial Econometrics Syllabus: Topics to be Covered - #2
Time-series tests of asset pricing models
The mathematics of the mean-variance frontier
Estimating expected asset returns
Estimating the covariance matrix of asset returns
Forming mean variance efficient portfolio, the Global Minimum Volatility Portfolio, and the minimum Tracking Error Volatility Portfolio.
8 Professor Doron Avramov, Financial Econometrics Syllabus: Topics to be Covered - #3
The Sharpe ratio: estimation and distribution
The Delta method
The Black-Litterman approach for estimating expected returns.
Principal component analysis.
9 Professor Doron Avramov, Financial Econometrics Syllabus: Topics to be Covered - #4
Risk management and downside risk measures: value at risk, shortfall probability, expected shortfall (also known as C-VaR), target semi-variance, downside beta, and drawdown.
Option pricing: testing the validity of the B&S formula
Model verification based on failure rates
10 Professor Doron Avramov, Financial Econometrics Syllabus: Topics to be Covered - #5
Predicting asset returns using time series regressions
The econometrics of back-testing
Understanding time varying volatility models including ARCH, GARCH, EGARCH, stochastic volatility, implied volatility (VIX), and realized volatility
11 Professor Doron Avramov, Financial Econometrics Session #1 – Overview
12 Professor Doron Avramov, Financial Econometrics Let us Start
This session is mostly an overview. Major contents: Why do we need a course in financial econometrics? Normal, Bivariate normal, and multivariate normal densities The Chi-squared, F, and Student t distributions Regression analysis Basic rules and operations applied to matrices Iterated expectations and variance decomposition
13 Professor Doron Avramov, Financial Econometrics Financial Econometrics
In previous courses in finance and economics you had mastered the concept of the efficient frontier.
A portfolio lying on the frontier is the highest expected return portfolio for a given volatility target.
Or it is the lowest volatility portfolio for a given expected return target.
14 Professor Doron Avramov, Financial Econometrics Plotting the Efficient Frontier
15 Professor Doron Avramov, Financial Econometrics However, how could you Practically Form an Efficient Portfolio? Problem: there are far TOO many parameters to estimate. For instance, investing in ten assets requires: 2 휇1 휎1 ...... 휎1,10 휇 2 2 휎1,2, 휎2 ...... 휇 2 10 휎1,10...... 휎10 which is about ten estimates for expected return, ten for volatility, and 45 for co-variances/correlations. Overall, 65 estimates are required. That is a lot given the limited amount of data.
16 Professor Doron Avramov, Financial Econometrics More generally, if there are N investable assets, you need: N estimates for means, N estimates for volatilities, 0.5N(N-1) estimates for correlations. Overall: 2N+0.5N(N-1) estimates are required! Mean, volatility, and correlation estimates are noisy as reflected through their standard errors. Beyond such parameter uncertainty there is also model uncertainty. It is about the uncertainty about the correct model underlying the evolution of expected returns, volatilities, and correlations.
17 Professor Doron Avramov, Financial Econometrics Sample Mean and Volatility
The volatility estimate is typically less noisy than the mean estimate (more later).
Consider T asset return observations: 푅1, 푅2, 푅3, … , 푅푇
When returns are IID, the mean and volatility are estimated as σ푇 푅 σ푇 푅 − 푅ത 2 푅ത = 푡=1 푡 Less Noise ⟶ 휎ො = 푡=1 푡 푇 푇 − 1
18 Professor Doron Avramov, Financial Econometrics Estimation Methods
One of the ideas here is to introduce various methods in which to estimate the comprehensive set of parameters.
We will discuss asset pricing models and the Black-Litterman approach for estimating expected returns.
We will further introduce several methods for estimating the large-scale covariance matrix of asset returns.
19 Professor Doron Avramov, Financial Econometrics Mean-Variance vs. Down Side Risk
We will comprehensively cover topics in mean variance analysis.
We will also depart from the mean variance paradigm and consider down side risk measures to form as well as evaluate investment strategies.
Why should one resort to down side risk measures?
20 Professor Doron Avramov, Financial Econometrics Down Side Risk
For one, investors typically assign higher weight to the downside risk of investments than to upside potential.
The practice of risk management as well as regulations of financial institutions are typically about downside risk – such as VaR, shortfall probability, and expected shortfall.
Moreover, there is a major weakness embedded in the mean variance paradigm.
21 Professor Doron Avramov, Financial Econometrics Drawback in the Mean-Variance Setup
To illustrate, consider two risky assets (be it stocks) A and B.
There are five states of nature in the economy.
Returns in the various states are described on the next page.
22 Professor Doron Avramov, Financial Econometrics Drawback in Mean-Variance
S1 S2 S3 S4 S5 A 5.00% 8.25% 15.00% 10.50% 20.05% B 3.05% 2.90% 2.95% 3.10% 3.01%
Stock A dominates stock B in every state of nature.
Nevertheless, a mean-variance investor may consider stock B because it may reduce the portfolio’s volatility.
Investment criteria based on down size risk measures could get around this weakness.
23 Professor Doron Avramov, Financial Econometrics The Normal Distribution In various applications in finance and economics, a common assumption is that quantities of interests, such as asset returns, economic growth, dividend growth, interest rates, etc., are normally (or log-normally) distributed.
The normality assumption is primarily done for analytical tractability.
The normal distribution is symmetric.
It is characterized by the mean and the variance.
24 Professor Doron Avramov, Financial Econometrics Dispersion around the Mean
Assuming that 푥 is a zero mean random variable. As 푥~푁(0, 휎2) the distribution takes the form:
When 휎 is small (big) the distribution is concentrated (dispersed)
25 Professor Doron Avramov, Financial Econometrics Probability Distribution Function
The Probability Density Function (pdf) of the normal distribution for a random variable r takes the form 1 1 푟 − 휇 2 푝푑푓 푟 = 푒푥푝 − 2휋휎2 2 휎2
1 Note that 푝푑푓 휇 = , and further 2휋휎2 1 if 휎 = 1, then 푝푑푓 휇 = 2휋 The Cumulative Density Function (CDF) is the integral of the pdf, e. g. , 푐푑푓 휇 = 0.5.
26 Professor Doron Avramov, Financial Econometrics Probability Integral Transform
Assume that x is normally distributed – what is the distribution of y=F(x), where F is a cumulative density Function?
Could one make a general statement here?
27 Professor Doron Avramov, Financial Econometrics Normally Distributed Return
Assume that the US excess rate of return on the market portfolio is normally distributed with annual expected return (equity premium) and volatility given by 8% and 20%, respectively.
That is to say that with a nontrivial probability the realized return can be negative. See figures on the next page.
The distribution around the sample mean return is also normal with the same expected return but smaller volatility.
28 Professor Doron Avramov, Financial Econometrics Confidence Level
Normality suggests that deviation of 2 SD away from the mean creates an 80% range (from -32% to 48%) for the realized return with approx. 95% confidence level (assuming 푟~푁(8%, (20%)2)
29 Professor Doron Avramov, Financial Econometrics Confidence Intervals for Annual Excess Return on the Market 푃푟표푏 0.08 − 0.2 < 푅 < 0.08 + 0.2 = 68% 푃푟표푏 0.08 − 2 × 0.2 < 푅 < 0.08 + 2 × 0.2 = 95% 푃푟표푏 0.08 − 3 × 0.2 < 푅 < 0.08 + 3 × 0.2 = 99%
The probability that the realization is negative 푅−0.08 0−0.08 푃푟표푏 푅 < 0 = 푃푟표푏 < = 푃푟표푏 푧 < −0.4 = 34.4% 0.2 0.2
30 Professor Doron Avramov, Financial Econometrics Higher Moments Skewness – the third moment is zero.
Kurtosis – the fourth moment is three times the variance.
Odd moments are all zero.
Even moments are (often complex) functions of the mean and the variance.
In the next slides, skewness and kurtosis are presented for other probability distribution functions.
31 Professor Doron Avramov, Financial Econometrics Skewness
The skewness can be negative (left tail) or positive (right tail).
32 Professor Doron Avramov, Financial Econometrics Kurtosis
Mesokurtic - A term used in a statistical context where the kurtosis of a distribution is similar, or identical, to the kurtosis of a normally distributed data set.
33 Professor Doron Avramov, Financial Econometrics The Essence of Skewness and Kurtosis
Positive skewness means nontrivial probability for large payoffs.
Kurtosis is a measure for how thick the distribution’s tails are. It can reflect the uncertainty about variation.
When is the skewness zero? In symmetric distributions.
34 Professor Doron Avramov, Financial Econometrics Bivariate Normal Distribution
Bivariate normal: 2 푥 휇푥 휎푥 , 휎푥푦 푦 ~푁 휇 , 2 푦 휎푥푦, 휎푦
The marginal densities of x and y are 2 푥~푁 휇푥, 휎푥 2 푦~푁 휇푦, 휎푦
What is the distribution of 푦 if 푥 is known?
35 Professor Doron Avramov, Financial Econometrics Conditional Distribution
The conditional distribution is still normal: 휎 휎2 푥푦 2 푥푦 푦 ∣ 푥~푁 휇푦 + 2 푥 − 휇푥 , 휎푦 − 2 휎푥 휎푥 푎푙푤푎푦푠 푛표푛−푛푒푔푎푡푖푣푒
If the correlation between 푥 and 푦 is positive and 푥 > 휇푥 then the conditional expectation is higher than the unconditional one.
36 Professor Doron Avramov, Financial Econometrics Conditional Moments If 푥 and 푦 are uncorrelated, then the conditional and unconditional expected return of 푦 are identical.
2 2 휎푥푦 That is, if 휎푥푦 = 0, then 휎푦∣푥 = 휎푦 − 2 = 휎푦 휎푥 meaning that the realization of 푥 does not say anything about the conditional distribution of 푦. . It should be noted that random variables can be uncorrelated yet dependent. . Dependence can be represented by a copula. . Under normality, zero correlation and independence coincide.
37 Professor Doron Avramov, Financial Econometrics Conditional Standard Deviation
Developing the conditional standard deviation further:
휎2 휎2 2 푥푦 2 푥푦 2 휎푦∣푥 = 휎푦 − 2 = 휎푦 1 − 2 2 = 휎푦 1 − 푅 휎푥 휎푥 ∙ 휎푦
When goodness of fit is higher the conditional standard deviation is lower. That makes a great sense: the realization of x gives substantial information about y.
38 Professor Doron Avramov, Financial Econometrics Multivariate Normal
푥1 푥2 푋 = . ~푁 휇 , σ 푚×1 . 푚×1 푚×푚 푥푚
Define Z=AX+B.
39 Professor Doron Avramov, Financial Econometrics Multivariate Normal
′ 푍 = 퐴푛×푚 ∙ 푋푚×1 + 퐵푛×1 ∼ 푁 퐴푛×푚휇푚×1 + 퐵푛×1, 퐴푛×푚σ푚×푚퐴푚×푛
Let us now make some transformations to end up with 푁(0, 퐼):
′ 푍 − 퐴푛×푚휇푚×1 + 퐵푛×1 ∼ 푁 0, 퐴푛×푚σ푚×푚퐴푚×푛
1 ′ − 퐴푛×푚σ푚×푚퐴푚×푛 2 푍 − 퐴푛×푚휇푚×1 − 퐵푛×1 ∼ 푁 0, 퐼
40 Professor Doron Avramov, Financial Econometrics Multivariate Normal Consider an N-vector of stock returns which are normally distributed:
푅푁×1~푁 휇푁×1, Σ푁×푁 Then: 푅 − 휇~푁 0, Σ 1 Σ−2 푅 − 휇 ~푁 0, 퐼
1 1 − − σ 2 ∙ σ 2 = σ−1 1 − 1 1 What does σ 2 mean? A few rules: σ2 ∙ σ2 = σ 1 1 − σ 2 ∙ σ2 = 퐼
41 Professor Doron Avramov, Financial Econometrics The Chi-Squared Distribution
2 2 If 푋1~푁(0,1) then 푋1 ~휒 1
If 푋1~푁 0,1 , 푋2~푁 0,1 , and 푋1 ⊥ 푋2 then
2 2 2 푋1 + 푋2 ~휒 2
42 Professor Doron Avramov, Financial Econometrics More about the Chi-Squared
2 푋1~휒 푚 2 Moreover if ൞푋2~휒 푛 푋1 ⊥ 푋2
2 then 푋1 + 푋2~휒 푚 + 푛
43 Professor Doron Avramov, Financial Econometrics The F Distribution
Gibbons, Ross, and Shanken (GRS) designated a finite sample asset pricing test that has the F - Distribution. The GRS test is one of the most well known and heavily used in the filed of asset pricing. To understand the F distribution notice that
2 푋1~휒 푚 2 If ൞푋2~휒 푛 푋1 ⊥ 푋2 푋1 ൗ푚 then 퐴 = 푋2 ~퐹 푚, 푛 ൗ푛
44 Professor Doron Avramov, Financial Econometrics The t Distribution
Suppose that 푟1, 푟2, … , 푟푇 is a sample of length T of stock returns which are normally distributed.
The sample mean and variance are 푇 푟 + ⋯ + 푟 1 푟ҧ = 1 푇 and 푠2 = 푟 − 푟ҧ 2 푇 푇 − 1 푡 푡=1
The following statistic has the t distribution with T-1 d.o.f
푟ҧ − 휇 푡 = 푠 ൗ 푇 − 1
45 Professor Doron Avramov, Financial Econometrics The t Distribution
The pdf of student-t is given by: 푣+1 − 1 푥2 2 푓 푥, 푣 = ∙ 1 + 푣 ∙ 퐵 푣Τ2 , 1Τ2 푣
where 퐵 푎, 푏 is beta function and 푣 ≥ 1 is the number of degrees of freedom
46 Professor Doron Avramov, Financial Econometrics The Student’s t Distribution
휈 is the number of degrees of freedom. When 휈 = +∞ the t- distribution becomes normal dist.
47 Professor Doron Avramov, Financial Econometrics The t Distribution
The t-distribution is the sampling distribution of the t-value when the sample consist of independently and identically distributed observations from a normally distributed population. It is obtained by dividing a normally distributed random variable by a square root of a Chi-squared distributed random variable when both random variables are independent. Indeed, later we will show that when returns are normally distributed the sample mean and variance are independent.
48 Professor Doron Avramov, Financial Econometrics Regression Analysis
Various applications in corporate finance and asset pricing require the implementation of a regression analysis.
We will estimate regressions using matrix notation.
For instance, consider the time series predictive regression
푅푡 = 훼 + 훽1푍1,푡−1 + 훽2푍2,푡−1 + 휀푡 푡 = 1,2, … , 푇
49 Professor Doron Avramov, Financial Econometrics Rewriting the system in a matrix form
푅1 1, 푍1,0, 푍2,0 휀1 푅 1, 푍 , 푍 훼 휀 2 1,1 2,1 .2 푅 = . 푋 = . 훾 = 훽1 휀 = 푇×1 푇×3 3×1 푇×1 . . . 훽2 푅푇 1, 푍1,푇−1, 푍2,푇−1 휀푇 yields R= 푋훾 + 휀 We will derive regression coefficients and their standard errors using OLS, MLE, and Method of Moments.
50 Professor Doron Avramov, Financial Econometrics Vectors and Matrices: some Rules 훼11 훼12 A is a column vector: 퐴 = . 푡×1 . 훼1푡
The transpose of A is a row vector: 퐴′1×푡 = 훼11, 훼12, … , 훼1푡
The identity matrix satisfies 퐼 ∙ 퐵 = 퐵 ∙ 퐼 = 퐵
51 Professor Doron Avramov, Financial Econometrics Multiplication of Matrices
푎11,푎12 푏11,푏12 푎11,푎21 퐴2×2 = 퐵2×2 = 퐴′2×2 = 푎21,푎22 푏21,푏22 푎12,푎22 ′ ′ ′ 푎11푏11 + 푎12푏21,푎11푏12 + 푎12푏22 (퐴퐵) = 퐵 퐴 퐴2×2퐵2×2 = , −1 −1 −1 푎21푏11 + 푎22푏21,푎21푏12 + 푎22푏22 (퐴퐵) = 퐵 퐴 ∗
The inverse of the matrix 퐴 is 퐴−1 which satisfies: 1,0 퐴 퐴−1 = 퐼 = 2×2 2×2 0,1
* Both A and B are invertible
52 Professor Doron Avramov, Financial Econometrics Solving Large Scale Linear Equations
퐴푚×푚푋푚×1 = 푏푚×1 푋 = 퐴−1푏
Of course, A has to be a square invertible matrix. That is, the number of equations must be equal to the number of unknowns and further all equations must be linearly independent.
53 Professor Doron Avramov, Financial Econometrics Linear Independence and Norm
The vectors 푉1, … , 푉푁 are linearly independent if there does not exist scalars 푐1, … , 푐푁 such that 푐1푉1 + 푐2푉2 + ⋯ + 푐푁푉푁 = 0 unless 푐1 = 푐2 … = 푐푁 = 0. In the context of financial economics – we will consider N risky assets such that the payoff of each asset is not a linear combination of the other N-1 assets. Then the covariance matrix of asset returns is positive definite (see next page) of rank N and hence is invertible. The norm of a vector V is 푉 = 푉′푉
54 Professor Doron Avramov, Financial Econometrics A Positive Definite Matrix An 푁 × 푁 matrix σ is called positive definite if
푉′ ⋅ σ ⋅ > 0 1×푁 푁×푁 푁×1 for any nonzero vector V.
Such matrix is invertible and it has N distinct Eigen-values and Eigen-vectors.
A non-positive definite matrix cannot be inverted and its determinant is zero.
55 Professor Doron Avramov, Financial Econometrics The Trace of a Matrix
2 휎1 … … … … … 휎1,푁 휎 , 휎2 … … … … … Let σ = 1,2 2 = σ 1 , … , σ 푁 푁×푁 … … … … … … … … 푁×1 푁×1 2 휎푁,1, … … … … … 휎푁 Trace of a matrix is the sum of diagonal elements
2 2 2 푡푟 σ = 휎1 + 휎2 + ⋯ + 휎푁
56 Professor Doron Avramov, Financial Econometrics Matrix Vectorization
휎 1 This is the vectorization operator – 2 it works for both square as well as 푉푒푐 σ = 휎 푁2×1 ⋮ non square matrices. 휎 푁
57 Professor Doron Avramov, Financial Econometrics The VECH Operator
Similar to 푉푒푐 σ but takes only the upper triangular elements of σ. 1 휎1 2 휎2 푉푒푐ℎ σ = ⋮ 푁+1 푁 ×1 휎2푁 2 ⋮ 2 휎푁
58 Professor Doron Avramov, Financial Econometrics Partitioned Matrices
A, a square nonsingular matrix, is partitioned as 퐴11 , 퐴12 푚 ×푚 푚 ×푚 퐴 = 1 1 1 2 퐴21 , 퐴22 푚2×푚1 푚2×푚2 Then the inverse of A is given by −1 −1 퐴 − 퐴 퐴−1퐴 , − 퐴 − 퐴 퐴−1퐴 퐴 퐴−1 퐴−1 = 11 12 22 21 11 12 22 21 12 22 −1 −1 −1 −1 −1 −퐴22 퐴21 퐴11 − 퐴12퐴22 퐴21 , (퐴22−퐴21퐴11 퐴12)
−1 Confirm that 퐴 ⋅ 퐴 = 퐼 푚1+푚2
59 Professor Doron Avramov, Financial Econometrics Matrix Differentiation
Y is an M-vector. X is an N-vector. Then 휕푌 휕푌 휕푌 1 , 1 , … , 1 휕푌 휕푋1 휕푋2 휕푋푁 = ⋮ 휕푋 휕푌 휕푌 휕푌 푀×푁 푀 , 푀 , … , 푀 휕푋1 휕푋2 휕푋푁 And specifically, if 푌 = 퐴 푋 푀×1 푀×푁 푁×1 Then: 휕푌 = 퐴 휕푋
60 Professor Doron Avramov, Financial Econometrics Matrix Differentiation
Let 푍 = 푌′ 퐴 푋 1×푀 푀×푁 푁×1
휕푍 = 푌′퐴 휕푋
휕푍 = 푋′퐴′ 휕푌
61 Professor Doron Avramov, Financial Econometrics Matrix Differentiation
Let 휃 = 푋′ 퐶 푋 1×푁 푁×푁 푁×1 휕휃 = 푋′ 퐶 + 퐶′ 휕푋 If C is symmetric, then 휕휃 = 2푋′퐶 휕푋
62 Professor Doron Avramov, Financial Econometrics Kronecker Product
It is given by 퐶푛푝×푚푔 = 퐴푛×푚 ⊗ 퐵푝×푔
푎11퐵 … … … … … , 푎1푚퐵 퐶푛푝×푚푔 = … … … … … … … … … 푎푛1퐵 … … … … … , 푎푛푚퐵
63 Professor Doron Avramov, Financial Econometrics Kronecker Product
For square matrices A and B, the following equalities apply
퐴 ⊗ 퐵 −1 = 퐴−1 ⊗ 퐵−1
푀 푁 퐴푀×푀 ⊗ 퐵푁×푁 = 퐴 퐵
퐴 ⊗ 퐵 퐶 ⊗ 퐷 = 퐴퐶 ⊗ 퐵퐷
64 Professor Doron Avramov, Financial Econometrics Operations on Matrices: More Advanced
푡푟(퐴 ⊗ 퐵) = 푡푟(퐴)푡푟(퐵) 푣푒푐(퐴 + 퐵) = 푣푒푐(퐴) + 푣푒푐(퐵) 푣푒푐ℎ(퐴 + 퐵) = 푣푒푐ℎ(퐴) + 푣푒푐ℎ(퐵) 푡푟(퐴퐵) = 푣푒푐(퐵′)′푣푒푐(퐴) 퐸(푋′퐴푋) = 퐸 푡푟(푋′퐴푋) = 퐸 푡푟(푋푋′)퐴 = 푡푟 퐸(푋푋′)퐴 = 휇′퐴휇 + 푡푟(Σ퐴) where 휇 = 퐸 푋 and Σ = 퐸(푋 − 휇)(푋 − 휇)′
65 Professor Doron Avramov, Financial Econometrics Using Matrix Notation in a Portfolio Choice Context The expectation and the variance of a portfolio’s rate of return in the presence of three stocks are formulated as
′ 휇푝 = 휔1휇1 + 휔2휇2 + 휔3휇3 = 휔 휇
2 2 2 2 2 2 2 휎푝 = 휔1휎1 + 휔2휎2 + 휔3휎3
+2휔1휔2휎1휎2휌12 + 2휔1휔3휎1휎3휌13 + 2휔2휔3휎2휎3휌23 = 휔′σ휔
66 Professor Doron Avramov, Financial Econometrics Using Matrix Notation in a Portfolio Choice Context where 2 휇1 휔1 휎1 , 휎12, 휎13 2 휇3×1 = 휇2 휔3×1 = 휔2 σ = 휎21, 휎2 , 휎23 3×3 휇3 휔3 2 휎31, 휎32, 휎3
67 Professor Doron Avramov, Financial Econometrics The Expectation, Variance, and Covariance of Sum of Random Variables
퐸(푎푥 + 푏푦 + 푐푧) = 푎퐸(푥) + 푏퐸(푦) + 푐퐸(푧)
푉푎푟 푎푥 + 푏푦 + 푐푧 = 푎2푉푎푟 푥 + 푏2푉푎푟 푦 + 푐2푉푎푟 푧 +2푎푏퐶표푣(푥, 푦) + 2푎푐퐶표푣(푥, 푧) + 2푏푐퐶표푣(푦, 푧)
퐶표푣 푎푥 + 푏푦, 푐푧 + 푑푤 = 푎푐퐶표푣 푥, 푧 + 푎푑퐶표푣 푥, 푤 +푏푐퐶표푣(푦, 푧) + 푏푑퐶표푣(푦, 푤)
68 Professor Doron Avramov, Financial Econometrics Law of Iterated Expectations (LIE)
The LIE relates the unconditional expectation of a random variable to its conditional expectation via the formulation
퐸 푌 = 퐸푥 퐸 푌 ∣ 푋
Paul Samuelson shows the relation between the LIE and the notion of market efficiency – which loosely speaking asserts that the change in asset prices cannot be predicted using current information.
69 Professor Doron Avramov, Financial Econometrics LIE and Market Efficiency
Under rational expectations, the time t security price can be written as the rational expectation of some fundamental value, conditional on information available at time t: ∗ ∗ 푃푡 = 퐸 푃 |퐼푡 = 퐸푡[푃 ] ∗ ∗ Similarly, 푃푡+1 = 퐸 푃 |퐼푡+1 = 퐸푡+1[푃 ]
The conditional expectation of the price change is ∗ ∗ 퐸[푃푡+1 − 푃푡|퐼푡] = 퐸푡 퐸푡+1[푃 ] − 퐸푡[푃 ] = 0 The quantity does not depend on the information.
70 Professor Doron Avramov, Financial Econometrics Variance Decomposition (VD) 푉푎푟 푦 can be represented as the sum of two components:
푉푎푟 푦 = 푉푎푟푥 퐸 푦|푥 + 퐸푥 푉푎푟 푦|푥
Shiller (1981) documents excess volatility in the equity market. We can use VD to prove it:
퐷 퐷 The theoretical stock price: 푃∗ = 푡+1 + 푡+2 ⋯ 푡 1+푟 1+푟 2 based on actual future dividends
∗ The actual stock price: 푃푡 = 퐸 푃푡 ∣ 퐼푡
71 Professor Doron Avramov, Financial Econometrics Variance Decomposition
∗ ∗ ∗ 푉푎푟 푃푡 = 푉푎푟 퐸 푃푡 ∣ 퐼푡 + 퐸 푉푎푟 푃푡 ∣ 퐼푡 푝표푠푖푡푖푣푒
∗ 푉푎푟 푃푡 = 푉푎푟 푃푡 + 푝표푠푖푡푖푣푒 ↓ ∗ 푉푎푟 푃푡 > 푉푎푟 푃푡
By variance decomposition, the variance of the theoretical price must be higher than that of the actual price. Data, however, implies the exact opposite. Either the present value formula (with constant discount factors) is away off, or asset prices are subject to behavioral biases.
72 Professor Doron Avramov, Financial Econometrics Session 2(part a) - Taylor Approximations in Financial Economics
73 Professor Doron Avramov, Financial Econometrics TA in Finance
Several major applications in finance require the use of Taylor series approximation.
The following table describes three applications.
74 Professor Doron Avramov, Financial Econometrics TA in Finance: Major Applications
Expected Utility Bond Pricing Option Pricing
First Order Mean Duration Delta
Second Order Volatility Convexity Gamma
Third Order Skewness
Fourth Order Kurtosis
75 Professor Doron Avramov, Financial Econometrics Taylor Approximation Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
Taylor approximation is written as: 1 푓 푥 = 푓 푥 + 푓′ 푥 푥 − 푥 + 0 1! 0 0 1 1 + 푓′′ 푥 푥 − 푥 2 + 푓′′′ 푥 푥 − 푥 3 + ⋯ 2! 0 0 3! 0 0
It can also be written with σ notation: ∞ 푛 푓 (푥0) 푛 푓(푥) = 푥 − 푥0 푛=0 푛!
