Capturing Tail Risk in a Risk Budgeting Model

DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2020

FILIP LUNDIN

MARKUS WAHLGREN

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

Capturing Tail Risk in a Risk Budgeting Model

FILIP LUNDIN

MARKUS WAHLGREN

Degree Projects in Financial Mathematics (30 ECTS credits) Master's Programme in Industrial Engineering and Management KTH Royal Institute of Technology year 2020 Supervisor at Nordnet AB: Gustaf Haag Supervisor at KTH: Anja Janssen Examiner at KTH: Anja Janssen

TRITA-SCI-GRU 2020:050 MAT-E 2020:016

Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Abstract

Risk budgeting, in contrast to conventional portfolio management strategies, is all about distributing the risk between holdings in a portfolio. The risk in risk budgeting is traditionally measured in terms of volatility and a Gaussian distribution is commonly utilized for modeling return data. In this thesis, these conventions are challenged by introducing diﬀerent risk measures, focusing on tail risk, and other probability distributions for modeling returns.

Two models for forming risk budgeting portfolios that acknowledge tail risk were chosen. Both these models were based on CVaR as a risk measure, in line with what previous researchers have used. The ﬁrst model modeled returns with their empirical distribution and the second with a Gaussian mixture model. The performance of these models was thereafter evaluated. Here, a diverse set of asset classes, several risk budgets, and risk targets were used to form portfolios. Based on the performance, measured in risk-adjusted returns, it was clear that the models that took tail risk into account in general had superior performance in relation to the standard model. Nevertheless, it should be noted that the superiority was signiﬁcantly higher for portfolios that constituted of mainly high-risk assets than for portfolios with more low-risk assets and also that the superior performance did not hold in all time periods considered. It was also clear that the model that used the empirical distribution to model returns performed better than the model based on an assumption of returns belonging to the Gaussian mixture model when the portfolio consisted of more assets with heavier tails.

Filip Lundin i Markus Wahlgren

En riskbudgeteringsmodell som tar hän- syn till svansrisk

Sammanfattning

Jämfört med konventionella portföljhanteringsstrategier handlar riskbudgete- ring mer om att fördela risken mellan innehav i en portfölj. Risken i riskbud- getering mäts traditionellt med avseende på volatilitet och en Gaussisk för- delning används normalt för att modellera avkastningsdata. I den här avhandlingen anlyseras andra modeller som istället fokuserar på svansrisk genom att införa andra riskmått och genom att använda andra sannolikhetsfördelningar för modellering av avkastningsdata.

Två modeller för att konstruera riskbudgeteringsportföljer som tar hänsyn till svansrisk har analyserats i den här avhandlingen. Båda dessa modeller använ- de sig av CVaR som ett riskmått, i linje med vad tidigare forskare har använt. Den första modellen modellerade avkastningar med den empiriska fördelning- en och den andra modellen med en Gaussisk blandningsmodell. Därefter ut- värderades hur de olika modellerna presterade. Här användes en mångfald av tillgångsklasser, ﬂera riskbudgetar och riskmål för att bilda portföljerna. Ba- serat på prestanda, mätt i termer av riskjusterad avkastning, var det tydligt att de modeller som tog hänsyn till svansrisk generellt presterade bättre än den konventionella modellen. Det bör emellertid noteras att för portföljer som hu- vudsakligen bestod av tillgångar med låg risk så var detta resultat mindre signi- ﬁkant och även att resultatet inte gällde för alla tidsperioder som analyserades. Det var också tydligt att modellen som använde den empiriska fördelningen för att modellera avkastningsdata fungerade bättre än den Gaussiska bland- ningsmodellen när portföljen till större del bestod av tillgångar med tyngre svansar.

Filip Lundin ii Markus Wahlgren

Acknowledgements

We want to thank Nordnet for presenting us with the opportunity to write this thesis and for providing the necessary data. We especially want to thank our supervisor at Nordnet, Gustaf Haag, for introducing us to the subject of risk budgeting and for helping us to pinpoint a subject to write about. Also, we want to thank our supervisor at KTH Royal Institute of Technology, Anja Janssen for the helpful guidance and feedback that we have received throughout the process of writing this thesis.

Filip Lundin & Markus Wahlgren Stockholm, May 13, 2020

Filip Lundin iii Markus Wahlgren

Contents

1 Introduction 1 1.1 Research Questions ...... 2 1.2 Scope & Limitations ...... 3 1.3 Related Work ...... 3 1.4 Outline ...... 6

2 Background 7 2.1 Distributions ...... 7 2.1.1 Gaussian Distribution ...... 7 2.1.2 Gaussian Mixture Distribution ...... 8 2.1.3 Empirical Distribution ...... 9 2.2 Parameter Estimation ...... 10 2.2.1 Gaussian Distribution ...... 10 2.2.2 Maximum Likelihood ...... 10 2.2.3 Gaussian Mixture Model ...... 11 2.3 Risk Measures ...... 14 2.3.1 Coherent Risk Measures ...... 14 2.3.2 Volatility ...... 15 2.3.3 Value-at-Risk ...... 16 2.3.4 Conditional Value-at-Risk ...... 17 2.4 Risk Budget Portfolios ...... 18 2.4.1 Existence and Uniqueness of the Portfolio ...... 19 2.4.2 Euler’s Theorem on Homogeneous Functions . . . . . 19 2.4.3 Volatility as a Risk Measure ...... 20 2.4.4 CVaR as a Risk Measure ...... 21 2.4.5 CVaR as a Risk Measure with Gaussian Mixture Model 23 2.5 Optimization ...... 28 2.5.1 Optimization Problem ...... 28 2.5.2 Quadratic Programming ...... 29

iv 2.5.3 Sequential Quadratic Programming ...... 29 2.6 Risk Targeting ...... 29 2.7 Performance Measures ...... 31 2.7.1 Sharpe Ratio ...... 31 2.7.2 Other Measures of Risk-Adjusted Returns ...... 31 2.7.3 Maximum Drawdown ...... 32

3 Data and Methodology 33 3.1 Data ...... 33 3.2 Risk Budgeting Portfolio Creation ...... 37 3.2.1 Portfolios with Leverage ...... 38 3.3 Portfolio Evaluation ...... 38

4 Results 40 4.1 Performance Graphs ...... 40 4.2 Performance Measures for the Full Period ...... 46 4.3 Performance Measures for Sub-Periods ...... 48

5 Discussion 52 5.1 Evaluation of the Findings ...... 52 5.2 Sources of Error ...... 55 5.3 Future Work ...... 56

6 Conclusions 57

7 Bibliography 59

A Model data 62 A.1 Capital Weights ...... 62 A.2 QQ-Plots for the Models ...... 68 A.3 Leverage ...... 70 A.4 Performance Measures for Sub-Periods ...... 73 List of Figures

2.1 Illustration of VaR for a density function fL of losses ...... 17 2.2 Illustration of CVaR for a density function fL of losses ...... 18 2.3 Illustration of maximum drawdown ...... 32

4.1 Performance for unleveraged risk parity portfolios ...... 41 4.2 Performance for low-risk portfolios for a daily empirical CVaR0.05 target of 0.5% ...... 43 4.3 Performance for mid-risk portfolios for a daily empirical CVaR0.05 target of1% ...... 44 4.4 Performance for high-risk portfolios for a daily empirical CVaR0.05 target of 1.5% ...... 45 4.5 The Sharpe ratio for the different periods for the risk parity portfolio . . 48 4.6 The Tail risk-adjusted return for the different periods for the risk parity portfolio ...... 49 4.7 The Sharpe ratio for the different periods for the high-risk portfolio . . . 50 4.8 The Tail risk-adjusted return for the different periods for the high-risk portfolio ...... 50

vi List of Tables

3.1 Portfolio risk budgets ...... 34 3.2 The daily empirical CVaR0.05 target for the diﬀerent risk levels . . . . . 34 3.3 Performance measures for individual assets ...... 36

4.1 Performance measures for risk parity portfolio ...... 46 4.2 Performance measures for low-risk portfolio ...... 46 4.3 Performance measures for mid-risk portfolio ...... 47 4.4 Performance measures for high-risk portfolio ...... 47

vii Chapter 1

Introduction

Historically, portfolio allocation strategies have often involved investing 60 % in equities and 40 % in bonds (60/40-portfolio) or applying Markowitz’s [1] mean-variance analysis to determine portfolio weights. The mean-variance analysis has allowed investors to ﬁnd the optimal capital allocation that maximizes the expected return for a given volatility level or minimize the volatility for a given level of expected return. Although both these strategies have been very popular, some criticism has been directed against them. For instance, the 60/40-portfolio is said to not diversify risks properly and the mean-variance method is highly sensitive to the input parameters. In the past decades, alternative approaches for portfolio construction have therefore gained traction in the investment com- munity, one of them being risk budgeting. Risk budgeting, in contrast to the previously discussed allocation strategies, is all about distributing the risk between holdings in a portfolio. The risk in risk budgeting is often measured by volatility, which is a commonly used measure of risk in ﬁnance. A special case of risk budgeting is risk parity or equal risk contribution (ERC) where the risk is allocated equally between assets.

The ﬁrst risk budgeting fund, called All-Weather, was developed by Bridgewater associates in 1996 and used a risk parity approach to establish an investment strategy that would perform well in all economic conditions [2]. Since then and particularly after the ﬁnancial crisis of 2008, when numerous institutional investors understood that their current models yielded riskier portfolios than expected, the popularity of risk budgeting and risk parity has increased considerably [3].

Risk budgeting as an investment strategy has been praised for its ability to generate true diversiﬁcation considering risks in portfolios. In for instance the 60/40-portfolio 90 % of the portfolio risk, measured in volatility, normally originates from the equity component since equities are signiﬁcantly riskier than bonds [4]. The risk in a risk parity portfolio

1 Chapter 1. Introduction 1.1. Research Questions

is, on the other hand, by deﬁnition set to be distributed equally among assets. Thus, the problem that one of the asset classes makes up for the majority of the overall portfolio risk is averted. Risk parity is however not in all instances preferable considering its re- strictiveness in the way that risk is allocated and therefore risk budgeting is consistently utilized to enable investors to set suitable budgets for the risk allocation themselves.

When estimating the risk that should be allocated in a risk budgeting portfolio, volatility is conventionally applied as the risk measure together with an assumption of Gaussian distributed returns. Returns of various financial assets have nevertheless been proven not to follow the Gaussian distribution. Particularly, it has been demonstrated that numerous financial asset returns possess heavier tails than what would be presumed under the assumption of the Gaussian distribution [5]. The aforementioned indicates that the risk of large losses may be neglected utilizing models that apply volatility as a risk measure and assume Gaussian distributed returns. It has also been shown that some financial asset returns exhibit skewness and are asymmetrical in their distribution [6][7]. This contra- dicts the assumption that returns follow the symmetrical Gaussian distribution.

In this Master thesis, the convention of employing volatility as a risk measure and assuming Gaussian distributed returns in risk budgeting models will be challenged. A risk budgeting model will be implemented that omits the assumption of Gaussian distributed returns and uses an alternative risk measure that is focused on measuring tail risk, i.e. the risk of substantial losses.

Nordnet AB, the provider of the project, currently uses risk budgeting strategies for its three Nordnet Smart mutual funds. The firm is interested in alternative methods for evaluating risks in risk budgeting models. Their portfolios consist of assets including stocks, interest rates, credits, commodities, inflation-protected assets, and lastly alternative risk premia which aim at exploiting anomalies in the market e.g. by buying shares with high momentum and selling shares with low momentum. In this thesis, portfolios will be formed using similar assets as used in Nordnet’s Smart funds when analyzing the risk budgeting strategy that incorporates the effects of tail risk.

1.1 Research Questions

The research questions for this master thesis are the following:

• How can a risk budgeting strategy be implemented that takes into account tail risk in asset returns?

Filip Lundin 2 Markus Wahlgren Chapter 1. Introduction 1.2. Scope & Limitations

• What is the diﬀerence in the performance of a risk budgeting strategy that acknowl- edges tail risk in returns and one that does not?

1.2 Scope & Limitations

This thesis will be limited by the data that will be considered in the analysis of the models. Data has been provided by Nordnet so that similar assets could be used in the portfolio construction process as used in their Smart Funds. However, the data consist of prices for several diﬀerent asset classes that could be used by any investment manager to form a portfolio and therefore the results will be relevant from a general perspective.

This thesis will not take into consideration the transaction costs that arise when the portfolio is rebalanced or the management fees for the examined assets. Other costs such as the interest costs from leveraging will also not be investigated in this thesis. The risk-free rate is therefore assumed to be equal to zero throughout the thesis.

This thesis aims to drop the established risk measure volatility and replace it with a risk measure that incorporates tail risk. There exist many diﬀerent types of risk measures that take tail risk into account but this thesis will only investigate one risk measure. Here, the risk measure that will be examined is the conditional value-at-risk (CVaR). The other aim of this thesis is to explore other assumptions than that returns are Gaussian distributed. This assumption will be challenged by calculating CVaR empirically and also through adapting a Gaussian mixture distribution.

1.3 Related Work

The introduction of alternative risk measures such as conditional value-at-risk, which is a risk measure that focuses on the tail risk, into risk budgeting is a subject that has been investigated by multiple researchers.

