A performance investigation and evaluation of selected portfolio optimization methods with varying assets and market scenarios

(1)

IN

DEGREE PROJECT MATHEMATICS, SECOND CYCLE, 30 CREDITS

STOCKHOLM SWEDEN 2016,

A performance investigation and evaluation of selected portfolio optimization methods with varying assets and market scenarios

GUSTAF CALLERT

FILIP HALÉN DAHLSTRÖM

KTH ROYAL INSTITUTE OF TECHNOLOGY

(2)

(3)

A performance investigation and evaluation of selected portfolio optimization methods with varying assets and market scenarios

G U S T A F C A L L E R T F I L I P H A L É N D A H L S T R Ö M

Master’s Thesis in Financial Mathematics (30 ECTS credits) Master Programme in Industrial Engineering and Management (120 credits)

Royal Institute of Technology year 2016 Supervisor at Nordea: Krister Alvelius Supervisor at KTH: Henrik Hult Examiner: Henrik Hult

TRITA-MAT-E 2016:50 ISRN-KTH/MAT/E--16/50--SE

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

(4)

(5)

Abstract

This study investigates and evaluates how different portfolio optimization methods perform when varying assets and financial market scenarios. Methods included are mean variance, Conditional Value-at-Risk, utility based, risk factor based and Monte Carlo optimization. Market scenarios are represented by stagnating, bull and bear market data from the Bloomberg database. In order to perform robust optimizations resampling of the Bloomberg data has been done hundred times. The evaluation of the methods has been done with respect to selected ratios and two benchmark portfolios. Namely an equally weighted portfolio and an equally weighted risk contributions portfolio. The study found that mean variance and Conditional Value-at-Risk optimization performed best when using linear assets in all the investigated cases. Considering non-linear assets such as options an equally weighted portfolio performs best.

Keywords: Portfolio optimization, Asset allocation, Evaluation ratios, Asset pricing, Risk

(6)

(7)

Sammanfattning

Den här studien undersöker och utvärderar hur olika portföljoptimeringsmetoder presterar med varierande finansiella tillgångsslag och marknadsscenarion. De metoder som har undersökts är: väntevärde-varians, villkorligt-värde-av-risk, nyttjande- och Monte Carlo baserad optimering. De marknadsscenarion som valts är: stagnerande, uppåt- samt nedåtgående scenarion där marknadsdata hämtats från Bloomberg för respektive till- gång. För att erhålla robusta optimeringsresultat har data omsamplats hundra gånger.

Utvärderingen av metoderna har gjorts med avseende på utvalda indikatorer och två jämförelseportföljer, en likaviktad portfölj och en likariskviktad portfölj. Studien fann att portföljer genererade av väntevärde-varians och villkorligt-värde-av-risk optimering visade bäst prestanda, när linjära tillgångar använts i samtliga scenarion. När ickelin- jära tillgångar såsom optioner har använts gav den likaviktade jämförelseportföljen bäst

(8)

(9)

Acknowledgements

First and foremost we would like to thank Professor Henrik Hult (KTH), Krister Alvelius (Nordea Markets) and Peter Seippel (Nordea Markets) for their continuous support throughout the project. Furthermore we dedicate our thoughts to our families and friends who have been with us during our five years at KTH.

Stockholm, August 16, 2016

Filip Halén Dahlström and Gustaf Callert

(10)

(11)

List of Figures

5.1 Historical prices of assets GB, AH, GO and US - Case 1 and Case 3 . . . 22

5.2 Historical prices of assets OM, GB, EI and EH - Case 2 and Case 4 (bull market) . . . 23

5.3 Historical prices of assets OM, GB, EI and EH - Case 2 and Case 4 (bear market) . . . 24

5.4 Implied volatility and option prices . . . 25

5.5 Historical prices of assets VS and VO - Case 5 (bull market) . . . 26

5.6 Historical prices of assets VS, HS and VO - Case 6 (bull market) . . . 27

6.1 Optimal portfolios – Case 1 . . . 29

6.2 Optimal portfolios – Case 2 (bull market) . . . 31

6.3 Optimal portfolios – Case 2 (bear market) . . . 32

6.4 Optimal portfolios – Case 3 . . . 34

6.6 Optimal portfolios – Case 4 (bear market) . . . 37

6.8 Monte Carlo simulation of asset prices VS and VO – Case 5 (bull market) 40 6.9 Distribution of Monte Carlo asset VS log returns – Case 5 (bull market) . 40 6.10 Distribution of Monte Carlo asset VO log returns – Case 5 (bull market) . 41 6.11 Optimal portfolios – Case 5 (bull market – in/out of sample) . . . 42