76 Professor Doron Avramov, Financial Econometrics Maximizing Expected Utility of Terminal Wealth
The invested wealth is 푊푇, the investment horizon is 퐾 periods. The terminal wealth, the wealth in the end of the investment horizon, is 푊푇+퐾 = 푊푇 1 + 푅 where 1 + 푅 = 1 + 푅푇+1 1 + 푅푇+2 … 1 + 푅푇+퐾 Applying “ln” on both sides of the equation: ln 1 + 푅 = ln 1 + 푅푇+1 1 + 푅푇+2 … 1 + 푅푇+퐾 = = ln 1 + 푅푇+1 + ln 1 + 푅푇+2 + ⋯ + ln 1 + 푅푇+퐾
77 Professor Doron Avramov, Financial Econometrics Transforming to Log Returns
Denote the log returns as: 푟푇+1 = ln ( 1 + 푅푇+1) 푟푇+2 = ln ( 1 + 푅푇+2) . . . . . 푟푇+퐾 = ln ( 1 + 푅푇+퐾) And: 푟 = ln ( 1 + 푅) ent horizon is:
2 + ⋯ + 푟푇+퐾 R is therefore:
(푇 exp ( 푟
78 Professor Doron Avramov, Financial Econometrics Power utility as a Function of Cumulative Log Return (CLR)
Assume that the investor has the power utility function (where 0 < 훾 < 1): 1 푢 푊 = 푊훾 푇+퐾 훾 푇+퐾 Then the utility function of the terminal wealth can be expressed as a function of CLR
1 1 1 푢 푊 = 푊훾 = 푊 exp ( 푟) 훾 = 푊 훾 exp ( 훾푟) 푇+퐾 훾 푇+푘 훾 푇 훾 푇
79 Professor Doron Avramov, Financial Econometrics Power utility as a Function of Cumulative Log Return
Note that 푟 is stochastic (unknown), since all future returns are unknown
We can use the Taylor Approximation to express the utility as a function of the moments of CLR.
80 Professor Doron Avramov, Financial Econometrics The Utility Function Approximation
푊훾 푊훾 푢(푊 ) = 푇 exp ( 휇훾) + 푇 exp ( 휇훾)(푟 − 휇)훾 + 푇+퐾 훾 훾 1 푊훾 1 푊훾 푇 exp ( 휇훾)(푟 − 휇)2훾2 + 푇 exp ( 휇훾)(푟 − 휇)3훾3 + 2 훾 6 훾 1 푊훾 푇 exp ( 휇훾)(푟 − 휇)4훾4 + ⋯ 24 훾
And the expected value of 푢(푊푇+퐾) is: 푊훾 1 퐸[푢(푊 )] ≈ 푇 exp ( 휇훾)[1 + 푉푎푟(푟)훾2 + 푇+퐾 훾 2 1 1 푆푘푒(푟)훾3 + 퐾푢푟(푟)훾4] 6 24
81 Professor Doron Avramov, Financial Econometrics Expected Utility
Let us assume that log return is normally distributed: 푟~푁 휇, 휎2 푆푘푒 = 0, 퐾푢푟 = 3휎4 Then: 푊훾 훾2 훾4 퐸 푢 푊 ≈ 푇 exp ( 훾휇) 1 + 휎2 + 휎4 푇+퐾 훾 2 8
The exact solution under normality is 푊훾 훾2 퐸 푢 푊 = 푇 exp 훾휇 + 휎2 푇+퐾 훾 2 Are approximated and exact solutions close enough?
82 Professor Doron Avramov, Financial Econometrics Taylor Approximation – Bond Pricing
Taylor approximation is also used in bond pricing.
푛 퐶퐹푖 The bond price is: 푃 푦0 = σ푖=1 푖 where 푦0 is the yield to 1+푦0 maturity.
Assume that 푦0 changes to 푦1
The delta (change) of the yield to maturity is written as: Δ푦 = 푦1 − 푦0
83 Professor Doron Avramov, Financial Econometrics Changes in Yields and Bond Pricing
Using Taylor approximation we get: 1 1 푃 푦 ≈ 푃 푦 + 푃′ 푦 푦 − 푦 + 푃′′ 푦 푦 − 푦 2 1 0 1! 0 1 0 2! 0 1 0
Therefore: 1 1 푃 푦 − 푃 푦 ≈ 푃′ 푦 푦 − 푦 + 푃′′ 푦 푦 − 푦 2 ≈ 1 0 1! 0 1 0 2! 0 1 0
′ 1 ′′ 2 ≈ 푃 푦0 ∆푦 + 2푃 푦0 ∙ ∆푦
84 Professor Doron Avramov, Financial Econometrics Duration and Convexity
Denote 푃 = 푃 푦0 . Dividing by 푃(푦0) yields
∆푃 푃′ 푦 ∙ ∆푦 푃′′ 푦 ∙ ∆푦 2 ≈ 0 + 0 푃 푃 푦0 2푃 푦0
푃′ 푦 Instead of: 0 we can write “-MD” (Modified Duration). 푃 푦0
푃′′ 푦 Instead of: 0 we can write “Con” (Convexity). 푃 푦0
85 Professor Doron Avramov, Financial Econometrics The Approximated Bond Price Change
It is given by
∆푃 퐶표푛 ∙ ∆푦2 ≈ −푀퐷 ∙ ∆푦 + 푃 2
What if the yield to maturity falls?
86 Professor Doron Avramov, Financial Econometrics The Bond Price Change when Yields Fall
The change of the bond price is: ∆푃 퐶표푛 ∙ ∆푦2 = −푀퐷 ∙ ∆푦 + 푃 2
According to duration - bond price should increase.
According to convexity - bond price should also increase.
The bond price clearly rises.
87 Professor Doron Avramov, Financial Econometrics The Bond Price Change when Yields Increase
And what if the yield to maturity increases?
Again, the change of the bond price is
∆푃 퐶표푛 ∙ ∆푦2 = −푀퐷 ∙ ∆푦 + 푃 2
According to duration – the bond price should decrease.
88 Professor Doron Avramov, Financial Econometrics The Bond Price Change when Yields Increase
According to convexity – the bond price should increase.
What is the overall effect?
The influence of duration is always stronger than that of convexity as the duration is a first order effect while the convexity is only second order.
So the bond price must fall.
89 Professor Doron Avramov, Financial Econometrics Option Pricing
A call price is a function of the underlying asset price.
What is the change in the call price when the underlying asset pricing changes? 휕퐶 1 휕2퐶 퐶(푃 ) ≈ 퐶(푃 ) + 푃 − 푃 + 푃 − 푃 2 푠 푡 휕푃 푠 푡 2 휕푃2 푠 푡 1 ≈ 퐶(푃 ) + Δ 푃 − 푃 + Γ 푃 − 푃 2 푡 푠 푡 2 푠 푡
Focusing on the first order term – this establishes the delta neutral trading strategy.
90 Professor Doron Avramov, Financial Econometrics Delta Neutral Strategy
Suppose that the underlying asset volatility increases. Suppose further that the implied volatility lags behind. The call option is then underpriced – buy the call. However, you take the risk of fluctuations in the price of the underlying asset. To hedge that risk you sell Delta units of the underlying asset. The same applies to trading strategies involving put options.
91 Professor Doron Avramov, Financial Econometrics Session 2(part b) - OLS, MLE, MOM
92 Professor Doron Avramov, Financial Econometrics Ordinary Least Squares (OLS) The goal is to estimate the regression parameters (least squares) ′ Consider the regression 푦푡 = 푥푡훽 + 휀푡 and assume homoskedasticity:
푦1 1, 푥11, … , 푥퐾1 휀1 푌 = . , 푋 = … … … … … , 푉 = . 푦푇 1, 푥1푇, … , 푥퐾푇 휀푇 Expressing in a matrix form: 푌 = 푋훽 + 푉 Or: V= 푌 − 푋훽 Define a function of the squares of the errors that we want to minimize:
푇 2 ′ ′ ′ ′ ′ ′ 푓 훽 = 휀푡 = 푉 푉 = 푌 − 푋훽 ′ 푌 − 푋훽 = 푌 푌 + 훽 푋 푋훽 − 2훽 푋 푌 푡=1
93 Professor Doron Avramov, Financial Econometrics OLS First Order Conditions
Let us differentiate the function with respect to beta and consider the first order condition:
휕푓 훽 = 2(푋′푋)훽 − 2푋′푌 = 0 휕훽 푋′푋 훽 = 푋′푌 ⇒ 훽መ = (푋′푋)−1푋′푌 Recall: X and Y are observations. ′ 푇 2 Could we choose other 훽 to obtain smaller 푉 푉 = σ푡=1 휀푡 ?
94 Professor Doron Avramov, Financial Econometrics The OLS Estimator
No as we minimize the quantity 푉′푉. We know: 푌 = 푋훽 + 푉 → 훽መ = (푋′푋)−1푋′(푋훽 + 푉) 훽መ = (푋′푋)−1푋′푋훽 + (푋′푋)−1푋′푉 훽መ = 훽 + (푋′푋)−1푋′푉 Is the estimator 훽መ unbiased for 훽 ?
퐸 훽መ − 훽 = 퐸 (푋′푋)−1푋′푉 = (푋′푋)−1푋′퐸 푉 = 0 , So – it is indeed unbiased.
95 Professor Doron Avramov, Financial Econometrics The Standard Errors of the OLS Estimates
What about the Standard Error of 훽? መ መ ′ σ훽 = 퐸 훽 − 훽 훽 − 훽
Reminder: 훽መ − 훽 = (푋′푋)−1푋′푉 ′ 훽መ − 훽 = 푉′푋(푋′푋)−1
96 Professor Doron Avramov, Financial Econometrics The Standard Errors of the OLS Estimates
Continuing: ′ −1 ′ ′ ′ −1 σ훽 = 퐸 (푋 푋) 푋 푉푉 푋(푋 푋) = 퐴 푠푐푎푙푎푟 ′ −1 ′ ′ ′ −1 ′ −1 ′ ′ −1 2 ′ −1 2 (푋 푋) 푋 퐸 푉푉 푋(푋 푋) = (푋 푋) 푋 푋(푋 푋) 휎ො휀 = (푋 푋) 휎ො휀
1 We get: σ = (푋′푋)−1휎ො2 where 휎ො2 = 푉 ′푉 and 퐾 is the number of 훽 휀 휀 푇−퐾−1 explanatory variables. Therefore, we can calculate σ훽: (푋′푋)−1푉 ′푉 σ = 훽 푇 − 퐾 − 1
97 Professor Doron Avramov, Financial Econometrics Maximum Likelihood Estimation (MLE)
We now turn to Maximum Likelihood as a tool for estimating parameters as well as testing models.
푖푖푑 2 Assume that 푟푡 ~ 푁 (μ, σ ) The goal is to estimate the distribution of the underlying parameters: 휇 휃 = . 휎2 Intuition: the data implies the “most likely” value of 휃.
MLE is an asymptotic procedure and it is a parametric approach in that the distribution of the regression residuals must be specified explicitly.
98 Professor Doron Avramov, Financial Econometrics Implementing MLE
Let us estimate 휇 and 휎2 using MLE; then derive the joint distribution of these estimates.
Under normality, the probability distribution function (pdf) of the rate of return takes the form 2 1 1 푟푡 − 휇 푝푑푓 푟푡 = exp − 2πσ2 2 휎2
99 Professor Doron Avramov, Financial Econometrics The Joint Likelihood
We define the “Likelihood function” 퐿 as the joint pdf Following Bayes Rule: 퐿 = 푝푑푓 푟1, 푟2, … , 푟푇 = 푝푑푓 푟1 ∣ 푟2 … 푟푇 × 푝푑푓 푟2 ∣ 푟3 … 푟푇 × ⋯ × 푝푑푓 푟푇 Since returns are assumed IID - it follows that 퐿 = 푝푑푓 푟1 × 푝푑푓 푟2 × ⋯ × 푝푑푓 푟푇 푇 푇 2 − 1 푟푡 − 휇 퐿 = 2휋휎2 2 exp − 2 휎 푡=1 Now take the natural log of the joint likelihood: 푇 푇 푇 1 푟 − 휇 2 ln 퐿 = − ln 2휋 − ln 휎2 − 푡 2 2 2 휎 푡=1
100 Professor Doron Avramov, Financial Econometrics MLE: Sample Estimates
Derive the first order conditions 푇 푇 휕 푙푛 퐿 푟 − 휇 1 = 푡 = 0 ⇒ 휇ො = 푟 휕휇 휎2 푇 푡 푡=1 푡=1 푇 푇 휕 푙푛 퐿 푇 1 푟 − 휇 2 1 = − + 푡 = 0 ⇒ 휎ො2 = 푟 − 휇ො 2 휕휎2 2휎2 2 휎4 푇 푡 푡=1 푡=1
Since 퐸 휎ො2 ≠ 휎2 - the variance estimator is not unbiased.
101 Professor Doron Avramov, Financial Econometrics MLE: The Information Matrix
Take second derivatives: 푇 휕2 푙푛 퐿 1 푇 휕2 푙푛 퐿 푇 = − = − ⇒ 퐸 = − 휕휇2 휎2 휎2 휕휇2 휎2 푡=1 푇 휕2 푙푛 퐿 푟 − 휇 휕2 푙푛 퐿 = − 푡 ⇒ 퐸 = 0 휕휇휕휎2 휎4 휕휇휕휎2 푡=1 푇 휕2 푙푛 퐿 푇 푟 − 휇 2 휕2 푙푛 퐿 푇 = − 푡 ⇒ 퐸 = − 휕휎2 2 2휎4 휎4 휕휎2 2 2휎4 푡=1
102 Professor Doron Avramov, Financial Econometrics MLE: The Covariance Matrix
휕2 푙푛 퐿 Set the information matrix 퐼 휃 = −퐸 휕휃휕휃′
The variance of 휃መ is: 푉푎푟 휃መ = 퐼(휃)−1
Or put differently, the asymptotic distribution of 휃 is: 휃 − 휃መ ∼ 푁 0, 퐼(휃)−1
[It is suggested that you find and understand the proof] In our context, the covariance matrix is derived from the information matrix as follows: 푇 푇 휎2 − , 0 , 0 , 0 휎2 푀푈퐿푇퐼푃퐿푌 "−1" 휎2 퐼푁푉퐸푅푆퐸 푇 푇 푇 2휎4 0, − 0, 0, 2휎4 2휎4 푇
103 Professor Doron Avramov, Financial Econometrics To Summarize
Multiply by 푇 and get: 푇 휃 − 휃መ ∼ 푁 0, 푇 ∙ 퐼(휃)−1 And more explicitly:
푇 1 휇 − 푟푡 푇 2 푡=1 0 휎 , 0 푇 푇 푇 ∼ 푁 , 4 1 1 0 0,2휎 휎2 − ( 푟 − 푟 )2 푇 푡 푇 푡 푡=1 푡=1
104 Professor Doron Avramov, Financial Econometrics The Sample Mean and Variance
Notice that when returns are normally distributed – the sample mean and the sample variance are independent. Departing from normality, the covariance between the sample mean and variance is nonzero. Notice also that the ratio obtained by dividing the variance of the variance (2휎4) by the variance of the mean (휎2) is smaller than one as long as volatility is below 70%. The mean return estimate is more noisy because the volatility is typically far below 70%, especially for well diversified portfolios.
105 Professor Doron Avramov, Financial Econometrics Method of Moments: departing from Normality
We know:
퐸 푟푡 = 휇 2 2 퐸 푟푡 − 휇 = 휎 If: 푟푡 − 휇 푔푡 휃 = 2 2 푟푡 − 휇 − 휎
Then: 0 퐸 푔 = 푡 0 That is, we set two momentum conditions.
106 Professor Doron Avramov, Financial Econometrics Method of Moments (MOM)
There are two parameters: 휇, 휎2
푟푡 − 휇 Stage 1: Moment Conditions 푔푡 = 2 2 푟푡 − 휇 − 휎
Stage 2: estimation 푇 1 푔ො = 0 푇 푡 푡=1
107 Professor Doron Avramov, Financial Econometrics Method of Moments
Continue estimation: 1 푟 − 휇Ƹ = 0 푡 = 1, … , 푇 푇 푡 푇 1 휇Ƹ = 푟 푇 푡 푡=1 푇 1 푟 − 휇Ƹ 2 − 휎ො2 = 0 푇 푡 푡=1 푇 1 휎ො2 = 푟 − 휇Ƹ 2 푇 푡 푡=1
108 Professor Doron Avramov, Financial Econometrics MOM: Stage 3
′ 푆 = 퐸 푔푡푔푡
2 3 2 푟푡 − 휇 , 푟푡 − 휇 − 휎 푟푡 − 휇 푆 = 퐸 3 2 4 2 2 4 푟푡 − 휇 − 휎 푟푡 − 휇 , 푟푡 − 휇 − 2휎 푟푡 − 휇 + 휎
↓
2 휎 , 휇3 푆 = 4 휇3, 휇4 − 휎
109 Professor Doron Avramov, Financial Econometrics MOM: stage 4
Memo: 푟푡 − 휇 푔푡 휃 = 2 2 푟푡 − 휇 − 휎
Stage 4: differentiate 푔푡 휃 w.r.t. 휃 and take the expected value
휕푔푡 휃 −1, 0 −1, 0 퐷 = 퐸 = 퐸 = 휕휃 −2 푟푡 − 휇 , −1 0 , −1
110 Professor Doron Avramov, Financial Econometrics MOM: The Covariance Matrix
Stage 5: ′ −1 −1 σ휃 = 퐷 푆 퐷
In this specific case we have: 퐷 = −퐼, therefore:
σ휃 = 푆
111 Professor Doron Avramov, Financial Econometrics The Covariance Matrices Denote the MLE covariance matrix by σ1 , and the MOM covariance 2 휃 matrix by σ휃: 2 2 1 휎 , 0 2 휎 , 휇3 σ휃 = 4 , σ휃 = 4 0, 2휎 휇3, 휇4 − 휎
3 휇3 = 푆퐾 = 푠푘 × 휎 (sk is the skewness of the standardized return) 4 휇4 = 퐾푅 = 푘푟 × 휎 (kr is the kurtosis of the standardized return) Sample estimates of the Skewness and Kurtosis are 푇 푇 1 1 휇Ƹ = 푟 − 푟lj 3 휇Ƹ = 푟 − 푟lj 4 3 푇 푡 4 푇 푡 푡=1 푡=1
112 Professor Doron Avramov, Financial Econometrics Under Normality the MOM Covariance Matrix Boils Down to MLE
2 2 휎 , 휇3 = 0 1 σ휃 = 4 4 = σ휃 휇3 = 0, 휇4 − 휎 = 2휎
113 Professor Doron Avramov, Financial Econometrics MOM: Estimating Regression Parameters Let us run the time series regression 푟푡 = 훼 + 훽 ⋅ 푟푚푡 + 휀푡 where: ′ 푥푡 = 1, 푟푚푡 휃 = 훼, 훽 ′
We know that 퐸 휀푡|푥푡 = 0
Given that 퐸 휀푡|푥푡 = 0, from the Law of Iterated Expectations (LIE) ′ it follows that 퐸 푥푡휀푡 = 푥푡 푟푡 − 푥푡 휃 = 0 That is, there are two moment conditions (stage 1):
퐸 휀푡 = 0 푟푡 − 훼 − 훽 ⋅ 푟푚푡 → 푔푡 = 퐸 푟푚푡 휀푡 = 0 푟푡 − 훼 − 훽 ⋅ 푟푚푡 푟푚푡
114 Professor Doron Avramov, Financial Econometrics MOM: Estimating Regression Parameters
Stage 2: estimation: 푇 푇 1 1 푥 푟 − 푥 푥′휃መ = 0 푇 푡 푡 푇 푡 푡 푡=1 푡=1
−1 푇 푇 መ ′ ′ −1 ′ 휃 = 푥푡 푥푡 푥푡 푟푡 = 푋 푋 푋 푅 푡=1 푡=1
1, 푟푚1 푟1 푋푇×2 = ⋯ ⋯ 푅 = … 1, 푟푚푇 푟푇
115 Professor Doron Avramov, Financial Econometrics MOM: Estimating Standard Errors
Estimation of the covariance matrix (assuming no serial correlation) Stage 3: 푇 푇 1 1 푆 = 푔 (휃መ)푔 (휃መ)′ = ( 푥 푥′)휀Ƹ2 푇 푇 푡 푡 푇 푡 푡 푡 푡=1 푡=1
Stage 4: 푇 푇 1 휕푔 (휃መ) 1 푋′푋 퐷 = 푡 = − ( 푥 푥′) = − 푇 푇 휕휃መ 푇 푡 푡 푇 푡=1 푡=1
116 Professor Doron Avramov, Financial Econometrics MOM: Estimating Standard Errors
Stage 5: the covariance matrix estimate is:
′ −1 −1 Σ휃 = (퐷푇푆푇 퐷푇) 푇 ′ −1 ′ 2 ′ −1 Σ휃 = 푇(푋 푋) ( 푥푡푥푡)휀푡Ƹ (푋 푋) 푡=1
Then, asymptotically we get
푇(휃 − 휃መ) ∼ 푁(0, Σ휃)
117 Professor Doron Avramov, Financial Econometrics Session #3: Hypothesis Testing
118 Professor Doron Avramov, Financial Econometrics Overview
A short brief of the major contents for today’s class: Hypothesis testing TESTS: Skewness, Kurtosis, Bera-Jarque Deriving test statistic for the Sharpe ratio
119 Professor Doron Avramov, Financial Econometrics Hypothesis Testing
Let us assume that a mutual fund invests in value stocks (e.g., stocks with high ratios of book-to-market). Performance evaluation is mostly about running the regression of excess fund returns on the market benchmark (often multiple benchmarks): 푒 푒 푅푖푡 = 훼푖 + 훽푖푅푚푡 + 휀푖푡 The hypothesis testing for examining performance is
H0: 훼푖 = 0 means no performance
H1: Otherwise (positive or negative performance)
120 Professor Doron Avramov, Financial Econometrics Hypothesis Testing
Errors emerge if we reject H0 while it is true, or when we do not reject H0 when it is wrong:
Don’t reject H0 Reject H0
Type 1 error H Good decision 0 훼 True state of world Type 2 error H Good decision 1 훽
121 Professor Doron Avramov, Financial Econometrics Hypothesis Testing - Errors
훼 is the first type error (size), while 훽 is the second type error (related to power).
The power of the test is equal to 1 − 훽.
We would prefer both 훼 and 훽 to be as small as possible, but there is always a trade-off.
When 훼 decreases → 훽 increases and vice versa.
The implementation of hypothesis testing requires the knowledge of distribution theory.
122 Professor Doron Avramov, Financial Econometrics Skewness, Kurtosis & Bera Jarque Test Statistics
We aim to test normality of stock returns.
We use three distinct tests to examine normality.
123 Professor Doron Avramov, Financial Econometrics Test I – Skewness
TEST 1 - Skewness (third moment) The setup for testing normality of stock return: 2 H0: 푅푡 ∼ 푁 휇, 휎
H1: otherwise Sample Skewness is 푇 3 1 푅 − 휇ො 퐻 6 푆 = 푡 ∼0 푁 0, 푇 휎ො 푇 푡=1
124 Professor Doron Avramov, Financial Econometrics Test I – Skewness
푇 푇 Multiplying 푆 by , we get 푆~푁 0,1 6 6 If the statistic value is higher (absolute value) than the critical value e.g., the statistic is equal to -2.31, then reject H0, otherwise do not reject the null of normality.
125 Professor Doron Avramov, Financial Econometrics TEST 2 - Kurtosis
Kurtosis estimate is: 푇 4 1 푅 − 휇ො 퐻 24 퐾 = 푡 ∼0 푁 3, 푇 휎ො 푇 푡=1
After transformation:
푇 퐻 퐾 − 3 ∼0 푁 0,1 24
126 Professor Doron Avramov, Financial Econometrics TEST 3 - Bera-Jarque Test
The statistic is: 푇 푇 퐵퐽 = 푆2 + 퐾 − 3 2 ∼ 휒2 2 6 24
Why 휒2(2) ?
127 Professor Doron Avramov, Financial Econometrics TEST 3 - Bera-Jarque Test
If 푋1~푁 0,1 , 푋2~푁 0,1 , and 푋1 ⊥ 푋2 then:
2 2 2 푋1 + 푋2 ∼ 휒 2
푇 푇 푆 and 퐾 − 3 are both standard normal and 6 24 they are independent random variables.
128 Professor Doron Avramov, Financial Econometrics Chi Squared Test
In financial economics, the Chi squared test is implemented quite frequently in hypothesis testing.
Let us derive it.
1 − Suppose that: 푦 = σ 2 푅 − 휇 ∼ 푁 0, 퐼 Then: 푦′푦 = 푅 − 휇 ′ σ−1 푅 − 휇 ∼ 휒2 푁
129 Professor Doron Avramov, Financial Econometrics Joint Hypothesis Test
You run the regression:
푦푡 = 훽0 + 훽1푥1푡 + 훽2푥2푡 + 휀푡, 푡 = 1,2, … , 푇
130 Professor Doron Avramov, Financial Econometrics Joint Hypothesis Test
퐸 휀푡 = 0 GMM would establish three orthogonal conditions: 퐸 휀푡푥1푡 = 0 퐸 휀푡푥2푡 = 0 Using matrix notation: 푦1 1, 푥11, 푥21 휀1 . 1, 푥 , 푥 . 푌 = 푋 = 12 22 퐸 = 훽 = 훽 , 훽 , 훽 ′ 푇×1 . 푇×3 … … … … 푇×1 . 3푥1 1 2 3 푦푇 1, 푥1푇, 푥2푇 휀푇
푌푇×1 = 푋푇×3훽3×1 + 퐸푇×1
131 Professor Doron Avramov, Financial Econometrics Joint Hypothesis Test
2 Let us assume that: 휀푡 ∼ 푁 0, 휎휀 휀1, 휀2, 휀3 … 휀푇 are iid ↓ መ ′ −1 2 훽 ∼ 푁 훽, 푋 푋 휎휀 훽መ = 푋′푋 −1푋′푌
Joint hypothesis testing:
퐻0: 훽0 = 1, 훽2 = 0 퐻1: 표푡ℎ푒푟푤푖푠푒
132 Professor Doron Avramov, Financial Econometrics Joint Hypothesis Test 1,0,0 1 Define: 푅 = , 푞 = 0,0,1 0
The joint test becomes: 퐻0: 푅훽 = 푞 훽0 1,0,0 훽0 H0 1 푅 → × 훽1 = = ← 푞 0,0,1 훽2 0 훽2 퐻1: 표푡ℎ푒푟푤푖푠푒
133 Professor Doron Avramov, Financial Econometrics Joint Hypothesis Test 퐻 : 푅훽 − 푞 = 0 Returning to the testing: 0 퐻1: 표푡ℎ푒푟푤푖푠푒 ↓ መ ′ 푅훽 − 푞 ∼ 푁 푅훽 − 푞, 푅σ훽푅 መ H0 ′ Under H0: 푅훽 − 푞 ~푁 0, 푅 σ훽 푅
መ ′ ′ −1 መ 2 Chi squared: 푅훽 − 푞 푅 σ훽 푅 푅훽 − 푞 ~휒 2 푡푒푠푡 푠푡푎푡푖푠푡푖푐
134 Professor Doron Avramov, Financial Econometrics Joint Hypothesis Test
Yet, another example: 푦푡 = 훽0 + 훽1푥1푡 + 훽2푥2푡 + 훽3푥3푡 + 훽4푥4푡 + 훽5푥5푡 + 휀푡 푡 = 1,2,3, … , 푇
1, 푥11, … , 푥51 훽 = 훽 , 훽 , … 훽 ′ 푦1 6×1 0 1 6 . 휀1 푌푇×1 = . , 푋푇×6 = , . 휀푇×1 = . 푦푇 1, 푥1푇, … , 푥5푇 휀푇
푌 = 푋훽 + 퐸
135 Professor Doron Avramov, Financial Econometrics Joint Hypothesis Test
Joint hypothesis test: 1 1 1 퐻 : 훽 = 1, 훽 = , 훽 = , 훽 = , 훽 = 7 0 0 1 2 2 3 3 4 5 퐻1: 표푡ℎ푒푟푤푖푠푒
Here is a receipt: 1. 훽መ = 푋′푋 −1푋′푌 2. 퐸 = 푌 − 푋훽መ 1 3. 휎ො2 = 퐸′ ⋅ 퐸 휀 푇−6 መ ′ −1 2 4. 훽 ∼ 푁 훽, 푋 푋 휎휀 = σ훽
136 Professor Doron Avramov, Financial Econometrics Joint Hypothesis Test
1 1,0,0,0,0,0 1 0,1,0,0,0,0 2 1 5. 푅 = 0,0,1,0,0,0 , 푞 = 3 0,0,0,1,0,0 1 0,0,0,0,0,1 4 7
6. 퐻0: 푅훽 − 푞 = 0 መ ′ 7. 푅훽 − 푞 ∼ 푁 푅훽 − 푞, 푅 σ훽 푅 መ 퐻0 ′ 8. 푅훽 − 푞 ∼ 푁 0, 푅 σ훽 푅 መ ′ ′ −1 መ 퐻0 2 9. 푅훽 − 푞 푅 σ훽 푅 푅훽 − 푞 ∼ 휒 5
137 Professor Doron Avramov, Financial Econometrics Joint Hypothesis Test
There are five degrees of freedom implied by the five restrictions on 훽0, 훽1, 훽2, 훽3, 훽5 The chi-squared distribution is always positive.