AllianceBernstein (2013) describe a model which they call tail risk parity. They suggest using risk parity with CVaR as a risk measure instead of volatility. In their model, the CVaR is acquired from the options markets by analyzing implied volatilities. The au- thors show that their tail risk parity strategy has a higher risk-adjusted return, when the risk is measured in terms of CVaR, compared to a conventional risk parity strategy. The strategy does however not show any improvement when risk-adjusted returns were measured by Sharpe ratio, where risk is measured in terms of volatility. In the article, there are no speciﬁc mathematical details on how the tail risk parity portfolio is constructed

Filip Lundin 3 Markus Wahlgren Chapter 1. Introduction 1.3. Related Work

[8].

Boudt, Carl and Peterson (2012) use Cornish–Fisher expansions and CVaR to form a strategy similar to a risk parity strategy, but where instead of allocating the risk equally among assets they decide the allocation by minimizing the greatest CVaR risk contribution from the assets. Utilizing Cornish-Fisher expansions together with CVaR they can account for heavy tails and skewness in returns [9].

Several articles that have investigated CVaR as a risk measure for the risk budgeting portfolio construction have estimated it non-parametrically. Cesarone and Colucci (2017) investigate the method by using empirical CVaR as the risk measure and setting the CVaR contribution to be equal among assets. They present methods for allocating the risk based on empirical CVaR that utilize both a naive approach where diversification is not taken into account which means that the portfolio weights are proportional to the assets’ inverse CVaR and a model that considers the effects of diversification. For the former, the calculations for the asset allocation are simple and for the latter, more ad- vanced method, the portfolio weights can be found by utilizing optimization methods [10]. This type of model has also been analyzed in a well-cited Master thesis by Ste- fanovits [11]. An alternative method for estimating the non-parametric CVaR is to use a bootstrap resampling procedure which is used in an article by Cagna and Casuccio [12]. This way of using bootstrap approaches to estimate a non-parametric CVaR is also used in another study by Colucci [13] where a filtered bootstrap approach is used to estimate CVaR and then the model is compared with the one that uses the ordinary volatility risk measure.

Some studies have developed models that also take skewness risk into account when creating risk parity portfolios. One example of introducing skewness risk into risk parity models is by dropping the assumption that returns are Gaussian distributed and instead to use a Gaussian mixture distribution to model returns. This model together with CVaR as risk measure is used by Bruder, Kostyuchyk and Roncalli [14] who have applied the developed model based on historical data and have concluded that skewness-based risk parity models yield better allocation compared to ordinary volatility-based risk parity strategies since it yields less turnover and therefore lower costs. However, the performance in terms of risk-adjusted returns, measured in Sharpe ratio, is similar between the strategies. This method is also investigated in a Master thesis by Vu [15] who applies a constrained Gaussian mixture model with CVaR as a risk measure and compares it with the conventional volatility risk measure.

Mausser and Romanko [16] investigate possible issues with the use of the empirical

Filip Lundin 4 Markus Wahlgren Chapter 1. Introduction 1.3. Related Work

CVaR as a risk measure instead of volatility. They do this by considering convex optimization to find long-only equal risk contribution portfolios (ERC) for a set of scenarios of asset returns that are equally likely. The first difficulty of using the empirical CVaR instead of volatility is that it can both be positive and negative. When it is negative it is not certain if any of the long-only ERC portfolios can be found using convex optimization. The second main problem is that the empirical CVaR is not continuously differentiable which indicates that there exists a possibility that a solution may not exist at all. In their paper, Mausser and Romanko present the conditions for these problems to occur and also suggest a heuristic method to find approximate ERC portfolios for these cases.

The existing literature on the subject indicates that there are only a few studied models and methods for implementing a new risk measure such as CVaR into risk budgeting. Most prominently, the Gaussian mixture model and the empirical distribution are used to model the distribution of the returns together with CVaR as a risk measure. What, however, is common for the current research is that all the papers investigate how alternative risk measures aﬀect the performance of the risk parity portfolio, which is a special case of the risk budgeting portfolio. What is lacking in the literature are papers that have implemented a new risk measure for a more general risk budgeting model and compared its performance to the standard model. Together with the more commonly considered asset types, such as stocks and interest rates, this thesis also examines other asset classes such as credits, real estate, commodities, and alternative risk premia which are not commonly used in previous studies.

The two most prominent methods for implementing CVaR as a risk measure are, like previously mentioned, using the empirical distribution or to fit a Gaussian mixture distribution and then to calculate the CVaR based on these distributions. Both these two methods will be implemented and compared in this thesis. This comparison of the two most commonly used methods will contribute to the existing research since a direct comparison between the models has not yet been performed. All the articles in the literature that compare the performance of a CVaR model and the volatility-based model have only investigated one way of implementing CVaR. A comparison of multiple methods for implementing a CVaR model is, therefore, lacking in the literature and this thesis seeks to fill this gap. Also, portfolios with different risk level targets will be considered that utilize leverage in order to achieve the risk target. This has not been examined in the current research and is therefore a further contribution from this thesis.

Filip Lundin 5 Markus Wahlgren Chapter 1. Introduction 1.4. Outline

1.4 Outline

The thesis will continue with a background chapter. Here, the relevant mathematical theory will be presented that is needed in order to understand the mathematical tools that are used to construct the risk budgeting portfolios. Thereafter, a chapter that describes the data and the methodology used for constructing the different portfolios is presented. Here, the real financial data that are used for evaluation of the performance are described. Also, the methodology for how the portfolios are formed using the data is presented. Then, the results are laid out, starting with graphs for the value development of the different portfolios and risk budgeting models. Apart from the graphs, this section also includes tables with performance measures for the full time period that are used to evaluate how well each model performs for instance with regard to risk-adjusted returns. Diagrams that show the performance measures for shorter time periods are also presented in order to show how these measure change through time. Thereafter, the results are discussed, focusing on describing how the results could be explained and the relevance of the results. Lastly, the thesis ends with a conclusion where the research questions are answered.

Filip Lundin 6 Markus Wahlgren Chapter 2

Background

This section will provide necessary and relevant background for this thesis. It will describe the theory of potential risk measures together with possible distributions and the estimation of their parameters. The background on how to construct the risk budgeting portfolios based on alternative risk measures will also be explained together with optimization methods. How speciﬁc risk levels are targeted for the portfolios will then be explained followed by a description of the performance measures that can be used for model comparison.

2.1 Distributions

Probability distributions describe how likely diﬀerent events are to occur. For a random variable X, the distribution function is deﬁned as P (X ≤ x), i.e. the probability that X takes on a value less or equal to the number x. If X is a continuous random variable, then there exists a density fX (t) such that

Z x FX (x) = P (X ≤ x) = fX (t)dt. −∞ Below the Gaussian distribution, which is used in the conventional risk budgeting models together with volatility as a risk measure, is described. Also, the Gaussian mixture model and the empirical distribution are presented which will be used to form a risk budgeting model that takes into account tail risk in ﬁnancial returns.

2.1.1 Gaussian Distribution A Gaussian or normal distribution is a continuous probability distribution that is commonly used for modeling natural phenomenons or, which is more relevant for this thesis,

7 Chapter 2. Background 2.1. Distributions

ﬁnancial asset returns. The probability density function (PDF) of the Gaussian distribution can be deﬁned as the following [17]

2 1 − (x−µ) f(x) = √ e 2σ2 , x ∈ R. σ 2π It is determined by the two parameters µ and σ. The ﬁrst parameter µ is the mean or expected value of the distribution and the second parameter σ is the standard deviation of the distribution.

In a multivariate setting the density function can be deﬁned as the following

− 1 (x−µ)>Σ−1(x−µ) e 2 fX (x) = , p(2π)n|Σ| where µ and Σ are a mean vector and a covariance matrix respectively [18]. A vector X with density fX has expected value E(X) = µ and covariance matrix Σ. Here |Σ| denotes the determinant of Σ.

2.1.2 Gaussian Mixture Distribution Mixture distributions combine several distributions in order to ﬁt empirical data better than individual distributions could. Mathematically, a probability density function f(x) of a mixture distribution with g distributions for a random vector X can be written as

g X f(x) = πifi(x), i=1 where fi(x) are component densities of the mixture and πi are the components’ weights in the distribution [19]. Naturally, the following conditions need to be fulﬁlled for the weights

0 ≤ πi ≤ 1, g X πi = 1. i=1 Bruder et al. [14] use a Gaussian mixture model with two Gaussian components to model returns when constructing a risk parity portfolio. Here, one component is considered a

Filip Lundin 8 Markus Wahlgren Chapter 2. Background 2.1. Distributions

continuous component and the other one a jump component. The continuous component is modelled with a Gaussian distribution with mean µ and covariance Σ while the jump component is modelled with a Gaussian distribution with mean µ+µe and covariance Σ+ Σe. These contributions are then used to form a Gaussian mixture model by introducing the probability (1 − λ) for the continuous component and the probability λ for the jump component. Mathematically, the density function for this Gaussian mixture model can, therefore, be described as the following, see [14],

1 − λ 1 > −1 f(x) = n 1 exp − (x − µ) (Σ) (x − µ) (2π) 2 |Σ| 2 2 λ 1 + exp − x − (µ + µ)>(Σ + Σ)−1x − (µ + µ) . n 1 e e e (2π) 2 |Σ + Σe| 2 2

Here, π1 = 1 − λ and π2 = λ following the notation from above.

2.1.3 Empirical Distribution The empirical distribution function is an estimate of an unknown distribution function based on observations and can be deﬁned in the following way [20].

Let the unknown distribution function be F (x) = P (X ≤ x) for the observations x1, x2, ..., xn of independent and identically distributed random variables X1,X2, ..., Xn. The empirical distribution function Fn(x) is then deﬁned by

n 1 X F (x) = n n I{xi≤x}, i=1 where I{xi≤x} is the indicator function which means that it will return a 1 if xi is less or equal to x and a 0 if xi is larger than x.

The empirical distribution function, like previously mentioned distributions, can be used to estimate tail risk and has the advantages that it is only based on actual information. It only considers the observations and does not have any distributional assumptions. The disadvantages of using the empirical distribution for estimating the tail risk in the form of the empirical value-at-risk and conditional value-at-risk, as described in section 2.3.3 and 2.3.4, is that the estimations that are outside the range of the sample will show greater variation compared to a correctly ﬁtted parametric distribution [21].

Filip Lundin 9 Markus Wahlgren Chapter 2. Background 2.2. Parameter Estimation

2.2 Parameter Estimation

In order to apply the previously mentioned distributions in the modeling, their parameters have to be estimated. Below the methods used for estimation of the diﬀerent distributions’ parameters are explained.

2.2.1 Gaussian Distribution The two parameters that need to be estimated for the one-dimensional Gaussian distribution are µ and σ. The parameter µ can be estimated by taking the arithmetic mean x¯ of the sample of n observations

n 1 X x¯ = x . n i i=1 The parameter σ can be estimated by taking the sample standard deviation s of a sample of n observations which is deﬁned as follows v u n u 1 X 2 s = t (xi − x¯) . n − 1 i=1

For the multivariate Gaussian distribution the covariance, σjk, is also necessary to estimate. The sample covariance is given by the following expression

n 1 X cov(x , x ) = (x − x¯ )(x − x¯ ). j k n − 1 j,i j k,i k i=1

2.2.2 Maximum Likelihood The maximum likelihood method can be used to estimate parameters for a distribution based on empirical data. Consider the one dimensional case with x1, ..., xn, an i.i.d sample from a certain distribution with CDF F (x; θ), where θ is an unknown parameter in the distribution. Then, if X is a continuous random variable with PDF f(x; θ), the likelihood function is given by

L(θ) = f(x1; θ) · f(x2; θ) ····· f(xn; θ).

Filip Lundin 10 Markus Wahlgren Chapter 2. Background 2.2. Parameter Estimation

The likelihood function represents the probability that the sample is occurring under the distribution f(x, θ). Thus, ﬁnding the θ∗ that yields the maximum value of the likelihood function corresponds to ﬁnding the distribution with this θ∗ as a parameter that the sample would most likely come from [17].

2.2.3 Gaussian Mixture Model The parameters of the Gaussian mixture model can be estimated by an expectation- maximization algorithm (EM). The EM algorithm is a method to ﬁnd maximum likelihood estimates when no closed form solution to maximizing L(θ) exists.

Bruder et al. [14] describes an EM-method to estimate the parameters of a Gaussian mixture model and it can be described in the following way.

They consider a Gaussian mixture model that has two Gaussian components and thus contains ﬁve parameters that need to be estimated, as described in section 2.1.2, i.e.

θ = λ, µ, Σ, µ,˜ Σ˜.

