FOUNDATIONS OF MANAGEMENT: THE AND THE CAPITAL MODEL EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ 푛

퐸 푋 = 휇푥 = 푃(푋 = 푋푖)( 푋푖) 푖=1

FOR TWO VARIABLES

퐸 푋 + 푌 = 푃 푋 = 푋푖, 푌 = 푌푗 푋푖 + 푌푗 = 푃푖푗 푋푖 + 푌푗 = 푖 푗 푖 푗

= 푃푖푗 푋푖 + 푃푖푗 푌푗 = 푃푖푋푖 + 푃푗푌푗 = 푬 푿 + 푬(풀) 푖 푗 푖 푗 푖 푖

푃푖푗 = 푃푖 푃푖푗 = 푃푗 푗 푖

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 2 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

The on portfolio is a weighted sum of returns on individual assets 푁

퐸 푅푝 = 휔푖 퐸 푅푖 푖=1

퐸 푅푝 – expected return on portfolio

푤푖 – weight of Asset i in portfolio 퐸 푅푖 – expected return on Asset i All weights should sum up to 100% of portfolio

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 3 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

Old (alternative approach) is the evaluate each investment opportunity on its own

N

σp = ωi σi i=1

Let’s consider the following example

Portfolio consist of two assets

$100 is invested in asset A (standard deviation σA = $4)

$100 is invested in asset B standard deviation σB = $6 N 1 1 σ = ω σ = × $4 + × $6 = $5 p i i 2 2 i=1

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 4 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

In 1950s Harry Markowitz provided a framework for measuring risk- reduction benefits of diversification; he concluded that, unless the returns of risky assets are perfectly positively correlated, risk is reduced by diversifying across assets

Markowitz used standard deviation as a measure of risk of the assets; it is still used as the best proxy

Diversification – process of including additional different assets in the portfolio in order to minimize risk (i.e. include bonds and ETFs to all- portfolio)

The modern portfolio theory (MPT) was founded in 1960s by several independent scientific studies

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 5 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

n 2 2 2 Var X = (σx) = E X − E X X − E X = E (X − E(X)) = P(Xi)(Xi − E(X)) i=1

FOR TWO VARIABLES

Var X + Y = E (X + Y − E(X + Y))2 =

= E (X + Y − E X − E(Y))2 = E (퐗 − 퐄 퐗 + 퐘 − 퐄(퐘))2 =

퐄 퐗 − 퐄 퐗 퐗 − 퐄 퐗 + 퐄 퐘 − 퐄 퐘 퐘 − 퐄 퐘 + 퐄 퐗 − 퐄 퐗 퐘 − 퐄 퐘 + 퐄 퐘 − 퐄 퐘 퐗 − 퐄 퐗 =

퐂퐨퐯 퐗, 퐗 + 퐂퐨퐯 퐘, 퐘 + 퐂퐨퐯 퐗, 퐘 + 퐂퐨퐯 퐘, 퐗 =

퐕퐚퐫 퐗 + 퐕퐚퐫 퐘 + ퟐ × 퐂퐨퐯 퐗, 퐘

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 6 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

Amount Expected Expected Correlation invested return ABC 20000 7% 15% 0.3 XYZ 30000 11% 22%

Investor has invested cash in two companies: ABC and XYZ Weights of companies in portfolio are 20000 ◦ 푤 = = 0.4 퐴퐵퐶 20000+30000 30000 ◦ 푤 = = 0.6 푋푌푍 20000+30000

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 7 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

Expected return

퐸 푅푝 = 푤퐴퐵퐶 ∙ 퐸 푅퐴퐵퐶 + 푤푋푌푍 ∙ 퐸 푅푋푌푍 = 0.4 ∙ 7% + 0.6 ∙ 11% = 9.4%

Expected volatility

2 2 2 2 𝜎푝 = 푤1 ∙ 𝜎1 + 푤2 ∙ 𝜎2 + 2 ∙ 푤1 ∙ 푤2 ∙ 𝜌1,2 ∙ 𝜎1 ∙ 𝜎2 = = 0.42 ∙ 0.152 + 0.62 ∙ 0.222 + 2 ∙ 0.4 ∙ 0.6 ∙ 0.3 ∙ 0.15 ∙ 0.22 = 16.05%

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 8 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

ABC and XYZ example shows only one allocation of capital between two assets Expected return and risk of portfolio depends on the allocation

ABC/ XYZ 100%/0% 80%/20% 60%/40% 50%/50% 40%/60% 20%/80% 0% /100%

퐸 푅푝 7.00% 7.80% 8.60% 9.00% 9.40% 10.20% 11.00%

𝜎푝 15.00% 13.97% 14.35% 15.06% 16.05% 18.72% 22.00%

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 9 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