138 Professor Doron Avramov, Financial Econometrics Estimating the Sample Sharpe Ratio You observe time series of returns on a stock, or a bond, or any investment vehicle (e.g., a mutual fund or a hedge fund): 푟1, 푟2, … , 푟푇
You attempt to estimate the mean and the variance of those returns, derive their distribution, and test whether the Sharpe Ratio of that investment is equal to zero.
Let us denote the set of parameters by 휃 = 휇, 휎2 ′
휇−푟 The Sharpe ratio is equal to 푆푅(휃) = 푓 휎
139 Professor Doron Avramov, Financial Econometrics MLE vs. MOM
To develop a test statistic for the SR, we can implement the MLE or MOM, depending upon our assumption about the return distribution. Let us denote the sample estimates by 휃መ
푖푖푑 푟푒푡푢푟푛푠 푑푒푝푎푟푡 푓푟표푚 푟 ∼ 푁 휇, 휎2 푡 푖푖푑 푛표푟푚푎푙 MLE MOM መ 푎 1 መ 푎 2 푇 휃 − 휃 ∼ 푁 0, σ휃 푇 휃 − 휃 ∼ 푁 0, σ휃
140 Professor Doron Avramov, Financial Econometrics MLE vs. MOM
As shown earlier, the asymptotic distribution using either MLE or MOM is normal with a zero mean but distinct variance covariance matrices:
መ 푀퐿퐸 1 መ 푀푂푀 2 푇 휃 − 휃 ∼ 푁 0, σ휃 푇 휃 − 휃 ∼ 푁 0, σ휃
141 Professor Doron Avramov, Financial Econometrics Distribution of the SR Estimate: The Delta Method
훼 2 We will show that 푇 푆푅 − 푆푅 ∼ 푁 0, 휎푆푅
2 We use the Delta method to derive 휎푆푅
The delta method is based upon the first order Taylor approximation.
142 Professor Doron Avramov, Financial Econometrics Distribution of the SR Estimate: The Delta Method
The first-order TA is 휕푆푅 푆푅 휃 = 푆푅 휃መ + ⋅ 휃 − 휃መ ⇒ 휕휃′ 1×2 2×1
휕푆푅 푆푅 휃 − 푆푅 휃መ = ⋅ 휃 − 휃መ 휕휃′ 1×2 2×1 The derivative is estimated at 휃መ
143 Professor Doron Avramov, Financial Econometrics Distribution of the Sample SR
휕푆푅 퐸 푆푅 휃 − 푆푅 휃መ = 퐸 휃 − 퐸 휃መ = 0 휕휃′ whereas ′ 푉퐴푅 휃መ = 퐸 휃መ − 휃 휃መ − 휃 2 푉퐴푅 푆푅 휃 − 푆푅 휃መ = 퐸 푆푅 휃 − 푆푅 휃መ
144 Professor Doron Avramov, Financial Econometrics The Variance of the SR
Continue:
휕푆푅 ′ 휕푆푅 퐸 휃 − 휃መ 휃 − 휃መ = 휕휃′ 휕휃
휕푆푅 ′ 휕푆푅 = 퐸 휃 − 휃መ 휃 − 휃መ = 휕휃′ 휕휃
휕푆푅 휕푆푅 = σ 휕휃′ 휃 휕휃
145 Professor Doron Avramov, Financial Econometrics First Derivatives of the SR
휇−푟 The SR is formulated as 푆푅 = 푓 휎2 0.5 Let us derive: 휕푆푅 1 = 휕휇 휎
1 2 −0.5 휕푆푅 − 휎 휇 − 푟푓 − 휇 − 푟푓 = 2 = 휕휎2 휎2 2휎3
146 Professor Doron Avramov, Financial Econometrics The Distribution of SR under MLE
Continue: 1 휕푆푅 휕푆푅 1 휇 − 푟푓 휎2, 0 휎 σ휃 = , − 휕휃′ 휕휃 휎 2휎3 0, 2휎4 휇 − 푟푓 − 2휎3 1 ⇒ 푇 푆푅 − 푆푅 ∼ 푁 0, 1 + 푆푅2 2
147 Professor Doron Avramov, Financial Econometrics The Delta Method in General
Here is the general application of the delta method
If 푇 휃መ − 휃 ~푁 0, σ휃 Then let 푑 휃 be some function of 휃 : መ ′ 푇 푑 휃 − 푑 휃 ∼ 푁 0, 퐷 휃 × σ휃 × 퐷 휃 where 퐷 휃 is the vector of derivatives of 푑 휃 with respect to 휃
148 Professor Doron Avramov, Financial Econometrics Hypothesis Testing
Does the S&P index outperform the Rf?
H0: SR=0
H1: Otherwise Under the null there is no outperformance. 1 2 푇푆푅 ∼ 푁(0,1 + 2푆푅 ) 푇푆푅 Thus, under the null ∼ 푁(0,1) 1 1+ 푆푅2 2
149 Professor Doron Avramov, Financial Econometrics Session #4: The Efficient Frontier and the Tangency Portfolio
150 Professor Doron Avramov, Financial Econometrics Testing Asset Pricing Models
Central to financial econometrics is the formation of test statistics to examine the validity of asset pricing models.
There are time series as well as cross sectional asset pricing tests.
In this course the focus is on time-series tests, while in the companion course for PhD students – Asset Pricing – also cross sectional tests are covered.
151 Professor Doron Avramov, Financial Econometrics Testing Asset Pricing Models Time series tests are only implementable when common factors are portfolio spreads, such as excess return on the market portfolio as well as the SMB (small minus big), the HML (high minus low), the WML (winner minus loser), the TERM (long minus short maturity), and the DEF (low minus high quality) portfolios.
The first four are equity while the last two are bond portfolios.
Cross sectional tests apply to both portfolio and non-portfolio based factors.
Consumption growth in the consumption based CAPM (CCAPM) is a good example of a factor that is not a return spread.
152 Professor Doron Avramov, Financial Econometrics Time Series Tests and the Tangency Portfolio
Interestingly, time series tests are directly linked to the notion of the tangency portfolio and the efficient frontier. Here is the efficient frontier, in which the tangency portfolio is denoted by T.
T
153 Professor Doron Avramov, Financial Econometrics Economic Interpretation of the Time Series Tests
Testing the validity of the CAPM entails the time series regressions: 푒 푒 푟1푡 = 훼1 + 훽1푟푚푡 + 휀1푡 … … … … … … … 푒 푒 푟푁푡 = 훼푁 + 훽푁푟푚푡 + 휀푁푡
The CAPM says: H0: 훼1 = 훼2 = ⋯ = 훼푁=0
H1: Otherwise
154 Professor Doron Avramov, Financial Econometrics Economic Interpretation of the Time Series Tests
The null is equivalent to the hypothesis that the market portfolio is the tangency portfolio.
Of course, even if the model is valid – the market portfolio WILL NEVER lie on the estimated frontier.
This is due to sampling errors; the efficient frontier is estimated.
The question is whether the market portfolio is close enough, up to a statistical error, to the tangency portfolio.
155 Professor Doron Avramov, Financial Econometrics What about Multi-Factor Models?
The CAPM is a one-factor model.
There are several extensions to the CAPM.
The multivariate version is given by the K-factor model:
156 Professor Doron Avramov, Financial Econometrics Testing Multifactor Models
푒 푟1푡 = 훼1 + 훽11푓1 + 훽12푓2 + ⋯ + 훽1퐾푓퐾 + 휀1푡 … … … … … … … 푒 푟푁푡 = 훼푁 + 훽푁1푓1 + 훽푁2푓2 + ⋯ + 훽푁퐾푓퐾 + 휀푁푡
The null is again: H0: 훼1 = 훼2 = ⋯ = 훼푁=0
H1: Otherwise In the multi-factor context, the hypothesis is that some particular combination of the factors is the tangency portfolio.
157 Professor Doron Avramov, Financial Econometrics The Efficient Frontier: Investable Assets
Consider N risky assets whose returns at time t are: 푅1푡 푅푡 = … 푁×1 푅푁푡
The expected value of return is denoted by: 퐸 푅1푡 휇1 퐸 푅푡 = … = … = 휇 퐸 푅푁푡 휇푁
158 Professor Doron Avramov, Financial Econometrics The Covariance Matrix
The variance covariance matrix is denoted by: 2 휎1 , … , … , … 휎1푁 2 푉 = 퐸 푅 − 휇 푅 − 휇 ′ = … , 휎2 , … , … 휎2푁 푡 푡 … … … … … … 2 … … … … … , 휎푁
159 Professor Doron Avramov, Financial Econometrics Creating a Portfolio
푤1 A portfolio is investing 푤 = … is N assets. 푁×1 푤푁
The return of the portfolio is: 푅푝 = 푤1푅1 + 푤2푅2 + ⋯ + 푤푁푅푁
The expected return of the portfolio is:
퐸 푅푝 = 푤1퐸 푅1 + 푤2퐸 푅2 + ⋯ + 푤푁퐸 푅푁 = 푁 ′ = 푤1휇1 + 푤2휇2 + ⋯ + 푤푁휇푁 = σ푖=1푤푖휇푖 = 푤 휇
160 Professor Doron Avramov, Financial Econometrics Creating a Portfolio
The variance of the portfolio is:
2 2 2 휎푝 = 푉퐴푅 푅푝 = 푤1 휎1 + 푤1푤2휎12 + ⋯ 푤1푤푁휎1푁 2 2 +푤1푤2휎12 + 푤2 휎2 + ⋯ + 푤2푤푁휎2푁 + ⋮ + 2 2 +푤1푤푁휎1푁 + 푤2푤푁휎2푁 + ⋯ + 푤푁휎푁
161 Professor Doron Avramov, Financial Econometrics Creating a Portfolio
2 푁 푁 Thus 휎푝 = σ푖=1 σ푗=1 푤푖 푤푗휎푖휎푗휌푖푗 where 휌푖푗 is the coefficient of correlation.
2 ′ Using matrix notation: σ푃 = 푤 푉 푤 1×1 1×푁 푁×푁 푁×1
162 Professor Doron Avramov, Financial Econometrics The Case of Two Risky Assets
To illustrate, let us consider two risky assets: 푅푝 = 푤1푅1 + 푤2푅2 2 2 2 2 2 We know: 휎푝 = 푉퐴푅 푅푝푡 = 푤1 휎1 + 2푤1푤2휎12 + 푤2 휎2
Let us check: 2 2 휎1 , 휎12 푤1 2 2 2 2 휎푝 = 푤1, 푤2 2 푤 = 푤1 휎1 + 2푤1푤2휎12 + 푤2 휎2 휎12, 휎2 2
So it works!
163 Professor Doron Avramov, Financial Econometrics Dominance of the Covariance
When the number of assets is large, the covariances define the portfolio’s volatility. To illustrate, assume that all assets have the same volatility and the same pairwise correlations. Then an equal weight portfolio’s variation is 푁 + 휌푁(푁 − 1) 푁→∞ 휎2 = 휎2 휎2휌 = cov 푝 푁2
164 Professor Doron Avramov, Financial Econometrics The Efficient Frontier: Excluding Risk-free Asset
The optimization program: min 푤′푉푤 푠. 푡 푤′휄 = 1 ′ 푤 휇 = 휇푝
1 where 휄 is the Greek letter iota ɇ푁×1 = … , 1 and where 휇푝 is the expected return target set by the investor.
165 Professor Doron Avramov, Financial Econometrics The Efficient Frontier: Excluding Risk-free Asset
Using the Lagrange setup: 1 퐿 = 푤′푉푤 + 휆 1 − 푤′휄 + 휆 휇 − 푤′휇 2 1 2 푝 휕퐿 = 푉푤 − 휆 휄 − 휆 휇 = 0 휕푤 1 2 −1 −1 푤 = 휆1푉 휄 + 휆2푉 휇
166 Professor Doron Avramov, Financial Econometrics The Efficient Frontier: Without Risk-free Asset
Let 푎 = 휄′푉−1휇 푏 = 휇′푉−1휇 푐 = 휄′푉−1휄 푑 = 푏푐 − 푎2 1 푔 = 푏푉−1휄 − 푎푉−1휇 푑 1 ℎ = 푐푉−1휇 − 푎푉−1휄 푑 ∗ The optimal portfolio is: 푤 = 푔 + ℎ 휇푝 푁×1 푁×1 푁×1
167 Professor Doron Avramov, Financial Econometrics Examine the Optimal Solution
That is, once you specify the expected return target, the optimal portfolio follows immediately.
Let us check whether the sum of weights is equal to 1.
168 Professor Doron Avramov, Financial Econometrics Examine the Optimal Solution
′ ′ ′ 휄 푤 = 휄 푔 + 휄 ℎ휇푝 1 1 푑 휄′푔 = 푏휄′푉−1휄 − 푎휄′푉−1휇 = 푏푐 − 푎2 = = 1 푑 푑 푑 1 1 휄′ℎ = 푐휄′푉−1휇 − 푎휄′푉−1휄 = 푎푐 − 푎푐 = 0 푑 푑
Indeed, 푤′휄 = 1.
169 Professor Doron Avramov, Financial Econometrics Examine the Optimal Solution
Let us now check whether the expected return on the portfolio is equal to 휇푝.
Recall: 푤 = 푔 + ℎ휇푝 ′ ′ ′ 푤 휇 = 푔 휇 + 휇푝 ℎ 휇
So we need to show 푔′휇 = 0 ℎ′휇 = 1
170 Professor Doron Avramov, Financial Econometrics Examine the Optimal Solution 1 1 푔′휇 = 푏휄′푉−1휇 − 푎휇′푉−1휇 = 푎푏 − 푎푏 = 0 푑 푑
Try it yourself: prove that ℎ′휇 = 1.
171 Professor Doron Avramov, Financial Econometrics Efficient Frontier
The optimization program also delivers the shape of the frontier.
A
172 Professor Doron Avramov, Financial Econometrics Efficient Frontier
Point A stands for the Global Minimum Variance Portfolio (GMVP).
The efficient frontier reflects the investment opportunities; this is the supply side in partial equilibrium.
Points below A are inefficient since they are being dominated by other portfolios that deliver better risk-return tradeoff.
173 Professor Doron Avramov, Financial Econometrics The Notion of Dominance
If 휎퐵 ≤ 휎퐴
휇퐵 ≥ 휇퐴
and there is at least one strong inequality, then portfolio B dominates portfolio A.
174 Professor Doron Avramov, Financial Econometrics The Efficient Frontier with Risk-free Asset
In practice, there is not really a risk-free asset. Why? Credit risk (see Greece, Spain, Iceland). Inflation risk. Interest rate risk and re-investment risk when the investment horizon is longer than the maturity of the supposedly risk-free instrument or when coupons are paid.
175 Professor Doron Avramov, Financial Econometrics Setting the Optimization in the Presence of Risk-free Asset The optimal solution is given by: min 푤′푉푤 ′ ′ ′ 푒 s. t 푤 휇 + 1 − 푤 휄 푅푓 = 휇푝 or, 푅푓 + 푤 휇 = 휇푝 and the tangency portfolio takes the form: −1 −1 푒 푉 휇 − 푅푓 ⋅ 휄 푉 휇 푤ഥ ∗ = = ′ −1 ′ −1 푒 휄 푉 휇 − 푅푓 ⋅ 휄 휄 푉 휇
The tangency portfolio is investing all the funds in risky assets.
176 Professor Doron Avramov, Financial Econometrics The Investment Opportunities
However, the investor could select any point in the line emerging from the risk-free rate and touching the efficient frontier in point T.
T
The location depends on the attitude toward risk.
177 Professor Doron Avramov, Financial Econometrics Fund Separation
Interestingly, all investors in the economy will mix the tangency portfolio and the risk-free asset.
The mix depends on preferences.
But the proportion of risky assets will be equal across the board.
One way to test the CAPM is indeed to examine whether all investors hold the same proportions of risky assets.
Obviously they don’t!
178 Professor Doron Avramov, Financial Econometrics Equilibrium
The efficient frontier reflects the supply side.
What about the demand?
The demand side can be represented by a set of indifference curves.
179 Professor Doron Avramov, Financial Econometrics Equilibrium
What is the slope of indifference curve positive? The equilibrium obtains when the indifference curve tangents the efficient frontier No risk-free asset:
A
180 Professor Doron Avramov, Financial Econometrics Equilibrium
With risk-free asset:
A
181 Professor Doron Avramov, Financial Econometrics Maximize Utility / Certainty Equivalent Return
In the presence of a risk-free security, the tangency point can be found by maximizing a utility function of the form 1 푈 = 휇 − 훾휎2 푝 2 푝
where 훾 is the relative risk aversion.
182 Professor Doron Avramov, Financial Econometrics Maximize Utility / Certainty Equivalent Return
Notice that utility is equal to expected return minus a penalty factor.
The penalty factor positively depends on the risk aversion (demand) and the variance (supply).
183 Professor Doron Avramov, Financial Econometrics Utility Maximization 1 푈 푤 = 푅 + 푤′휇푒 − 훾 ⋅ 푤′푉푤 푓 2 휕푈 = 휇푒 − 훾푉푤 = 0 휕푤 1 푤∗ = 푉−1휇푒 훾 The utility maximization yields the same tangency portfolio −1 ∗ 푉 휇푒 푤ഥ = ′ −1 휄 푉 휇푒 where 푤∗ reflects the fraction of weights invested in risky assets. The rest is invested in the risk-free asset.
184 Professor Doron Avramov, Financial Econometrics Mixing the Risky and Risk-free Assets 1 푤∗ = 푉−1휇푒 훾
So if 훾 = 휄′푉−1휇푒 then all funds (100%) are invested in risky assets.
If 훾 > 휄′푉−1휇푒 then some fraction is invested in risk-free asset.
If 훾 < 휄′푉−1휇푒 then the investor borrows money to leverage his/her equity position.
185 Professor Doron Avramov, Financial Econometrics The Exponential Utility Function
The exponential utility function is of the form 푈 푊 = − exp −휆푊 where 휆 > 0
Notice that 푈′ 푊 = 휆 exp −휆푊 > 0 푈′′ 푊 = −휆2 exp −휆푊 < 0
That is, the marginal utility is positive but it diminishes with an increasing wealth.
186 Professor Doron Avramov, Financial Econometrics The Exponential Utility Function 푈′′ 퐴푅퐴 = − = 휆 푈′
Indeed, the exponential preferences belong to the class of constant absolute risk aversion (CARA).
For comparison, power preferences belong to the class of constant relative risk aversion (CRRA).
187 Professor Doron Avramov, Financial Econometrics Exponential: The Optimization Mechanism
The investor maximizes the expected value of the exponential utility where the decision variable is the set of weights w and subject to the wealth evolution.
That is
max 퐸 푈 푊푡+1 |푊푡 푤 ′ 푒 푠. 푡. 푊푡+1 = 푊푡 1 + 푅푓 + 푤 푅푡+1
188 Professor Doron Avramov, Financial Econometrics Exponential: The Optimization Mechanism
푒 푒 Let us assume that 푅푡+1 ∼ 푁 휇 , 푉
′ 푒 2 ′ Then 푊푡+1 ∼ 푁 푊푡 1 + 푅푓 + 푤 휇 , 푊푡 푤 푉푤 푚푒푎푛 푣푎푟푖푎푛푐푒
189 Professor Doron Avramov, Financial Econometrics Exponential: The Optimization Mechanism
2 It is known that for 푥 ∼ 푁 휇푥, 휎푥 1 퐸 exp 푎푥 = exp 푎휇 + 푎2휎2 푥 2 푥
190 Professor Doron Avramov, Financial Econometrics Exponential: The Optimization Mechanism
Thus,
1 −퐸 exp −휆푊 = − exp −휆퐸 푊 + 휆2 ⋅ 푉퐴푅 푊 푡+1 푡+1 2 푡+1 1 = − exp −휆푊 1 + 푅 + 푤′휇푒 + 휆2푊2푤′푉푤 푡 푓 2 푡 1 = − exp −휆푊 1 + 푅 exp −휆푊 푤′휇푒 − 휆푊 푤′푉푤 푡 푓 푡 2 푡
191 Professor Doron Avramov, Financial Econometrics Exponential: The Optimization Mechanism
Notice that 훾 = 휆푊푡
where 훾 is the relative risk aversion coefficient.
192 Professor Doron Avramov, Financial Econometrics Exponential: The Optimization Mechanism
So the investor ultimately maximizes
1 max 푤′휇푒 − 훾푤′푉푤 푤 2
1 The optimal solution is 푤∗ = 푉−1휇푒. 훾
The tangency portfolio is the same as before
푉−1휇푒 푤ഥ ∗ = 휄′푉−1휇푒
193 Professor Doron Avramov, Financial Econometrics Exponential: The Optimization Mechanism
Conclusion: The joint assumption of exponential utility and normally distributed stock return leads to the well-known mean variance solution.
194 Professor Doron Avramov, Financial Econometrics Quadratic Preferences
The quadratic utility function is of the form 푏 푈 푊 = 푎 + 푊 − 푊2 where 푏 > 0 2
휕푈 = 1 − 푏푊 휕푊
휕2푈 = −푏 < 0 휕푊2
1 Notice that the first derivative is positive for 푏 < 푊
195 Professor Doron Avramov, Financial Econometrics Quadratic Preferences
The utility function looks like
푈 푊
푊 1 푏
196 Professor Doron Avramov, Financial Econometrics Quadratic Preferences
It has a diminishing part – which makes no sense – because we always prefer higher than lower wealth
Utility is thus restricted to the positive slope part
푈′′ 푏푊 Notice that 훾 = 푅푅퐴 = − 푊 = 푈′ 1−푏푊
The optimization formulation is given by
max 퐸 푈 푊푡+1 |푊푡 푤 ′ 푒 푠. 푡. 푊푡+1 = 푊푡 1 + 푅푓푡 + 푤 푅푡+1
197 Professor Doron Avramov, Financial Econometrics Quadratic Preferences
Avramov and Chordia (2006 JFE) show that the optimization could be formulated as 1 ′ −1 max 푤′휇푒 − 푤′ 푉 + 휇푒휇푒 푤 푤 1 2 − 푅 훾 푓푡
The solution takes the form 1 푉−1휇푒 푤∗ = − 푅 훾 푓푡 1 + 휇푒′ 푉−1휇푒
198 Professor Doron Avramov, Financial Econometrics Quadratic Preferences
The tangency portfolio is 푉−1휇푒 푤ഥ ∗ = 휄′푉−1휇푒
The only difference from previously presented competing specifications is the composition of risky and risk-free assets.
But in all solutions the proportions of risky assets are identical.
199 Professor Doron Avramov, Financial Econometrics The Sharpe Ratio of the Tangency Portfolio
′ Notice that 휇푒 푉−1휇푒 is actually the squared Sharpe Ratio of the tangency portfolio.
Let us prove it 푉−1휇푒 푤ഥ ∗ = 휄′푉−1휇푒 ′ 휇푒 푉−1휇푒 휇 − 푅 = 푤ഥ ∗′휇푒 + 1 − 푤ഥ ∗′휄 푅 = 푤ഥ ∗휇푒 = 푇푃 푓 푓 휄′푉−1휇푒 ′ 휇푒 푉−1휇푒 휎2 = 푤ഥ ∗′푉푤ഥ ∗ = 푇푃 휄′푉−1휇푒 2
200 Professor Doron Avramov, Financial Econometrics The Sharpe Ratio of the Tangency Portfolio
2 2 휇푇푃−푅푓 푒′ −1 푒 Thus, 푆푅푇푃 = 2 = 휇 푉 휇 휎푇푃
TP is a subscript for the tangency portfolio.
201 Professor Doron Avramov, Financial Econometrics Session #5: Testing Asset Pricing Models: Time Series Perspective
202 Professor Doron Avramov, Financial Econometrics Why Caring about Asset Pricing Models?
An essential question that arises is why would both academics and practitioners invest huge resources in developing and testing asset pricing models.
It turns out that pricing models have crucial roles in various applications in financial economics – both asset pricing as well as corporate finance.
In the following, I list five major applications.
203 Professor Doron Avramov, Financial Econometrics 1 – Common Risk Factors
Pricing models characterize the risk profile of a firm.
In particular, systematic risk is no longer stock return volatility – rather it is the loadings on risk factors.
For instance, in the single factor CAPM the market beta – or the co-variation with the market – characterizes the systematic risk of a firm.
204 Professor Doron Avramov, Financial Econometrics 1 – Common Risk Factors
Likewise, in the single factor (C)CAPM the consumption growth beta – or the co-variation with consumption growth – characterizes the systematic risk of a firm.
In the multi-factor Fama-French (FF) model there are three sources of risk – the market beta, the SMB beta, and the HML beta.
205 Professor Doron Avramov, Financial Econometrics 1 – Common Risk Factors
Under FF, other things being equal (ceteris paribus), a firm is riskier if its loading on SMB beta is higher.
Under FF, other things being equal (ceteris paribus), a firm is riskier if its loading on HML beta is higher.
206 Professor Doron Avramov, Financial Econometrics 2 – Moments for Asset Allocation
Pricing models deliver moments for asset allocation.
For instance, the tangency portfolio takes on the form
푉−1휇푒 푤 = 푇푃 휄′푉−1휇푒
207 Professor Doron Avramov, Financial Econometrics 2 – Asset Allocation
Under the CAPM, the vector of expected returns and the covariance matrix are given by: 푒 푒 휇 = 훽휇푚 ′ 2 푉 = 훽훽 휎푚 + Σ
where σ is the covariance matrix of the residuals in the time- series asset pricing regression. We denoted by Ψ the residual covariance matrix in the case wherein the off diagonal elements are zeroed out.
208 Professor Doron Avramov, Financial Econometrics 2 – Asset Allocation
The corresponding quantities under the FF model are
푒 푒 휇 = 훽푀퐾푇휇푚 + 훽푆푀퐿휇푆푀퐿 + 훽퐻푀퐿휇퐻푀퐿
′ 푉 = 훽σ퐹훽 + σ
where σ퐹 is the covariance matrix of the factors.
209 Professor Doron Avramov, Financial Econometrics 3 – Discount Factors
Expected return is the discount factor, commonly denoted by k, in present value formulas in general and firm evaluation in particular: 푇 퐶퐹 푃푉 = 푡 1 + 푘 푡 푡=1
In practical applications, expected returns are typically assumed to be constant over time, an unrealistic assumption.
210 Professor Doron Avramov, Financial Econometrics 3 – Discount Factors
Indeed, thus far we have examined models with constant beta and constant risk premiums 휇푒 = 훽′휆 where 휆 is a K-vector of risk premiums.
When factors are return spreads the risk premium is the mean of the factor.
Later we will consider models with time varying factor loadings.
211 Professor Doron Avramov, Financial Econometrics 4 – Benchmarks
Factors in asset pricing models serve as benchmarks for evaluating performance of active investments.