A sample Rt = (R1,t, ..., Rn,t) of i.i.d asset returns for n assets where Ri,t is the return of asset i observed at time t is considered. The log-likelihood function is then

T X `(θ) = ln f(Rt) t=1 where f(y) is the multivariate probability function for a Gaussian mixture model with two Gaussian components as deﬁned in section 2.1.2. Thus in this case it is deﬁned as the following ˜ f(y) = π1φn(y; µ, Σ) + π2φn(y; µ +µ, ˜ Σ + Σ) where π1 = 1 − λ, π2 = λ and φn(y; µ, Σ) represents the PDF of the multivariate Gaussian distribution with parameters µ and Σ. The notations µ1 = µ, Σ1 = Σ, µ2 = ˜ µ +µ ˜ and Σ2 = Σ + Σ are introduced and the log-likelihood function becomes the following

T 2 X X `(θ) = ln πjφn(Rt; µj, Σj). t=1 j=1

Filip Lundin 11 Markus Wahlgren Chapter 2. Background 2.2. Parameter Estimation

The partial derivative of the log-likelihood `(θ) with respect to µj is

T ∂`(θ) X πjφn(Rt; µj, Σj) = Σ−1(R − µ ). ∂µ P2 j t j j t=1 s=1 πsφn(Rt; µs, Σs) The ﬁrst-order condition is, therefore

T X −1 πj,tΣj (Rt − µj) = 0 t=1 where π φ (R ; µ , Σ ) π = j n t j j . j,t P2 s=1 πsφn(Rt; µs, Σs)

The expression for the estimator µˆj then is the following

PT π R µˆ = t=1 j,t t . (2.1) j PT t=1 πj,t

To express the partial derivative of the likelihood function l(θ) with respect to Σj, con- −1 sider the function g(Σj ) which is deﬁned as follows

1 1 > −1 −1 − 2 (Rt−µj ) Σj (RT −µj ) g(Σj ) = n/2 1/2 e (2π) |Σj| −1 1/2 |Σj | − 1 trace Σ−1(R −µ )(R −µ )> . = e 2 j T j t j (2π)n/2 It then follows that the partial derivative is

−1 −1 −1/2 −1 ∂g(Σj ) 1 |Σj | |Σj |Σj 1 −1 > − 2 trace Σj (Rt−µj )(Rt−µj ) −1 = n/2 · e ∂Σj 2 (2π) −1 1/2 1 > |Σj | − 1 trace Σ−1(R −µ )(R −µ )> − (R − µ )(R − µ ) · e 2 j t j t j 2 t j t j (2π)n/2 Σ − (R − µ )(R − µ )> 1 1 −1 > j t j t j − 2 (Rt−µj )Σj (Rt−µj ) = n/2 1/2 · e . (2π) |Σj| 2 Then it is clear that

T ∂`(θ) 1 X πjφn(Rt; µj, Σj) = Σ − (R − µ )(R − µ )> . ∂Σ−1 2 P2 j t j t j j t=1 s=1 πsφn(Rt; µs, Σs)

Filip Lundin 12 Markus Wahlgren Chapter 2. Background 2.2. Parameter Estimation

The expression for the ﬁrst order condition is equal to

T X > πj,t Σj − (Rt − µj)(Rt − µj) = 0. t=1 ˆ Thus, the estimator Σj is given by

PT π (R − µˆ )(R − µˆ )> Σˆ = t=1 j,t t j t j . (2.2) j PT t=1 πj,t

For the mixture probabilities πj, the following expression for the partial derivative with respect to πj is yielded from the ﬁrst-order condition

T ∂`(θ) X φn(Rt; µj, Σj) = . ∂π P2 j t=1 s=1 πsφn(Rt; µs, Σs)

It can be concluded that it is not possible to deﬁne the estimator πˆj directly since the numerator cannot take on the value zero. Therefore, another approach has to be utilized in order to obtain the ML estimators.

Deﬁne πˆj,t as the estimator for the posterior probability of the regime j at time t, given as π φ (R ; µ , Σ ) πˆ = j n t j j . (2.3) j,t P2 s=1 πsφn(Rt; µs, Σs)

If πˆj,t is known, the estimator πˆj can be given by

PT πˆ πˆ = t=1 j,t . (2.4) j T The EM algorithm for the Gaussian mixture distribution can then be described in the following steps.

(0) (0) (0) 1. Set the initial starting values πj , µj and Σj for k = 0.

2. The E-step, where the posterior distributions πj,t are updated for all the observations using equation (2.3) which yields the expression

(k) (k) (k) πj φn Rt; µj , Σj π(k) = . j,t P2 (k) (k) (k) s=1 πs φn Rt; µs , Σs

Filip Lundin 13 Markus Wahlgren Chapter 2. Background 2.3. Risk Measures

ˆ 3. The M-step, where the estimator πˆ, µˆj and Σj are updated using equations (2.4), (2.1) and (2.2) yielding the following expressions

PT π(k) π(k+1) = t=1 j,t j T PT (k) π Rt µ(k+1) = t=1 j,t j PT (k) t=1 πj,t > PT (k) (k+1) (k+1) t=1 πj,t Rt − µj Rt − µj Σ(k+1) = . j PT (k) t=1 πj,t 4. Iterate step 2 and 3 until the estimator converges.

(∞) (∞) ˆ (∞) 5. Finally the estimated parameters are yielded as πˆj = πj , µˆj = µj and Σj = Σj .

2.3 Risk Measures

In this section risk measures and their properties are described. The most commonly used risk measures in ﬁnance are introduced which can then be utilized for constructing risk budgeting portfolios.

2.3.1 Coherent Risk Measures

Let X be a linear vector space of random variables X which represent the values of different portfolios at time 1. A function R that assigns real values to each X in X is called a risk measure. The risk of X is thereby denoted R(X) and can be interpreted to be the amount of capital that needs to be invested into a reference instrument with percentage return R0 at time 0 in order to yield an acceptable position. The reference instrument can, in this case, be considered to be the risk-free zero-coupon bond that matures at time 1. No capital is needed to be added to a portfolio for the position to be acceptable if R(X) ≤ 0.

In order for a risk measure R to be coherent, it needs to have the following properties [20]:

i) Monotonicity: X2 ≤ X1, then R(X1) ≤ R(X2)

ii) Subadditivity: R(X1 + X2) ≤ R(X1) + R(X2)

iii) Positive homogeneity: R(λX) = λR(X) for all λ ≥ 0

Filip Lundin 14 Markus Wahlgren Chapter 2. Background 2.3. Risk Measures

iv) Translation invariance: R(X + cR0) = R(X) − c for a real number c

The property monotonicity indicates that if the current time is 0 and a position X1 has a greater value than another position X2 at time 1 for sure, then the position X1 with the greater value will have less risk compared to the other position X2.

Subadditivity rewards diversiﬁcation by ensuring that the total risk of the portfolio X1 + X2 cannot be larger than the sum of the risk of X1 and X2 separately.

Positive homogeneity means that if we scale a portfolio with a factor λ we also scale the total risk with the same factor. It also implies that R(0) = 0.

Translation invariance indicates that if an amount of cash of value c is added to the portfolio and is invested in the reference instrument, a risk-free zero-coupon bond in this case, it will decrease the risk by the same amount c [20].

2.3.2 Volatility Volatility is a common measure of risk in ﬁnancial settings that is normally measured in terms of standard deviation. Formally, the historical volatility can therefore be calculated with the following expression v u N u 1 X σ = t (r − r¯)2 N − 1 t t=1 where N is the number of observed returns, rt the return at time t and r¯ the arithmetic mean of all N returns [22].

Another commonly used measure for volatility is the implied volatility which can be derived from the options market using the well-known Black and Scholes formula. Thus, this is a measure of how the market anticipates future volatility in contrast to the historical volatility which considers backward-looking data [23].

Volatility has been criticized as a risk measure in ﬁnancial settings. To begin with, some argue that, since the volatility increases both when an asset experiences large losses and gains, it is sub-optimal since many would appreciate the gain but not the loss [24]. Also, the volatility measure is often used with an assumption of Gaussian distributed data. It has however been proven that many ﬁnancial asset returns have heavier tails than what would be expected by the Gaussian distribution [5].

Filip Lundin 15 Markus Wahlgren Chapter 2. Background 2.3. Risk Measures

2.3.3 Value-at-Risk Value-at-risk is a risk measure that considers the tail of the distribution of losses, that is the most negative outcomes, compared to volatility, which considers overall deviations from the mean. The value-at-risk VaRp(X) for a risk level p and a portfolio with value X at time 1 can be deﬁned as the following [20] VaRp(X) = min m : P (L ≤ m) ≥ 1 − p

X where L is the discounted portfolio loss L = − and R0 is the return of a risk-free R0 asset in percent between time 0 and 1. VaRp(X) can then be described as the minimal value m such that the probability of the discounted portfolio loss L being no more than m is at least 1 − p.

Value-at-risk can statistically be deﬁned as the (1−p)-quantile of the discounted portfolio loss L, i.e.

−1 VaRp(X) = FL (1 − p).

The (1 − p)-quantile of a random variable L that has the distribution function FL is deﬁned in the expression

−1 FL (1 − p) := min m : FL(m) ≥ 1 − p .

−1 If FL is strictly increasing then FL is the regular inverse of the distribution function FL. −1 If FL is both strictly increasing and continuous then FL is the unique value m such that FL(m) = 1 − p.

The value-at-risk can be estimated using empirical data, without assuming any parametric distribution for the data. This empirical estimate of VaR calculated for a sample {L1,L2, .., Ln} of n observations of the discounted portfolio loss L is given by

VaR\p(X) = Lbnpc+1,n.

Where L1,n ≥ L2,n ≥ ... ≥ Ln,n is the ordered sample of {L1, ..., Ln} and bzc is the integer part of z. It is therefore only the empirical (1−p)-quantile of the sample of L [20].

A signiﬁcant disadvantage of using VaR as a risk measure is that it is not coherent since it is not sub-additive. This means that the total risk of a portfolio of multiple assets could be larger than the sum of individual assets’ risk [25]. Diversiﬁcation does therefore not

Filip Lundin 16 Markus Wahlgren Chapter 2. Background 2.3. Risk Measures

necessarily provide a reduction in risk when the VaR is used as a risk measure. A further disadvantage with using VaR as a risk measure is that it does not capture the events that happen in the end of the tail. This is a problem since a distribution could for instance have events with low probability but with a much greater outcome than the VaR at the chosen risk level. This problem is visualised in ﬁgure below, where the right tail corresponds to the most negative outcomes and where the tail is heavier on the right than the left.

Figure 2.1: Illustration of VaR for a density function fL of losses

2.3.4 Conditional Value-at-Risk Value-at-risk only considers the distribution up to a certain point in the tail and does therefore not take into account the risk in the far end of the tail as mentioned in the previous section. In order to cope with this, another risk measure denoted conditional value-at-risk is often used. Practically, the conditional value-at-risk is the average of the value-at-risk between the risk level and the end of the tail in the distribution and will therefore incorporate all the events in the tail which is illustrated in ﬁgure 2.2. Formally, the conditional value-at-risk at the risk level p can thus be written as

1 Z p CVaRp(X) = VaRu(X)du. p 0 Similarly, as with value-at-risk, the CVaR can also be estimated directly from empirical data. Then the empirical CVaR can be described by the following expression

Filip Lundin 17 Markus Wahlgren Chapter 2. Background 2.4. Risk Budget Portfolios

Z p bnpc 1 1 X Lk,n bnpc CVaR\(X) = L du = + p − L p p bnuc+1,n p n n bnpc+1,n 0 k=1 where the L1,n ≥ L2,n ≥ ... ≥ Ln,n is the ordered sample of losses as described above.

CVaR is also called expected shortfall (ES). This risk measure is, in contrast to value- at-risk, a coherent risk measure [20].

Figure 2.2: Illustration of CVaR for a density function fL of losses

2.4 Risk Budget Portfolios

Risk budgeting is an alternative way of creating portfolios which instead of concentrat- ing on capital-distribution concentrates on the distribution of risk among assets. The idea is that each asset should contribute with some predetermined amount of risk to the total risk of the portfolio. To form this portfolio, it is thus ﬁrst necessary to determine how much risk each asset should contribute to the portfolio and then calculate the asset allocation in terms of capital. A special case of this is when each asset contributes with the same amount of risk. This is usually called a risk parity portfolio or equally weighted risk contribution (ERC) portfolio.

To formulate this mathematically the risk budgeting portfolio can be deﬁned by the following system of nonlinear equations, see [26],

Filip Lundin 18 Markus Wahlgren Chapter 2. Background 2.4. Risk Budget Portfolios

 RCi(x) = biR(x)  bi > 0  xi ≥ 0 (2.5)  n P b = 1  i=1 i Pn i=1 xi = 1 where bi is the risk budget of asset i which has the constraint that it cannot be set to zero, i.e. bi > 0, which is necessary for the risk budget portfolio to be unique. Furthermore, xi is the portfolio weight of asset i and x = (x1, x2, ..., xn) is the vector of portfolio weights for a portfolio with n assets. The portfolio weight xi has the constraint xi ≥ 0 which means that long-only portfolios are the only portfolios that are considered. RCi(x) is the risk contribution that asset i has to the total risk of the portfolio x which is measured using the risk measure R(x).

2.4.1 Existence and Uniqueness of the Portfolio In the remainder of this report, the coherent risk measure CVaR will be utilized when constructing risk budgeting portfolios and then the following restriction must be imposed in order for the risk budgeting portfolio to exist [14] R(x) ≥ 0, for all x. The risk measure needs to be positive in order for the portfolio to exist and be unique. A coherent risk measure has, as described in section 2.3.1, the property of positive homogeneity. This property means that R(λx) = λR(x) where λ is a positive scalar. If there is a portfolio where R < 0 it would mean that the portfolio could be leveraged with a scaling factor λ > 1 such that R(λx) < R(x) < 0. It also follows that lim R(λx) = −∞. λ→∞ This is the reason why the risk measure R(x) needs to be positive. In order for a risk budget portfolio to exist and be unique the risk measure thus needs to be positive and the risk budgets bi needs to be positive as well, as described in section 2.4.