PORTFOLIO POSSIBILITIES CURVE (RETURN) 12,00%

11,00% 11,00% 10,20% 10,00% 9,40% 9,00% 8,60% 9,00% 8,00% 7,80% 7,00% 7,00%

6,00% 10,00% 15,00% 20,00% 25,00%

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 10 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

PORTFOLIO POSSIBILITIES CURVE (RISK) 12,00%

11,00% 22,00%

10,00% 18,72% 16,05% 9,00% 14,35% 15,06% 8,00% 13,97% 7,00% 15,00%

6,00% 10,00% 15,00% 20,00% 25,00%

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 11 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

Minimum variance portfolio is a portfolio that has the minimum variance among all possible allocation of capital between assets.

Allocation weights are solution of following problem 2 2 2 2 2 𝜎푝 = 푤1 ∙ 𝜎1 + 푤2 ∙ 𝜎2 + 2 ∙ 푤1 ∙ 푤2 ∙ 𝜌1,2 ∙ 𝜎1 ∙ 𝜎2 → 푚𝑖푛 푤1 + 푤2 = 1

푤1 = 75.34%, 푤2 = 24.66%

MINIMUM VARIANCE PORTFOLIO 12,00%

11,00%

10,00%

9,00% E(R)=7.99%, 8,00% s.d.=13.93% 7,00%

6,00% 10,00% 15,00% 20,00% 25,00%

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 12 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

Correlation impact on PPC 12,00%

11,00% Corr = 1 10,00% Corr = 0.7 9,00% Corr = 0.3 Corr = 0 8,00% Corr = -0.5 7,00% Corr = -1

6,00% 0,00% 5,00% 10,00% 15,00% 20,00% 25,00%

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 13 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

Expected return

INEFFICIENT PORTFOLIOS

Risk-free

Portfolio risk (σ)

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 14 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

. All portfolios on efficient frontier are made up with risky assets . Risk-free assets earns some return (at risk-free rate) and this return is expected to have zero volatility . Combination of risk-free asset and portfolio gives a new set of portfolios that form a line Expected return of combinations 퐸 푅퐶 = 푤퐹푅퐹 + 푤푃퐸 푅푃 Volatility of returns 2 2 2 2 2 𝜎퐶 = 푤퐹 𝜎퐹 + 푤푃𝜎푃 + 2푤퐹푤푃퐶표푣퐹,푃 Volatility of risk-free asset is zero, so its variance and with risky portfolio are zero as well 2 2 2 𝜎퐶 = 푤푃𝜎푃 ; 𝜎푐 = 푤푃𝜎푝

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 15 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

Expected return CAL (C)

CAL (B)

(B) CAL (A)

(C)

(A) Risk-free

Portfolio risk (σ)

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 16 EXPLAIN MODERN PORTFOLIO THEORY AND INTERPRET THE MARKOWITZ EFFICIENT FRONTIER

Expected return and volatility of combination of risk-free asset and risky portfolio are linear functions Relationship of expected return and volatility is also linear 퐸 푅퐶 = 푤퐹푅퐹 + 푤푃퐸 푅푃 𝜎푐 = 푤푃𝜎푝 푤퐹 + 푤푃 = 1

𝜎 푤 = 퐶 푃 𝜎푃 푤퐹 = 1 − 푤푃 𝜎퐶 𝜎퐶 퐸 푅퐶 = 1 − 𝜎푃 푅퐹 + 𝜎푃 ∙ 퐸 푅푃

푬 푹푷 − 푹푭 푬 푹푪 = 푹푭 + ∙ 흈푪 흈푷

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 17 INTERPRET THE LINE

Newly formed line that is tangent to efficient frontier is called (CML) • the intercept equals a risk-free rate • the tangency point is known as a • the slope equals a reward-to-risk ratio of risky (market) portfolio If have same expectations of risk and return for assets, all they will hold combination of risk-free asset and market portfolio • More risk-averse investors will buy less part of market portfolio and lend cash at risk-free rate • More risk-tolerant investors will borrow cash and buy more market portfolio

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 18 INTERPRET THE CAPITAL MARKET LINE

Expected return CML

퐸 푟푚 − 푟푓 퐶푀퐿: 퐸 푟 = 푟푓 + 𝜎 𝜎푀 Tangency portfolio (Market Portfolio) 퐸 푅푀 − 푅퐹 Risk-free 푇ℎ푒 푠푙표푝푒 표푓 퐶푀퐿 = 𝜎푀

Portfolio risk (σ)

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 19 INTERPRET THE CAPITAL MARKET LINE

Differential Borrowing and Lending Rates. Most investors can lend unlimited amounts at the risk-free rate by buying government securities, but they must pay a premium relative to the prime rate when borrowing money. The effect of this differential is that there will be two different lines going to the Markowitz efficient frontier. Expected return

New Tangency Portfolio

Borrowing rate (Market Portfolio)

Risk-free

Portfolio risk (σ)

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 20 UNDERSTAND THE DERIVATION AND COMPONENTS OF THE CAPM DESCRIBE THE ASSUMPTIONS UNDERLYING THE CAPM The CAPM is an equilibrium model that predicts the expected return on a stock, given the expected return on the market, the stock's coefficient, and the risk-free rate.