In particular, performance is the intercept (alpha) in the time series regression of excess fund returns on a set of benchmarks (typically four benchmarks in mutual funds and more so in hedge funds):
푒 푒 푟푡 = 훼 + 훽푀퐾푇 × 푟푀퐾푇,푡 + 훽푆푀퐵 × 푆푀퐵푡 +훽퐻푀퐿 × 퐻푀퐿푡 + 훽푊푀퐿 × 푊푀퐿푡 + 휀푡
212 Professor Doron Avramov, Financial Econometrics 5 – Corporate Finance
There is a plethora of studies in corporate finance that use asset pricing models to risk adjust asset returns.
Here are several examples:
Examining the long run performance of IPO firm.
Examining the long run performance of SEO firms.
Analyzing abnormal performance of stocks going through dividend initiation, dividend omission, splits, and reverse splits.
213 Professor Doron Avramov, Financial Econometrics 5 – Corporate Finance
Analyzing mergers and acquisitions
Analyzing the impact of change in board of directors.
Studying the impact of corporate governance on the cross section of average returns.
Studying the long run impact of stock/bond repurchase.
214 Professor Doron Avramov, Financial Econometrics Time Series Tests Time series tests are designated to examine the validity of asset pricing models in which factors are return spreads.
Example: the market factor is the return difference between the market portfolio and the risk-free asset.
Consumption growth is not a return spread.
Thus, the consumption CAPM cannot be tested using time series regressions, unless you form a factor mimicking portfolio (FMP) for consumption growth.
215 Professor Doron Avramov, Financial Econometrics Time Series Tests FMP is a convex combination of returns on some basis assets having the maximal correlation with consumption growth.
The statistical time series tests have an appealing economic interpretation. In particular:
Testing the CAPM amounts to testing whether the market portfolio is the tangency portfolio.
Testing multi-factor models amounts to testing whether some particular combination of the factors is the tangency portfolio.
216 Professor Doron Avramov, Financial Econometrics Testing the CAPM
Run the time series regression: 푒 푒 푟1푡 = 훼1 + 훽1푟푚푡 + 휀1푡 ⋮ 푒 푒 푟푁푡 = 훼푁 + 훽푁푟푚푡 + 휀푁푡
The null hypothesis is: 퐻0: 훼1 = 훼2 = ⋯ = 훼푁 = 0
217 Professor Doron Avramov, Financial Econometrics Testing the CAPM
In the following, I will introduce four times series test statistics:
WALD.
Likelihood Ratio.
GRS (Gibbons, Ross, and Shanken (1989)).
GMM.
218 Professor Doron Avramov, Financial Econometrics The Distribution of 훼. Recall, 훼 is asset mispricing.
The time series regressions can be rewritten using a vector form as: 푒 푒 푟푡 = Ƚ + Ⱦ ⋅ 푟푚푡 + ε푡 푁×1 푁×1 푁×1 1×1 푁×1
Let us assume that 푖푖푑 ε푡 ~ 푁 0, σ for 푡 = 1,2,3, … , 푇 푁×1 푁×푁
Let 휃 = 훼′, 훽′, 푣푒푐ℎ σ ′ ′ be the set of all parameters.
219 Professor Doron Avramov, Financial Econometrics The Distribution of 훼.
Under normality, the likelihood function for 휀푡 is 1 − 1 퐿 휀 |휃 = 푐 σ 2exp − 푟푒 − 훼 − 훽푟푒 ′σ−1 푟푒 − 훼 − 훽푟푒 푡 2 푡 푚푡 푡 푚푡 where 푐 is the constant of integration (recall the integral of a probability distribution function is unity).
220 Professor Doron Avramov, Financial Econometrics The Distribution of 훼.
Moreover, the IID assumption suggests that
푇 푇 − 퐿 휀1, 휀2, … , 휀푁|휃 = 푐 σ 2
1 × exp − σ푇 푟푒 − 훼 − 훽푟푒 ′σ−1 푟푒 − 훼 − 훽푟푒 2 푡=1 푡 푚푡 푡 푚푡
Taking the natural log from both sides yields 푇 1 ln 퐿 ∝ − ln Σ − σ푇 푟푒 − 훼 − 훽푟푒 ′σ−1 푟푒 − 훼 − 훽푟푒 2 2 푡=1 푡 푚푡 푡 푚푡
221 Professor Doron Avramov, Financial Econometrics The Distribution of 훼.
Asymptotically, we have 휃 − 휃መ ∼ 푁 0, Σ(휃)
where −1 휕2푙푛퐿 Σ 휃 = −퐸 휕휃휕휃′
222 Professor Doron Avramov, Financial Econometrics The Distribution of 훼. Let us estimate the parameters
푇 휕 푙푛 퐿 −1 푒 푒 = σ σ 푟푡 − 훼 − 훽푟푚푡 휕훼 푡=1
푇 휕 푙푛 퐿 −1 푒 푒 푒 = σ σ 푟푡 − 훼 − 훽푟푚푡 × 푟푚푡 휕훽 푡=1
푇 휕 푙푛 퐿 푇 −1 1 −1 ′ −1 = − σ + σ σ 휀푡휀푡 σ 휕σ 2 2 푡=1
223 Professor Doron Avramov, Financial Econometrics The Distribution of 훼. Solving for the first order conditions yields
푒 መ 푒 훼ො = 휇ො − 훽 ⋅ 휇ො푚
σ푇 푟푒 − 휇ො푒 푟푒 − 휇ො푒 መ 푡=1 푡 푚푡 푚 훽 = 푇 푒 푒 2 σ푡=1 푟푚푡 − 휇ො푚
224 Professor Doron Avramov, Financial Econometrics The Distribution of 훼. Moreover, 푇 1 Σ = 휀Ƹ 휀Ƹ′ 푇 푡 푡 푡=1 푇 1 휇ො푒 = 푟푒 푇 푡 푡=1 푇 1 휇푒 = 푟푒 푚 푇 푚푡 푡=1
225 Professor Doron Avramov, Financial Econometrics The Distribution of 훼.
Recall our objective is to find the standard errors of 훼ො.
Standard errors could be found using the information matrix.
226 Professor Doron Avramov, Financial Econometrics The Distribution of 훼.
The information matrix is constructed as follows
휕2 푙푛 퐿 휕2 푙푛 퐿 휕2 푙푛 퐿 , , 휕훼휕훼′ 휕훼휕훽′ 휕훼휕 σ′ 휕2 푙푛 퐿 휕2 푙푛 퐿 휕2 푙푛 퐿 퐼 휃 = − 퐸 , , 휕훽휕훼′ 휕훽휕훽′ 휕훽휕 σ′ 휕2 푙푛 퐿 휕2 푙푛 퐿 휕2 푙푛 퐿 , , 휕 σ 휕 훼′ 휕 σ 휕 훽′ 휕 σ 휕 σ′
227 Professor Doron Avramov, Financial Econometrics The Distribution of the Parameters
Exercise: establish the information matrix.
Notice that 훼ො and 훽መ are independent of σ - thus, you can ignore the second derivatives with respect to σ in the information matrix if your objective is to find the distribution of 훼ො or 훽መ or both.
If you aim to derive the distribution of σ then focus on the bottom right block of the information matrix.
228 Professor Doron Avramov, Financial Econometrics The Distribution of 훼.
We get: 2 1 휇ො푒 훼ො ∼ 푁 훼, 1 + 푚 Σ 푇 휎ො푚 Moreover, 1 1 መ 훽 ∼ 푁 훽, ⋅ 2 Σ 푇 휎ො푚
푇Σ ∼ 푊 푇 − 2, Σ
229 Professor Doron Avramov, Financial Econometrics The Distribution of 훼.
Notice that 푊 (푥, 푦) stands for the Wishart distribution with 푥 = 푇 − 2 degrees of freedom and a parameter matrix 푦 = σ.
230 Professor Doron Avramov, Financial Econometrics The Wald Test
Recall, if 푋 ∼ 푁 휇, Σ then 푋 − 휇 Σ−1 푋 − 휇 ∼ 휒2 푁 Here we test
퐻0: 훼ො = 0 퐻0 where 훼ො ~푁 0, Σ훼 퐻1: 훼ො ≠ 0
′ −1 2 The Wald statistic is 훼ො Σ훼 훼ො ∼ 휒 푁
231 Professor Doron Avramov, Financial Econometrics The Wald Test
which becomes: 2 −1 휇ො푒 훼ො′Σ−1훼ො 푚 ′ −1 퐽1 = 푇 1 + 훼ො Σ 훼ො = 푇 2 휎ො푚 1 + 푆푅푚
where 푆푅푚 is the Sharpe ratio of the market factor.
232 Professor Doron Avramov, Financial Econometrics Algorithm for Implementation
The algorithm for implementing the statistic is as follows: Run separate regressions for the test assets on the common factor:
푒 푒 푟1 = 푋 θ1 + ε1 1, 푟푚1 푇×1 푇×22×1 푇×1 푇×2 = ⋮ ⋮ where 푒 푒 1, 푟푚푇 푟푁 = 푋 θ푁 + ε푁 푇×2 ′ 푇×1 2×1 푇×1 휃푖 = 훼푖, 훽푖
233 Professor Doron Avramov, Financial Econometrics Algorithm for Implementation
Retain the estimated regression intercepts ′ 훼ො = 훼ො1, 훼ො2, … , 훼ො푁 and 휀Ƹ = 휀1Ƹ , ⋯ , 휀푁Ƹ 푇×푁
Compute the residual covariance matrix 1 Σ = 휀Ƹ′휀Ƹ 푇
234 Professor Doron Avramov, Financial Econometrics Algorithm for Implementation
Compute the sample mean and the sample variance of the factor.
Compute 퐽1.
235 Professor Doron Avramov, Financial Econometrics The Likelihood Ratio Test
We run the unrestricted and restricted specifications: 푒 푒 un: 푟푡 = 훼 + 훽푟푚푡 + 휀푡 휀푡 ∼ 푁 0, Σ 푒 ∗ 푒 ∗ ∗ ∗ res: 푟푡 = 훽 푟푚푡 + 휀푡 휀푡 ∼ 푁 0, Σ
Using MLE, we get:
236 Professor Doron Avramov, Financial Econometrics The Likelihood Ratio Test
σ푇 푟푒 푟푒 መ∗ 푡=1 푡 푚푡 훽 = 푇 푒 2 σ푡=1 푟푚푡 푇 1 Σ∗ = 휀Ƹ∗ 휀Ƹ∗′ 푇 푡 푡 푡=1 1 1 መ∗ 훽 ∼ 푁 훽, 푒 2 2 Σ 푇 휇ො푚 + 휎ො푚 푇Σ∗ ∼ 푊 푇 − 1, Σ
237 Professor Doron Avramov, Financial Econometrics The LR Test 푇 퐿푅 = ln 퐿∗ − ln 퐿 = − ln Σ∗ − ln Σ 2 ∗ 2 퐽2 = −2퐿푅 = 푇 ln Σ − ln Σ ∼ 휒 푁 Using some algebra, one can show that 퐽 퐽 = 푇 exp 2 − 1 1 푇 Thus, 퐽 퐽 = 푇 ⋅ ln 1 + 1 2 푇
238 Professor Doron Avramov, Financial Econometrics GRS (1989)
Theorem: let ~푁 0, Σ 푁×1
let A ~푊 휏, Σ where 휏 ≥ 푁 푁×푁
and let A and X be independent then 휏 − 푁 + 1 푋′퐴−1푋 ∼ 퐹 푁 푁,휏−푁+1
239 Professor Doron Avramov, Financial Econometrics GRS (1989) In our context: 1 − 푒 2 2 휇ො 퐻 푋 = 푇 1 + 푚 훼ො ∼0 푁 0, Σ 휎ො푚 퐴 = 푇Σ ∼ 푊 휏, Σ where 휏 = 푇 − 2 Then: −1 푒 2 푇 − 푁 − 1 휇ො푚 ′ −1 퐽3 = 1 + 훼ො Σ 훼ො ∼ 퐹 푁, 푇 − 푁 − 1 푁 휎ො푚 This is a finite-sample test.
240 Professor Doron Avramov, Financial Econometrics GMM
I will directly give the statistic without derivation:
−1 ′ ′ −1 −1 ′ 퐻0 2 퐽4 = 푇훼ො 푅 퐷푇푆푇 퐷푇 푅 ⋅ 훼ො ∼ 휒 (푁)
where
푅 = 퐼푁 , 0 푁×2푁 푁×푁 푁×푁
푒 1, 휇ො푚 퐷푇 = − 푒 푒 2 2 ⊗ 퐼푁 휇ො푚, 휇ො푚 + 휎ො푚
241 Professor Doron Avramov, Financial Econometrics GMM
Assume no serial correlation but Heteroskedasticity: 푇 1 푆 = 푥 푥′ ⊗ 휀Ƹ 휀Ƹ′ 푇 푇 푡 푡 푡 푡 푡=1 where 푒 푥푡 = 1, 푟푚푡 ′
Under homoscedasticity and serially uncorrelated moment conditions: 퐽4 = 퐽1.
That is, the GMM statistic boils down to the WALD.
242 Professor Doron Avramov, Financial Econometrics The Multi-Factor Version of Asset Pricing Tests
푒 푟푡 = Ƚ + Ⱦ ⋅ 퐹푡 + ε푡 푁×1 푁×1 푁×퐾 퐾×1 푁×1 ′ −1 −1 ′ −1 퐽1 = 푇 1 + 휇ො퐹Σ퐹 휇ො퐹 훼ො Σ 훼ො ∼ 휒 푁
퐽2 follows as described earlier.
푇 − 푁 − 퐾 −1 퐽 = 1 + 휇ො′ Σ−1휇ො 훼ො′Σ−1훼ො ∼ 퐹 3 푁 퐹 퐹 퐹 푁,푇−푁−퐾
where 휇ො퐹 is the mean vector of the factor based return spreads.
243 Professor Doron Avramov, Financial Econometrics The Multi-Factor Version of Asset Pricing Tests σ퐹 is the variance covariance matrix of the factors.
For instance, considering the Fama-French model: 푒 휎ො2 , 휎ො , 휎ො 휇ො푚 푚 푚,푆푀퐵 푚,퐻푀퐿 2 휇ො퐹 = 휇ො푆푀퐵 Σ퐹 = 휎ො푆푀퐵,푚, 휎ො푆푀퐵, 휎ො푆푀퐵,퐻푀퐿 휇ො 2 퐻푀퐿 휎ො퐻푀퐿,푚, 휎ො퐻푀퐿,푆푀퐵, 휎ො퐻푀퐿
244 Professor Doron Avramov, Financial Econometrics The Current State of Asset Pricing Models
The CAPM has been rejected in asset pricing tests.
The Fama-French model is not a big success.
Conditional versions of the CAPM and CCAPM display some improvement.
Should decision-makers abandon a rejected CAPM?
245 Professor Doron Avramov, Financial Econometrics Should a Rejected CAPM be Abandoned?
Often a misspecified model can improve estimation. In particular, assume that expected stock return is given by
휇푖 = 훼푖 + 푅푓 + 훽푖 휇푚 − 푅푓 where 훼푖 ≠ 0
You estimate 휇푖 using the sample mean and the misspecified CAPM: 1 휇ො 1 = σ푇 푅 푖 푇 푡=1 푖푡 2 መ 휇ො푖 = 푅푓 + 훽푖 휇ො푚 − 푅푓
246 Professor Doron Avramov, Financial Econometrics Mean Squared Error (MSE)
The quality of estimates is evaluated based on the Mean Squared Error (MSE) 2 1 1 푀푆퐸 = 퐸 휇ො푖 − 휇푖 2 2 2 푀푆퐸 = 퐸 휇ො푖 − 휇푖
247 Professor Doron Avramov, Financial Econometrics MSE, Bias, and Noise of Estimates
Notice that 푀푆퐸 = 푏푖푎푠2 + 푉푎푟 푒푠푡푖푚푎푡푒
Of course, the sample mean is unbiased thus 1 1 푀푆퐸 = 푉푎푟 휇ො푖
However, the CAPM is rejected, thus 2 2 2 푀푆퐸 = 훼푖 + 푉푎푟 휇ො푖
248 Professor Doron Avramov, Financial Econometrics The Bias-Variance Tradeoff
It might be the case that 푉푎푟 휇ො 2 is significantly lower than 푉푎푟 휇ො 1 - thus even when the CAPM is rejected, still zeroing out 훼푖 could produce a smaller mean square error.
249 Professor Doron Avramov, Financial Econometrics When is the Rejected CAPM Superior? 1 1 푉푎푟 휇ො 1 = 휎2 푅 = 훽2휎2 푅 + 휎2 휀 푖 푇 푖 푇 푖 푚 푖 2 መ 푒 푉푎푟 휇ො푖 = 푉푎푟 훽푖휇ො푚
250 Professor Doron Avramov, Financial Econometrics When is the Rejected CAPM Superior?
Using variance decomposition 2 መ 푒 መ 푒 푒 መ 푒 푒 푉푎푟 휇ො푖 = 푉푎푟 훽푖휇ො푚 = 퐸 푉푎푟 훽푖휇ො푚|휇ො푚 + 푉푎푟 퐸 훽푖휇ො푚|휇ො푚
2 푒 2 2 푒 2 휎 휀푖 1 푒 1 2 휇ෝ푚 2 휎ෝ푚 = 퐸 휇ො푚 2 ⋅ + 푉푎푟 훽푖휇ො푚 = 휎 휀푖 퐸 2 + 훽푖 휎ෝ푚 푇 푇 휎ෝ푚 푇 1 = 푆푅 휎2 휀 + 훽2휎ො2 푇 푚 푖 푖 푚 where
푒 2 휇ෝ푚 푆푅푚 = 퐸 2 휎ෝ푚
251 Professor Doron Avramov, Financial Econometrics When is the Rejected CAPM Superior?
Then 2 2 2 2 푉푎푟 휇ො푖 푆푅 휎 휀 + 훽 휎 = 푚 푖 푖 푚 = 푆푅 1 − 푅2 + 푅2 1 2 푚 휎 푅푖 푉푎푟 휇ො푖 where 푅2 is the 푅 squared in the market regression.
Since 푆푅푚 is small -- the ratio of the variance estimates is smaller than 1.
252 Professor Doron Avramov, Financial Econometrics Example
Let 2 휎 푅푖 = 0.01 2 휇ො푒 퐸 푚 = 0.05 휎ො푚 푅2 = 0.3
For what values of 훼푖 ≠ 0 it is still preferred to use the CAPM?
Find 훼푖 such that the MSE of the CAPM is smaller.
253 Professor Doron Avramov, Financial Econometrics Example
푀푆퐸 2 훼2 = 푆푅 1 − 푅2 + 푅2 + 푖 < 1 푀푆퐸 1 푚 푀푆퐸 1 훼2 0.05 × 0.7 + 0.3 + 푖 < 1 1 ⋅ 0.01 60 1 훼2 < ⋅ 0.01 × 0.665 푖 60 훼푖 < 0.01528 = 1.528%
254 Professor Doron Avramov, Financial Econometrics Economic versus Statistical Factors
Factors such as the market portfolio, SMB, HML, WML, TERM, SPREAD are pre-specified.
Such factors are considered to be economically based.
For instance, Fama and French argue that SMB and HML factors are proxying for underlying state variables in the economy.
255 Professor Doron Avramov, Financial Econometrics Economic versus Statistical Factors
Statistical factors are derived using econometric procedures on the covariance matrix of stock return. Two prominent methods are the factor analysis and the principal component analysis (PCA). Such methods are used to extract common factors. The first factor typically has a strong (about 96%) correlation with the market portfolio. Later, I will explain the PCA.
256 Professor Doron Avramov, Financial Econometrics The Economics of Time Series Test Statistics
Let us summarize the first three test statistics:
훼ො′Σ−1훼ො 퐽1 = 푇 2 1 + 푆푅푚
퐽 퐽 = 푇 ⋅ ln 1 + 1 2 푇
푇 − 푁 − 1 훼ො′Σ−1훼ො 퐽3 = 2 푁 1 + 푆푅푚
257 Professor Doron Avramov, Financial Econometrics The Economics of Time Series Test Statistics
The 퐽4 statistic, the GMM based asset pricing test, is actually a Wald test, just like J1, except that the covariance matrix of asset mispricing takes account of heteroscedasticity and often incorporates potential serial correlation.
Notice that all test statistics depend on the quantity 훼ො′Σ−1훼ො
258 Professor Doron Avramov, Financial Econometrics The Economics of the Time Series Tests
GRS show that this quantity has a very insightful representation.
Let us provide the steps.
259 Professor Doron Avramov, Financial Econometrics Understanding the Quantity 훼ො′σ−1훼ො. Consider an investment universe that consists of 푁 + 1 assets - the 푁 test assets as well as the market portfolio.
The expected return vector of the 푁 + 1 assets is given by 푒 푒′ ′ λ = μො푚, μො 푁+1 ×1 1×1 1×푁 푒 where 휇ො푚 is the estimated expected excess return on the market portfolio and 휇ො푒 is the estimated expected excess return on the 푁 test assets.
260 Professor Doron Avramov, Financial Econometrics Understanding the Quantity 훼ො′σ−1훼ො. The variance covariance matrix of the 푁 + 1 assets is given by
휎ො2 , 훽መ′휎ො2 Φ = 푚 푚 푁+1 × 푁+1 መ 2 훽휎ො푚, 푉
where 2 휎ො푚 is the estimated variance of the market factor. 훽መ is the N-vector of market loadings and 푉 is the covariance matrix of the N test assets.
261 Professor Doron Avramov, Financial Econometrics Understanding the Quantity 훼ො′σ−1훼ො. Notice that the covariance matrix of the N test assets is
መ መ′ 2 푉 = 훽훽 휎ො푚 + Σ
The squared tangency portfolio of the 푁 + 1 assets is
2 መ′ −1 መ 푆푅푇푃 = 휆 Φ 휆
262 Professor Doron Avramov, Financial Econometrics Understanding the Quantity 훼ො′σ−1훼ො. Notice also that the inverse of the covariance matrix is
휎ො2 −1 + 훽መ′Σ−1훽መ, −훽መ′Σ−1 Φ −1 = 푚 −Σ−1훽መ, Σ−1
Thus, the squared Sharpe ratio of the tangency portfolio could be represented as
263 Professor Doron Avramov, Financial Econometrics Understanding the Quantity 훼ො′σ−1훼ො.
푒 2 2 휇ෝ푚 푒 መ 푒 ′ −1 푒 መ 푒 푆푅푇푃 = + 휇ො − 훽휇ො푚 Σ 휇ො − 훽휇ො푚 휎ෝ푚 2 2 ′ −1 푆푅푇푃 = 푆푅푚 + 훼ො Σ 훼ො or ′ −1 2 2 훼ො Σ 훼ො = 푆푅푇푃 − 푆푅푚
264 Professor Doron Avramov, Financial Econometrics Understanding the Quantity 훼ො′σ−1훼ො. In words, the 훼ො′Σ−1훼ො quantity is the difference between the squared Sharpe ratio based on the 푁 + 1 assets and the squared Sharpe ratio of the market portfolio.
If the CAPM is correct then these two Sharpe ratios are identical in population, but not identical in sample due to estimation errors.
265 Professor Doron Avramov, Financial Econometrics Understanding the Quantity 훼ො′σ−1훼ො. The test statistic examines how close the two sample Sharpe ratios are.
Under the CAPM, the extra N test assets do not add anything to improving the risk return tradeoff.
The geometric description of 훼ො′Σ−1훼ො is given in the next slide.
266 Professor Doron Avramov, Financial Econometrics Understanding the Quantity 훼ො′σ−1훼ො.
휇
푇푃
푀
Φ2 푅푓 Φ1
휎 ′ −1 2 2 훼ො Σ 훼ො = Φ1 − Φ2
267 Professor Doron Avramov, Financial Econometrics Understanding the Quantity 훼ො′σ−1훼ො. So we can rewrite the previously derived test statistics as
2 2 푆푅푇푃 − 푆푅푚 2 퐽1 = 푇 2 ~휒 푁 1 + 푆푅푚
2 2 푇 − 푁 − 1 푆푅푇푃 − 푆푅푚 퐽3 = × 2 ~퐹 푁, 푇 − 푁 − 1 푁 1 + 푆푅푚
268 Professor Doron Avramov, Financial Econometrics Session #6: Asset Pricing Models with Time Varying Beta
269 Professor Doron Avramov, Financial Econometrics Asset Pricing Models with Time Varying Beta
We consider for simplicity only the one factor CAPM – extensions follow the same vein.
Let us model beta variation with the lagged dividend yield or any other macro variable – again for simplicity we consider only one information, predictive, macro, or lagged variable.
270 Professor Doron Avramov, Financial Econometrics Asset Pricing Models with Time Varying Beta
Typically, the set of predictive variables contains the dividend yield, the term spread, the default spread, the yield on a T-bill, inflation, lagged market return, market volatility, market illiquidity, etc.
271 Professor Doron Avramov, Financial Econometrics Conditional Models
Here is a conditional asset pricing specification: 푒 푒 푟푖푡 = 훼푖 + 훽푖푡푟푚푡 + 휀푖푡
훽푖푡 = 훽푖0 + 훽푖1푧푡−1
푧푡 = 푎 + 푏푧푡−1 + 휂푡 푒 푒 퐸(푟푚푡|푧푡−1) = 휇푚
cov ( 휀푖푡, 휂푡) = 0
272 Professor Doron Avramov, Financial Econometrics Conditional Asset Pricing Models
Substituting beta back into the asset pricing equation yields. 푒 푒 푒 푟푖푡 = 훼푖 + 훽푖0푟푚푡 + 훽푖1푟푚푡푧푡−1 + 휀푖푡
Interestingly, the one factor conditional CAPM becomes a two factor unconditional model – the first factor is the market portfolio, while the second is the interaction of the market with the lagged variable.
273 Professor Doron Avramov, Financial Econometrics Conditional Asset Pricing Models
You can use the statistics 퐽1 through 퐽4 to test such models.
If we have 퐾 factors and 푀 predictive variables then the K-conditional factor model becomes a 퐾 + 퐾푀 - unconditional factor model.
If you only scale the market beta, as is typically the case, we have an 푀 + 퐾 unconditional factor model.
274 Professor Doron Avramov, Financial Econometrics Conditional Moments
Suppose you are at time 푡 – what is the discount factor for time 푡 + 2? 푒 푒 푒 푟푖푡+2 = 훼푖 + 훽푖0푟푚푡+2 + 훽푖1푟푚푡+2푧푡+1 + 휀푖푡+2 푒 푒 = 훼푖 + 훽푖0푟푚푡+2 + 훽푖1푟푚푡+2 × 푎 + 푏푧푡 + 휂푡+1 + 휀푖푡+2 푒 푒 푒 푒 = 훼푖 + 훽푖0푟푚푡+2 + 푎훽푖1푟푚푡+2 + 훽푖1푏푟푚푡+2푧푡 + 훽푖1푟푚푡+2휂푡+1 + 휀푖푡+2
275 Professor Doron Avramov, Financial Econometrics Conditional Moments
푒 푒 푒 푒 퐸 푟푖푡+2ห푧푡 = 훼푖 + 훽푖0휇푚 + 푎훽푖1휇푚 + 훽푖1푏휇푚푧푡 푒 푒 = 훼푖 + 훽푖0 + 푎훽푖1 휇푚 + 훽푖1푏휇푚푧푡
Notice that here the discount factor, or the conditional expected return, is no longer constant through time.
Rather, it varies with the macro variable.
276 Professor Doron Avramov, Financial Econometrics Conditional Moments
Could you derive a general formula – in particular – you are at time 푡 what is the expected return for time 푡 + 푇 as a function of the model parameters as well as 푧푡?
277 Professor Doron Avramov, Financial Econometrics Conditional Moments
Next, the conditional covariance matrix – the covariance at time 푡 + 1 given 푧푡 – is given by
푒 ′ 2 푉 푟푡+1ห푧푡 = 훽0 + 훽1푧푡 훽0 + 훽1푧푡 휎푚 + Σ
where 훽0 and 훽1 are the N-asset versions of 훽푖0 and 훽푖1 respectively.