2.4.2 Euler’s Theorem on Homogeneous Functions An important theorem needed for determining the risk contribution of speciﬁc assets is Euler’s theorem on homogeneous functions. This theorem is explained below [27].

Filip Lundin 19 Markus Wahlgren Chapter 2. Background 2.4. Risk Budget Portfolios

A function f : U → R,U ⊂ Rn, is homogeneous of degree τ if the following equation holds for any λ > 0

f(λx) = λτ f(x), x ∈ U.

Thus, the positive homogeneity property for a coherent risk measure in section 2.3.1 is for a degree of 1.

Now let U ⊂ Rn be an open set and f : U → R be a continuously diﬀerentiable function. Then f is homogeneous of degree τ if and only if it satisﬁes the following equation

n X ∂f(u) τf(u) = u . i ∂u i=1 i

If the function f(u) is the risk measure R(x) and is continuously diﬀerentiable and homogeneous of degree 1, the equation becomes

n n X ∂R X R(x) = x = RC . i ∂x i i=1 i i=1

The marginal risk contribution each asset i has to the total risk measure R(x) when increasing the portfolio weight xi can then be described by the partial derivative of the risk measure with respect to the weight of asset i. The sum of all assets’ risk contribution then adds up to the risk measure R(x).

2.4.3 Volatility as a Risk Measure The established risk measure for risk budget portfolios is volatility and the use of this risk measure for risk budgeting portfolios is also conventionally based on an assumption of Gaussian distributed asset returns. If the volatility σ is used as risk measure it means that

√ q T 2 2 R(x) = σ(x) = x Σx = Σixi σi + ΣiΣj6=ixixjσij

2 where Σ is the covariance matrix, σi is the variance of the return of asset i and σij is the covariance of the returns of assets i and j. Here, σ(x) describes the total risk of the portfolio measured in volatility. The marginal risk contribution can then be deﬁned as the following, see [28],

Filip Lundin 20 Markus Wahlgren Chapter 2. Background 2.4. Risk Budget Portfolios

∂R(x) ∂σ(x) (Σx) x σ2 + Σ x σ = = √ i = i i √ j6=i j ij , ∂xi ∂xi xT Σx xT Σx which represents the change in volatility of the total portfolio that comes from a change th of the portfolio weight xi of asset i. Here, (Σx)i denotes the i component of the vector issued from the product of Σ with x.

Then the risk contribution RCi(x) of asset i can be deﬁned by ∂σ(x) RCi(x) := σi(x) = xi · . ∂xi This then leads to the following Euler decomposition, as described in section 2.4.2, which indicates that the total risk of the portfolio can be seen as a sum of the risk contributions of the assets i = 1, 2, ..., n.

n n X X ∂σ(x) σ (x) = x · i i ∂x i=1 i=1 i n X (Σx)i = xi · √ T i=1 x Σx Σx = xT · √ T √ x Σx = xT Σx = σ(x)

For the case of a risk parity or ERC portfolio the risk contributions would be equal for all assets such that σi(x) = σj(x) for all i, j = 1, 2, 3, ..., n. This can also be expressed in the following way

σ(x) σ (x) = . i n

2.4.4 CVaR as a Risk Measure The methods for creating a risk budgeting model can also be adjusted by considering conditional value-at-risk (CVaR) as a risk measure instead of volatility. The use of CVaR in

Filip Lundin 21 Markus Wahlgren Chapter 2. Background 2.4. Risk Budget Portfolios

risk budgeting can be described in the following way.

To begin with, the CVaR at risk level p of the full portfolio can be written

n n X ∂CVaRp(x) X CVaR R(x) = CVaR (x) = x = RC p (x) p i ∂x i i=1 i i=1

CVaRp ∂CVaRp(x) where RC (x) = xi , which is the risk contribution in terms of CVaRp, to i ∂xi the portfolios total risk from asset i. This is in accordance with the Euler decomposition described in section 2.4.2 [10].

If we replace the unknown true distribution function with the empirical one, the latter leads to

bpT c 1 X R(x) = CVaR (x) = L(k)(x) p bpT c p k=1

(1) (2) (T ) (k) where Lp (x) ≥ Lp (x)... ≥ Lp (x) are sorted portfolio losses. Clearly Lp can be written as the following where N is the number of assets in the portfolio

N (k) X (k) Lp (x) = xili(x) , i=1

(k) (k) where li(x) is the loss of asset i that corresponds to the portfolio loss Lp at obser- (1) vation k. Thus if the worst portfolio loss, Lp , occurs for instance at observation 200, (1) li(x) will be the individual asset losses that occur at the same observation 200. The partial derivative of CVaRp(x) with respect to xi, which is the marginal risk contribution of asset i, then becomes

bpT c ∂CVaRp(x) 1 X (k) = li(x) . ∂xi bpT c k=1

CVaR The risk contribution RCi of asset i is then the following expression [10]

bpT c CVaR ∂CVaRp(x) 1 X (k) RCi (x) = xi = xi li(x) . (2.6) ∂xi bpT c k=1

Filip Lundin 22 Markus Wahlgren Chapter 2. Background 2.4. Risk Budget Portfolios

2.4.5 CVaR as a Risk Measure with Gaussian Mixture Model In order to form a risk budgeting model utilizing conditional value-at-risk as a risk measure and Gaussian mixture models to model returns, an analytical expression for the marginal risk contributions from the assets can be derived.

Bruder et al. [14] derives an expression for the conditional value-at-risk for the full portfolio under the distribution speciﬁed above in section 2.1.2 in their appendix 4. Then they derive a deﬁnition of the marginal conditional value-at-risk. Below the derivations are displayed following the steps in this article.

Start by setting

ϕ(a) = E[I{a ≤ Y }· Y ] where I is the indicator function and Y a Gaussian random variable such that Y ∼ N(µ, σ2). From this it is clear that the following expression holds, where φ(x) is the PDF and Φ(x) is the CDF of the standard Gaussian distribution.

Z ∞ y y − µ x − µ ϕ(a) = φ dy change of variable t = a σ σ σ Z ∞ = (µ + σt)φ(t)dt σ−1(a−µ) Z ∞ ∞ σ 1 2 = µ[Φ(t)]σ−1(a−µ) + √ t exp(− t )dt 2π σ−1(a−µ) 2 a − µ σ 1 ∞ = µ 1 − Φ + √ − exp − t2 σ 2π 2 σ−1(a−µ) a − µ a − µ = µΦ − + σφ . σ σ Let us consider now the loss of the portfolio L(x) where thus L(x) = −R(x) if R(x) is the portfolio returns. Then the conditional value-at-risk, at risk level p, can be deﬁned as CVaRp(x) = E L(x)|VaRp(x) ≤ L(x) .

This can be rewritten using the indicator function and its expectation according to below

Filip Lundin 23 Markus Wahlgren Chapter 2. Background 2.4. Risk Budget Portfolios

1 h i CVaR (x) = VaR (x) ≤ L(x) · L(x) p pE I p 1 Z ∞ = yg(y)dy, p VaRp(x) where the density of L(x) is denoted g(y). By the expression for the density in section 2.1.2 and the symmetry of the Gaussian distribution it is given by

1 y + µ (x) 1 y + µ (x) g(y) = (1 − λ) φ 1 + λ φ 2 σ1(x) σ1(x) σ2(x) σ2(x)

T T 2 T 2 T where µ1(x) = x µ, µ2(x) = x (µ+µe), σ1(x) = x Σx and σ2(x) = x (Σ+Σe)x.

2 2 Let us then denote L1(x) ∼ N − µ1(x), σ1(x) and L2(x) ∼ N − µ2(x), σ2(x) , where L(x) again is the portfolio loss such that L(x) = −R(x). This then yields

1 Z ∞ CVaRp(x) = yg(y)dy p VaRp(x) 1 = (1 − λ) · {VaR (x) ≤ L (x)}· L (x)+ p E I p 1 1 + λ · E I{VaRp(x) ≤ L2(x)}· L2(x) 1 − λ VaRp(x) + µ1(x) = σ1(x)φ (2.7) p σ1(x) VaRp(x) + µ1(x) − µ1(x)Φ − σ1(x) λ VaRp(x) + µ2(x) + σ2(x)φ p σ2(x) VaRp(x) + µ2(x) − µ2(x)Φ − . σ2(x) Thus, an expression for the CVaR under this Gaussian mixture model has been obtained. However, the VaR is included in the expression which means that it is necessary to show how it can be calculated as well. By deﬁnition it is clear that

P L(x) ≥ VaRp(x) = p.

Filip Lundin 24 Markus Wahlgren Chapter 2. Background 2.4. Risk Budget Portfolios

Then the following can be concluded that

Z VaRp(x) g(y)dy = 1 − p −∞ and the value-at-risk can be found using a numerical method such as the bisection algorithm in order to solve the following equality

VaR (x) + µ (x) VaR (x) + µ (x) (1 − λ) · Φ p 1 + λ · Φ p 2 = 1 − p. σ1(x) σ2(x) From this, the marginal conditional value-at-risk from each of the assets can be calculated. To do this, at ﬁrst, the marginal contributions to the VaR will be considered. Following [14] as above, let us deﬁne

VaRp(x) + µi(x) hi(x) = . (2.8) σi(x) Then the value-at-risk fulﬁlls the following as stated above

(1 − λ) · Φ(h1(x)) + λ · Φ(h2(x)) = 1 − p.

Taking the derivative on both sides of the equation and using the chain rule this can be written as

(1 − λ) · φ(h1(x))∂xh1(x) + λ · φ(h2(x))∂xh2(x) = 0.

The partial derivatives of hi, i = 1, 2 are given by

∂xVaRp(x) + µ h1(x) ∂xh1(x) = − 2 Σx (2.9) σ1(x) σ1(x)

∂xVaRp(x) + µ + µ˜ h2(x) ˜ ∂xh2(x) = − 2 (Σ + Σ)x. (2.10) σ2(x) σ2(x) From this, the marginal contribution to the value-at-risk can be written

h1(x) h2(x) γ1(x)( Σx − µ) + γ2(x)( (Σ + Σ˜)x − (µ + µ˜)) σ1(x) σ2(x) ∂xVaRp(x) = γ1(x) + γ2(x)

Filip Lundin 25 Markus Wahlgren Chapter 2. Background 2.4. Risk Budget Portfolios

for

πiφ(hi(x)) γi(x) = , σi(x) where π1 = 1 − λ, π2 = λ. Then the marginal CVaR can be derived by taking the derivative of the expression of CVaR from above 2.7 and using the newly introduced notations. Thus,

1 − λ ∂ CVaR (x) = (∂ σ (x)φ(h (x)) − σ (x)h (x)φ(h (x))∂ h (x)) x p p x 1 1 1 1 1 x 1 1 − λ − (∂ µ (x)Φ(−h (x)) − µ (x)φ(h (x))∂ h (x)) p x 1 1 1 1 x 1 λ + (∂ σ (x)φ(h (x)) − σ (x)h (x)φ(h (x))∂ h (x)) p x 2 2 2 2 2 x 2 λ − (∂ µ (x)Φ(−h (x)) − µ (x)φ(h (x))∂ h (x)). p x 2 2 2 2 x 2 The expression can then be rewritten as the following using equation 2.9 and 2.10 and using the expressions for the partial derivatives ∂xµi(x) and ∂xσi(x)

1 − λ ∂xCVaRp(x) = Σxφ(h1(x)) pσ1(x) 1 − λ ∂xVaRp(x) + µ h1(x) − σ1(x)h1(x)φ(h1(x))( − 2 Σx) p σ1(x) σ1(x) 1 − λ − µΦ(−h (x)) p 1

1 − λ ∂xVaRp(x) + µ h1(x) + µ1(x)φ(h1(x))( − 2 Σx) p σ1(x) σ1(x) λ ˜ + (Σ + Σ)xφ(h2(x)) pσ2(x) λ ∂xVaRp(x) + µ + µ˜ h2(x) ˜ − σ2(x)h2(x)φ(h2(x))( − 2 (Σ + Σ)x) p σ2(x) σ2(x) λ − (µ + µ˜)Φ(−h (x)) p 2 λ ∂xVaRp(x) + µ + µ˜ h2(x) ˜ + µ2(x)φ(h2(x))( − 2 (Σ + Σ)x). p σ2(x) σ2(x) Rewriting this using the equation 2.8 then yields

Filip Lundin 26 Markus Wahlgren Chapter 2. Background 2.4. Risk Budget Portfolios

1 − λ ∂xCVaRp(x) = Σxφ(h1(x)) pσ1(x) λ ˜ + (Σ + Σ)xφ(h2(x)) pσ2(x) 1 − λ ∂xVaRp(x) + µ h1(x) − (VaRp(x) + µ1(x))φ(h1(x))( − 2 Σx) p σ1(x) σ1(x) 1 − λ ∂xVaRp(x) + µ h1(x) + µ1(x)φ(h1(x))( − 2 Σx) p σ1(x) σ1(x) λ ∂xVaRp(x) + µ + µ˜ h2(x) ˜ − (VaRp(x) + µ2(x))φ(h2(x))( − 2 (Σ + Σ)x) p σ2(x) σ2(x) λ ∂xVaRp(x) + µ + µ˜ h2(x) ˜ + µ2(x)φ(h2(x))( − 2 (Σ + Σ)x) p σ2(x) σ2(x) 1 − λ − µΦ(−h (x)) p 1 λ − (µ + µ˜)Φ(−h (x)). p 2 Some of the terms will cancel out which then yields

1 − λ λ ˜ ∂xCVaRp(x) = Σxφ(h1(x)) + (Σ + Σ)xφ(h2(x)) pσ1(x) pσ2(x) 1 − λ ∂xVaRp(x) + µ h1(x) − VaRp(x)φ(h1(x))( − 2 Σx) p σ1(x) σ1(x) λ ∂xVaRp(x) + µ + µ˜ h2(x) ˜ − VaRp(x)φ(h2(x))( − 2 (Σ + Σ)x) p σ2(x) σ2(x) 1 − (1 − λ)µΦ − h (x) + λ(µ + µ˜)Φ − h (x) . p 1 2 This ﬁnally can be written as

γ (x) γ (x) ∂ CVaR (x) = 1 δ (x) + 2 δ (x) x p p 1 p 2 1 − (1 − λ)µΦ − h (x) + λ(µ + µ˜)Φ − h (x) p 1 2 where

h1(x) δ1(x) = 1 + VaRp(x) Σx − VaRp(x) ∂xVaRp(x) + µ σ1(x)

Filip Lundin 27 Markus Wahlgren Chapter 2. Background 2.5. Optimization

and

h2(x) ˜ δ2(x) = 1 + VaRp(x) (Σ + Σ)x − VaRp(x) ∂xVaRp(x) + µ + µ˜ . σ2(x)

2.5 Optimization

In order to ﬁnd the capital weights for risk budgeting portfolios, it is necessary to apply optimization methods. Below, the optimization problem that needs to be solved and the utilized optimization techniques are explained.