6.12 Efficient frontier – Case 5 (bull market) . . . 43

6.14 Monte Carlo simulation of asset prices VS, HS and VO – Case 5 (bull market) . . . 45

6.15 Distribution of Monte Carlo asset HS log returns – Case 6 (bull market) . 46 6.16 Optimal portfolios – Case 6 (bull market - in/out of sample) . . . 47

6.17 Efficient frontier – Case 6 (bull market) . . . 48

7.1 Flow chart – Non complex portfolio . . . 53

7.2 Flow chart – Standard portfolio . . . 54

7.3 Flow chart – Complex portfolio . . . 54

(14)

(15)

List of Tables

1.1 Gantt Chart . . . 3

5.1 Considered assets . . . 21

5.2 Historical parameters - Case 1 and Case 3 . . . 21

5.3 Historical parameters - Case 2 and Case 4 (bull market) . . . 22

5.4 Historical parameters - Case 2 and Case 4 (bear market) . . . 23

5.5 Historical parameters - Case 5 (bull market) . . . 25

5.6 Historical parameters - Case 6 (bull market) . . . 26

6.1 Results non-complex – Case 1 . . . 28

6.2 Results non-complex with bootstrapping – Case 1 . . . 29

6.3 Results non-complex – Case 2 (bull market) . . . 30

6.4 Results non-complex with bootstrapping – Case 2 (bull market) . . . 31

6.5 Results non-complex – Case 2 (bear market) . . . 32

6.6 Results non-complex with bootstrapping – Case 2 (bear market) . . . 33

6.7 Results standard portfolio – Case 3 . . . 33

6.8 Results standard with bootstrapping – Case 3 . . . 34

6.9 Results standard portfolio – Case 4 (bull market) . . . 35

6.10 Results standard with bootstrapping – Case 4 (bull market) . . . 36

6.11 Results standard portfolio – Case 4 (bear market) . . . 36

6.12 Results standard portfolio with bootstrapping – Case 4 (bear market) . . 37

6.13 Results complex - Case 5 (bull market) . . . 38

6.14 Results complex portfolio with bootstrapping – Case 5 (bull market) . . . 39

6.15 Results complex – Case 5 (bull market, in/out of sample) . . . 41

6.16 Results complex – Case 6 (bull market) . . . 44

6.17 Results complex portfolios with bootstrapping – Case 6 (bull market) . . . 45

6.18 Results complex portfolio – Case 6 (bull market, in/out of sample) . . . . 46

(16)

(17)

Chapter 1

Introduction

1.1 Background

A master thesis accounts for 30 ECTS credits and could be written at either an university institution or a company, during the final semester of a two year Master’s Program. This particular project was executed and written on behalf of Nordea Markets in Stockholm where Krister Alvelius and Peter Seippel have been supervisors. The supervisors have provided the authors with help on formulating the problem as well as feedback, brain- storming and knowledge from an industrial point of view. At KTH, professor Henrik Hult has been supervisor and he has contributed with his deep mathematical knowledge during plenty of occasions.

1.2 Purpose

Different investors have different goals when investing, some might be risk takers who allow for larger volatility in riskier assets while some are risk avert and look for safer returns. How much risk an investor is willing to take on (the risk appetite) plays a large role in determining the asset universe in which he or she can invest in. Two other important factors are the time horizon of the investment and how active the investor is.

For a given asset universe the investor can thus chose infinitely many combinations.

In the framework of this thesis the investor is choosing optimally over assets with a given risk appetite. The answer to what is optimal is in this framework given by five different methods, namely mean variance, Conditional Value-at-Risk, utility based, Monte Carlo and risk-factor based optimization. The results are evaluated against an equally weighted portfolio and an equally weighted risk contributions portfolio. Furthermore the results are evaluated by three different ratios; Sharpe, drawdown and Calmar ratio. Ideally one would like to have a high risk-adjusted return which is reflected by a high Sharpe and Calmar ratio and low drawdown ratio.

(18)

CHAPTER 1. INTRODUCTION

When carrying out optimizations and presenting solutions to clients it is important to know your customer. Depending on how sophisticated the client is, how their investment policy is formulated (defining the asset universe) and risk appetite Nordea Markets would like to recommend a particular portfolio generated by a certain optimization method, which is the ultimate goal of this thesis.