278 Professor Doron Avramov, Financial Econometrics Conditional Moments
Could you derive a general formula – in particular – you are at time 푡 what is the conditional covariance matrix for time 푡 + 푇 as a function of the model parameters as well as 푧푡?
Could you derive general expressions for the conditional moments of cumulative return?
279 Professor Doron Avramov, Financial Econometrics Conditional versus Unconditional Models
There are different ways to model beta variation. Here we used lagged predictive variables; other applications include using firm level variables such as size and book market to scale beta as well as modeling beta as an autoregressive process.
280 Professor Doron Avramov, Financial Econometrics Conditional versus Unconditional Models
You can also model time variation in the risk premiums in addition to or instead of beta variations.
Asset pricing tests show that conditional models typically outperform their unconditional counterparts.
281 Professor Doron Avramov, Financial Econometrics Different Ways to Model Beta Variation
The base case: beta is constant, or time invariant.
Case II: beta varies with macro conditions
훽푖푡 = 훽푖0 + 훽푖1푧푡−1
푧푡 = 푎 + 푏푧푡−1 + 휀푡
282 Professor Doron Avramov, Financial Econometrics Different Ways to Model Beta Variation
Case III: beta varies with firm-level size and the book-to-market ratio
훽푖푡 = 훽푖0 + 훽푖1푠푖푧푒푖,푡−1 + 훽푖2푏푚푖,푡−1
Case IV: beta is some function of both macro and firm-level variables as well as their interactions:
훽푖푡 = 푓 푧푡−1, 푠푖푧푒푖,푡−1, 푏푚푖,푡−1
283 Professor Doron Avramov, Financial Econometrics Different Ways to Model Beta Variation
Case V: beta follows an auto-regressive AR(1) process
훽푖푡 = 푎 + 푏훽푖,푡−1 + 푣푖푡
284 Professor Doron Avramov, Financial Econometrics Single vs. Multiple Factors
Notice that we focus on a single factor single macro variable.
We can expand the specification to include more factors and more macro and firm-level variables.
Even if we expand the number of factors it is common to model variation only in the market beta, while the other risk loadings are constant.
Some scholars model time variations in all factor loadings.
285 Professor Doron Avramov, Financial Econometrics Testing Conditional Models
You can implement the 퐽1 − 퐽4 test statistics only to those cases where beta is either constant or it varies with macro variables.
Those specifications involving firm-level characteristics require cross sectional tests.
The last specification (AR(1)) requires Kalman filtering methods involving state space representations.
286 Professor Doron Avramov, Financial Econometrics Session #7: GMVP, Tracking Error Volatility, Large scale covariance matrix
287 Professor Doron Avramov, Financial Econometrics GMVP
Of particular interest to academics and practitioners is the Global Minimum Volatility Portfolio. For two distinct reasons: 1. No need to estimate the notoriously difficult to estimate 휇. 2. Low volatility stocks have been found to outperform high volatility stocks.
288 Professor Doron Avramov, Financial Econometrics GMVP Optimization 푚푖푛 푤′푉푤 푠. 푡 푤′휄 = 1
푉−1휄 Solution: 푤 = 퐺푀푉푃 휄′푉−1휄
No analytical solution in the presence of portfolio constraints – such as no short selling.
289 Professor Doron Avramov, Financial Econometrics GMVP Optimization
Ex ante, the GMVP is the lowest volatility portfolio among all efficient portfolios.
Ex ante, it is also the lowest mean portfolio, but ex post it performs reasonably well in delivering high payoffs. That is related to the volatility anomaly: low volatility stocks have delivered, on average, higher payoffs than high volatility stocks.
290 Professor Doron Avramov, Financial Econometrics The Tracking Error Volatility (TEV) Portfolio
Actively managed funds are often evaluated based on their ability to achieve high return subject to some constraint on their Tracking Error Volatility (TEV).
In that context, a managed portfolio can be decomposed into both passive and active components.
TEV is the volatility of the active component.
291 Professor Doron Avramov, Financial Econometrics The Tracking Error Volatility (TEV) Portfolio
The passive component is the benchmark portfolio.
The benchmark portfolio changes with the investment objective.
For instance, if you invest in S&P500 stocks the proper benchmark would be the S&P index.
292 Professor Doron Avramov, Financial Econometrics Tracking Error Volatility: The Benchmark and Active Portfolios Let 푞 be the vector of weights of the benchmark portfolio.
Then the expected return and variance of the benchmark portfolio are given by ′ 휇퐵 = 푞 휇 2 ′ 휎퐵 = 푞 푉푞
where, as usual, μ and 푉 are the vector of expected return and the covariance matrix of stock returns.
293 Professor Doron Avramov, Financial Econometrics Tracking Error Volatility: The Benchmark and Active Portfolios The 푉 matrix can be estimated in different methods – most prominent of which will be discussed here.
The active fund manager attempts to outperform this benchmark.
Let 푥 be the vector of deviations from the benchmark, or the active part of the managed portfolio.
Of course, the sum of all the components of 푥, by construction, must be equal to zero.
294 Professor Doron Avramov, Financial Econometrics The Mathematics of TEV
So the fund manager invests 푤 = 푞 + 푥 in stocks, 푞 is the passive part of the portfolio and x is the active part.
2 ′ Notice that 휎휉 = 푥 푉푥 is the tracking error variance.
Also notice that the expected return and volatility of the chosen portfolio are ′ 휇푝 = 휇퐵 + 푥 휇 2 ′ 2 ′ 2 휎푝 = 푤 푉푤 = 휎퐵 + 2푞 푉푥 + 휎휉
295 Professor Doron Avramov, Financial Econometrics The Mathematics of TEV
The optimization problem is formulated as max 푥′ 휇 푥 푠. 푡. 푥′휄 = 0 푥′푉푥 = 휗
296 Professor Doron Avramov, Financial Econometrics The Mathematics of TEV
The resulting active part of the portfolio, 푥, is given by 휗 푎 푥 = ± 푉−1 휇 − 휄 푒 푐 where 휇′푉−1휇 = 푏 휇′푉−1휄 = 푎 휄′푉−1휄 = 푐 푎 − 푏2/푐 = 푒
297 Professor Doron Avramov, Financial Econometrics Caveats about the Minimum TEV Portfolio Richard Roll (distinguished UCLA professor) points out that the solution is independent on the benchmark.
Put differently, the active part of the portfolio 푥 is totally independent of the passive part 푞.
Of course, the overall portfolio 푞 + 푥 is impacted by 푞.
The unexpected result is that the active manager pays no attention to the assigned benchmark. So it does not really matter if the benchmark is S&P or any other index.
298 Professor Doron Avramov, Financial Econometrics TEV with total Volatility Constraint (based on Jorion – an Expert in Risk Management)
Given the drawbacks underlying the TEV portfolio we add one more constraint on the total portfolio volatility.
The derived active portfolio displays two advantages.
First, its composition does depend on the benchmark.
Second, the systematic volatility of the portfolio is controlled by the investor.
299 Professor Doron Avramov, Financial Econometrics TEV with total volatility constraint
The optimization is formulated as max 푥′ 휇 푥 푠. 푡. 푥′휄 = 0 푥′푉푥 = 휗 ′ 2 (푞 + 푥) 푉(푞 + 푥) = 휎푝 Home assignment: derive the optimal solution.
300 Professor Doron Avramov, Financial Econometrics Estimating the Large Scale Covariance Matrix
301 Professor Doron Avramov, Financial Econometrics Estimating the Covariance Matrix:
There are various applications in financial economics which use the covariance matrix as an essential input.
The Global Minimum Variance Portfolio, the minimum tracking error volatility portfolio, the mean variance efficient frontier, and asset pricing tests are good examples.
In what follows I will present the most prominent estimation methods of the covariance matrix.
302 Professor Doron Avramov, Financial Econometrics The Sample Covariance Matrix (Denoted S)
This method uses sample estimates.
Need to estimate 푁(푁 + 1)/2 parameters which is a lot.
You can use excel to estimate all variances and co-variances which is tedious and inefficient.
Here is a much more efficient method.
Consider 푇 monthly returns on 푁 risky assets.
We can display those returns in a 푇 by 푁 matrix 푅.
303 Professor Doron Avramov, Financial Econometrics The Sample Covariance Matrix (Denoted S)
Estimate the mean return of the 푁 assets – and denote the N-vector of the mean estimates by 휇ො.
Next, compute the deviations of the return observations from their sample means: 퐸 = 푅 − 휄휇ො′
where 휄푇 is a T vector of ones. Then the sample covariance matrix is estimated as 1 푆 = 퐸′퐸 푇
304 Professor Doron Avramov, Financial Econometrics The Equal Correlation Based Covariance Matrix (Denoted F): Estimate all 푁(푁 − 1)/2 pair-wise correlations between any two securities and take the average.
Let 휌ҧ be that average correlation, let 푠푖푖 be the estimated variance of asset 푖, and let 푠푗푗 be the estimated variance of asset j, both estimates are the i-th and j-th elements of the diagonal of S.
Then the matrix F follows as 퐹 푖, 푖 = 푠푖푖 퐹 푖, 푗 = 휌ҧ 푠푖푖푠푗푗
305 Professor Doron Avramov, Financial Econometrics The Factor Based Covariance Matrix:
Consider the time series regression
푟푡 = 훼 + 훽 × 퐹푡 + 푒푡
where 푟푡 is an N vector of returns at time t and 퐹푡 is a set of 퐾 factors. Factor means are denoted by 휇퐹
Notice that the mean return is given by
휇 = 훼 + 훽 × 휇퐹
306 Professor Doron Avramov, Financial Econometrics The Factor Based Covariance Matrix
Thus deviations from the means are given by 푟푡 − 휇 = 훽 × 퐹푡 − 휇퐹 + 푒푡 The factor based covariance matrix is estimated by መ መ′ 푉 = 훽σ퐹퐹훽 + Ψ Here, 훽መ is an 푁 by 퐾 matrix of factor loadings and Ψ is a diagonal matrix with each element represents the idiosyncratic variance of each of the assets.
307 Professor Doron Avramov, Financial Econometrics The Factor Based Covariance Matrix – Number of Parameters This procedure requires the estimation of 푁퐾 betas as well as 퐾 variances of the factors, 퐾(퐾 − 1)/2 correlations of those factors, and 푁 firm specific variances.
Overall, you need to estimate 푁퐾 + 퐾 + 퐾(퐾 − 1)/2 + 푁 parameters, which is considerably less than 푁(푁 + 1)/2 since 퐾 is much smaller than 푁.
For instance, using a single factor model – the number of parameters to be estimated is only 2푁 + 1.
308 Professor Doron Avramov, Financial Econometrics Steps for Estimating the Factor Based Covariance Matrix 1. Run the MULTIVARIATE regression of stock returns on asset pricing factors 푅 = 휄 훼′ + 퐹 훽′ + 퐸 = 퐹 ⋅ 휃′ + 퐸 푇×푁 푇×11×푁 푇×퐾퐾×푁 푇×푁 푇× 퐾+1 퐾+1 ×푁 푇×푁
where 퐹 = 휄푇, 퐹 휃 = 훼, 훽
309 Professor Doron Avramov, Financial Econometrics Steps for Estimating the Factor Based Covariance Matrix 2. Estimate 휃 −1 ′ −1 휃መ = 퐹′퐹 퐹′푅 = 푅′퐹 퐹′퐹 = 훼ො, 훽መ and retain 훽 only.
310 Professor Doron Avramov, Financial Econometrics Steps for Estimating the Factor Based Covariance Matrix 3. Estimate the covariance matrix of 퐸 1 Δ = 퐸′ 퐸 푁×푁 푇 푁×푇 푇×푁
4. Let Ψ be a diagonal matrix with the (i,i)-th component being equal to the (i,i)-th component of Δ.
311 Professor Doron Avramov, Financial Econometrics Steps for Estimating the Factor Based Covariance Matrix
5. Compute 푉 = 퐹 - 휄 ⋅ 휇ො′푓 푇×퐾 푇×퐾 푇×1 1×퐾
where 휇ො푓 is the mean return of the factors.
1 ′ 6. Estimate: σ퐹 = 푉 ⋅ 푉 퐾×퐾 푇 퐾×푇 푇×퐾
312 Professor Doron Avramov, Financial Econometrics Steps for Estimating the Factor Based Covariance Matrix መ 7. 푉 = 훽 σ푓 훽′ + Ψ 푁×푁 푁×퐾 퐾×퐾퐾×푁 푁×푁 This is the estimated covariance matrix of stock returns.
8. Notice – there is no need to run N individual regressions! Use multivariate specifications.
313 Professor Doron Avramov, Financial Econometrics A Shrinkage Approach – Based on a Paper by Ledoit and Wolf (LW) There is a well perceived paper (among Wall Street quants) by LW demonstrating an alternative approach to estimating the covariance matrix.
It had been claimed to deliver superior performance in reducing tracking errors relative to benchmarks as well as producing higher Sharpe ratios.
Here are the formal details.
314 Professor Doron Avramov, Financial Econometrics A Shrinkage Approach – Based on a Paper by Ledoit and Wolf (LW) Let 푆 be the sample covariance matrix, let 퐹 be the equal correlation based covariance matrix, and let 훿 be the shrinkage intensity. 푆 and 퐹 were derived earlier.
The operational shrinkage estimator of the covariance matrix is given by 푉ത = 훿መ∗ × 퐹 + (1 − 훿መ∗) × 푆
Notice that F is the shrinkage target.
315 Professor Doron Avramov, Financial Econometrics The Shrinkage Intensity
LW propose the following shrinkage intensity, based on optimization: 푘 훿መ∗ = max 0, min , 1 푇 휋−휂 where 푇 is the sample size and 푘 is given as 푘 = 훾 푁 푁 2 and where 훾 = σ푖=1 σ푗=1(푓푖푗 − 푠푖푗) with 푠푖푗 being the (푖, 푗) component of 푆 and 푓푖푗 is the (푖, 푗) component of 퐹.
316 Professor Doron Avramov, Financial Econometrics The Shrinkage Intensity
푁 푁
휋 = 휋푖푗 푖=1 푗=1 푇 1 _ 휋 = {(푟 − 푟ҧ )(푟 − 푟 ) − 푠 }2 푖푗 푇 푖푡 푖 푗푡 푗 푖푗 푡=1 푁 푁 푁 휌ҧ 푠푗푗 푠푖푖 휂 = 휋푖푖 + 휗푖푖,푖푗 + 휗푗푗,푖푗 2 푠푖푖 푠푗푗 푖=1 푖=1 푗=1,푗≠푖 푇 1 _ _ _ 휗 = (푟 − 푟 )2−푠 (푟 − 푟 ) (푟 − 푟 ) − 푠 푖푖,푖푗 푇 푖푡 푖 푖푖 푖푡 푖 푗푡 푗 푖푗 푡=1 푇 1 _ _ 휗 = (푟 − 푟ҧ )2−푠 (푟 − 푟 ) (푟 − 푟 ) − 푠 푗푗,푖푗 푇 푗푡 푗 푗푗 푖푡 푖 푗푡 푗 푖푗 푡=1 and with 푟푖푡 and 푟푖ҧ being the time 푡 return on asset 푖 and the time-series average of return on asset 푖, respectively.
317 Professor Doron Avramov, Financial Econometrics The Shrinkage Intensity – A Naïve Method
If you get overwhelmed by the derivation of the shrinkage intensity it would still be useful to use a naïve shrinkage approach, which often even works better. For instance, you can take equal weights: 1 1 푉ത = 퐹 + 푆 2 2
318 Professor Doron Avramov, Financial Econometrics Backtesting
We have proposed several methods for estimating the covariance matrix.
Which one dominates?
We can backtest all specifications.
That is, we can run a “horse race” across the various models searching for the best performer.
There are two primary methods for backtesting – rolling versus recursive schemes.
319 Professor Doron Avramov, Financial Econometrics The Rolling Scheme
You define the first estimation window.
It is well received to use the first 60 sample observations as the first estimation window.
Based on those 60 observations derive the GMVP under each of the following methods:
320 Professor Doron Avramov, Financial Econometrics Competing Covariance Estimates
The sample based covariance matrix
The equal correlation based covariance matrix
Factor model using the market as the only factor
Factor model using the Fama French three factors
Factor model using the Fama French plus Momentum factors
The LW covariance matrix – either the full or the naïve method.
321 Professor Doron Avramov, Financial Econometrics Out of Sample Returns
Then given the GMVPs compute the actual returns on each of the derived strategies.
For instance, if the derived strategy at time t is 푤푡 then the realized return at time t+1 would be ′ 푅푝,푡+1 = 푤푡 × 푅푡+1
where 푅푡+1 is the realized return at time 푡 + 1 on all the 푁 investable assets.
322 Professor Doron Avramov, Financial Econometrics Out of Sample Returns Suppose you rebalance every six months – derive the out of sample returns also for the following 5 months ′ 푅푝,푡+2 = 푤푡 × 푅푡+2 ′ 푅푝,푡+3 = 푤푡 × 푅푡+3 ′ 푅푝,푡+4 = 푤푡 × 푅푡+4 ′ 푅푝,푡+5 = 푤푡 × 푅푡+5 ′ 푅푝,푡+6 = 푤푡 × 푅푡+6
Then at time t+6 you re-derive the GMVPs.
323 Professor Doron Avramov, Financial Econometrics The Recursive Scheme
A recursive scheme is using an expanding window.
That is, you first estimate the GMVPs based on the first 60 observations, then based on 66 observations, and so on, while in the rolling scheme you always use the last 60 observations.
Pros: the recursive scheme uses more observations.
Cons: since the covariance matrix may be time varying perhaps you better drop initial observations.
324 Professor Doron Avramov, Financial Econometrics Out of Sample Returns
So you generate out of sample returns on each of the strategies starting from time 푡 + 1 till the end of sample, which we typically denote by 푇.
Next, you can analyze the out of sample returns.
For instance, you can form the table on the next page and examine which specification has been able to deliver the best performance.
325 Professor Doron Avramov, Financial Econometrics Out of Sample Returns
Rolling Scheme Recursive Scheme
FF+ FF+ S F MKT FF LW S F MKT FF LW MOM MOM
Mean
STD
SR
SP (5%)
alpha
IR
326 Professor Doron Avramov, Financial Econometrics Out of Sample Returns In the above table: Mean is the simple mean of the out of sample returns
STD is the volatility of those returns
SR is the associated Sharpe ratio obtained by dividing the difference between the mean return and the mean risk free rate by STD.
SP is the shortfall probability with a 5% threshold applied to the monthly returns.
327 Professor Doron Avramov, Financial Econometrics Out of Sample Returns
In the above table: alpha is the intercept in the regression of out of sample EXCESS returns on the contemporaneous market factor (market return minus the risk-free rate).
IR is the information ratio – obtained by dividing alpha by the standard deviation of the regression error, not the STD above.
Of course, higher SR, higher alpha, higher IR are associated with better performance.
328 Professor Doron Avramov, Financial Econometrics High Frequency Data
Let 푟푡 denote the continuously compounded one-year return for 2 year 푡, where 푟푡 are independent variables from 푁 휇, 휎 (iid).
Based on 푇 annual observations, we can estimate 휇 and 휎2:
1 1 휇ො = σ푇 푟 , and 휎ො2 = σ푇 푟 − 휇ො 2. 푇 푡=1 푡 푇−1 푡=1 푡
휎ෝ2 2휎ෝ4 We have 푉푎푟 휇ො = and 푉푎푟 휎ො2 = . 푇 푇
329 Professor Doron Avramov, Financial Econometrics High Frequency Data – continued
Now, partition each year to 푁 equally-separated intervals (e.g., 5 minutes), so that we have 푁 returns for each year 푡: 푁 푟푡,1, 푟푡,2, … , 푟푡,푁 so that 푟푡 = σ푛=1 푟푡,푛.
Overall, we have 푇 ∙ 푁 observations, and denote: 휇푛 = 퐸 푟푡,푛 (the expected return of one interval). 휇ෝ 휎ෝ2 Using the iid property, we can estimate: 휇ො = and 휎ො 2 = . 푛 푁 푛 푁 2 2 2 2 And we have: 푉푎푟 휇ො = 푁 푉푎푟 휇ො푛 and 푉푎푟 휎ො = 푁 푉푎푟 휎ො푛 .
330 Professor Doron Avramov, Financial Econometrics High Frequency Data – continued – The variance of 휇Ƹ and of 휎ො2
휎ෝ 2 As usual, 푉푎푟 휇ො = 푛 푛 푇푁 Therefore we have (also using the results from previous slide): 푁2휎ෝ 2 푁휎ෝ 2 휎ෝ2 푉푎푟 휇ො = 푁2푉푎푟 휇ො = 푛 = 푛 = . 푛 푇푁 푇 푇 The conclusion regarding 휇ො: its variance is the same whether it is estimated based on 푇 annual observations or whether it is estimated using 푇 ∙ 푁 high-frequency observations. There is no gain in using high-frequency observations. We will see a different result for 휎ො2.
331 Professor Doron Avramov, Financial Econometrics High Frequency Data – continued – The variance of 휇Ƹ and of 휎ො2 What is the variance of 휎ො2 based on 푇 ∙ 푁 high-frequency observations? 2휎ෝ 4 As usual, 푉푎푟 휎ො 2 = 푛 . 푛 푇푁 2푁2휎ෝ 4 1 2휎ෝ4 푉푎푟 휎ො2 = 푁2푉푎푟 휎ො 2 = 푛 = ∙ . 푛 푇푁 푁 푇 The variance of 휎ො2 is much smaller with the partitioning ! Moreover, the larger the partition is (i.e., 푁 is larger), the smaller the variance of the variance is.
332 Professor Doron Avramov, Financial Econometrics Session #8: Principal Component Analysis (PCA)
333 Professor Doron Avramov, Financial Econometrics PCA
The aim is to extract 퐾 common factors to summarize the information of a panel of rank 푁.
In particular, we have a 푇 × 푁 panel of stock returns where 푇 is the time dimension and 푁 (< 푇) is the number of firms – of course 퐾 << 푁 = 푅1, … , 푅푁 푇×푁
The PCA is an operation on the sample covariance matrix of stock returns.
334 Professor Doron Avramov, Financial Econometrics PCA
You have returns on 푁 stocks for 푇 periods
푅1 , … , 푅푁 푇×1 푇×1
푅෨1 = 푅1 − 휇ො1 … 푅෨푁 = 푅푁 − 휇ො푁 let 푅෨ = 푅෨1, 푅෨2, … , 푅෨푁 1 푉 = 푅෨′푅෨ 푇
335 Professor Doron Avramov, Financial Econometrics PCA
Extract K-Eigen vectors corresponding to the largest K-Eigen values.
Each of the Eigen vector is an N by 1 vector.
The extraction mechanism is as follows.
The first Eigen vector is obtained as ′ max 푤1푉푤1 푊1 ′ 푠. 푡 푤1푤1 = 1
336 Professor Doron Avramov, Financial Econometrics PCA
푤1 is an Eigen vector since 푉푤1 = 휆1푤1 Moreover ′ 휆1 = 푤1푉푤1
휆1 is therefore the highest Eigen value.
337 Professor Doron Avramov, Financial Econometrics PCA
Extracting the second Eigen vector
′ max 푤2푉푤2 푤2 ′ 푠. 푡 푤2푤2 = 1 ′ 푤1푤2 = 0
338 Professor Doron Avramov, Financial Econometrics PCA
The optimization yields:
푉푤2 = 휆2푤2 ′ 휆2 = 푤2푉푤2 < 휆1
The second Eigen value is smaller than the first due to the presence of one extra constraint in the optimization – the orthogonality constraint.
339 Professor Doron Avramov, Financial Econometrics PCA
The K-th Eigen vector is derived as ′ max 푤퐾푉푤퐾 ′ 푠. 푡 푤퐾푤퐾 = 1 ′ 푤퐾푤1 = 0 ′ 푤퐾푤2 = 0 ⋮ ′ 푤퐾푤퐾−1 = 0
340 Professor Doron Avramov, Financial Econometrics PCA
The optimization yields: 푉푤퐾 = 휆퐾푤퐾
′ 휆퐾 = 푤퐾푉푤퐾 < 휆퐾−1 < ⋯ < 휆2 < 휆1
341 Professor Doron Avramov, Financial Econometrics PCA
Then - each of the K-Eigen vectors delivers a unique asset pricing factor.
Simply, multiply excess stock returns by the Eigen vectors: 푒 퐹1 = 푅 ⋅ 푤1 푇×1 푇×푁 푁×1 ⋮ 푒 퐹퐾 = 푅 ⋅ 푤퐾 푇×1 푇×푁 푁×1
342 Professor Doron Avramov, Financial Econometrics PCA
Recall, the basic idea here is to replace the original set of 푁 variables with a lower dimensional set of K-factors (퐾 << 푁).
The contribution of the 푗-th Eigen vector to explain the covariance matrix of stock returns is
휆푗 푁 σ푖=1 휆푖
343 Professor Doron Avramov, Financial Econometrics PCA
Typically the first three Eigen vectors explain over and above 95% of the covariance matrix.
What does it mean to “explain the covariance matrix”? Coming up soon!
344 Professor Doron Avramov, Financial Econometrics Understanding the PCA: Digging Deeper
The covariance matrix can be decomposed as
′ 휆1, … , 0 푤1 푉 = 푤1, … , 푤푁 ⋮ ⋱ ⋮ ′ 0, … , 휆푁 푤푁
345 Professor Doron Avramov, Financial Econometrics Understanding the PCA: Digging Deeper
If some of the 휆 − 푠 are either zero or negative – the covariance matrix is not properly defined -- it is not positive definite.
In fixed income analysis – there are three prominent Eigen vectors, or three factors.
The first factor stands for the term structure level, the second for the term structure slope, and the third for the curvature of the term structure.
346 Professor Doron Avramov, Financial Econometrics Understanding the PCA: Digging Deeper
In equity analysis, the first few (up to three) principal components are prominent.
Others are around zero.
The attempt is to replace the sample covariance matrix by the matrix 푉෨ which mostly summarizes the information in the sample covariance matrix.
347 Professor Doron Avramov, Financial Econometrics Understanding the PCA: Digging Deeper
The matrix 푉෨ is given by ′ 휆1, … , 0 푤1 푉෨ = 푤1, … , 푤푘 ⋮ ⋱ ⋮ 푁×푁 ′ 0, … , 휆푘 푤푘
Of course, the dimension of 푉෨ is 푁 by 푁.
However, its rank is 퐾, thus the matrix is not invertible.
348 Professor Doron Avramov, Financial Econometrics Is 푉 close to 푉෨? This is the same as asking: what does it mean to explain the sample covariance matrix?
349 Professor Doron Avramov, Financial Econometrics Is 푉 close to 푉෨? Let us represent returns as
푟1푡 = 훼1 + Ⱦ11푓1푡 + Ⱦ12푓2푡 + ⋯ + Ⱦ1퐾푓퐾푡 + 휀1푡 ∀푡 = 1, … , 푇 푟1ǁ 푡 ⋮ 푟푁푡 = 훼푁 + Ⱦ푁1푓1푡 + Ⱦ푁2푓2푡 + ⋯ + Ⱦ푁퐾푓퐾푡 + 휀푁푡 ∀푡 = 1, … , 푇 푟ǁ푁푡
350 Professor Doron Avramov, Financial Econometrics Is 푉 close to 푉෨?
where 푓1푡 … 푓퐾푡 are the principal component based factors and 훽푖1 … 훽푖퐾 are the exposures of firm i to those factors.
351 Professor Doron Avramov, Financial Econometrics Is 푉 close to 푉෨? 푉 is closed enough to 푉෨ - if 1. The variances of the residuals cannot be dramatically reduced by adding more factors. 2. The pairwise cross-section correlations of the residuals cannot be considerably reduced by adding more factors.