2.5.1 Optimization Problem The risk budgeting portfolio should fulﬁll the following system of equations as explained in section 2.4.  RCi(x) = biR(x)  bi > 0  xi ≥ 0  n P b = 1  i=1 i Pn i=1 xi = 1. When constructing a risk budgeting portfolio the portfolio manager decides on an allocation of the risk b = (b1, b2, ..., bn) between n number of assets in the portfolio. Then, as the risk budgets b is only a vector with ﬁxed numbers, the portfolio weights can be found by solving the following minimization problem,

n 2 X RCi(x) min − 1 x b R(x) i=1 i

subject to xi ≥ 0 n X xi = 1. i=1 Here the function to be minimized is the sum of the square of the actual risk contribution divided by the desired risk contribution with 1 subtracted. This modiﬁed version of the least-squares function minimizes the percentage deviation the actual risk contribution has from the target risk contribution instead of minimizing the absolute deviation as in the conventional least square function. This modiﬁcation is used since the assets

Filip Lundin 28 Markus Wahlgren Chapter 2. Background 2.6. Risk Targeting

will have risk contributions of diﬀerent scale and it would therefore not make sense to minimize the absolute error. The reason behind why the problem needs to be formulated as an optimization problem is that there are no closed-form equations for ﬁnding the capital weights except for the most simple cases as for uncorrelated assets.

2.5.2 Quadratic Programming Quadratic programming (QP) is an optimization method for solving quadratic optimization problems with linear constraints. The problem consists of maximizing or minimizing a quadratic function of multiple variables that is subject to linear constraints which can both be equality or inequality constraints. A quadratic minimization problem with n variables and m inequality constraints can then be defined in the following way where the objective is to find a solution x∗ [29] 1 min q(x) = x>Gx + g>x x 2 subject to A>x ≤ b where q(x) is the objective function to be minimized, x is a n-dimensional vector, G is an n × n symmetric matrix, g is a n-dimensional vector with real values, A is a n × m real matrix and b is a m-dimensional real vector. The solution x∗ is a global solution if the matrix G is positive semi-definite and the solution is also unique if the matrix is strictly positive definite.

2.5.3 Sequential Quadratic Programming Sequential quadratic programming (SQP) is an optimization method that can solve nonlinear optimization problems (NLP) with both equality and inequality constraints. It is able to solve these problems by solving a sequence of sub-problems where each sub- problem is a linearly constrained quadratic optimization problem. Quadratic programming problems are further explained in the previous section.

Practically, this algorithm solves a quadratic programming sub-problem for a given input xk, k ∈ N and subsequently utilizes this value to ﬁnd a new value xk+1. By repeating this, a local minimum x∗ of the objective function is found when k → ∞ [30].

2.6 Risk Targeting

Risk targeting can be implemented for a portfolio in order to achieve a predetermined constant level of risk. This constant risk can be achieved by using leverage to scale the

Filip Lundin 29 Markus Wahlgren Chapter 2. Background 2.6. Risk Targeting

risk. If the portfolio enters a period of low risk the leverage will be increased in order to keep the risk at the same level and if the portfolio instead enters a period of high risk the leverage will be decreased [31].

Risk targeting is especially relevant for risk budgeting strategies. Risk budgeting strategies often lead to portfolios that have a higher capital allocation to less risky asset classes such as bonds and lower capital allocation to more risky assets such as stocks. This strategy is supposed to yield portfolios with a higher risk-adjusted return but the overall return will still be lower compared to the more conventional portfolios. The normal way to increase the overall return would be to increase the allowed risk contribution for the riskier assets which would then increase the risk and overall return of the portfolio but would also decrease the risk-adjusted return. An alternative method would instead be to use risk targeting which would increase the return and risk of the portfolio with the use of leverage. The advantage of this strategy is that the risk-adjusted return would not decrease since asset allocation remains the same.

The amount of leverage that is needed in order to keep the risk at a constant level RT arget can be calculated through the expression

R Leverage factor = T arget RActual where RT arget is the actual historical risk, measured in term of any homogeneous risk measure, for some chosen data. The leverage factor is the factor that the portfolio needs to be scaled with in order to reach the target risk. Thus, if the target risk, for instance, is equal to the actual risk the ratio will be equal to one and no leverage will be needed. On the other hand, if the ratio instead is equal to 2, the leverage will be 100% of the current portfolio value meaning that the current portfolio value will be doubled by borrowing extra cash and investing them in the portfolio.

Volatility is the more conventionally used risk measure for risk targeting which would lead to the following expression

σ Leverage factor = T arget σActual where σ is the volatility of the portfolio. The choice of volatility as a risk measure together with the assumption of Gaussian distributed returns has been somewhat criticized, which is discussed in section 2.3. Another risk measure that can be used is CVaR which

Filip Lundin 30 Markus Wahlgren Chapter 2. Background 2.7. Performance Measures

is a risk measure that better captures tail risk compared to volatility. The needed leverage for CVaR as a risk measure with a risk level p can then be formulated as follows [32]

T arget CVaRp Leverage factor = Actual . CVaRp

2.7 Performance Measures

This section will present diﬀerent performance measures that can be used to compare the diﬀerent risk budgeting models that are examined in this thesis.

2.7.1 Sharpe Ratio The Sharpe ratio is the most commonly used performance measure for comparing the risk-adjusted returns of different assets or portfolios and was first introduced by Sharpe in 1966 as the reward-to-variability ratio [33]. It can be defined by the following expression R − R Sharpe ratio = p f σ where Rp is the return of the portfolio, Rf is the risk-free rate and σ is the standard deviation of the excess return.

2.7.2 Other Measures of Risk-Adjusted Returns The purpose of this thesis is to compare risk budgeting portfolios that use the two different risk measures volatility and conditional value-at-risk. To supplement the existing performance measure that is based on the volatility, that is the Sharpe ratio, a new tail risk-based performance measure is introduced. This measure is similar to the Sharpe ratio in the way that it represents the risk-adjusted return of an asset or portfolio. The change is that the risk is measured in conditional value-at-risk instead of the conventional volatility. This tail risk-based performance measure, here denoted Tail Risk-Adjusted Return (TRAR), can be deﬁned in the following way, R − R TRAR = p f , CVaRp where Rp is the return of the portfolio, Rf is the risk-free rate and CV aRp is the conditional value-at-risk at the risk level p of the excess return of the portfolio. This type of risk measure has been proposed earlier, e.g. by Maillard in [34].

Filip Lundin 31 Markus Wahlgren Chapter 2. Background 2.7. Performance Measures

2.7.3 Maximum Drawdown Maximum drawdown is a risk measure that aims at describing the maximum loss an investor could have obtained by buying at a peak and selling at the bottom following that peak [35]. If measured in percentages this can be deﬁned mathematically as

Vk − min Vj j=k+1,...,n Maximum Drawdown = max , k=1,...,n Vk where Vk are prices.

Figure 2.3: Illustration of maximum drawdown

Filip Lundin 32 Markus Wahlgren Chapter 3

Data and Methodology

In this chapter, the data and methodology used for evaluating the different models for constructing risk budgeting portfolios will be presented. A standard risk parity portfolio will first be considered followed by three portfolios with different risk levels that utilize risk targeting through leverage, explained more in detail in section 3.2. The portfolios will be tested using real financial data presented in section 3.1 below. Lastly, a description will be provided for how the performance of the portfolios will be evaluated in section 3.3.

3.1 Data

The evaluation and comparison of the performance for the different risk budgeting models will be based on a diverse set of asset classes. In this thesis, the asset classes that will be examined are stocks, interest rates, credits, commodities, real estate, and alternative risk premia. These asset classes will be represented by different indices that contain assets of each type. The original data have been provided by the project partner and consist of daily price data for different indices that track these assets between 2007-03-28 to 2019-12-20.

Three portfolios with different risk levels and risk budgets will be investigated. These portfolios are called the low-, mid-, and high-risk portfolios. A risk parity portfolio will also be examined in addition to these three portfolios. The specific risk budget, i.e. the bi:s in equation 2.5, for each asset class and type of portfolio is displayed in table 3.1. The most significant difference between the portfolios is that the portfolios with higher risk level have a higher proportion of stocks compared to the less risky portfolios which instead have a higher proportion of interest rates.

33 Chapter 3. Data and Methodology 3.1. Data

The reason why several portfolios with different risk budgets are investigated is, mainly, to make sure that the same performance results hold for different risk budgets. The best performing model can be identified for all the different portfolios and it can be concluded if the best performing model is the same model for every portfolio or if it changes de- pending on the portfolio. This thesis will, therefore, provide a better understanding of the performance of the models by investigating several portfolios of different risk budgets.

Risk parity Low-risk Mid-risk High-risk Stocks 16.66% 25% 50% 70% Interest Rates 16.66% 40% 20% 5% Credits 16.66% 5% 5% 5% Commodities 16.66% 5% 10% 10% Real Estate 16.66% 5% 5% 5% Alternative Risk Premia 16.66% 20% 10% 5%

Table 3.1: Portfolio risk budgets

These low-, mid- and high-risk portfolios will, in addition to having different risk budgets, also be leveraged so that they will remain at a constant level of risk which will be described in more detail in section 3.2.1. The risk parity portfolio will not target a specific risk level and will therefore not use any leverage. The empirical CVaR will be used as a risk measure for the risk targeting and the specific CVaR targets for the three risk levels are displayed in table 3.2 below. Leverage and risk targeting is often used in practice in order for the investor to achieve the desired risk level for an investment. If the investor chooses a high-risk level, higher returns can be expected and vice versa.

CVaR0.05 target Low-risk 0.5% Mid-risk 1% High-risk 1.5%

Table 3.2: The daily empirical CVaR0.05 target for the diﬀerent risk levels

Each asset class will consist of speciﬁc assets, where the asset class stocks, for example, will consist of two stock indices. All the speciﬁc assets and their weights within each asset class are displayed in table below.

Filip Lundin 34 Markus Wahlgren Chapter 3. Data and Methodology 3.1. Data 90 % 10 % 80 % 20 % 100 % 100 % 100 % 100 % Weight Description US equity REITs with a low credit score with a high credit score emerging market equities developed market equities Investment grade bonds from momentum and value factors. Includes the size, volatility, quality, 24 developed and emerging markets Credit risk against 125 US companies Credit risk against 100 US companies A broadly diversiﬁed commodity index Long/Short multifactor alternative risk premia. Market capitalization weighted total return index of Market capitalization weighted total return index of Index (BCOMTR) (NDUEEGF) (LEGATRUU) Estate (FNERTR) Not publicly available Aggregate Total Return Bloomberg Commodity MSCI Emerging Markets MSCI World (NDDUWI) FTSE NAREIT U.S. Real Bloomberg Barclays Global Excess Return (ERINCLIG) Excess Return (ERINCLHY) Markit CDX.NA.IG Long 5Y Markit CDX.NA.HY Long 5Y Asset Stocks Premia Credits Real Estate Commodities Interest Rates US High Yield Alternative Risk Emerging Markets Developed Markets US Investment Grade

Filip Lundin 35 Markus Wahlgren Chapter 3. Data and Methodology 3.1. Data

Most of the asset types that are used in this thesis are well known except for alternative risk premia which is a fairly new asset class and therefore might need an explanation. It is a strategy that takes advantage of anomalies in the market by taking long and short positions in companies that are exposed to speciﬁc factors to yield high risk-adjusted returns. These strategies, at least the ones considered in this report, are constructed so that the market risk is mitigated. An example would be the size factor where the strategy then could be to buy small-cap companies and sell large-cap companies. Other commonly used factors are volatility, momentum, value, and quality, which are included in the multifactor index described in the above table.