1.3 Problem formulation

This master thesis seeks to investigate and evaluate the performance of different portfolio optimization methods with a varying set of asset classes and market scenarios. These studies are carried out in order to decide which optimization method to use given a specific set of assets in a particular market.

1.4 The indented reader

It is assumed that the reader has a background within financial mathematics and or mathematical statistics. I.e basic understanding in probability theory, optimization, computational simulations as well as an intermediate understanding of financial instruments.

2

(19)

CHAPTER 1. INTRODUCTION

1.5 Method

This project was conducted during the spring and summer of 2016 and in Table 1.1 outlined below the rough time line of the project is presented, containing the main blocks of the project. The numbers in the upper part of the Gantt Chart represent months, i.e 3 corresponds to March, 4 to April and so forth.

Table 1.1: Gantt Chart

Master thesis, second cycle 2016

3 4 5 6 7 8

Problem formulation Literature study Data fetching and method implementation Simulation and analysis Report writing Submission

Some short notes on the blocks follows.

• Problem formulation: Conducted while reviewing particular mathematical areas of interest, the final problem and scope (Section 1.3) was formulated together with Nordea Markets.

• Literature study: Course literature, scientific papers, books and websites. See references list on page 56.

• Data fetching and method implementation: The Bloomberg data license has been used when fetching time series of interest to Excel. The methods have been implemented in Matlab which fetches data from Excel and perform the calculations.

• Simulation and analysis: Solely made in Matlab

• Report writing: Solely made in L^ATEX.

(20)

(21)

Chapter 2

Evaluation of optimization methods

This chapter outlines how the portfolio performance is analyzed and evaluated using asset allocation models and ratios.

2.1 Performance evaluation by asset allocation models

There are numerous of different allocation models relevant in this setting, however bounded by the size of the scope and time, two models have been chosen. Both being commonly used by hedge funds and industrial professionals when evaluating performance of investment vehicles. One model is of more straight forward character, namely the equally weighted portfolio and one somewhat more complex, the equally weighted risk contributions portfolio. These two models each representing a portfolio in every case, have been used as benchmark portfolios against the other portfolios in this thesis.

2.1.1 Equal allocation – equally weighted portfolio

As the name suggest, the holdings in an equally weighted portfolio (EWP) are determined by w_i = 1/N , where w_i is the weight of the i:th asset and N is the total number of assets, giving each of the holdings the same percentage weight. Note that this allocation model has an alternative variant where the holdings are equally weighted in terms of the assets market value. This is not the model used in this thesis.

2.1.2 Allocation by equal risk contribution¹

The idea of an equally weighted risk contributions (ERC) portfolio is that each asset wi in a portfolio of N assets has an equal amount of risk σ_i to the total portfolio. The formula for the weights w_i are derived below.

The risk of a portfolio is rewritten as:

σ =

√

w^TΣw =X

i

w²_iσ_i²+X

i

X

j6=i

w_iw_jσ_ij (2.1)

(22)

CHAPTER 2. EVALUATION OF OPTIMIZATION METHODS

Now let the marginal risk contribution ∂_w_iσ(w) be defined and rewritten as:

∂wiσ(w) = ∂σ(w) σwi

= w_iσ²+P

j6=iw_jσ_ij

σ(w) (2.2)

Since the volatility function satisfies Euler’s theorem the individual risk contribution is written as σ_i(w) = w_i∂_w_iσ(w) and the total risk of the portfolio is then σ =PN

i=1σ_i(w) for N number of assets.

Now let w_i ∈ [0, 1], PN

i w_i = 1 and simplify by assuming equal correlations i.e ρ_ij = ρ.

The individual risk contribution is then rewritten as:

σ_i(w) = w_i²σ_i²+ ρP

j6=iwiwjσiσj

σ = wiσi((1 − ρ)wiσi+ ρP

jwjσj)

σ (2.3)

Using σ_i(w) = σ_j(w), ∀i, j and ρ ≥ −1/(N − 1) it can be shown that w_iσ_i= x_jσ_j, which together withP

iwi = 1 yield the sought expression for the asset weights:

w_i = σ⁻¹_i Pn

j=1σ_j⁻¹ (2.4)

2.2 Performance evaluation by ratios

In order to evaluate the portfolios given by the asset allocation models and the ones generated by the optimization methods, a selection of ratios have been chosen. This has been done in order to compare the portfolios with one another in a standardized fashion.