352 Professor Doron Avramov, Financial Econometrics The Contribution of the PC-S to Explain Portfolio Variation
′ Let Ⱦ푖 = 훽푖1, … , 훽푖퐾 be the exposures of firm 푖 to the 퐾 퐾×1 common factors.
Let 푤푝 be an N-vector of portfolio weights: 푤푝 = 푤1푝, 푤2푝, … , 푤푁푝 ′
Recall, 푅 is a 푇 × 푁 matrix of stock returns.
353 Professor Doron Avramov, Financial Econometrics The Contribution of the PC-S to Explain Portfolio Variation The portfolio’s rate of return is 푅푝 = ⋅ 푤푝 푇×1 푇×푁 푁×1
Moreover, the portfolio time t return is given by ′ 푅푝푡 = 푤1푝푟1푡 + 푤2푝푟2푡 + ⋯ + 푤푁푝푟푁푡 ≡ 푤푝 ⋅ 푟푡 1×푁 푁×1
354 Professor Doron Avramov, Financial Econometrics The Contribution of the PC-S to Explain Portfolio Variation We can approximate the portfolio’s rate of return as:
푅෨푝푡 = 푤1푝 훽11푓1푡 + 훽12푓2푡 + ⋯ + 훽1퐾푓퐾푡
+푤2푝 훽21푓1푡 + 훽22푓2푡 + ⋯ + 훽2퐾푓퐾푡 + … … … … … … … … … … … … …
+푤푁푝 훽푁1푓1푡 + 훽푁2푓2푡 + ⋯ + 훽푁퐾푓퐾푡
355 Professor Doron Avramov, Financial Econometrics The Contribution of the PC-S to Explain Portfolio Variation
Thus, 푅෨푝푡 = 훿1푓1푡 + 훿2푓2푡 + ⋯ + 훿퐾푓퐾푡 where ′ 훿1 = 푤푝 ⋅ 훽1 1×푁 푁×1 ′ 훿2 = 푤푝 ⋅ 훽2 1×푁 푁×1 ⋮ ′ 훿퐾 = 푤푝 ⋅ 훽푘 1×푁 푁×1
356 Professor Doron Avramov, Financial Econometrics The Contribution of the PC-S to Explain Portfolio Variation Notice that
훿1 … 훿퐾 are the 퐾 loadings on the common factors,
or they are the risk exposures, while 푓1푡 … 푓퐾푡 are the K-realizations of the factors at time t.
357 Professor Doron Avramov, Financial Econometrics Explain the Portfolio Variance
The actual variation is 2 ′ 휎 (푅푝푡) = 푤푝푉푤푝 The approximated variation is 2 2 2 2 휎 (푅෨푝푡) = 훿1 var ( 푓1푡) + 훿2 var ( 푓2푡) + ⋯ + 훿퐾 var ( 푓퐾푡)
Both quantities are quite similar.
358 Professor Doron Avramov, Financial Econometrics Explain the Portfolio Variance
The contribution of the i-th PC to the overall portfolios variance is:
2 훿푖 푣푎푟 ( 푓푖) 푘 2 σ푗=1 훿푗 푣푎푟 ( 푓푗)
359 Professor Doron Avramov, Financial Econometrics Asy. PCA: What if N>T?
Then create a 푇 × 푇 matrix 푉 and extract 퐾 Eigen vectors – those Eigen vectors are the factors 1 푉 = 퐸′퐸 푁 where 퐸 = 푅 − 휇ො ⋅ 휄′푁 푇×푁 푇×푁 푇×1 1×푁
and 휇ො is the T-vector of cross sectional (across- stocks) mean of returns.
360 Professor Doron Avramov, Financial Econometrics Other Applications
PCA can be implemented in a host of other applications.
For instance, you want to predict economic growth with many predictors, say 푀 where 푀 is large.
You have a panel of 푇 × 푀 predictors, where 푇 is the time- series dimension.
361 Professor Doron Avramov, Financial Econometrics Other Applications
In a matrix form 푍11, 푍12, … , 푍1푀 = ⋮ 푇×푀 푍푇1, 푍푇2, … , 푍푇푀
where 푍푡푚 is the m-the predictor realized at time 푡.
362 Professor Doron Avramov, Financial Econometrics Other Applications
If 푇 > 푀 compute the covariance matrix of 푍 – then extract 퐾 principal components such that you summarize the M-dimension of the predictors with a smaller subspace of order 퐾 << 푀.
You extract 푤1 , … , 푤퐾 Eigen vectors. 푀×1 푀×1
363 Professor Doron Avramov, Financial Econometrics Other Applications
Then you construct K predictors:
푍1 = ⋅ 푤1 푇×1 푇×푀 푀×1 ⋮ 푍퐾 = ⋅ 푤퐾 푇×1 푇×푀 푀×1
364 Professor Doron Avramov, Financial Econometrics What if 푀 > 푇?
If 푀 > 푇 then you extract 퐾 Eigen vectors from the T×T matrix.
In this case, the 퐾-predictors are the extracted 퐾 Eigen vectors.
Be careful of a look-ahead bias in real time prediction.
365 Professor Doron Avramov, Financial Econometrics The Number of Factors in PCA
An open question is: how many factors/Eigen vectors should be extracted?
Here is a good mechanism: set 퐾max - the highest number of factors.
Run the following multivariate regression for 퐾 = 1,2, … , 퐾max 푅 = 휄 + 훼′ + 퐹 훽′ + 퐸 푇×푁 푇×1 1×푁 푇×퐾 퐾×푁 푇×푁
366 Professor Doron Avramov, Financial Econometrics The Number of Factors in PCA
Estimate first the residual covariance matrix and then the average of residual variances: 1 푉 = 퐸′퐸 푁×푁 푇 − 퐾 − 1
푡푟 푉 휎ො2 퐾 = 푁
where 푡푟 푉 is the sum of diagonal elements in 푉
367 Professor Doron Avramov, Financial Econometrics The Number of Factors in PCA
Compute for each chosen 퐾
푁 + 푇 푁푇 푃퐶 퐾 = 휎ො2 퐾 + 퐾휎ො2 퐾 ⋅ ln max 푁푇 푁 + 푇
and pick 퐾 which minimizes this criterion.
Notice that with more factors the first component (goodness of fit) diminishes but the second component (penalty factor) rises. There is a tradeoff here.
368 Professor Doron Avramov, Financial Econometrics Session #9: Bayesian Econometrics
369 Professor Doron Avramov, Financial Econometrics Bayes Rule
Let x and 푦 be two random variables Let 푃 푥 and 푃 푦 be the two marginal probability distribution functions of x and y Let 푃 푥 푦 and 푃 푦 푥 denote the corresponding conditional pdfs Let 푃 푥, 푦 denote the joint pdf of x and 푦 It is known from the law of total probability that the joint pdf can be decomposed as 푃 푥, 푦 = 푃 푥 푃 푦 푥 = 푃 푦 푃 푥 푦 Therefore 푃 푦 푃 푥 푦 푃 푦 푥 = 푃 푥 = 푐푃 푦 푝 푥 푦 where c is the constant of integration (see next page) The Bayes Rule is described by the following proportion
푃 푦 푥 ∝ 푃 푦 푃 푥 푦
370 Professor Doron Avramov, Financial Econometrics Bayes Rule
Notice that the right hand side retains only factors related to y, thereby excluding 푃 푥
푃 푥 , termed the marginal likelihood function, is 푃 푥 = න 푃 푦 푃 푥 푦 푑푦
= න 푃 푥, 푦 푑푦
as the conditional distribution 푃 푦 푥 integrates to unity. The marginal likelihood 푃 푥 is an essential ingredient in computing the model posterior probability, or the probability that a model is correct. The marginal likelihood obtains by integrating out y from the joint density 푃 푥, 푦 Similarly, if the joint distribution is 푃 푥, 푦, 푧 and the pdf of interest is 푃 푥, 푦 one integrates 푃 푥, 푦, 푧 with respect to z.
371 Professor Doron Avramov, Financial Econometrics Bayes Rule
The essence of Bayesian econometrics is the Bayes Rule. Ingredients of Bayesian econometrics are parameters underlying a given model, the sample data, the prior density of the parameters, the likelihood function describing the data, and the posterior distribution of the parameters. A predictive distribution is often involved. Indeed, in the Bayesian setup parameters are stochastic while in the classical (non Bayesian) approach parameters are unknown constants. Decision making is based on the posterior distribution of the parameters or the predictive distribution of the data as described below. On the basis of the Bayes rule, in what follows, y stands for unknown stochastic parameters, x for the data, 푃 푦 푥 for the posetior distribution, 푃 푦 for the prior, and 푃 푥 푦 for the likelihood Then the Bayes rule describes the relation between the prior, the likelihood, and the posterior
푃 푦 푥 ∝ 푃 푦 푃 푥 푦 Zellner (1971) is an excellent reference
372 Professor Doron Avramov, Financial Econometrics Bayes Econometrics in Financial Economics
You observe the returns on the market index over T months: 푟1, … , 푟푇
Let 푅: 푟1, … , 푟푇 ’ denote the 푇 × 1 vector of all return realizations 2 Assume that 푟푡~푁 휇, 휎0 for 푡 = 1, … , 푇 where µ is a stochastic random variable denoting the mean return 2 휎0 is the volatility which , at this stage, is assumed to be a known constant and returns are IID (independently and identically distributed) through time. By Bayes rule 2 2 푃 휇 푅, 휎0 ∝ 푃 휇 푃 푅 휇, 휎0 where 2 푃 휇 푅, 휎0 is the posterior distribution of µ 푃 휇 is the prior distribution of µ 2 and 푃 푅 휇, 휎0 is the joint likelihood of all return realizations.
373 Professor Doron Avramov, Financial Econometrics Bayes Econometrics: Likelihood
. The likelihood function of a normally distributed return realization is given by
2 1 1 2 푃 푟푡 휇, 휎0 = 2 푒푥푝 − 2 푟푡 − 휇 2휋휎0 2휎0 Since returns are assumed to be IID, the joint likelihood of all realized returns is
푇 − 1 2 2 2 푇 2 푃 푅 휇, 휎0 = 2휋휎0 푒푥푝 − 2 σ푡=1 푟푡 − 휇 2휎0 Notice: 2 2 σ 푟푡 − 휇 = σ 푟푡 − 휇Ƹ + 휇Ƹ − 휇 = ν푠2 + 푇 휇 − 휇Ƹ 2 since the cross product is zero, and ν = 푇 − 1 1 푠2 = 푟 − 휇Ƹ 2 푇 − 1 푡
1 휇Ƹ = σ 푟 푇 푡 374 Professor Doron Avramov, Financial Econometrics Prior
The prior is specified by the researcher based on economic theory, past experience, past data, current similar data, etc. Often, the prior is diffuse or non-informative For the next illustration, it is assumed that 푃 휇 ∝ 푐 ,that is, the prior is diffuse, non- informative, in that it apparently conveys no information on the parameters of interest. I emphasize “apparently” since innocent diffuse priors could exert substantial amount of information about quantities of interest which are non-linear functions of the parameters. Informative priors with sound economic appeal are well perceived in financial economics. For instance, Kandel and Stambaugh (1996), who study asset allocation when stock returns are predictable, entertain informative prior beliefs weighted against predictability. Pastor and Stambaugh (1999) introduce prior beliefs about expected stock returns which consider factor model restrictions. Avramov, Cederburg, and Kvasnakova (2015) study prior beliefs about predictive regression parameters which are centered around values implied by either prospect theory or the long run risk model of Bansal and Yaron (2004). Computing posterior probabilities of competing models (e.g., Avramov (2002)) necessitates the use of informative priors. Here, diffuse priors won’t do the work.
375 Professor Doron Avramov, Financial Econometrics The Posterior Distribution
With diffuse prior and normal likelihood, the posterior is
2 1 2 2 푃 휇 푅, 휎0 ∝ 푒푥푝 − 2 ν푠 + 푇 휇 − 휇Ƹ 2휎0
푇 2 ∝ 푒푥푝 − 2 휇 − 휇Ƹ 2휎0 The bottom relation follows since only factors related to µ are retained The posterior distribution of the mean return is given by
2 2 휎0 ൗ 휇|푅, 휎0 ~푁 휇Ƹ, 푇 In classical econometrics: 2 2 휎0 휇Ƹ|푅, 휎0 ~푁 휇, ൗ푇 That is, in classical econometrics, the sample estimate of µ is stochastic while µ itself is an unknown constant.
376 Professor Doron Avramov, Financial Econometrics Informative Prior
The prior on the mean return is often modeled as 1 − 1 2 2 푃 휇 ∝ 휎푎 푒푥푝 − 2 휇 − 휇푎 2휎푎
where 휇푎 and 휎푎 are prior parameters to be specified by the researcher The posterior obtains by combining the prior and the likelihood: 2 2 푃 휇 푅, 휎0 ∝ 푃 휇 푃 푅 휇, 휎0 2 1 휇−휇푎 푇 2 ∝ 푒푥푝 − 2 + 2 휇 − 휇Ƹ 2 휎푎 휎0
1 휇−휇 2 ∝ 푒푥푝 − 2 휎2 The bottom relation obtains by completing the square on µ Notice, in particular, 2 2 휇 푇 2 휇 2 + 2 휇 = 2 휎푎 휎0 휎 1 푇 1 2 + 2 = 2 휎푎 휎0 휎 377 Professor Doron Avramov, Financial Econometrics The Posterior Mean
Hence, the posterior variance of the mean is −1 2 1 1 휎 = 2 + 2 휎푎 휎0 ൗ푇 = (prior precision + likelihood precision)−1 Similarly, the posterior mean of 휇 is
2 휇0 푇휇Ƹ 휇 = 휎 2 + 2 휎푎 휎0 = 푤1휇0 + 푤2휇Ƹ where 1 휎 2 prior precision 푤 = 푎 = 1 1 1 prior precision + likelihood precision 2 + 2 휎푎 휎0 ൗ 푇 푤2 = 1 − 푤1 So the posterior mean of µ is the weighted average of the prior mean and the sample mean with weights depending on the prior and likelihood precisions, respectively. 378 Professor Doron Avramov, Financial Econometrics What if σ is unknown? – The case of Diffuse Prior
Bayes: 푃 휇, 휎 푅 ∝ 푃 휇, 휎 푃 푅 휇, 휎 The non-informative prior is typically modeled as 푃 휇, 휎 ∝ 푃 휇 푃 휎 푃 휇 ∝ 푐 푃 휎 ∝ 휎−1 Thus, the joint posterior of µ and σ is 1 푃 휇, 휎 푅 ∝ 휎− 푇+1 푒푥푝 − ν푠2 + 푇 휇 − 휇Ƹ 2 2휎2 The conditional distribution of the mean follows straightforwardly
휎2 푃 휇 휎, 푅 is 푁 휇Ƹ, ൗ푇 More challenging is to uncover the marginal distributions, which are obtained as 푃 휇 푅 = න 푃 휇, 휎 푅 푑휎
푃 휎 푅 = න 푃 휇, 휎 푅 푑휇
379 Professor Doron Avramov, Financial Econometrics Solving the Integrals: Posterior of 휇 Let Ƚ = ν푠2 + 푇 휇 − 휇Ƹ 2
Then, ∞ − 푇+1 훼 푃 휇 푅 ∝ න 휎 푒푥푝 − 2 푑휎 휎=0 2휎 We do a change of variable 훼 푥 = 2 2휎 1 1 1 푑휎 −1 −1 = −2 2훼2푥 2 푑푥 푇+1 − 훼 2 휎−푇+1 = 2푥 푇−2 푇 푇 − ∞ −1 Then 2 2 2 푃 휇 푅 ∝ 2 훼 푥=0 푥 푒푥푝 −푥 푑푥 푇 ∞ −1 푇 Notice 푥 2 푒푥푝 −푥 푑푥 = Γ 푥=0 2 Therefore, 푇−2 푇 푇 − 푃 휇 푅 ∝ 2 2 Γ 훼 2 2 ν+1 − ∝ ν푠2 + 푇 휇 − 휇Ƹ 2 2 휇−휇ෝ We get 푡 = 푠 ~푡 ν , corresponding to the Student t distribution with ν degrees of freedom. ൗ 푇
380 Professor Doron Avramov, Financial Econometrics The Marginal Posterior of σ
1 The posterior on 휎 푃 휎 푅 ∝ 휎− 푇+1 푒푥푝 − ν푠2 + 푇 휇 − 휇Ƹ 2 푑휇 2휎2 ν푠2 푇 휎− 푇+1 푒푥푝 − 푒푥푝 − 휇 − 휇Ƹ 2 푑휇 ∝ 2휎2 2휎2 푇 휇−휇ෝ Let 푧 = , then 휎 푑푧 1 = 푇 푑휇 휎
ν푠2 2 푃 휎 푅 ∝ 휎−푇푒푥푝 − 푒푥푝 − 푧 ൗ 푑푧 2휎2 2 ν푠2 ∝ 휎−푇푒푥푝 − 2휎2 ν푠2 ∝ 휎− ν+1 푒푥푝 − 2휎2 which corresponds to the inverted gamma distribution with ν degrees of freedom and parameter s The explicit form (with constant of integration) of the inverted gamma is given by ν 2 ν푠2 ൗ2 ν푠2 푃 휎 ν, 푠 = 휎− ν+1 푒푥푝 − ν 2 2휎2 Γ 2 381 Professor Doron Avramov, Financial Econometrics The Multiple Regression Model
The regression model is given by 푦 = 푋훽 + 푢 where y is a 푇 × 1 vector of the dependent variables X is a 푇 × 푀 matrix with the first column being a 푇 × 1 vector of ones β is an M × 1 vector containing the intercept and M-1 slope coefficients and u is a 푇 × 1 vector of residuals. 2 We assume that 푢푡~푁 0, 휎 ∀ 푡 = 1, … , 푇 and IID through time The likelihood function is 1 푃 푦 푋, 훽, 휎 ∝ 휎−푇푒푥푝 − 푦 − 푋훽 ′ 푦 − 푋훽 2휎2 1 ′ ∝ 휎−푇푒푥푝 − ν푠2 + 훽 − 훽መ 푋′푋 훽 − 훽መ 2휎2
382 Professor Doron Avramov, Financial Econometrics The Multiple Regression Model
where ν = 푇 − 푀 훽መ = 푋′푋 −1푋′푦 1 ′ 푠2 = 푦 − 푋훽መ 푦 − 푋훽መ ν It follows since ′ 푦 − 푋훽 ′ 푦 − 푋훽 = 푦 − 푋훽መ − 푋 훽 − 훽መ 푦 − 푋훽መ − 푋 훽 − 훽መ ′ ′ = 푦 − 푋훽መ 푦 − 푋훽መ + 훽 − 훽መ 푋′푋 훽 − 훽መ while the cross product is zero
383 Professor Doron Avramov, Financial Econometrics Assuming Diffuse Prior
The prior is modeled as 1 푃 훽, 휎 ∝ 휎 Then the joint posterior of β and σ is 1 ′ 푃 훽, 휎 푦, 푋 ∝ 휎− 푇+1 푒푥푝 − ν푠2 + 훽 − 훽መ 푋′푋 훽 − 훽መ 2휎2 The conditional posterior of β is
1 ′ 푃 훽 휎, 푦, 푋 ∝ 푒푥푝 − 훽 − 훽መ 푋′푋 훽 − 훽መ 2휎2
which obeys the multivariable normal distribution
푁 훽መ, 푋′푋 −1휎2
384 Professor Doron Avramov, Financial Econometrics Assuming Diffuse Prior
What about the marginal posterior for β ? 푃 훽 푦, 푋 = න 푃 훽, 휎 푦, 푋 푑휎
푇 ′ − ൗ2 ∝ ν푠2 + 훽 − 훽መ 푋′푋 훽 − 훽መ
which pertains to the multivariate student t with mean 훽መ and T-M degrees of freedom What about the marginal posterior for 휎? 푃 휎 푦, 푋 = න 푃 훽, 휎 푦, 푋 푑훽
ν푠2 ∝ 휎− ν+1 푒푥푝 − 2휎2 which stands for the inverted gamma with T-M degrees of freedom and parameter s . You can simulate the distribution of β in two steps without solving analytically the integral, drawing first 휎 from its inverted gamma distribution and then drawing from the conditional of β which is normal as shown earlier.
385 Professor Doron Avramov, Financial Econometrics Bayesian Updating/Learning
Suppose the initial sample consists of 푇1 observations of 푋1 and 푦1. Suppose further that the posterior distribution of 훽, 휎 based on those observations is given by: 1 푃 훽, 휎 푦 , 푋 ∝ 휎−(푇1+1)푒푥푝 − 푦 − 푋 훽 ′ 푦 − 푋 훽 1 1 2휎2 1 1 1 1 1 ∝ 휎−(푇1+1)푒푥푝 − ν 푠 2 + 훽 − 훽መ ′푋 ′푋 훽 − 훽መ 2휎2 1 1 1 1 1 1 where ν1 = 푇1 − 푀 መ −1 훽1 = 푋1′푋1 푋1푦1 2 መ ′ መ ν1푠1 = 푦1 − 푋1훽1 푦1 − 푋1훽1
You now observe one additional sample 푋2 and 푦2 of length 푇2 observations The likelihood based on that sample is 1 푃 푦 , 푋 훽, 휎 ∝ 휎−푇2푒푥푝 − 푦 − 푋 훽 ′ 푦 − 푋 훽 2 2 2휎2 2 2 2 2
386 Professor Doron Avramov, Financial Econometrics Bayesian Updating/Learning
Combining the posterior based on the first sample (which becomes the prior for the second sample) and the likelihood based on the second sample yields: 1 푃 훽, 휎 푦 , 푦 , 푋 , 푋 ∝ 휎− 푇1+푇2+1 푒푥푝 − 푦 − 푋 훽 ′ 푦 − 푋 훽 + 푦 − 푋 훽 ′ 푦 − 푋 훽 1 2 1 2 2휎2 1 1 1 1 2 2 2 2 1 ′ ∝ 휎− 푇1+푇2+1 푒푥푝 − ν푠2 + 훽 − 훽෨ ϻ 훽 − 훽෨ 2휎2 where ′ ′ ϻ = 푋1푋1 + 푋2푋2 ෨ −1 ′ ′ 훽 = ϻ 푋1푦1 + 푋2푦2 2 ෨ ′ ෨ ෨ ′ ෨ ν푠 = 푦1 − 푋1훽 푦1 − 푋1훽 + 푦2 − 푋2훽 푦2 − 푋2훽 ν = 푇1 + 푇2 − 푀 Then the posterior distributions for β and σ follow using steps outlined earlier With more observations realized you follow the same updating procedure Notice that the same posterior would have been obtained starting with diffuse priors and ′ then observing the two samples jointly Y=[푦1′, 푦2′]′ and 푋 = [푋1′, 푋2]′.
387 Professor Doron Avramov, Financial Econometrics The Black-Litterman way of Estimating Mean Returns The well-known BL approach to estimating mean return exhibits some Bayesian appeal.
It is notoriously difficult to propose good estimates for mean returns.
The sample means are quite noisy – thus asset pricing models -even if misspecified -could give a good guidance.
To illustrate, you consider a K-factor model (factors are portfolio spreads) and you run the time series regression 푒 푟푡 = 훼 + 훽1 푓1푡 + 훽2 푓2푡 + ⋯ + Ⱦ퐾 푓퐾푡 + 푒푡 푁×1 푁×1 푁×1 푁×1 푁×1 푁×1
388 Professor Doron Avramov, Financial Econometrics Estimating Mean Returns
Then the estimated excess mean return is given by
푒 መ መ መ 휇ො = 훽1μො푓1 + 훽2휇ො푓2 + ⋯ + 훽퐾휇ො푓퐾
መ መ መ where 훽1, 훽2 … 훽퐾 are the sample estimates of factor loadings,
and 휇ො푓1, 휇ො푓2 … 휇ො푓퐾 are the sample estimates of the factor mean returns.
389 Professor Doron Avramov, Financial Econometrics The BL Mean Returns
The BL approach combines a model (CAPM) with some views, either relative or absolute, about expected returns.
The BL vector of mean returns is given by
−1 −1 ′ −1 −1 푒푞 ′ −1 푣 μ퐵퐿 = ɒ σ + 푃 Ω 푃 ⋅ ɒ σ μ + 푃 Ω μ 푁×1 1×1 푁×푁 푁×퐾 퐾×퐾 퐾×푁 1×1 푁×푁 푁×1 푁×퐾 퐾×퐾 퐾×1
390 Professor Doron Avramov, Financial Econometrics Understanding the BL Formulation
We need to understand the essence of the following parameters, which characterize the mean return vector
σ, 휇푒푞, 푃, 휏, Ω, 휇푣
Starting from the σ matrix – you can choose any of the specifications derived in the previous meetings – either the sample covariance matrix, or the equal correlation, or an asset pricing based covariance, or you could rely on the LW shrinkage approach – either the complex or the naïve one.
391 Professor Doron Avramov, Financial Econometrics Constructing Equilibrium Expected Returns The 휇푒푞, which is the equilibrium vector of expected return, is constructed as follows.
Generate 휔푀퐾푇, the 푁 × 1 vector denoting the weights of any of the 푁 securities in the market portfolio based on market capitalization. Of course, the sum of weights must be unity.
휇푚−푅푓 2 Then, the price of risk is 훾 = 2 where 휇푚 and 휎푚 are the 휎푚 expected return and variance of the market portfolio.
Later, we will justify this choice for the price of risk.
392 Professor Doron Avramov, Financial Econometrics Constructing Equilibrium Expected Returns
One could pick a range of values for 훾 going from 1.5 to 2.5 and examine performance of each choice.
If you work with monthly observations, then switching to the annual frequency does not change 훾 as both the numerator and denominator are multiplied by 12.
Having at hand both 휔푀퐾푇 and 훾, the equilibrium return vector is given by 푒푞 휇 = 훾Σ휔푀퐾푇
393 Professor Doron Avramov, Financial Econometrics Constructing Equilibrium Expected Returns
This vector is called neutral mean or equilibrium expected return.
To understand why, notice that if you have a utility function that generates the tangency portfolio of the form σ−1 휇푒 푤 = 푇푃 휄′ σ−1 휇푒
then using 휇푒푞 as the vector of excess returns on the 푁 assets would deliver 휔푀퐾푇 as the tangency portfolio.
394 Professor Doron Avramov, Financial Econometrics What if you Directly Apply the CAPM?
The question being – would you get the same vector of equilibrium mean return if you directly use the CAPM?
Yes, if…
395 Professor Doron Avramov, Financial Econometrics The CAPM based Expected Returns
Under the CAPM the vector of excess returns is given by
푒 푒 휇 = 훽 휇푚 푁×1 푁×1
푒 푒 푒 푒 ′ cov 푟 , 푟푚 cov 푟 , 푟 푤푀퐾푇 σ 푤푀퐾푇 훽 = 2 = 2 = 2 휎푚 휎푚 휎푚
σ 푒 푤푀퐾푇 푒 퐶퐴푃푀: 휇 = 2 휇푚 = 훾σ푤푀퐾푇 푁×1 휎푚
396 Professor Doron Avramov, Financial Econometrics What if you use Directly the CAPM?
푒 푒 ′ 푒 푒 ′ Since 휇푚 = 휇 푤푀퐾푇 and 푟푚 = 푟 푤푀퐾푇
푒 푒 휇푚 푒푞 then 휇 = 2 σ 푤푀퐾푇 = 휇 휎푚
397 Professor Doron Avramov, Financial Econometrics What if you use Directly the CAPM?
So indeed, if you use (i) the sample covariance matrix, rather than any other specification, as well as (ii)
휇푚 − 푅푓 훾 = 2 휎푚
then the BL equilibrium expected returns and expected returns based on the CAPM are identical.
398 Professor Doron Avramov, Financial Econometrics The 푃 Matrix: Absolute Views
In the BL approach the investor/econometrician forms some views about expected returns as described below.
푃 is defined as that matrix which identifies the assets involved in the views.