In the table below the performance measures for each of the individual assets that are examined in this thesis over the full period 2007-2019 are displayed in order to show their speciﬁc characteristics. The annualized return and volatility are shown for each asset together with the daily CVaR 5%. Also, the Sharpe ratio and the tail risk-adjusted return (TRAR) ratio are presented. Note that the TRAR ratio is calculated using daily returns and daily CVaR 5% as a risk measure and that it is then multiplied by 100 in order to yield more readable results. Lastly, the ratio between the daily CVaR 5% and daily volatility is displayed to show the relationship between the diﬀerent risk measures for each asset.

Return Volatility Sharpe CVaR 5 % TRAR (100x) CVaR/Volatility DM Stocks 6.61 16.79 0.38 2.67 0.92 2.57 EM Stocks 2.56 19.82 0.13 2.96 0.33 2.41 Global Rates 2.53 5.16 0.48 0.72 1.32 2.27 US IG Credits 1.55 2.32 0.66 0.33 1.77 2.32 US HY Credits 5.54 7.86 0.69 1.18 1.75 2.42 Commodities -7.52 16.53 -0.47 2.56 -1.17 2.50 REITs 8.72 33.39 0.25 5.25 0.61 2.54 Multifactor ARP 4.26 4.38 0.95 0.60 2.66 2.22

Table 3.3: Performance measures for individual assets

There exists a large diﬀerence in risk-adjusted performance between the assets. The multifactor ARP has the best risk-adjusted return, both when considering CVaR and volatility as a risk measure. The lowest risk-adjusted return is for commodities which is the only asset that had a negative annualized return. REITs, emerging and developed markets stocks also display a low risk-adjusted return for both risk measures.

It is also clear that developed markets stocks, REITs and commodities have the largest

Filip Lundin 36 Markus Wahlgren Chapter 3. Data and Methodology 3.2. Risk Budgeting Portfolio Creation

CVaR/Volatility ratio which means that these asset’s CVaR 5% is larger in relation to the asset’s volatility than for the other assets in the table. The assets that have the lowest CVaR/Volatility ratio are multifactor ARP and global rates.

3.2 Risk Budgeting Portfolio Creation

As mentioned above the risk budgeting portfolios are tested using historical data for several ﬁnancial assets. In order to perform the backtesting of the portfolios, several assumptions have to be made. These assumptions are e.g. about how often the portfolio should be rebalanced and how much historical data that should be used when performing calculations used to decide the portfolio structure. These kinds of assumptions will be the same for all portfolios tested in this thesis and are described below. Apart from this, some assumptions and methodologies are unique for each portfolio type. These portfolio unique assumptions will also be described in detail below.

To begin with, the historic data for which the calculations of portfolio weights are based on needs to be decided upon. Here, it has been decided to use data for the past year, approximated as 261 trading days. Thus, the input data for each calculation of the portfolio allocation will be the past 261 returns for the assets.

Also, the portfolio needs to be rebalanced in order to have the correct risk weights. It is not reasonable to assume that the rebalancing should be done at each time step, that is every day, since the trading cost would be too extensive with such a strategy. Instead, it has been decided that the rebalancing should be done each month which is approximated as each 21st trading day. At a rebalancing date, assets in the portfolio are bought and sold such that the risk weights calculated using the past year’s data are accurate.

When implementing the models that use CVaR as a risk measure the risk level has been determined to be 5%. This is to ensure that there are enough observations for the model that uses empirical CVaR as a risk measure.

In order to create the models that are based on the diﬀerent risk measures one needs to estimate the expression for the risk contribution for each of the risk measures, as described in section 2.4.3, 2.4.4 and 2.4.5. Estimation of parameters is necessary for some models, such as for the volatility-based model and the CVaR model based on the Gaus- sian mixture distribution. These parameters can be estimated as described in section 2.2. The optimization problem in section 2.5.1 then has to be solved using the risk contribution for each of the diﬀerent risk measures. This optimization problem is solved using

Filip Lundin 37 Markus Wahlgren Chapter 3. Data and Methodology 3.3. Portfolio Evaluation

sequential quadratic programming as described in section 2.5.3.

For the volatility based model the risk contribution can be calculated through the expression in section 2.4.3 where µ and Σ is estimated through the mean and covariance matrix, as explained in section 2.2.1.

The risk contribution for the model that is based on the empirical CVaR is obtained from equation 2.6 in section 2.4.4. This expression for the risk contribution can then be used to solve the optimization problem.

The Gaussian mixture model is a model that is composed of two Gaussian distributions. For this distribution, the parameters can be estimated by the methods found in section 2.2.3. Here, the expression for the risk contributions in terms of CVaR for this model, which is necessary to solve the optimization problem, can be yielded through the procedure found in section 2.4.5.

3.2.1 Portfolios with Leverage For the low-, mid- and high-risk portfolio, leverage is used in order to achieve a constant pre-speciﬁed risk level. Here, CVaR is used as a risk measure for all models. The CVaR is measured at risk level 5 % daily and measured empirically when calculating the leverage. Other methods for calculating the risk that is used for the risk targeting have also been tested, e.g. using the Gaussian mixture model to calculate the CVaR. These other methods have yielded similar results to the empirical CVaR and will therefore not be shown. In the same way, as for the risk weights, the leverage is recalculated and adjusted each month.

In the portfolios with leverage, the interest rate for the borrowed cash is set to zero. Although this is an approximation it should not aﬀect the comparison between the models signiﬁcantly if comparisons are done for each portfolio separately since the leverage turned out to be rather similar between models. Also, the borrowing costs during the investigated period have been rather low.

3.3 Portfolio Evaluation

The portfolios will be evaluated based on several performance measures which are described in section 2.7. All performance measures for the portfolios will be calculated using the portfolios’ daily return data for the full period. For the Sharpe ratio and tail

Filip Lundin 38 Markus Wahlgren Chapter 3. Data and Methodology 3.3. Portfolio Evaluation

risk-adjusted return the interest rate is set to zero. The historical value development of the diﬀerent strategies for each portfolio will also be presented in order to get a visual representation of the performance.

Filip Lundin 39 Markus Wahlgren Chapter 4

Results

In this chapter, the results of the thesis will be presented. The portfolios will be evaluated based on their performance which is measured through risk-adjusted returns. To start with, graphs of the performance of the diﬀerent models will be displayed. Diﬀerent performance measures will then be shown in tables to be able to directly compare results.

4.1 Performance Graphs

The performance of the created portfolios based on the diﬀerent models is compared graphically through the plots displayed below. Each portfolio is given a start value of 100 and is calculated for the period 2008-03-17 to 2019-12-20. The period is decreased by a year compared to the complete asset price data set since a full year of data is needed before any portfolio value calculations can be made.

All portfolios except the risk parity portfolio utilize risk targeting to make the comparisons relevant. This means that the value development of the risk parity portfolio might not be fairly compared between the models in the graphs since each model could have diﬀerent risks. The risk parity model is still displayed in order to show a base case with the simplest type of risk budgeting and without applying leverage. However, it is shown in the tables below in this section that the risk levels turned out to be rather similar even for the risk parity models in terms of CVaR0.05, thus making the comparisons rather reasonable. For the other portfolios, diﬀerent risk levels are targeted. The low-risk portfolio has a daily CVaR0.05 target of 0.5% while the mid-risk portfolio has a target of 1% and the high-risk portfolio has a target of 1.5%.

The value development for a comparable 100% stock portfolio and a comparable 60/40- portfolio is also displayed for each type of portfolio (risk parity, low-, mid- and high-risk).

40 Chapter 4. Results 4.1. Performance Graphs

The asset that is used for the 100% stock portfolio is the developed stock markets index that was introduced in above table covering all assets in the risk budgeting portfolios. This index is also used for the 60/40-portfolio together with the global interest rates index that was introduced in the same table. The 100% stock portfolio and 60/40-portfolio are leveraged to achieve the same amount of risk for each of the diﬀerent portfolios. Thus, these portfolios are unleveraged for the risk parity portfolio and are leveraged to achieve the same CVaR0.05 target for the other portfolios. These two portfolios are displayed in order to compare the risk budgeting strategies to more conventional strategies.

Figure 4.1: Performance for unleveraged risk parity portfolios

Filip Lundin 41 Markus Wahlgren Chapter 4. Results 4.1. Performance Graphs

The value development for the unleveraged risk parity portfolio is very similar for all the different models. A significant difference is however visible when comparing the risk parity models, that is the volatility, empirical CVaR and GMM CVaR model, to the more conventional models, the 100% stock and 60/40-portfolio. The conventional models exhibit substantially larger losses and are more volatile compared to the risk parity models which perform better during the financial crisis in 2008 and display a more smooth value development throughout the full period. The conventional models do however have a higher overall return for the full period compared to the risk parity models.

Filip Lundin 42 Markus Wahlgren Chapter 4. Results 4.1. Performance Graphs

Figure 4.2: Performance for low-risk portfolios for a daily empirical CVaR0.05 target of 0.5%

The value development for the low-risk portfolio with a daily CVaR0.05 target of 0.5% indicates that the model using empirical CVaR and the model using the Gaussian mixture model together with CVaR performs best and the model using volatility as a risk measure is lagging behind, especially in the middle to the end of the period. All the risk budgeting models perform substantially better than the conventional models when they are leveraged to target the same risk.

Filip Lundin 43 Markus Wahlgren Chapter 4. Results 4.1. Performance Graphs

Figure 4.3: Performance for mid-risk portfolios for a daily empirical CVaR0.05 target of 1%

The value development for the mid-risk portfolio with a risk target, measured with CVaR0.05, of 1% shows that the model based on empirical CVaR performs best, followed by the Gaussian mixture CVaR model and that the Gaussian model with volatility as risk measure performs worst of the risk budgeting models. The diﬀerences in performance between the risk budgeting portfolios and the conventional models have somewhat decreased when the risk level has been increased from low- to mid-risk.

Filip Lundin 44 Markus Wahlgren Chapter 4. Results 4.1. Performance Graphs

Figure 4.4: Performance for high-risk portfolios for a daily empirical CVaR0.05 target of 1.5%

The value development for the high-risk model with a CVaR0.05 target of 1.5% yields similar results as the previous portfolio but with an even greater diﬀerence in performance between the models. The comparison of performance between the risk budgeting and the conventional models yields a similar conclusion but with a decreased diﬀerence in performance compared to the mid-risk portfolio.

The key takeaway from all the portfolios is that the CVaR models perform better than

Filip Lundin 45 Markus Wahlgren Chapter 4. Results 4.2. Performance Measures for the Full Period

the volatility-based model. Increasing the risk level of the portfolios through leveraging and through the modification of the risk budgets yields a greater difference in performance between the risk budgeting models which is seen when moving from the low-risk portfolio to the mid- and high-risk portfolios. Also, it is clear that the CVaR model based on the empirical distributions performs better than the CVaR model based on the Gaussian mixture models for the portfolios with higher risk targets. When comparing the risk budgeting models to the conventional models it is evident that the largest difference in performance is exhibited for the portfolios with low risk while the difference in performance decreases when the risk target is increased.

4.2 Performance Measures for the Full Period

Performance measures for the diﬀerent models and portfolios are presented in the tables below. The measures are calculated using data for the period 2008-03-17 - 2019-12-20 as in the previous section. Note that the tail risk-adjusted ratio (TRAR) has been multiplied by 100 to yield more readable results.

Gaussian Volatility Empirical CVaR GMM CVaR Annualized Mean Return 2.62 2.56 2.67 Annualized Volatility 3.27 3.13 3.19 Daily CVaR 5 % 0.48 0.45 0.47 Maximum Drawdown 11.82 10.81 11.50 Sharpe Ratio 0.80 0.82 0.83 TRAR (100x) 2.09 2.16 2.19

Table 4.1: Performance measures for risk parity portfolio

Gaussian Volatility Empirical CVaR GMM CVaR Annualized Mean Return 3.54 3.83 3.67 Annualized Volatility 3.82 3.93 3.86 Daily CVaR 5 % 0.56 0.57 0.56 Maximum Drawdown 9.87 9.52 9.67 Sharpe Ratio 0.92 0.97 0.94 TRAR (100x) 2.52 2.65 2.58

Table 4.2: Performance measures for low-risk portfolio

Filip Lundin 46 Markus Wahlgren Chapter 4. Results 4.2. Performance Measures for the Full Period

Gaussian Volatility Empirical CVaR GMM CVaR Annualized Mean Return 6.02 7.09 6.60 Annualized Volatility 7.36 7.68 7.49 Daily CVaR 5 % 1.09 1.13 1.10 Maximum Drawdown 21.31 20.92 21.29 Sharpe Ratio 0.81 0.92 0.88 TRAR (100x) 2.18 2.49 2.37

Table 4.3: Performance measures for mid-risk portfolio

Gaussian Volatility Empirical CVaR GMM CVaR Annualized Mean Return 7.51 9.73 8.71 Annualized Volatility 10.75 11.18 10.91 Daily CVaR 5 % 1.64 1.67 1.65 Maximum Drawdown 31.05 30.16 31.22 Sharpe Ratio 0.69 0.86 0.79 TRAR (100x) 1.80 2.30 2.09

Table 4.4: Performance measures for high-risk portfolio

When examining the Sharpe ratio and tail risk-adjusted return for the diﬀerent portfolios it is evident that the models based on CVaR have a better performance than the model based on volatility. This is the case for all the portfolios and is also in line with the results of the previous section which also showed that the models based on CVaR performed better. It can also be noted that the performance, in terms of risk-adjusted returns, is better for the empirical CVaR model than the CVaR model based on the Gaussian mixture model for portfolios with higher risk levels.