푅푖 = 푅푓 + 훽푖(푅푚 − 푅푓)

푅푖 − 푟푒푡푢푟푛 표푛 푎푠푠푒푡 𝑖 푅푓 − 푟푒푡푢푟푛 표푛 푟𝑖푠푘 − 푓푟푒푒 푎푠푠푒푡 푅푚 − 푟푒푡푢푟푛 표푛 푚푎푟푘푒푡 푝표푟푡푓표푙𝑖표 훽푖 − 푠푒푛푠𝑖푡𝑖푣𝑖푡푦 표푓 푡ℎ푒 푎푠푠푒푡′푠 푟푒푡푢푟푛푠 푡표 푚푎푟푘푒푡 푟푒푡푢푟푛푠

The assumptions of the CAPM

■ All investors are risk averse and ■ All investors have homogeneous utility maximizing expectations

■ Markets are frictionless ■ All investments are infinitely divisible

■ All investors have the same one- ■ All investors are price takers. Their period time horizon trades cannot affect prices.

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 21 UNDERSTAND THE DERIVATION AND COMPONENTS OF THE CAPM DESCRIBE THE ASSUMPTIONS UNDERLYING THE CAPM

푅푖 − 푅푓 = 훽푖 푅푚 − 푅푓 + 푒푖 2 2 2 2 𝜎푖 = 훽푖 𝜎푚 + 𝜎푒 + 2퐶표푣(훽푖푅푚, 푒푖)

2 2 2 2 Assuming 퐶표푣 훽푖푅푚, 푒푖 = 0 → 𝜎푖 = 훽푖 𝜎푚 + 𝜎푒

SYSTEMATIC NON-SYSTEMATIC VARIANCE VARIANCE

퐶표푣 푅푖, 푅푚 훽푖 = 푉푎푟(푅푚)

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 22 UNDERSTAND THE DERIVATION AND COMPONENTS OF THE CAPM DESCRIBE THE ASSUMPTIONS UNDERLYING THE CAPM

TOTAL RISK = SYSTEMATIC (MARKET) RISK + NON-SYSTEMATIC (SPECIFIC) RISK

cannot be diversified can be diversified

*Note that non-systematic (specific) risk is not rewarded as it can be eliminated for free by diversification

Portfolio risk (σ) NON- SYSTEMATIC RISK

Number of securities 30 FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL APPLY THE CAPM IN CALCULATING THE EXPECTED RETURN ON AN ASSET

We have following data about the company Chocolove, Inc.

Expected market return 4%

Risk-free rate 1.5%

Chocolove beta 1.25

Required return for Chocolove according to CAPM

퐸 푅푖 = 푅퐹 + 퐸 푅푀 − 푅퐹 ∙ 훽푖 = 1.5% + 4% − 1.5% × 1.25 = 4.625%

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 24 APPLY THE CAPM IN CALCULATING THE EXPECTED RETURN ON AN ASSET

SML or to compare the relationship between risk and return. Unlike the CML, which uses standard deviation as a risk measure on the X axis, the SML uses the market beta, or the relationship between a security and the marketplace.

UNDERVALUED Expected return SECURITIES

SML

푅푚

푅 푓 OVERVALUED SECURITIES Beta

1.0

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 25 INTERPRET BETA AND CALCULATE THE BETA OF A SINGLE ASSET OR PORTFOLIO

Beta of an investment is a measure of the risk arising from exposure to general market movements.

퐶표푣 푅푖, 푅푚 훽푖 = 푉푎푟(푅푚)

Portfolio beta is a weighted sum of individual asset betas

Weight Beta Auto Inc 30% 1.5 Berryville 20% 0.7 Chipside Ltd 50% 1.1 Total 100% 1.14

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 26 CALCULATE, COMPARE, AND INTERPRET THE FOLLOWING PERFORMANCE MEASURES: THE SHARPE PERFORMANCE INDEX, THE TREYNOR PERFORMANCE INDEX, THE JENSEN PERFORMANCE INDEX, THE , INFORMATION RATIO, AND

Jensen's is used to determine the abnormal return of a security or portfolio of securities over the theoretical expected return.