To illustrate, consider two "absolute" views only.
The first view says that stock 3 has an expected return of 5% while the second says that stock 5 will deliver 12%.
399 Professor Doron Avramov, Financial Econometrics The 푃 Matrix: Absolute Views
In general the number of views is 퐾.
In our case 퐾 = 2.
Then 푃 is a 2 × 푁 matrix.
The first row is all zero except for the fifth entry which is one.
Likewise, the second row is all zero except for the fifth entry which is one.
400 Professor Doron Avramov, Financial Econometrics The 푃 Matrix: Relative Views
Let us consider now two "relative views".
Here we could incorporate market anomalies into the BL paradigm.
Market anomalies are cross sectional patterns in stock returns unexplained by the CAPM.
Example: price momentum, earnings momentum, value, size, accruals, credit risk, dispersion, and volatility.
401 Professor Doron Avramov, Financial Econometrics Black-Litterman: Momentum and Value Effects
Let us focus on price momentum and the value effects.
Assume that both momentum and value investing outperform.
The first row of 푃 corresponds to momentum investing.
The second row corresponds to value investing.
Both the first and second rows contain 푁 elements.
402 Professor Doron Avramov, Financial Econometrics Winner, Loser, Value, and Growth Stocks Winner stocks are the top 10% performers during the past six months.
Loser stocks are the bottom 10% performers during the past six months.
Value stocks are 10% of the stocks having the highest book-to- market ratio.
Growth stocks are 10% of the stocks having the lowest book-to- market ratios.
403 Professor Doron Avramov, Financial Econometrics Momentum and Value Payoffs
The momentum payoff is a return spread – return on an equal weighted portfolio of winner stocks minus return on equal weighted portfolio of loser stocks.
The value payoff is also a return spread – the return differential between equal weighted portfolios of value and growth stocks.
404 Professor Doron Avramov, Financial Econometrics Back to the 푃 Matrix
Suppose that the investment universe consists of 100 stocks The first row gets the value 0.1 if the corresponding stock is a winner (there are 10 winners in a universe of 100 stocks). It gets the value -0.1 if the corresponding stock is a loser (there are 10 losers). Otherwise it gets the value zero. The same idea applies to value investing. Of course, since we have relative views here (e.g., return on winners minus return on losers) then the sum of the first row and the sum of the second row are both zero.
405 Professor Doron Avramov, Financial Econometrics Back to the 푃 Matrix
More generally, if N stocks establish the investment universe and moreover momentum and value are based on deciles (the return difference between the top and bottom deciles) then the winner stock is getting 10/푁 while the loser stock gets −10/푁.
The same applies to value versus growth stocks.
Rule: the sum of the row corresponding to absolute views is one, while the sum of the row corresponding to relative views is zero.
406 Professor Doron Avramov, Financial Econometrics Computing the 휇푣 Vector
It is the 퐾 × 1 vector of 퐾 views on expected returns.
Using the absolute views above 휇푣 = 0.05,0.12 ′
Using the relative views above, the first element is the payoff to momentum trading strategy (sample mean); the second element is the payoff to value investing (sample mean).
407 Professor Doron Avramov, Financial Econometrics The Ω Matrix
Ω is a 퐾 × 퐾 covariance matrix expressing uncertainty about views.
It is typically assumed to be diagonal.
In the absolute views case described above Ω 1,1 denotes uncertainty about the first view while Ω 2,2 denotes uncertainty about the second view – both are at the discretion of the econometrician/investor.
408 Professor Doron Avramov, Financial Econometrics The Ω Matrix
In the relative views described above: Ω 1,1 denotes uncertainty about momentum. This could be the sample variance of the momentum payoff.
Ω 2,2 denotes uncertainty about the value payoff. This is the could be the sample variance of the value payoff.
409 Professor Doron Avramov, Financial Econometrics Deciding Upon 휏
There are many debates among professionals about the right value of 휏.
From a conceptual perspective it could be 1/푇 where 푇 denotes the sample size.
You can pick 휏 = 0.01
You can also use other values and examine how they perform in real-time investment decisions.
410 Professor Doron Avramov, Financial Econometrics Maximizing Sharpe Ratio
The remaining task is to run the maximization program
푤′휇 max 퐵퐿 푤 푤′훴푤
Preferably, impose portfolio constraints, such that each of the 푤 elements is bounded below by 0 and subject to some agreed upon upper bound, as well as the sum of the 푤 elements is equal to one.
411 Professor Doron Avramov, Financial Econometrics Extending the BL Model to Incorporate Sample Moments Consider a sample of size 푇, e.g., 푇 = 60 monthly observations. Let us estimate the mean and the covariance (푉) of our 푁 assets based on the sample. Then the vector of expected return that serves as an input for asset allocation is given by
−1 −1 −1 −1 −1 휇 = Δ + (푉푠푎푚푝푙푒/푇) Δ 휇퐵퐿 + (푉푠푎푚푝푙푒/푇) 휇푠푎푚푝푙푒
where Δ = (휏Σ)−1+푃′Ω−1푃 −1
412 Professor Doron Avramov, Financial Econometrics Session #10: Risk Management: Down Side Risk Measures
413 Professor Doron Avramov, Financial Econometrics Downside Risk Downside risk is the financial risk associated with losses.
Downside risk measures quantify the risk of losses, whereas volatility measures are both about the upside and downside outcomes.
Volatility treats symmetrically up and down moves (relative to the mean).
Or volatility is about the entire distribution while down side risk concentrates on the left tail.
414 Professor Doron Avramov, Financial Econometrics Downside Risk
Example of downside risk measures Value at Risk (VaR) Expected Shortfall Semi-variance Maximum drawdown Downside Beta Shortfall probability
We will discuss below all these measures.
415 Professor Doron Avramov, Financial Econometrics Value at Risk (VaR)
The 푉푎푅95% says that there is a 5% chance that the realized return, denoted by 푅, will be less than-푉푎푅95%. More generally, Pr 푅 ≤ −푉푎푅1−훼 = 훼
5%
−푉푎푅95% 휇
416 Professor Doron Avramov, Financial Econometrics Value at Risk (VaR) −푉푎푅 − 휇 1−훼 = Φ−1 훼 ⇒ 푉푎푅 = − 휇 + 휎Φ−1 훼 휎 1−훼 where
Φ−1 훼 , the critical value, is the inverse cumulative distribution function of the standard normal evaluated at 훼.
Let 훼 = 5% and assume that 푅 ∼ 푁 휇, 휎2 The critical value is Φ−1 0.05 = −1.64
417 Professor Doron Avramov, Financial Econometrics Value at Risk (VaR)
Therefore −1 푉푎푅1−훼 = − 휇 + Φ 훼 휎 = −휇 + 1.64휎 Check: If 푅 ∼ 푁 휇, 휎2 Then 푅 − 휇 −푉푎푅 − 휇 Pr 푅 ≤ −푉푎푅 = Pr ≤ 훼 1−훼 휎 휎 푅 − 휇 휇 − 1.64휎 − 휇 = Pr < 휎 휎 = Pr 푧 < −1.64 = Φ −1.64 = 0.05
418 Professor Doron Avramov, Financial Econometrics Example: The US Equity Premium
Suppose: 푅 ∼ 푁 0.08, 0.202 ⇒ 푉푎푅0.95 = − 0.08 − 1.64 ⋅ 0.20 = 0.25 That is to say that we are 95% sure that the future equity premium won’t decline more than 25%.
If we would like to be 97.5% sure – the price is that the threshold loss is higher:
푉푎푅0.975 = − 0.08 − 1.96 ⋅ 0.20 = 0.31
419 Professor Doron Avramov, Financial Econometrics VaR of a Portfolio
Evidently, the VaR of a portfolio is not necessarily lower than the combination of individual VaR-s – which is apparently at odds with the notion of diversification.
However, VaR is a downside risk measure while volatility (a risk measure) does diminish with better diversification.
420 Professor Doron Avramov, Financial Econometrics Backtesting the VaR The VaR requires the specification of the exact distribution and its parameters (e.g., mean and variance).
Typically the normal distribution is chosen.
Mean could be the sample average.
Volatility estimates could follow ARCH, GARCH, EGARCH, stochastic volatility, and realized volatility, all of which are described later in this course.
We can examine the validity of VaR using backtesting.
421 Professor Doron Avramov, Financial Econometrics Backtesting the VaR
Assume that stock returns are normally distributed with mean and variance that vary over time 2 푟푡 ∼ 푁 휇푡, 휎푡 ∀푡 = 1,2, … , 푇
The sample is of length 푇. Receipt for backtesting is as follows. Use the first, say, sixty monthly observations to estimate the mean and volatility and compute the VaR. If the return in month 61 is below the VaR set an indicator function 퐼 to be equal to one; otherwise, it is zero.
422 Professor Doron Avramov, Financial Econometrics Backtesting the VaR
Repeat this process using either a rolling or recursive schemes and compute the fraction of time when the next period return is below the VaR.
If 훼 = 5% - only 5% of the returns should be below the computed VaR.
Suppose we get 5.5% of the time – is it a bad model or just a bad luck?
423 Professor Doron Avramov, Financial Econometrics Model Verification Based on Failure Rates
To answer that question – let us discuss another example which requires a similar test statistic.
Suppose that 푌 analysts are making predictions about the market direction for the upcoming year. The analysts forecast whether market is going to be up or down.
After the year passes you count the number of wrong analysts. An analyst is wrong if he/she predicts up move when the market is down, or predict down move when the market is up.
424 Professor Doron Avramov, Financial Econometrics Model Verification Based on Failure Rates
Suppose that the number of wrong analysts is 푥.
Thus, the fraction of wrong analysts is 푃 = 푥/푌 – this is the failure rate.
425 Professor Doron Avramov, Financial Econometrics The Test Statistic
The hypothesis to be tested is
퐻0: 푃 = 푃0
퐻1: 푂푡ℎ푒푟푤푖푠푒
Under the null hypothesis it follows that 푦 푓 푥 = 푃푥 1 − 푃 푦−푥 푥 0 0
426 Professor Doron Avramov, Financial Econometrics The Test Statistic
Notice that 퐸 푥 = 푃0푦 푉푎푅 푥 = 푃0 1 − 푃0 푦 Thus 푥 − 푃 푦 푍 = 0 ∼ 푁 0,1 푃0 1 − 푃0 푦
427 Professor Doron Avramov, Financial Econometrics Back to Backtesting VaR: A Real Life Example
In its 1998 annual report, JP Morgan explains: In 1998, daily revenue fell short of the downside (95%VaR) band on 20 trading days (out of 252) or more than 5% of the time (252×5%=12.6).
Is the difference just a bad luck or something more systematic? We can test the hypothesis that it is a bad luck.
퐻 : 푥 = 12.6 20 − 12.6 0 → 푍 = = 2.14 퐻1: 푂푡ℎ푒푟푤푖푠푒 0.05 ⋅ 0.95 ⋅ 252
428 Professor Doron Avramov, Financial Econometrics Back to Backtesting VaR: A Real Life Example
Notice that you reject the null since 2.14 is higher than the critical value of 1.96.
That suggests that JPM should search for a better model.
They did find out that the problem was that the actual revenue departed from the normal distribution.
429 Professor Doron Avramov, Financial Econometrics Expected Shortfall (ES): Truncated Distribution
ES is the expected value of the loss conditional upon the event that the actual return is below the VaR.
The ES is formulated as
퐸푆1−훼 = −퐸 푅|푅 ≤ −푉푎푅1−훼
430 Professor Doron Avramov, Financial Econometrics Expected Shortfall (ES) and the Truncated Normal Distribution Assume that returns are normally distributed: 푅 ∼ 푁 휇, 휎2 −1 ⇒ 퐸푆1−훼 = −퐸 푅|푅 ≤ 휇 + Φ 훼 휎
휙 훷−1 훼 ⇒ 퐸푆 = −휇 + 휎 1−훼 훼
431 Professor Doron Avramov, Financial Econometrics Expected Shortfall (ES) and the Truncated Normal Distribution where 휙 ∙ is the pdf of the standard normal density
e.g. 휙 −1.64 = 0.103961
This formula for ES is about the expected value of a truncated normally distributed random variable.
432 Professor Doron Avramov, Financial Econometrics Expected Shortfall (ES) and the Truncated Normal Distribution Proof: 푥 ∼ 푁 휇, 휎2
−1 휎휑 휅0 휑 훷 훼 퐸 푥|푥 ≤ −푉푎푅1−훼 = 휇 − = 휇 − 휎 훷 휅0 훼
since −푉푎푅 − 휇 휅 = 1−훼 = Φ−1 훼 0 휎
433 Professor Doron Avramov, Financial Econometrics Expected Shortfall: Example
Example: 휇 = 8% 휎 = 20% 휑 −1.64 0.10 퐸푆 = −0.08 + 0.20 ≈ −0.08 + 0.20 = 0.32 95% 0.05 0.05 휑 −1.96 0.06 퐸푆 = −0.08 + 0.20 ≈ −0.08 + 0.20 = 0.40 97.5% 0.025 0.25
434 Professor Doron Avramov, Financial Econometrics Expected Shortfall (ES) and the Truncated Normal Distribution Previously, we got that the VaRs corresponding to those parameters are 25% and 31%. Now the expected losses are higher, 32% and 40%. Why? The first lower figures (VaR) are unconditional in nature relying on the entire distribution. In contrast, the higher ES figures are conditional on the existence of shortfall – realized return is below the VaR.
435 Professor Doron Avramov, Financial Econometrics Expected Shortfall in Decision Making
The mean variance paradigm minimizes portfolio volatility subject to an expected return target.
Suppose you attempt to minimize ES instead subject to expected return target.
436 Professor Doron Avramov, Financial Econometrics Expected Shortfall with Normal Returns
If stock returns are normally distributed then the ES chosen portfolio would be identical to that based on the mean variance paradigm.
No need to go through optimization to prove that assertion.
Just look at the expression for ES under normality to quickly realize that you need to minimize the volatility of the portfolio subject to an expected return target.
437 Professor Doron Avramov, Financial Econometrics Target Semi Variance
Variance treats equally downside risk and upside potential. The semi-variance, just like the VaR, looks at the downside. The target semi-variance is defined as: 휆 ℎ = 퐸 min 푟 − ℎ, 0 2 where ℎ is some target level.
For instance, ℎ = 푅푓 Unlike the variance, 휎2 = 퐸 푟 − 휇 2
438 Professor Doron Avramov, Financial Econometrics Target Semi Variance
The target semi-variance:
1. Picks a target level as a reference point instead of the mean.
2. Gives weight only to negative deviations from a reference point.
439 Professor Doron Avramov, Financial Econometrics Target Semi Variance
Notice that if 푟 ∼ 푁 휇, 휎2 2 ℎ − 휇 ℎ − 휇 ℎ − 휇 ℎ − 휇 휆 ℎ = 휎2 휑 + 휎2 + 1 Φ 휎 휎 휎 휎
where 휙 and Φ are the PDF and CDF of a 푁 0,1 variable, respectively Of course if ℎ = 휇 휎2 then 휆 ℎ = 2
440 Professor Doron Avramov, Financial Econometrics Maximum Drawdown (MD)
The MD (M) over a given investment horizon is the largest M- month loss of all possible M-month continuous periods over the entire horizon.
Useful for an investor who does not know the entry/exit point and is concerned about the worst outcome.
It helps determine the investment risk.
441 Professor Doron Avramov, Financial Econometrics Down Size Beta
I will introduce three distinct measures of downsize beta – each of which is valid and captures the down side of investment payoffs.
Displayed are the population betas.
Taking the formulations into the sample – simply replace the expected value by the sample mean.
442 Professor Doron Avramov, Financial Econometrics Downside Beta
퐸 푅푖 − 푅푓 min 푅푚 − 푅푓 , 0 1 훽푖푚 = 2 퐸 min 푅푚 − 푅푓 , 0 The numerator in the equation is referred to as the co-semi- variance of returns and is the covariance of returns below 푅푓 on the market portfolio with return in excess of 푅푓 on security 푖.
It is argued that risk is often perceived as downside deviations below a target level by market participants and the risk-free rate is a replacement for average equity market returns.
443 Professor Doron Avramov, Financial Econometrics Downside Beta
(2) 퐸 푅푖 − 휇푖 min 푅푚 − 휇푚 , 0 훽푖푚 = 2 퐸 min 푅푚 − 휇푚 , 0
where 휇푖 and 휇푚 are security 푖 and market average return respectively. One can modify the down side beta as follows:
(3) 퐸 min 푅푖 − 휇푖 , 0 min 푅푚 − 휇푚 , 0 훽푖푚 = 2 퐸 min 푅푚 − 휇푚 , 0
444 Professor Doron Avramov, Financial Econometrics Shortfall Probability
We now turn to understand the notion of shortfall probability.
While VaR specifies upfront the probability of undesired outcome and then finds the threshold level, shortfall probability gives a threshold level and seeks for the probability that the outcome is below that threshold.
We will thoroughly study the implications of shortfall probability for long horizon investment decisions.
445 Professor Doron Avramov, Financial Econometrics Shortfall Probability in Long Horizon Asset Management Let us denote by 푅 the cumulative return on the investment over several years (say 푇 years).
Rather than finding the distribution of R we analyze the distribution of 푟 = ln 1 + 푅 which is the continuously compounded return over the investment horizon.
446 Professor Doron Avramov, Financial Econometrics Shortfall Probability in Long Horizon Asset Management The investment value after 푇 years is 푉푇 = 푉0 1 + 푅1 1 + 푅2 … 1 + 푅푇
Dividing both sides of the equation by 푉0 we get 푉푇 = 1 + 푅1 1 + 푅2 … 1 + 푅푇 푉0 Thus 1 + 푅 = 1 + 푅1 1 + 푅2 … 1 + 푅푇
447 Professor Doron Avramov, Financial Econometrics Shortfall Probability in Long Horizon Asset Management Taking natural log from both sides we get 푟 = 푟1 + 푟2 + ⋯ + 푟푇
Assuming that 퐼퐼퐷 2 푟푡 ∼ 푁 휇, 휎 ∀푡 = 1, . . . , 푇
Then using properties of the normal distribution, we get 푟 ∼ 푁 푇휇, 푇휎2
448 Professor Doron Avramov, Financial Econometrics Shortfall Probability and Long Horizon
The normality assumption for log return implies the log normal distribution for the cumulative return – more later.
Let us understand the concept of shortfall probability.
We ask: what is the probability that the investment yields a return smaller than a threshold level (e.g., the risk-free rate)?
To answer this question we need to compute the value of a risk- free investment over the 푇 year period.
449 Professor Doron Avramov, Financial Econometrics Shortfall Probability and Long Horizon
The value of such a risk-free investment is
푇 푉푟푓 = 푉0 1 + 푅푓
= 푉0 exp 푇푟푓
where 푟푓 is the continuously compounded risk free rate.
450 Professor Doron Avramov, Financial Econometrics Shortfall Probability and Long Horizon
Essentially we ask: what is the probability that 푉푇 < 푉푟푓
This is equivalent to asking what is the probability that 푉 푉푟푓 푇 < 푉0 푉0
This, in turn, is equivalent to asking what is the probability that 푉 푉푟푓 ln 푇 < ln 푉0 푉0
451 Professor Doron Avramov, Financial Econometrics Shortfall Probability and Long Horizon So we need to work out 푝 푟 < 푇푟푓
Subtracting 푇휇 and dividing by 푇휎 both sides of the inequality we get 푟푓 − 휇 푃 푧 < 푇 휎
We can denote this probability by 푟푓 − 휇 푆ℎ표푟푡푓푎푙푙 푝푟표푏푎푏푖푙푖푡푦 = Φ 푇 휎
452 Professor Doron Avramov, Financial Econometrics Shortfall Probability and Long Horizon
Typically 푟푓 < 휇 which means the probability diminishes the larger 푇.
Notice that the shortfall probability can be written as a function of the Sharpe ratio of log returns:
푆푃 = Φ − 푇 푆푅
453 Professor Doron Avramov, Financial Econometrics Example
Take 푟푓=0.04, μ=0.08, and σ=0.2 per year. What is the Shortfall Probability for investment horizons of 1, 2, 5, 10, and 20 years?
Use the excel norm.dist function. If T=1 SP=0.42 If T=2 SP=0.39 If T=5 SP=0.33 If T=10 SP=0.26 If T=20 SP=0.19
454 Professor Doron Avramov, Financial Econometrics Cost of Insuring against Shortfall
Let us now understand the mathematics of insuring against shortfall.
Without loss of generality let us assume that 푉0 = 1
The investment value at time 푇 is a given by the random variable 푉푇
455 Professor Doron Avramov, Financial Econometrics Cost of Insuring against Shortfall
Once we insure against shortfall the investment value after T years becomes
If 푉푇 > exp 푇푟푓 you get 푉푇
If 푉푇 < exp 푇푟푓 you get exp 푇푟푓
456 Professor Doron Avramov, Financial Econometrics Cost of Insuring against Shortfall So you essentially buy an insurance policy
The policy pays 0 if 푉푇 > exp 푇푟푓 while it pays
exp 푇푟푓 − 푉푇 if 푉푇 < exp 푇푟푓
You ultimately need to price a contract with terminal payoff given by max 0, exp 푇푟푓 − 푉푇
457 Professor Doron Avramov, Financial Econometrics Cost of Insuring against Shortfall
This is a European put option expiring in 푇 years with 1. S=1
2. 퐾 = exp 푇푟푓 .
3. Risk-free rate given by 푟푓 4. Volatility given by 휎 5. Dividend yield given by 훿 = 0
458 Professor Doron Avramov, Financial Econometrics Cost of Insuring against Shortfall
The B&S formula indicates that
푃푢푡 = 퐾 exp −푇푟푓 푁 −푑2 − 푆 exp −훿푇 푁 −푑1
Given the parameter outlined above the put price becomes
1 1 푃푢푡 = 푁 휎 푇 − 푁 − 휎 푇 2 2
459 Professor Doron Avramov, Financial Econometrics Cost of Insuring against Shortfall
To show the pricing formula of the put use the following:
1 푙푛(푆/퐾)+(푟−훿+ 휎2)푇 d = 2 and 푑 = 푑 − 휎 푇 1 휎 푇 2 1
while 푁 −푑1 = 1 − 푁 푑1
푁 −푑2 = 1 − 푁 푑2
460 Professor Doron Avramov, Financial Econometrics Cost of Insuring against Shortfall
The B&S option-pricing model gives the current put price 푃 as
푃푢푡 = 푁 푑1 − 푁 푑2 where 휎 푇 푑 = 1 2
푑2 = −푑1
and 푁 푑 is 푝푟표푏 푧 < 푑
461 Professor Doron Avramov, Financial Econometrics Cost of Insuring against Shortfall
For 휎 = 0.2 (per year) T (years) P 1 0.08
5 0.18
10 0.25
20 0.35
30 0.42
50 0.52
462 Professor Doron Avramov, Financial Econometrics Cost of Insuring against Shortfall
We have found out that the cost of the insurance increases in 푇, even when the probability of shortfall decreases in 푇 (as long as the Sharpe ratio is positive).
To get some idea about this apparently surprising outcome it would be essential to discuss the expected value of the investment payoff given the shortfall event.
It is a great opportunity to understand down side risk when the underlying distribution is log normal rather than normal.
463 Professor Doron Avramov, Financial Econometrics The Expected Value of Cumulative Return
Two proper questions emerge at this stage: 1. What is the expected value of cumulative return during the investment horizon 퐸 푉푇 ? 2. What is the conditional expectation – conditional on shortfall 퐸 푉푇 ∣ 푉푇 < exp 푇푟푓 ? We assume, without loss of generality, that the initial invested wealth is one.
464 Professor Doron Avramov, Financial Econometrics The Expected Value of Cumulative Return
Notice that, given the tools we have acquired thus far, finding the conditional expectation is a nontrivial task since 푉푇 is not normally distributed – rather it is log-normally distributed since. 2 ln 푉푇 ~푁 푇휇, 푇휎
Thus, let us first display some properties of the log normal distribution.
465 Professor Doron Avramov, Financial Econometrics The Log Normal Distribution
Suppose that x has the log normal distribution. Then the parameters 휇 and 휎 are, respectively, the mean and the standard deviation of the variable’s natural logarithm, which means 푥 = 푒휇+휎푧 where 푧 is a standard normal variable.
The probability density function of a log-normal distribution is
466 Professor Doron Avramov, Financial Econometrics The Log Normal Distribution
1 푙푛 푥−휇 2 − 2 푓푥 푥, 휇, 휎 = 푒 2휎 , 푥 > 0 푥휎 2훱 If 푥 is a log-normally distributed variable, its expected value and variance are given by 1 휇+ 휎2 퐸 푥 = 푒 2 2 2 2 푉푎푟 푥 = 푒휎 − 1 푒2휇+휎 = 푒휎 − 1 퐸 푥 2
467 Professor Doron Avramov, Financial Econometrics The Mean and Variance of 푉푇 Using moments of the log normal distribution, the mean and variance of 푉푇 are 1 퐸(푉 ) = exp ( 푇휇 + 푇휎2) 푇 2 2 2 푉푎푟(푉푇) = exp ( 2푇휇 + 푇휎 ) exp ( 푇휎 ) − 1
Next, we aim to find the conditional mean.
468 Professor Doron Avramov, Financial Econometrics The Mean of a Variable that has the Truncated Log Normal Distribution
∞ 1 푙푛 푥 −휇 2 1 − 퐹ത c E(x|x > c) = න x 푒 2 휎 푑푥 c x휎 2휋 where 1/퐹ത c is a normalizing constant.
Let us make change of variables: 푙푛(x) − 휇 = t ⇒ x = et휎+휇 and dx = 휎et휎+휇dt 휎
469 Professor Doron Avramov, Financial Econometrics The Mean of a Variable that has the Truncated Log Normal Distribution then: ∞ 1 1 − 푡2 퐹ത c E x x > c = න 푒 2 휎푒푡휎+휇푑푡 = 푙푛 푐 −휇 2훱 휎 ∞ 1 1 − 푡2+푡휎+휇 = න 푒 2 푑푡 2훱 푙푛 푐 −휇 휎 ∞ 1 1 − 푡−휎 2+ 휇+0.5휎2 = න 푒 2 푑푡 2훱 푙푛 푐 −휇 휎 ∞ 2 1 1 2 = 푒 휇+0.5휎 න 푒−2 푡−휎 푑푡 2훱 푙푛 푐 −휇 휎
470 Professor Doron Avramov, Financial Econometrics The Mean of a Variable that has the Truncated Log Normal Distribution Let us make another change of variables: v = t − 휎 ⇒ dv = dt
∞ 1 2 1 − 푣2 퐹ത c E(x|x > c) = 푒 휇+0.5휎 න 푒 2 푑푣 푙푛 푐 −휇 2훱 −휎 휎
The integral is the complement CDF of the standard normal random variable.