If the Sharpe ratio and tail risk-adjusted returns are compared for the diﬀerent portfolios instead of the models, the tables indicate that the low-risk portfolio performs best and that the ratios decrease when increasing the risk level of the portfolios. The high-risk portfolio, therefore, has the worst ratios of all the leveraged portfolios. It can here also be noted that the actual CVaR is slightly higher than the CVaR targets for all leveraged portfolios.

Filip Lundin 47 Markus Wahlgren Chapter 4. Results 4.3. Performance Measures for Sub-Periods

4.3 Performance Measures for Sub-Periods

Here, the performance measures for the diﬀerent models and portfolios are also calculated for shorter time periods instead of the full period as above. The full period is divided into six sub-periods where the performance measures for each period are calculated. Then the performance measures of all the periods are compared to see if the performance of the models changes throughout the diﬀerent periods. In the diagrams below only the Sharpe ratio and the TRAR are shown since they are the most important for the analysis of the performance and only the measures for the risk parity and high- risk portfolios are shown here in the results section. The same diagrams for the other portfolios can be found in section A.4 in the appendix. For the diagrams, the start date for each period is displayed below the x-axis where the periods end before the start date of the next period. Note that the tail risk-adjusted return ratio has, in line with above, been multiplied with 100 in order to yield more readable results.

Figure 4.5: The Sharpe ratio for the diﬀerent periods for the risk parity portfolio

Filip Lundin 48 Markus Wahlgren Chapter 4. Results 4.3. Performance Measures for Sub-Periods

Figure 4.6: The Tail risk-adjusted return for the diﬀerent periods for the risk parity portfolio

The risk parity portfolio is the portfolio that displays the most inconsistent results. In the ﬁrst period, the Gaussian mixture model performs the worst followed by the empirical CVaR model and then the Gaussian volatility model. In the next two periods, the empirical CVaR model is the best performer followed by the Gaussian mixture and Gaus- sian volatility model. This relationship then changes for the last three periods where the empirical CVaR model performs the worst of the three models. The Gaussian mixture model performs the best for the periods 2014-01-01 to 2015-12-31 and 2018-01-01 to 2019-12-20 while the Gaussian volatility model performs the best for the period 2016- 01-01 to 2017-12-31.

Filip Lundin 49 Markus Wahlgren Chapter 4. Results 4.3. Performance Measures for Sub-Periods

Figure 4.7: The Sharpe ratio for the diﬀerent periods for the high-risk portfolio

Figure 4.8: The Tail risk-adjusted return for the diﬀerent periods for the high-risk portfolio

It is clear that the two ﬁgures for the high-risk portfolio yield similar results for both of the performance measures to what has been concluded for the full period. The empirical CVaR model seems to perform the best for the majority of the periods with some ex-

Filip Lundin 50 Markus Wahlgren Chapter 4. Results 4.3. Performance Measures for Sub-Periods

ceptions followed by the Gaussian mixture CVaR model and then the Gaussian volatility model. Here, the difference in performance is quite significant in the first two periods where the period from 2010-01-01 to 2012-01-01 has the largest difference. After that, the performance difference is decreased and the empirical and Gaussian mixture CVaR models yield more similar ratios. The similarity between the models is greatest in the period 2016-01-01 to 2017-12-31 where all the models’ ratios are almost the same. The last period again displays a greater difference in performance between the models.

Filip Lundin 51 Markus Wahlgren Chapter 5

Discussion

This chapter will start with a general discussion of the results shown in the previous chapter. It will then be followed by a section where possible sources of error are discussed and a section where possible future research is proposed.

5.1 Evaluation of the Findings

It was clear from the results presented in the previous chapter that the performance in terms of both Sharpe ratios and tail risk-adjusted returns were superior for the risk budgeting models that apply CVaR as a measure of risk and disregards the traditional Gaus- sian distribution for modeling returns. For all considered portfolios, the model based on CVaR and the empirical distribution yielded the most superior risk-adjusted returns except for the risk parity portfolio where the Gaussian mixture model was slightly better. The model that used CVaR as a risk measure with the Gaussian Mixture model as a probability distribution for returns thus generally performed worse than the empirical model but better than the model based on the Gaussian distribution and volatility as a risk measure. For the risk parity and low-risk portfolios, the Gaussian mixture model and the empirical model performed rather similar, but when increasing the risk levels of the portfolios the superiority of performance of the empirical model compared to the Gaussian mixture model was increased.

Also, when considering the diﬀerent sub-periods of time in section 4.3, rather similar conclusions as mentioned above could be drawn. However, it can be noted that for some particular sub-periods the conventional Gaussian model with volatility as a measure was superior to the models based on CVaR when considering tail risk-adjusted returns and Sharpe ratios as the performance measure. For the higher risk portfolios, this was the case only for a couple of time periods, while the volatility-based model performed better

52 Chapter 5. Discussion 5.1. Evaluation of the Findings

than the CVaR models in several time periods for the lower risk portfolios.

The improvement in tail risk-adjusted returns for the models based on CVaR can, naturally, be explained by the diﬀerence in the asset allocations between the models. The individual assets’ performance measures are available in Table 3.3 in Section 3.1 where it was clear that e.g. Multifactor ARP has a low CVaR/Volatility-ratio which explain why more capital was invested in this asset in the models based on CVaR compared to the model based on volatility, as seen in A.1. These assets, like alternative risk premia and global rates, that have a low CVaR/Volatility-ratio also have a high tail risk-adjusted return ratio. This means that since the models using CVaR allocate more capital to these assets, compared to the model based on volatility, they will have higher tail risk-adjusted returns. Another reason for the improvement of tail risk-adjusted returns are, of course, that these models yield lower CVaR since they allocate the capital such that the combined assets yield portfolios with low CVaR since the tail correlation of the assets is taken into account when forming portfolios. Since more capital is allocated to low-risk assets, such as global rates, the CVaR models will allow more leverage in order to achieve the risk target in relation to the volatility model. This can be seen in the graphs of the leverage over time in Appendix A.3.

As mentioned, the Sharpe ratios also turned out to be higher for the models based on CVaR. This was more surprising since the Sharpe ratio is based on volatility and not CVaR, but can be explained by the fact that most assets with high tail risk-adjusted returns also had superior Sharpe ratios. The result of the improved Sharpe ratio is not consistent with current research. In for instance, [14] [10] [8], the Sharpe ratio is not shown to be improved by taking into account tail risk in risk budgeting models. The difference in results could be due to the fact that diﬀerent data sets are used. For instance, this thesis considers both risk budgeting and risk parity when others only have investigated risk parity and this thesis has also used more asset classes in the portfolios than other researchers have.

The volatility based model implicitly assumes that returns are Gaussian distributed. This is something that has been proven to be inaccurate and can be seen in the QQ-plot in figure A.13 (a) in section A.2 in the Appendix. This QQ-plot shows the quantiles for a fitted Gaussian distribution compared to the empirical quantiles for the asset developed markets stocks. From this QQ-plot, it is clear that the Gaussian distribution is a bad fit and that the empirical data has heavier tails compared to the fitted distribution. This is another reason as to why the model that is based on volatility as a risk measure and that assumes a Gaussian distribution performs the worst of the examined risk budgeting models.

Filip Lundin 53 Markus Wahlgren Chapter 5. Discussion 5.1. Evaluation of the Findings

There also existed a difference in performance between the two models that used CVaR as a risk measure. The model based on empirical CVaR performed better than the model based on a Gaussian mixture model, except for the risk parity portfolio where the Gaus- sian mixture model performed slightly better. This can be explained by the fact that the empirical CVaR model will capture the heavy tails of the data better compared to the Gaussian mixture model. The QQ-plot shown in figure A.13 (b) in section A.2 in the Appendix compares the quantiles of the fitted Gaussian mixture model to the empirical quantiles for the asset developed markets stocks. From this QQ-plot, it is clear that the Gaussian mixture distribution is a better fit to the empirical data compared to the Gaus- sian distribution. What however is visible in this QQ-plot is that the empirical quantiles exhibit heavier tails compared to the adapted Gaussian mixture model. This indicates that the Gaussian mixture model is not able to fully capture the heavy tails exhibited in the data. The model based on CVaR that uses the Gaussian mixture distribution will, therefore, underestimate the risk of heavy tailed assets such as stocks compared to the model that is based on the empirical CVaR and this can explain the difference in performance between the models.

The fact that the difference in performance between the empirical model and Gaussian mixture models changes when altering the risk budgets can also be explained by the fact that the empirical model is better able to capture the heavy tails of the data compared to the Gaussian mixture model. The difference between the portfolios of different risk levels is first that they are targeting different risk levels and secondly that they have different risk budgets. Here, the portfolios with a higher risk level have a higher risk allocation to riskier assets such as stocks while portfolios with lower risk levels have a higher concentration of risk towards less risky assets such as interest rates. A higher risk allocation of an asset clearly leads to a higher capital allocation which is also seen in A.1. The QQ- plots for a Gaussian and Gaussian mixture distribution for these two assets are displayed in section A.2. From these QQ-plots, it is clear that global rates exhibit less heavy tails than the developed markets stocks. It is also evident that the Gaussian mixture model is a better fit to the data for global rates compared to developed markets stocks. The fact that the low-risk portfolio has a large portion of the risk concentrated to rates compared to stocks and the fact that the Gaussian mixture model is a good fit to the data for global rates explains the smaller difference in performance between the empirical and Gaussian mixture model for the risk parity and low-risk portfolios. Also, the empirical model is better than the Gaussian mixture model at capturing heavy tails, but this is not as important for the risk parity and low-risk portfolios which mainly consists of assets with less heavy tails. It is therefore expected that the empirical model will perform better and better compared to the Gaussian mixture model when more of the risk is allocated to the

Filip Lundin 54 Markus Wahlgren Chapter 5. Discussion 5.2. Sources of Error

assets which display heavier tails, which is what is seen in the results.

The result that the low-risk portfolio yielded higher Sharpe and tail risk-adjusted return ratio can be explained by the diﬀerence in asset allocation between the portfolios. This portfolio was the one that had the highest concentration of risk and capital to the less risky assets. Table 3.3 showed that these less risky assets, such as alternative risk premia and global rates, had higher Sharpe and tail risk-adjusted return ratio compared to the more riskier assets. This explains why the low-risk portfolio had the highest ratios and why the ratios decrease when the risk level of the portfolios are increased.

Also, it was concluded that all risk budgeting models performed better than a portfolio of only stocks and a 60/40-portfolio. This was clear from the graphs of the value development presented in the previous chapter where all investments are adjusted to have the same risk levels. The result was in line with what previous researchers have shown and are often the motivation behind the use of risk budgeting portfolios.

5.2 Sources of Error

There exist sources of error in this thesis that could compromise the accuracy of the answers to the research questions. One of these sources of error is that the results in this thesis are based on a specific historic period with specific market conditions. These results are limited in the sense that it is not certain that the relationships and findings gathered from the past will hold in the future. One specific market condition that has been present in the data that has been used for this thesis is that interest rates have mainly been declining throughout the period. This phenomenon of declining interest rates are for example not certain to continue in the future.

Also, costs in different forms have not been considered in this thesis. Of course, in the financial markets, there will be cost involved when trading and thus when rebalancing the portfolios. Further, there are fees taken out by financial institutions for providing instruments that fund managers can invest in that follows a particular index. There are certainly also costs involved with borrowing money, which is done in order to achieve the leverage used in some of the portfolios considered. For the particular time period that this thesis investigates the interest rates have been low, but in the future this might not be the case and thus could have a more significant impact on results.

Filip Lundin 55 Markus Wahlgren Chapter 5. Discussion 5.3. Future Work

5.3 Future Work

Possible areas of future research could be to address the sources of error that were mentioned in the previous section. To begin with, it would be possible to investigate the error that could arise when basing models on historic data. Instead of using historic data to calculate the CVaR one could e.g. use information from the options-market to get an implied CVaR. This could lead to a model that would better predict events instead of reacting to events and could prove to be better at protecting against tail risk. A similar approach to this has been done by AllianceBernstein [8] as mentioned in Section 1.3. This, however, is only the case if the market is better at predicting the future tail risk than the historical data.

Another suggestion of future research, which also was a source of error mentioned in the previous section, is to study how costs would aﬀect the performance of these models and portfolios. This aspect of the performance has been determined to be outside the scope of this thesis but it would be interesting to investigate the impact of costs. Costs like transaction costs of trading, interest rate costs from leveraging and managing fees could be analyzed.

Filip Lundin 56 Markus Wahlgren Chapter 6

Conclusions

In this report, it has been shown how risk budgeting strategies can be implemented that take into account tail risk and how these strategies performed in relation to conventional risk budgeting models based on a Gaussian distribution to model returns and with volatility as a risk measure.

To begin with, two models for constructing risk budgeting portfolios that considered tail risk were presented. Both these models were based on CVaR as a risk measure, in line with what previous researchers have used in order to create similar models. The ﬁrst model modeled returns with their empirical distribution, where thus no assumption of a parametric distribution was made. For the second model, a multivariate Gaussian mixture model to model returns was introduced based on the work by Bruder, Kostyuchyk and Roncalli [14]. This model ﬁtted a mixture of Gaussian distributions to the historical returns in order to take into account both the skewness and heavy tails of the data. There are of course many ways of constructing risk budgeting models that consider tail risk, but these two were presented and used in the report since they were the ones most recurring in previous research on the topic.