As in the case with the asset allocation models we have looked at what is commonly used in the industry and selected a limited number of variations within that range.

2.2.1 Standardized risk and return^2,3

The two most fundamental parameters in modern portfolio theory, risk (standard deviation) σ and return µ are here defined as:

σ = ˆσ(R)

√

T (2.5)

µ = (ˆµ(R) + 1)^T − 1 (2.6)

Where R is the series of daily log returns generated by the portfolio, ˆσ(R) is the sample standard deviation, ˆµ(R) is the sample mean and T is the number of trading days in a year. Typically 250 days when presented on an annual basis and 20 days when presented on a monthly basis. By defining risk and return in this way, the performance from different portfolios can easily be compared and benchmarked. The aim is to have high risk-adjusted return.

5

(23)

CHAPTER 2. EVALUATION OF OPTIMIZATION METHODS

2.2.2 Sharpe ratio⁴

The Sharpe ratio provides the investor with a quick overview of how a particular investment vehicle yields risk-adjusted returns over some time period (usually a full trading year). The ratio measures excess return or risk premium, per unit of deviation in an investment. Ideally one would like to have low standard deviation relative to the return, reflected in a high Sharpe ratio – loosely speaking a Sharpe ratio above 1 is considered to be good. Below follows the mathematical representation of the ratio:

R − R₀

σ(R) (2.7)

Where R − R₀ is the return of the investment minus the risk-free rate and σ(R) is the standard deviation of the investment. Note here that the Sharpe ratio can be negative.

Reflecting that R > R₀ or R < 0. Interpreted as the risk-free rate being higher than the return on the investment or that the return on the investment simply is non positive.

2.2.3 Drawdown⁵

The drawdown ratio is the maximum percentage drop in a given period for an investment and gives the investor an intuition about the downside risk of an invesment. Drawdown DD, is defined by:

DD = P_peak− P_trough

P_peak (2.8)

Where P_peak and P_trough is the highest and lowest portfolio value respectively, in the specific period. Note that if the time series is strictly increasing over the period the drawdown ratio is zero, DD = 0. However (in the realistic case) the drawdown ratio is most likely positive DD > 0, since the portfolio value fluctuates.

2.2.4 Calmar ratio⁴

The Calmar ratio CR or drawdown ratio, is a common and important ratio used by hedge funds and asset managers where annual return R_annual is evaluated against the maximum drawdown DD, as defined in (2.8). The Calmar ratio is typically evaluated using data from three years or more and here it is defined as:

CR = R_annual

DD (2.9)

(24)

(25)

Chapter 3

Optimization methods and simulations

In the following chapter the chosen portfolio optimization methods are presented from a theoretical point of view along with some brief notes on pros and cons of the respective methods. The chapter begins by an outline of how risk measures are constructed and later on incorporated in the actual optimization. In the latter part some notes on simulations and data generation can be found.

3.1 Risk measures

Throughout time the global financial markets have undergone several different crises – the Asian financial crisis of 1997, the global meltdown of 2008 and the European debt crisis of 2011 – to name a few. These crises have resulted in banks defaulting and enor- mous individual losses. This has lead to a stricter regulatory frameworks and a need for incorporating adequate risk measures.

In order for professional investors and institutions to constrain risk in their investments and debt origination, regulators control certain risk measures (among other things) to make sure that these stakeholders are in line with what is required to be a prudent par- ticipant on the market.

Much could be written on this topic however it is not the main scope of this thesis and here two essential risk measures are presented, namely Value-at-Risk (VaR) and expected shortfall or Conditional Value-at-Risk (CVaR) – which were both formulated in the late 1980’s and later on VaR was implemented in the Basel II accords of 1999.