471 Professor Doron Avramov, Financial Econometrics The Mean of a Variable that has the Truncated Log Normal Distribution Thus, the formula is reduced to: 2 2 푙푛 푐 − 휇 − 휎 Fሜ c E x x > c = 푒 휇+0.5휎 1 − Φ LN 휎 2 2 − 푙푛 푐 + 휇 + 휎 = 푒 휇+0.5휎 Φ 휎
472 Professor Doron Avramov, Financial Econometrics The Mean of a Variable that has the Truncated Log Normal Distribution In the same way we can show that: 푙푛 푐 −휇 푐 1 푙푛 푥 −휇 2 1 − 휎 1 2 2 휎 − 푡 +푡휎+휇 퐹ത c ⋅ ELN(x|x ≤ c) = න 푥 푒 푑푥 = න 푒 2 푑푡 0 푥휎 ⋅ 2훱 −∞ 푙푛 푐 −휇 푙푛 푐 −휇 1 −휎 1 2 1 휎 − 푡−휎 2 2 1 휎 − 푣2 = 푒 휇+0.5휎 න 푒 2 푑푡 = 푒 휇+0.5휎 න 푒 2 푑푣 2훱 −∞ 2훱 −∞ 2 2 푙푛 푐 − 휇 − 휎 = 푒 휇+0.5휎 ⋅ Φ 휎
473 Professor Doron Avramov, Financial Econometrics Punch Lines
− 푙푛 푐 + 휇 + 휎2 훷 휎 E (x|x > c) = E (x) ⋅ LN LN − 푙푛 푐 + 휇 훷 휎
푙푛 푐 − 휇 − 휎2 훷 휎 E (x|x ≤ c) = E (x) ⋅ LN LN 푙푛 푐 − 휇 훷 휎
474 Professor Doron Avramov, Financial Econometrics The Expected Value given Shortfall
The expected value given shortfall is 푇푟 − 푇휇 − 푇휎2 훷 푓 1 푇휇+ 푇휎2 푇휎 퐸 푉 |푠ℎ표푟푡푓푎푙푙 = 푒 2 ⋅ 푇 푇푟 − 푇휇 훷 푓 푇휎 or
1 훷 − 푇 푆푅 + 휎 푇휇+ 푇휎2 퐸 푉푇|푠ℎ표푟푡푓푎푙푙 = 푒 2 ⋅ 훷 − 푇 ⋅ 푆푅
475 Professor Doron Avramov, Financial Econometrics The Expected Value given Shortfall
Thus,
푃푟표푏 푠ℎ표푟푡푓푎푙푙 × 퐸 푉푇|푠ℎ표푟푡푓푎푙푙 = 퐸[푉푡]Φ − 푇 푆푅 + 휎
Which means that the shortfall probability times the expected value given shortfall is equal to the unconditional expected value times a factor smaller than one.
That factor diminishes with higher Sharpe ratio and/or with higher volatility.
476 Professor Doron Avramov, Financial Econometrics The Horizon Effect
Numerical example
Let’s take 푟푓 = 5%, 휎 = 10% . For different values of 휇 > 푟푓 the conditional expectation over horizon 푇 looks like:
477 Professor Doron Avramov, Financial Econometrics The Horizon Effect
Previously we have shown that even when the shortfall probability diminishes with the investment horizon, the cost of insuring against shortfall rises.
Notice that the insured amount is exp 푇푟푓 − 푉푇
The expected value of that insured amount given shortfall sharply rises with the investment horizon, which explains the increasing value of the put option.
478 Professor Doron Avramov, Financial Econometrics Value at Risk with Log Normal Distribution
We have analyzed VaR when quantities of interest are normally distributed.
It is challenging to extend the analysis to the case wherein the log normal distribution is considered.
Analytics follow.
479 Professor Doron Avramov, Financial Econometrics VaR with Log Normal Distribution
We are looking for threshold, VaR, such that 훼 = Pr 푉푇 ∣ 푉0 < 푉푎푟 ∣ 푉0 = 퐶퐷퐹 푉푎푅 ∣ 푉0
Then in order to find the threshold we need to calculate quantile of lognormal distribution: −1 2 푉푎푅 = 푉0 ⋅ 퐶퐷퐹 훼; 푇휇, 푇휎 푇휇+ 푇휎⋅Φ−1 훼 = 푉0 ⋅ 푒
where Φ−1 훼 is as defined earlier.
480 Professor Doron Avramov, Financial Econometrics VaR with Log Normal Distribution
Specifically,
훼 = Pr 푉푇 ∣ 푉0< 푉푎푟 ∣ 푉0 = Pr ln 푉푇 ∣ 푉0 < ln 푉푎푅 ∣ 푉0
푙푛 푉 ∣ 푉 − 푇휇 푙푛 푉푎푅 ∣ 푉 − 푇휇 = Pr 푇 0 < 0 푇휎 푇휎
푙푛 푉푎푅 ∣ 푉 − 푇휇 = Φ 0 푇휎
481 Professor Doron Avramov, Financial Econometrics VaR with Log Normal Distribution
and then
푙푛 푉푎푅 ∣ 푉 − 푇휇 0 = Φ−1 훼 푇휎
푇휇+ 푇휎⋅Φ−1 훼 푉푎푅 = 푉0푒
That is to say that there is a 훼% probability that the investment value at time T will be below that VaR.
482 Professor Doron Avramov, Financial Econometrics VaR with 푡 Distribution
Suppose now that stock returns have a 푡 distribution with 휈 degrees of freedom and expected return and volatility given by μ and 휎
The pdf of stock return is formulated as 푣+1 − 1 (푥 − 휇)2 2 푓 푥|휇, 휎, 푣 = ⋅ 1 + 휎 푣 ⋅ 퐵 푣/2,1/2 푣
483 Professor Doron Avramov, Financial Econometrics Partial Expectation
푥−휇 Let 푌 = -standardized r.v distribution with 퐹 푥|0,1, 푣 . 휎 Than, 푣+1 − 푧 푧 1 푥2 2 푃퐸 푋|푋 ≤ 푧 = න 푥 ⋅ 푓 푥, 푣 푑푥 = න ⋅ 1 + 푑푥 = −∞ −∞ 푣 ⋅ 퐵 푣/2,1/2 푣
푣−1 − 1 −푣 푥2 2 = ⋅ 1 + |푧 = 푣 ⋅ 퐵 푣/2,1/2 푣 − 1 푣 −∞
푣+1 − +1 1 −푣 푧2 2 푣 + 푧2 ⋅ 1 + = −푓 푧, 푣 ⋅ 푣 ⋅ 퐵 푣/2,1/2 푣 − 1 푣 푣 − 1
484 Professor Doron Avramov, Financial Econometrics Partial Expectation
And thus 2 푓 푞1−훼, 푣 푣 + 푞1−훼 퐸푆푇 훼 = 휇 − 휎 ⋅ 푞1−훼 푣 − 1
Where 푞푥 = 푉푎푅 1 − 푥 is 푥 quantile of 푇 ∼ 푡푛
485 Professor Doron Avramov, Financial Econometrics Long Run Return when Periodic Return has the 푡-Distribution The sum of independent t-distributed random variables is not 푡-distributed. So we have no nice formula for expected shortfall in the long run. However, it can be approximated by normal with zero mean variance: 푣 푟 ∼ 푁 휇, 휎 푓표푟 푣 ≥ 2 푎푝푝푟표푥. 푣 − 2
486 Professor Doron Avramov, Financial Econometrics Long Run Return when Periodic Return has the 푡-Distribution Approximation makes sense for large v’s when 푡 coincides with normal distribution.
However, simulation studies show that for sufficient number of periods this approximation works well enough.
487 Professor Doron Avramov, Financial Econometrics Long Run Return when Periodic Return has the 푡-Distribution However simulations shows that for sufficient number of periods this approximation works well enough.
Let 휇푡 = 0.01; 휎푡 = 0.05. The next graphs show normal curve fit to the sum of 푡 r.v.s (over 푇 periods); sample estimates vs. predicted parameters are includes
488 Professor Doron Avramov, Financial Econometrics Long Run Return when Periodic Return has the 푡-Distribution
푇 = 10 900
800
700 휇Ƹ = 0.102 휇푁 = 0.1 휎ො = 0.273 휎푁 = 0.274 600
500 푣 = 3 400
300
200
100
0 -3 -2 -1 0 1 2 3
489 Professor Doron Avramov, Financial Econometrics Long Run Return when Periodic Return has the 푡-Distribution
푇 = 100 400
350 휇Ƹ = 1.011 휇 = 1 300 푁 휎ො = 0.856 휎푁 = 0.866 250
푣 = 3 200
150
100
50
0 -4 -3 -2 -1 0 1 2 3 4 5
490 Professor Doron Avramov, Financial Econometrics Long Run Return when Periodic Return has the 푡-Distribution
푇 = 10 350
300
휇Ƹ = 0.099 휇푁 = 0.1 250 휎ො = 0.174 휎푁 = 0.177
200 푣 = 10 150
100
50
0 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
491 Professor Doron Avramov, Financial Econometrics Long Run Return when Periodic Return has the 푡-Distribution
푇 = 100 350
300
휇Ƹ = 0.994 휇푁 = 1 250 휎ො = 0.566 휎푁 = 0.559
200 푣 = 10 150
100
50
0 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
492 Professor Doron Avramov, Financial Econometrics Session #11 (part a): Testing the Black Scholes Formula
493 Professor Doron Avramov, Financial Econometrics Option Pricing
The B&S call Option price is given by −훿T −rT C(S, K, 휎, r, T, 훿) = Se N(d1) − Ke N(d2)
The put Option price is −rT −훿T P(S, K, 휎, r, T, 훿) = Ke N(−d2) − Se N(−d1)
1 푙푛(푆/퐾)+(푟−훿+ 휎2)푇 where d = 2 and d = d − 휎 T 1 휎 푇 2 1
494 Professor Doron Avramov, Financial Econometrics Option Pricing
There are six inputs required:
S - Current price of the underlying asset. K - Exercise/Strike price. r - Continuously compounded risk-free rate. T - Time to expiration. 휎 - Volatility. 훿 - Continuously compounded dividend yield.
495 Professor Doron Avramov, Financial Econometrics The B&S Economy
The B&S formula is derived under several assumptions:
The stock price follows a geometric Brownian motion (continuous path and continuous time).
The dividend is paid continuously and uniformly over time.
The interest rate is constant over time.
496 Professor Doron Avramov, Financial Econometrics The B&S Economy
The underlying asset volatility is constant over time and it does not change with the option maturity or with the strike price.
You can short sell or long any amount of the stock.
You can borrow and lend in the risk-free rate.
There are no transactions costs.
497 Professor Doron Avramov, Financial Econometrics Testing the B&S Formula Mark Rubinstein analyzes call options that are deep out of the money.
He considers matched pairs: options with the same striking price, on the same underlying asset (stock), but with different time to maturity (expiration date).
He examines overall 373 pairs.
If B&S is correct then the implied volatility (IV) of the matched pair is equal. Time to maturity plays no role.
498 Professor Doron Avramov, Financial Econometrics Testing the B&S Formula
However, Rubinstein finds that our of the 373 examined matched pairs – shorter maturity options had higher IV.
Under the null – the expected value of such an outcome is 373/2=186.5.
Is the difference statistically significant?
499 Professor Doron Avramov, Financial Econometrics The Failure Rate based Test Statistic
Use the failure rate test developed earlier to show that time to expiration does play a major role.
That is to say that the constant volatility assumption is strongly violated in the data.
500 Professor Doron Avramov, Financial Econometrics The Volatility Smile for Foreign Currency Options
Implied
Volatility
Strike
Price
501 Professor Doron Avramov, Financial Econometrics Implied Distribution for Foreign Currency Options
Both tails are heavier than the lognormal distribution.
It is also “more peaked” than the lognormal distribution.
502 Professor Doron Avramov, Financial Econometrics The Volatility Smile for Equity Options
Implied
Volatility
Strike
Price
503 Professor Doron Avramov, Financial Econometrics Implied Distribution for Equity Options
The left tail is heavier and the right tail is less heavy than the lognormal distribution.
504 Professor Doron Avramov, Financial Econometrics Ways of Characterizing the Volatility Smiles
Plot implied volatility against 퐾/푆0 (The volatility smile is then more stable).
Plot implied volatility against 퐾/퐹0 (Traders usually define an option as at-the-money when 퐾 equals the forward price, 퐹0, not when it equals the spot price 푆0). Plot implied volatility against delta of the option (This approach allows the volatility smile to be applied to some non- standard options. At-the money is defined as a call with a delta of 0.5 or a put with a delta of −0.5. These are referred to as 50- delta options).
505 Professor Doron Avramov, Financial Econometrics Possible Causes of Volatility Smile
Asset price exhibits jumps rather than continuous changes.
Volatility for asset price is stochastic: In the case of an exchange rate volatility is not heavily correlated with the exchange rate. The effect of a stochastic volatility is to create a symmetrical smile. In the case of equities volatility is negatively related to stock prices because of the impact of leverage. This is consistent with the skew that is observed in practice.
506 Professor Doron Avramov, Financial Econometrics Volatility Term Structure
In addition to calculating a volatility smile, traders also calculate a volatility term structure.
This shows the variation of implied volatility with the time to maturity of the option.
The volatility term structure tends to be downward sloping when volatility is high and upward sloping when it is low
507 Professor Doron Avramov, Financial Econometrics Example of a Volatility Surface
퐾Τ푆0
0.90 0.95 1.00 1.05 1.10 1 month 14.2 13.0 12.0 13.1 14.5 3 months 14.0 13.0 12.0 13.1 14.2 6 months 14.1 13.3 12.5 13.4 14.2 1 year 14.7 14.0 13.5 14.0 14.8 2 years 15.0 14.4 14.0 14.5 15.1 5 years 14.8 14.6 14.4 14.7 15.0
508 Professor Doron Avramov, Financial Econometrics Session #11 (part b): Time Varying Volatility Models
509 Professor Doron Avramov, Financial Econometrics Volatility Models We describe several volatility models commonly applied in analyzing quantities of interest in finance and economics: ARCH GARCH EGARCH Stochastic Volatility Realized and implied Volatility
510 Professor Doron Avramov, Financial Econometrics Volatility Models All such models attempt to capture the empirical evidence that volatility is time varying (rather than constant) as well as persistent.
The EGARCH captures the asymmetric response of volatility to advancing versus diminishing markets.
In particular, volatility tends to be higher (lower) during down (up) markets.
511 Professor Doron Avramov, Financial Econometrics ARCH(1)
푟푡 = 휇 + 휀푡
휀푡 = 휎푡푒푡 where 푒푡~푁 0,1 2 2 휎푡 = 푤 + 훼휀푡−1 2 2 2 2 퐸푡−1 휀푡 = 퐸푡−1 휎푡 푒푡 = 휎푡 so 2 휎푡 is the conditional variance.
512 Professor Doron Avramov, Financial Econometrics ARCH(1)
2 2 퐸 휀푡−1 = 휎ത is the unconditional variance. 2 2 퐸 휎푡 = 퐸 푤 + 훼휀푡−1 2 = 푤 + 훼퐸 휀푡−1 2 2 = 푤 + 훼퐸 휎푡−1 퐸 푒푡−1 2 = 푤 + 훼퐸 휎푡−1 2 2 2 2 ⇒ 휎lj = 퐸 휎푡 = 퐸 휎푡−1 = 퐸 휎푡−2 푤 퐸 휎2 = 푡 1−훼
513 Professor Doron Avramov, Financial Econometrics ARCH(1)
Fat tail? 4 4 퐸 휀푡 퐸 퐸푡−1 휀푡 휇4 = 2 2 = 2 2 2 퐸 휀푡 퐸 퐸푡−1 푒푡 휎푡 4 4 4 퐸 퐸푡−1 푒푡 휎푡 퐸 3휎푡 = 2 = 2 퐸 퐸 푒2휎2 2 푡−1 푡 푡 퐸 휎푡 4 퐸 휎푡 = 3 2 ≥ 3 2 퐸 휎푡
514 Professor Doron Avramov, Financial Econometrics ARCH(1)
The last step follows because: 2 4 2 2 푉푎푟 휀푡 = 퐸 휀푡 − 퐸 휀푡 ≥ 0
so 4 퐸 휀푡 2 2 ≥ 1 퐸 휀푡
Yes – fat tail!
515 Professor Doron Avramov, Financial Econometrics GARCH(1,1)
푟푡 = 휇 + 휀푡
휀푡 = 휎푡푒푡 where 푒푡~푁 0,1
2 2 2 휎푡 = 푤 + 훼휀푡−1 + 훽휎푡−1
2 2 2 퐸 휎푡 = 푤 + 훼퐸 휀푡−1 + 훽퐸 휎푡−1
516 Professor Doron Avramov, Financial Econometrics GARCH(1,1)
휎ത2 = 푤 + 훼휎ത2 + 훽휎ത2
푤 휎ത2 = 1−훼−훽
3 1+훼+훽 1−훼−훽 휇 = > 3 4 1−2훼훽−3훼2−훽2
517 Professor Doron Avramov, Financial Econometrics EGARCH
푟푡 = 휇 + 휀푡 where 푒푡~푁 0,1
휀푡 = 휎푡푒푡
2 휀푡−1 2 휀푡−1 2 ln 휎푡 = 푤 + 훼 − − 훾 + 훽 ln 휎푡−1 휎푡−1 휋 휎푡−1
휀푡−1 2 2 The first component − = 푒푡−1 − 휎푡−1 휋 휋
is the absolute value of a normally distribution variable 푒푡−1 minus its expectation.
518 Professor Doron Avramov, Financial Econometrics EGARCH
휀푡−1 The second component is 훾 = 훾푒푡−1 휎푡−1
Notice that the two normal shocks behave differently.
The first produces a symmetric rise in the log conditional variance.
The second creates an asymmetric effect, in that, the log conditional variance rises following a negative shock.
519 Professor Doron Avramov, Financial Econometrics EGARCH
More formally if 푒푡−1 < 0 then the log conditional variance rises by 훼 + 훾.
If 푒푡−1 > 0 then the log conditional variance rises by 훼 − 훾
This produces the asymmetric volatility effect – volatility is higher during down market and lower during up market.
520 Professor Doron Avramov, Financial Econometrics Stochastic Volatility (SV)
There is a variety of SV models.
A popular one follows the dynamics
푟푡 = 휇푡 + 휎푡휀푡 where 휀푡 ∼ 푁 0,1
ln 휎푡 = 훾0 + 훾1 ln 휎푡−1 + 휂푡푣푡 where 푣푡~푁 0,1
cov 푣푡, 휀푡 = 0
Notice, that unlike ARCH, GARCH, and EGARCH, here the volatility itself has a stochastic component.
521 Professor Doron Avramov, Financial Econometrics Realized Volatility
The realized volatility (RV) is a very tractable way to measure volatility.
It essentially requires no parametric modeling approach.
Suppose you observe daily observations within a trading month on the market portfolio.
RV is the average of the squared daily returns within that month.
522 Professor Doron Avramov, Financial Econometrics Realized Volatility
Of course, volatility varies on the monthly frequency but it is assumed to be constant within the days of that particular month.
If you observe intra-day returns (available for large US firms) then daily RV is the sum of squared of five minute returns.
You can use AR(1) to model log realized variance and then predict future values.
523 Professor Doron Avramov, Financial Econometrics Implied Volatility (IV)
The B&S call Option price is given by −훿T −rT C(S, K, 휎, r, T, 훿) = Se N(d1) − Ke N(d2)
The put Option price is −rT −훿T P(S, K, 휎, r, T, 훿) = Ke N(−d2) − Se N(−d1)
1 푙푛(푆/퐾)+(푟−훿+ 휎2)푇 Where d = 2 and d = d − 휎 T 1 휎 푇 2 1
524 Professor Doron Avramov, Financial Econometrics Implied Volatility (IV)
In the traditional option pricing practice one inserts into the formula all the six parameters, i.e., the stock price, the strike price, the time to expiration, the cc risk-free rate, the cc dividend yield, and stock return volatility.
IV is that volatility that if inserted into the B&S formula would yield the market price of the call or put option.
As noted earlier, IV in not constant across maturities or across strike prices.
525 Professor Doron Avramov, Financial Econometrics Session #11 (part c): Stock Return Predictability, Model Selection, and Model Combination
526 Professor Doron Avramov, Financial Econometrics Return Predictability
If log returns are IID – there is no way you can deliver better prediction for stock return than the current mean return.
푖푖푑 2 That is, if 푟푡 = 휇 + 휀푡 where 휀푡 ~ = 푁 0, 휎
퐸 푟푡+1|퐼푡 = 휇 2 푉푎푟 푟푡+1|퐼푡 = 휎
where 푟푡 is the continuously compounded return and 퐼푡 is the set of information available at time 푡.
527 Professor Doron Avramov, Financial Econometrics Return Predictability
Also note that the variance of a two-period return 푟푡 + 푟푡+1 is equal to
2 푉푎푟 푟푡 + 푉푎푟 푟푡+1 + 2 cov 푟푡, 푟푡+1 = 2휎
That is, variance grows linearly with the investment horizon, while volatility grows in the rate square root.
528 Professor Doron Avramov, Financial Econometrics Variance Ratio Tests
However, is it really the case?
Perhaps stock returns are auto-correlated, or cov 푟푡, 푟푡+1 ≠ 0 then:
푉푎푟 푟푡 + 푟푡+1 푉푎푟 푟푡 + 푉푎푟 푟푡+1 + 2 푐표푣 푟푡, 푟푡+1 푉푅2 = = 2푉푎푟 푟푡 2푉푎푟 푟푡 푐표푣 푟 ,푟 = 1 + 푡 푡+1 = 1 + 휌 휎 푟푡 휎 푟푡+1
529 Professor Doron Avramov, Financial Econometrics Variance Ratio Tests
Test: 퐻0: 휌 = 0
퐻1: 휌 ≠ 0
The test statistic is 푑 푇 푉푅2 − 1 →푁 0,1
530 Professor Doron Avramov, Financial Econometrics Variance Ratio Tests
More generally, 푔 푉푎푟 푟푡 + 푟푡+1 + ⋯ + 푟푡+푔 푠 푉푅푔 = = 1 + 2 1 − 휌푠 푔 + 1 푉푎푟 푟푡 푔 + 1 푠=1
퐻0: 푉푅푔 = 1 no auto correlation
퐻1: 푂푡ℎ푒푟푤푖푠푒
531 Professor Doron Avramov, Financial Econometrics Variance Ratios
Test statistic: 푔−1 2 푑 푠 푇 푉푅 − 1 →푁 0, 4 1 − 푔 푔 푠+1 e.g. 푔 = 2 푑 푇 푉푅2 − 1 →푁 0,1 푔 = 3 푑 20 푇 푉푅 − 1 →푁 0, 3 9
532 Professor Doron Avramov, Financial Econometrics Predictive Variables
In the previous specification, we used lagged returns to forecast future returns or future volatility.
You can use a bunch of other predictive variables, such as: The term spread. The default spread. Inflation. The aggregate dividend yield
533 Professor Doron Avramov, Financial Econometrics Predictive Variables
The aggregate book-to-market ratio.
The market volatility.
The market illiquidity
534 Professor Doron Avramov, Financial Econometrics Predictive Regressions
To examine whether stock returns are predictable, we can run a predictive regression.
This is the regression of future excess log or gross return on predictive variables.
It is formulated as: 푟푡+1 = 푎 + 푏1푧1푡 + 푏2푧2푡 + ⋯ + 푏푀푧푀푡 + 휀푡+1
535 Professor Doron Avramov, Financial Econometrics Predictive Regressions
To examine whether either of the macro variables can predict future returns test whether either of the slope coefficients is different from zero.
Use the t-statistic or F-statistic for the regression R-squared.
There is a small sample bias if (i) the predictive variables are highly persistent, (ii) the contemporaneous correlation between the predictive regression residual and the innovation of the predictor is high, or (iii) the sample is small.
536 Professor Doron Avramov, Financial Econometrics Long Horizon Predictive Regressions
′ 푟푡+1,푡+퐾 = 훼 + 푏 푧푡 + 휀푡+1,푡+퐾 The dependent variable is the sum of log excess return over the investment horizon, which is 퐾 periods.
Since the residuals are auto correlated compute the standard errors for the slope coefficient accounting for serial correlation and often for heteroscedasticity.
For instance you can use the Newey-West correction.
537 Professor Doron Avramov, Financial Econometrics Newey-West Correction
Rewriting the long horizon regression
′ 푟푡+1,푡+퐾 = 푥푡 훽 + 휀푡+1,푡+퐾
′ ′ 푥푡 = 1, 푧푡
훽′ = 푎, 푏′
538 Professor Doron Avramov, Financial Econometrics Newey-West Correction
The estimation error of the regression coefficient is represented by 푉푎푟 훽መ 푉푎푟 훽መ = 푋′푋 −1푆መ where 푆መ is the Newey-West given by serially correlated adjusted estimator 퐾 푇 퐾 − 푗 1 푆መ = ⋅ 휀 휀 퐾 푇 푡 푡−푗 푗=−퐾 푡=푗+1
539 Professor Doron Avramov, Financial Econometrics Long Horizon Predictive Regressions
Tradeoff: Higher 퐾 – better coverage of dependence.
But we loose degrees of freedom.
Feasible solution: 1 퐾∞푇3
540 Professor Doron Avramov, Financial Econometrics Long Horizon Predictive Regressions
E.g. K=1: 푗 = −1, 푗 = 0, 푗 = 1
1 푆መ = σ푇 휀2 푇 푡=1 푡
Here we have no serial correlation.
541 Professor Doron Avramov, Financial Econometrics Long Horizon Predictive Regressions
E.g. K=2: 푗 = −2, 푗 = −1, 푗 = 0, 푗 = 1, 푗 = 2
푇−1 푇 푇 1 1 1 1 1 푆መ = ⋅ 휀 휀 + 휀2 + ⋅ 휀 휀 2 푇 푡 푡+1 푇 푡 2 푇 푡 푡−1 푡=1 푡=−1 푡=2
푇 푇−1 1 = 휀2 + 휀 휀 푇 푡 푡 푡+1 푡=1 푡=1
542 Professor Doron Avramov, Financial Econometrics In the Presence of Heteroskedasticity
−1 −1 푇 푇 1 1 1 푉푎푟 훽መ = 푥 푥 ′ 푆መ 푥 푥 ′ 푇 푇 푡 푡 푇 푡 푡 푡=1 푡=1
퐾 푇−퐾+1 퐾 − 푗 1 푆መ = ⋅ ⋅ 휀 푥 푥 ′휀 퐾 푇 푡 푡 푡−푗 푡−푗 푗=−퐾 푡=1
543 Professor Doron Avramov, Financial Econometrics Out of Sample Predictability
There is ample evidence of in-sample predictability, but little evidence of out-of-sample predictability.
Consider the two specifications for the stock return evolution
푀1: 푟푡 = 푎 + 푏푧푡−1 + 휀푡 푀2: 푟푡 = 휇 + 휀푡
Which one dominates? If 푀1 then there is predictability otherwise, there is no.
544 Professor Doron Avramov, Financial Econometrics Out of Sample Predictability
One way to test predictability is to compute the out of sample 푅2: 푇 2 2 σ푡=1 푟푡 − 푟푡Ƹ ,1 푅푂푂푆 = 1 − 푇 2 σ푡=1 푟푡 − 푟푡lj
Where 푟푡Ƹ ,1 is the return forecast assuming the presence of predictability, and 푟푡ҧ is the sample mean (no predictability).
Can compute the MSE (Mean Square Error) for both models.
545 Professor Doron Avramov, Financial Econometrics Model Selection
When 푀 variable are potential candidates for predicting stock returns there are 2푀 linear combinations of predictive models.
In the extreme, the model that drops all predictors is the no- predictability or IID model. The one that retains all predictors is the all inclusive model.
Which model to use?
One idea (bad) is to implement model selection criteria.
546 Professor Doron Avramov, Financial Econometrics Model Selection Criteria
퐴퐼퐶 = 2푚 − 2 ln 퐿 where 퐿 is the maximized value of the likelihood function. 퐵퐼퐶 = 푚 ln 푇 − 2 ln 퐿 푇−1 푚 푅ത2 = 1 − 1 − 푅2 = 푅2 − 1 − 푅2 푇−푚−1 푇−푚−1
Bayesian posterior probability
547 Professor Doron Avramov, Financial Econometrics Model Selection
Notice that all criteria are a combination of goodness of fit and a penalty factor.
You choose only one model and disregard all others.
Model selection criteria have been shown to exhibit very poor out of sample predictive power.
548 Professor Doron Avramov, Financial Econometrics Model Combination
The other approach is to combine models.
Bayesian model averaging (BMA) computes posterior probabilities for each model then it uses the posterior probabilities as weights to compute the weighted model.
There are more naive combinations.
Such combination methods produce quite robust predictors not only in sample but also out of sample.
549 Professor Doron Avramov, Financial Econometrics