Jensen′s alpha measure = Rp − Rf + βp Rm − Rf

UNDERVALUED Expected return SECURITIES SML

JENSEN′S ALPHA 푅푚

푅푓 OVERVALUED SECURITIES Beta

1.0

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 27 CALCULATE, COMPARE, AND INTERPRET THE FOLLOWING PERFORMANCE MEASURES: THE SHARPE PERFORMANCE INDEX, THE TREYNOR PERFORMANCE INDEX, THE JENSEN PERFORMANCE INDEX, THE TRACKING ERROR, INFORMATION RATIO, AND SORTINO RATIO

When we evaluate the performance of a portfolio with risk that differs from that of a benchmark, we need to adjust the portfolio returns for the risk of the portfolio

(푅푝 − 푅푓) 푇ℎ푒 푆ℎ푎푟푝푒 푟푎푡𝑖표 = 𝜎푝

The of a portfolio is its excess returns per unit of total portfolio risk, and higher Sharpe ratios indicate better risk-adjusted portfolio performance

The Treynor measure is risk-adjusted returns based on systematic risk (beta) rather than total risk

(푅푝 − 푅푓) The Treynor measure = 훽푝

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 28 CALCULATE, COMPARE, AND INTERPRET THE FOLLOWING PERFORMANCE MEASURES: THE SHARPE PERFORMANCE INDEX, THE TREYNOR PERFORMANCE INDEX, THE JENSEN PERFORMANCE INDEX, THE TRACKING ERROR, INFORMATION RATIO, AND SORTINO RATIO

Let’s calculate ratios for Chocolove, Inc

Expected Chocolove return 4.875%

Chocolove vol 7%

Risk-free rate 1.5%

Chocolove beta 1.25 Expected market return 4%

퐸 푅푝 − 푅푓 4.8% − 1.5% 푆ℎ푎푟푝푒 푟푎푡𝑖표 = = ≈ 0.47 𝜎푝 7% 퐸 푅푝 − 푅푓 4.8% − 1.5% 푇푟푒푦푛표푟 푟푎푡𝑖표 = = ≈ 2.64 훽푝 1.25 훼푝 = 퐸 푅푝 − 푅푓 + 퐸 푅푚 − 푅푓 ∙ 훽푖 = 4.8% − 4.625% = 0.175%

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 29 CALCULATE, COMPARE, AND INTERPRET THE FOLLOWING PERFORMANCE MEASURES: THE SHARPE PERFORMANCE INDEX, THE TREYNOR PERFORMANCE INDEX, THE JENSEN PERFORMANCE INDEX, THE TRACKING ERROR, INFORMATION RATIO, AND SORTINO RATIO

As tries to earn excess return over the benchmark, the difference in returns varies over time.

훼 = 푅푝 − 푅퐵

Tracking error is the standard deviation of difference between portfolio return and benchmark return

푡푟푎푐푘𝑖푛푔 푒푟푟표푟 = 𝜎훼

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 30 CALCULATE, COMPARE, AND INTERPRET THE FOLLOWING PERFORMANCE MEASURES: THE SHARPE PERFORMANCE INDEX, THE TREYNOR PERFORMANCE INDEX, THE JENSEN PERFORMANCE INDEX, THE TRACKING ERROR, INFORMATION RATIO, AND SORTINO RATIO

Information Ratio The ratio computes the surplus return relative to the surplus risk taken. The variability in the surplus return is a measure of the risk taken to achieve the surplus.

The higher information ratio, the better performance is.

푅푝 − 푅퐵 훼 퐼푅퐴 = = 𝜎푝 −퐵 푡푟푎푐푘𝑖푛푔 푒푟푟표푟

Where:

푅푝 - average portfolio return 푅퐵 - average benchmark return 𝜎푝 −퐵 - standard deviation of excess return over benchmark

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 31 CALCULATE, COMPARE, AND INTERPRET THE FOLLOWING PERFORMANCE MEASURES: THE SHARPE PERFORMANCE INDEX, THE TREYNOR PERFORMANCE INDEX, THE JENSEN PERFORMANCE INDEX, THE TRACKING ERROR, INFORMATION RATIO, AND SORTINO RATIO

Sortino ratio is similar to Sharpe ratio, but it measures not to risk, but excess return to semi-variance of returns

Down deviation is computed on observations when portfolio return falls below min acceptable return 2 푅푃 <푅푚푖푛(푅푃푡 − 푅푚푖푛) 푀푆퐷 = 푡 푚푖푛 푁

퐸 푅푃 − 푅푚푖푛 푆표푟푡𝑖푛표 푟푎푡𝑖표 = 푀푆퐷푚푖푛

FRM® PART I THE MODERN PORTFOLIO THEORY AND THE CAPITAL ASSET PRICING MODEL 32