3.1.1 Value-at-Risk⁶

Consider here an investment with value X at some future time T = 1, then the Value- at-Risk (VaR) is the smallest amount of monetary unit m that if added to the position

(26)

CHAPTER 3. OPTIMIZATION METHODS AND SIMULATIONS

today and invested in the risk-free asset ensures that the probability of a strictly negative value at time 1 is not greater than the chosen risk level p. Usually p is 5% or less. VaR is commonly defined as:

VaR_p(X) = min{m : P (mR₀+ X < 0) ≤ p}, p ∈ (0, 1) (3.1) where p is the chosen risk level of the portfolio value X at time 1 and R₀ is the return of the risk-free asset. One can rewrite the definition of VaR, to obtain a more intuitive sound expression:

VaR_p(X) = min{m : P (mR₀+ X < 0) ≤ p} (3.2)

= min{m : P (−X/R₀> m) ≤ p} (3.3)

= min{m : 1 − P (−X/R0 ≤ m) ≤ p} (3.4)

= min{m : P (−X/R0≤ m) ≥ 1 − p} (3.5) Set the loss function to L = −X/R₀, where X is the net gain from the investment to get the following:

VaR_p(X) = min{m : P (L ≤ m) ≥ 1 − p} (3.6) The u-quantile of a random variable L with distribution function F_L is defined as:

F_L⁻¹(u) = min{m : FL(m) ≥ u} (3.7) Finally we have that V aR_p(X) is the (1 − p) quantile of a random variable L and can write the Value-at-Risk as:

VaR_p(X) = F_L⁻¹(1 − p) (3.8)

3.1.2 Conditional Value-at-Risk⁶

Similar to Value-at-Risk Conditional Value-at-Risk (CVaR) or expected shortfall, presents the riskiness of an investment. This risk measure can be interpreted as the expected Value-at-Risk taking into account not only the quantile but also the tail. Here it is defined as:

CVaR_p(X) = 1 p

Z p 0

VaR_u(X)du = 1 p

Z p 1−p

F_L⁻¹(u)du (3.9) With the same definition of VaR as above for the same asset or portfolio X. Letting L = −_R^X

0 and expanding the integral, the above can be represented as:

CVaR_p(X) = −1 p

Z p 0

F_X/R⁻¹

0(u)du = −F_X/R⁻¹

0(p) 1 −F_X/R₀(F_X/R⁻¹

0(p) p

!

(3.10)

CVaR_p(X) = min

c − c +1

pE[(c − X/R0)+] (3.11) For some constant c ∈ R.

8

(27)

3.2 Portfolio optimization

Reviewing the optimization methods covered in the course Portfolio Theory and Risk Management (SF2942) at KTH, a selection has been made where the most relevant methods in this setting has been implemented. In the beginning of this section a sweep over some basic optimization theory is made, before going in to the specifics of the portfolio optimization methods.

3.2.1 Convex optimization⁶

As it turns out many optimization problems related to investments and finance have convex objective functions f and constraints g_k, i.e f : C 7→ R and gk : C 7→ R for k = 1, 2, ..., m and where C ⊂ R is a convex set.

Let the Lagrangian multiplier λ ∈ [0, 1] and the variables x, y ∈ C then it holds that λx + (1 − λ)y ∈ C and moreover: f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y), g_k(x) + (1 − λ)g_k(y).

Now let f (x) be minimized with respect to g_k(x) ≤ c_k, where c_k, k = 1, 2, ..., m are constants. That is:

" min

x f (x) s.t. g_k(x ≤ c_k)

(3.12) Which now is said to be a convex optimization problem, where f and g_k are convex and differentiable functions. An optimal solution x^∗∈ C has the property f(x^∗) ≤ f (y) ∀y, where y is the feasible solutions to the problem. Furthermore x^∗ satisfies the conditions (i)Of (x^∗) +P_m

K=1λ_kg_k(x^∗) = 0, (ii) g_k(x^∗) ≤ c_k, (iii) λ ≥ 0 and (iv) λ_k(g_k(x^∗) − c_k) = 0. Where the Lagrangian multiplier is used and the Lagrangian function defined by L(x^∗) = f (x^∗) +Pm

K=1λk(gk(x^∗) − ck).

3.2.2 The Markowitz framework and mean variance analysis⁶

Modern portfolio theory as presented by Harry Markowitz in 1952, is a framework in which the investor maximizes returns for a certain level of risk. This way of reasoning is commonly referred to as mean variance analysis, where investors are assumed to be risk averse i.e seeking the portfolio with lowest risk (standard deviation) when the return is similar for different portfolios. The minimum variance problem is convex and formulated

as: 



 minw

1

2w^TΣw

s.t. w₀R₀+ w^Tµµµ ≥ µ₀V₀ w₀+ w^T1 ≤ V₀

(3.13)

Where w is the portfolio weights in risky assets, w₀ the portfolio weighting in risk-free asset, Σ the covariance matrix, µµµ the return vector, R₀ the risk-free return, V₀ the port-

(28)

folio value at time 0 and µ₀ some targeted rate of return.