Next on the performance of the different models was evaluated. Here, a diverse set of asset classes, several risk budgets, and risk targets were used to form portfolios for the different models. Based on the performance, measured in terms of risk-adjusted returns, it was clear that the models that took into account tail risk had superior performance in relation to the conventional model based on an assumption of Gaussian distributed returns and volatility as a risk measure when considering the full-time period for the data. The superiority of these models was present both when the risk-adjusted returns were measured with Sharpe ratio and a measure that uses tail risk to measure the risk. How- ever, it should be noted that the superiority was significantly higher for portfolios that

57 Chapter 6. Conclusions

constituted of mainly high-risk assets, e.g. stocks, than for portfolios with more low-risk assets such as bonds. For the portfolios with the highest degree of low-risk assets, there were not any signiﬁcant improvements from using the models based on CVaR in relation to the model based on volatility. Even though the results were unambiguous for the full- time period it was clear that the performance of the models could diﬀer in shorter time periods. In some cases, the Gaussian conventional model had superior performance in relation to the models considering tail risk. This was especially notable for the lower risk portfolios.

It was also clear that the model that used the empirical distribution to model returns performed better than the model based on an assumption of the returns belonging to the Gaussian mixture model when the portfolio consisted more of assets with heavier tails, such as stocks. An explanation for this could be that the Gaussian mixture model was not able to fully take into account the existence of heavy tails in many of the assets in the considered data, while the empirical CVaR model was able to do so since it only uses the actual returns without any parametric assumption for them.

Filip Lundin 58 Markus Wahlgren Chapter 7

Bibliography

[1] Harry Markowitz. Portfolio selection. The journal of ﬁnance, 7(1):77–91, 1952.

[2] P Podolsky, R Johnson, and O Jennings. The all weather story. White paper, Bridgewater Associates, 2012.

[3] Thierry Roncalli. Introduction to risk parity and budgeting. Chapman and Hall/CRC, 2016.

[4] Edward Qian. Risk parity and diversiﬁcation. The Journal of Investing, 20(1):119– 127, 2011.

[5] Ronald Huisman, Kees G Koedijk, and Rachel AJ Pownall. Var-x: Fat tails in ﬁnancial risk management. Journal of risk, 1(1):47–61, 1998.

[6] Thierry Roncalli. Alternative risk premia: What do we know? In Factor Investing, pages 227–264. Elsevier, 2017.

[7] Lakshman A Alles and John L Kling. Regularities in the variation of skewness in asset returns. Journal of ﬁnancial Research, 17(3):427–438, 1994.

[8] Ashwin Alankar, Michael DePalma, and Myron Scholes. An introduction to tail risk parity: Balancing risk to achieve downside protection. AllianceBernstein, 2013.

[9] Kris Boudt, Peter Carl, and Brian G Peterson. Asset allocation with conditional value-at-risk budgets. Journal of Risk, 15(3):39–68, 2012.

[10] Francesco Cesarone and Stefano Colucci. Minimum risk versus capital and risk diversiﬁcation strategies for portfolio construction. Journal of the operational research society, pages 1–18, 2017.

59 Chapter 7. Bibliography

[11] David Stefanovits. Equal contributions to risk and portfolio construction. Master’s thesis, ETH Zurich, 2010.

[12] Elisabetta Cagna, Giulio Casuccio, et al. Equally-weighted risk contribution portfolios: an empirical study using expected shortfall. Technical report, Center for Research on Pensions and Welfare Policies, Turin (Italy), 2014.

[13] Stefano Colucci. How expected shortfall can simplify the equally-weighted risk contribution portfolio. Available at SSRN 1967871, 2011.

[14] Benjamin Bruder, Nazar Kostyuchyk, and Thierry Roncalli. Risk parity portfolios with skewness risk: An application to factor investing and alternative risk premia. Available at SSRN 2813384, 2016.

[15] Hoang Vu. Risk Parity with Constrained Gaussian Mixture Models. Master’s thesis, University of Oxford, 2018.

[16] Helmut Mausser and Oleksandr Romanko. Long-only equal risk contribution portfolios for cvar under discrete distributions. Quantitative Finance, 18(11):1927– 1945, 2018.

[17] Gunnar Blom. Probability and statistics: theory and applications. Springer, 1989.

[18] Allan Gut. An intermediate course in probability. Springer-Vlg, New York, 1995.

[19] Geoﬀrey J McLachlan and David Peel. Finite mixture models. John Wiley & Sons, 2000.

[20] Henrik Hult, Filip Lindskog, Ola Hammarlid, and Carl Johan Rehn. Risk and portfolio analysis: Principles and methods. Springer Science & Business Media, 2012.

[21] Kris Boudt, Brian G Peterson, and Christophe Croux. Estimation and decomposition of downside risk for portfolios with non-normal returns. Journal of Risk, 11(2):79–103, 2008.

[22] G William Schwert. Stock market volatility. Financial analysts journal, 46(3):23– 34, 1990.

[23] Leonard CG Rogers, Stephen E Satchell, and Y Yoon. Estimating the volatility of stock prices: a comparison of methods that use high and low prices. Applied Financial Economics, 4(3):241–247, 1994.

[24] Giora Hanoch and Haim Levy. The eﬃciency analysis of choices involving risk. The Review of Economic Studies, 36(3):335–346, 1969.

Filip Lundin 60 Markus Wahlgren Chapter 7. Bibliography

[25] Carlo Acerbi and Dirk Tasche. On the coherence of expected shortfall. Journal of Banking & Finance, 26(7):1487–1503, 2002.

[26] Thierry Roncalli. Introducing Expected Returns into Risk Parity Portfolios: A New Framework for Asset Allocation. Bankers, Markets & Investors, (138):18– 28, September 2015.

[27] Dirk Tasche. Euler allocation: Theory and practice. Technical report, Citeseer, 2007.

[28] Sébastien Maillard, Thierry Roncalli, and Jérôme Teïletche. The properties of equally weighted risk contribution portfolios. The Journal of Portfolio Manage- ment, 36(4):60–70, 2010.

[29] Roger Fletcher. Practical methods of optimization. John Wiley & Sons, 2013.

[30] Jorge Nocedal and Stephen Wright. Numerical optimization. Springer Science & Business Media, 2006.

[31] Campbell R Harvey, Edward Hoyle, Russell Korgaonkar, Sandy Rattray, Matthew Sargaison, and Otto Van Hemert. The impact of volatility targeting. The Journal of Portfolio Management, 45(1):14–33, 2018.

[32] Lars Rickenberg. Tail risk targeting: Target var and cvar strategies. Available at SSRN 3444999, 2019.

[33] William F Sharpe. Mutual fund performance. The Journal of business, 39(1):119– 138, 1966.

[34] Didier Maillard. Tail risk adjusted sharpe ratio. Amundi Working Paper WP-071, 2018.

[35] Carl R Bacon. Practical portfolio performance measurement and attribution, vol- ume 546. John Wiley & Sons, 2008.

Filip Lundin 61 Markus Wahlgren Appendix A

Model data

A.1 Capital Weights

The figures in this section display the capital weights of every model and for every portfolio throughout the period 2008 to 2019. For the portfolios that uses leverage in order to target a specific risk two figures are shown. The normalized weights are shown together with the actual leveraged weights.

Figure A.1: Risk parity based on volatility

62 Appendix A. Model data A.1. Capital Weights

Figure A.2: Risk parity based on empirical CVaR 5%

Figure A.3: Risk parity based on Gaussian mixture CVaR 5%

Filip Lundin 63 Markus Wahlgren Appendix A. Model data A.1. Capital Weights

(a) Normalized (b) Leveraged

Figure A.4: Low-risk portfolio based on volatility.

(a) Normalized (b) Leveraged

Figure A.5: Low-risk portfolio based on empirical CVaR 5%

Filip Lundin 64 Markus Wahlgren Appendix A. Model data A.1. Capital Weights

(a) Normalized (b) Leveraged

Figure A.6: Low-risk portfolio based on Gaussian mixture CVaR 5%.

(a) Normalized (b) Leveraged

Figure A.7: Mid-risk portfolio based on volatility.

Filip Lundin 65 Markus Wahlgren Appendix A. Model data A.1. Capital Weights

(a) Normalized (b) Leveraged

Figure A.8: Mid-risk portfolio based on empirical CVaR 5%

(a) Normalized (b) Leveraged

Figure A.9: Mid-risk portfolio based on Gaussian mixture CVaR 5%

Filip Lundin 66 Markus Wahlgren Appendix A. Model data A.1. Capital Weights

(a) Normalized (b) Leveraged

Figure A.10: High-risk portfolio based on volatility.

(a) Normalized (b) Leveraged

Figure A.11: High-risk portfolio based on empirical CVaR 5%

Filip Lundin 67 Markus Wahlgren Appendix A. Model data A.2. QQ-Plots for the Models

(a) Normalized (b) Leveraged

Figure A.12: High-risk portfolio based on Gaussian mixture CVaR 5%

A.2 QQ-Plots for the Models

This section will present QQ-plots for the diﬀerent models. The QQ-plots are shown to illustrate how well the diﬀerent models capture heavy tails in the data.

A QQ-plot comparing the quantiles of the adapted Gaussian distribution and the empirical data is shown below in Figure A.13 (a) for the asset developed markets stocks. It is evident from the QQ-plot that the Gaussian distribution is a bad ﬁt for the data and that the empirical data has heavier tails compared to the adapted distribution. The QQ-plot in Figure A.13 (b) instead compares the quantiles of an adapted Gaussian mixture model to the quantiles of the empirical data. The QQ-plot indicates that the adapted distribution is a better ﬁt to the data compared to the Gaussian distribution. It is however clear that the empirical data still has heavier tails compared to the adapted distribution.

Filip Lundin 68 Markus Wahlgren Appendix A. Model data A.2. QQ-Plots for the Models

(a) Gaussian (b) GMM

Figure A.13: QQ-plots comparing the quantiles of an adapted Gaussian distribution and a Gaussian mixture distribution to the empirical quantiles for the asset developed markets stocks

The QQ-plots below are also for the Gaussian and the Gaussian mixture model but for the asset global rates instead. What is clear from looking at the Gaussian QQ-plot is that global rates exhibit less heavy tails compared to developed markets stocks since the points in the QQ-plots deviate less from the 45-degree line. It is also evident from the Gaussian mixture QQ-plot that the distribution is a better ﬁt to the data compared to the developed markets stocks QQ-plot since the points here also deviates less from the 45- degree line. This would indicate that global rates have less heavy tails than developed markets stocks and that the Gaussian mixture model is a better ﬁt to the data for global rates than it is for developed markets stocks.

Filip Lundin 69 Markus Wahlgren Appendix A. Model data A.3. Leverage

(a) Gaussian (b) GMM

Figure A.14: QQ-plots comparing the quantiles of an adapted Gaussian distribution and a Gaussian mixture distribution to the empirical quantiles for the asset global rates

A.3 Leverage

The ﬁgures in this section displays the leverage each model has for each portfolio throughout the period 2008-2019. If the leverage factor is 1 it means that the portfolio does not have any leverage and if it for example is 2 it mean that the portfolio implements 100% leverage.

Filip Lundin 70 Markus Wahlgren Appendix A. Model data A.3. Leverage

Figure A.15: Leverage for low-risk portfolios with CVaR0.05 target of 0.5%

Filip Lundin 71 Markus Wahlgren Appendix A. Model data A.3. Leverage

Figure A.16: Leverage for mid-risk portfolios with CVaR0.05 target of 1.0%

Filip Lundin 72 Markus Wahlgren Appendix A. Model data A.4. Performance Measures for Sub-Periods

Figure A.17: Leverage for high-risk portfolios with CVaR0.05 target of 1.5%

A.4 Performance Measures for Sub-Periods

This section contains the same type of ﬁgures that were shown in section 4.3, i.e. the Sharpe ratio and TRAR-ratio for 6 sub-periods, but here for all of the portfolios. Note again that the tail risk-adjusted return has been multiplied with 100 in order to yield more readable results.

Filip Lundin 73 Markus Wahlgren Appendix A. Model data A.4. Performance Measures for Sub-Periods

Figure A.18: The Sharpe ratio for the diﬀerent periods for the risk parity portfolio

Figure A.19: The Tail risk-adjusted return for the diﬀerent periods for the risk parity portfolio

Filip Lundin 74 Markus Wahlgren Appendix A. Model data A.4. Performance Measures for Sub-Periods

Figure A.20: The Sharpe ratio for the diﬀerent periods for the low-risk portfolio

Figure A.21: The Tail risk-adjusted return for the diﬀerent periods for the low-risk portfolio

Filip Lundin 75 Markus Wahlgren Appendix A. Model data A.4. Performance Measures for Sub-Periods

Figure A.22: The Sharpe ratio for the diﬀerent periods for the mid-risk portfolio

Figure A.23: The Tail risk-adjusted return for the diﬀerent periods for the mid-risk portfolio

Filip Lundin 76 Markus Wahlgren Appendix A. Model data A.4. Performance Measures for Sub-Periods

Figure A.24: The Sharpe ratio for the diﬀerent periods for the high-risk portfolio

Figure A.25: The Tail risk-adjusted return for the diﬀerent periods for the high-risk portfolio

Filip Lundin 77 Markus Wahlgren

TRITA -SCI-GRU 2020:050