Using the conditions outlined in Section 3.2.1 the following system of linear equations is obtained with Lagrangian multipliers λ₁ and λ₂:

Σw − λ₁µµµ + λ₂1 = 0 (3.14)

− λ₁R₀+ λ₂= 0 (3.15)

w^Tµµµ + w0R0 = µ0V0 (3.16)

w^T1 + w₀= V₀ (3.17)

Inserting (3.15) into (3.14) one gets w = λ₁Σ⁻¹(µµµ − R₀1) (i) and λ11 ₂ = λ₁R₀. Inserting (i) in (3.16) and (3.17) one can by some simple manipulation end up with:

λ1 = µ0− R₀

(µµµ − R0111)Σ⁻¹(µµµ − R0111) (3.18) Now that an expression for λ₁ is obtained the optimal solution is solved for:

w = V₀(µ₀− R₀) Σ⁻¹(µµµ − R0111)

(µµµ − R₀111)^TΣ⁻¹(µµµ − R₀111) (3.19) Note here that if µ₀ − R₀ > 0 ⇒ λ₁ > 0 the optimal solution is according to (3.19). If µ0− R₀≤ 0 however, then w = 000 meaning that the investor should should not invest in any risky asset at all.

Limitations - Pros and Cons

This method is intuitive and often easy to implement. In the case of market turmoil (e.g. a financial crisis) holdings in a portfolio tend to move in the same direction leading to increased correlation and a risk that is not as easy to diversify as the Markowitz theory suggests. Furthermore this method does not capture non-linear dependence when introducing non-linear derivatives in the asset universe. Since the covariance matrix only accounts for linear dependence.

3.2.3 Optimization of Conditional Value-at-Risk^6,7,8

As defined in Section 3.1.2 Conditional Value-at-Risk (CVaR) is a commonly used risk measure that accounts for losses over the entire loss distribution. The CVaR function is also a convex function, thus making it suitable for certain optimization problems. Using (3.11) in Section 3.1.2 the optimization problem is formulated and solved in the same manner as in the mean variance problem, Section 3.2.2:

10

(29)







minc − c + 1

pE[(c − X/R₀)₊] s.t. w0R0+ w^Tµµµ ≥ µ0V0

w0+ w^T111 ≤ V0

p ≥ FL(u) − c p ≥ 0

(3.20)

Where the same abbreviations as before are being used. The analytical loss function FL(u) is approximated using historical simulations for p = 0.05.

As stated before CVaR is a good measure of risk since it captures the whole distribution of the loss function. However this method does not capture non-linear dependence when introducing non-linear derivatives in the asset universe. Since the covariance matrix only accounts for linear dependence.

3.2.4 Optimization of expected utility⁶

An important concept in ecnonomics and finance is the so called utility function u which reflects how the investor favour certain goods or in this case assets making up a portfolio. A portfolio value is said to have value V₀ at time zero and V₁ at time 1. Then E[u(V₁)] would be the investor’s expected utility of the portfolio value V₁ at time 1. As an investor seeks to maximize the future value of the portfolio V₁, u is an increasing function and assumed to be concave. I.e for a fixed λ ∈ [0, 1] we have u(λx + (1 − λ)y) ≥ λu(x) + (1 − λ)u(y) and furthermore if u is concave and twice differentiable, then: u⁰(x) ≥ 0, u⁰⁰(x) ≤ 0. There exists a wide range of concave utility functions leading to nice convex optimization problems (as laid out in Section 3.2.1).

A commonly used utility function is the exponential utility function family u_E, which is presented below. The first equation being the general case for some x and the second one for a portfolio given by w₀R₀+ w^TR.

uE(x) = −τ e^−x/τ (3.21)

u_E(w₀R₀+ w^TR) = −τ e^−(w⁰^R⁰^+w^T^µ^µ^{µ)/τ +w}^T^Σw/(2τ²⁾ (3.22) Here the parameter τ is chosen depending on the particular risk appetite and characteristics of the investor in question.

Suppose now that an investor has an assigned utility function u and there exists a vector R of returns for some assets. The investor would like to know the optimal position

(30)

(w0, w), i.e the positions in risk-less and risky assets that maximizes the expected utility.

The optimization problem is formulated as:

" max

w E[u(w₀R0+ w^TR − L)]

s.t. w₀+ w^T111 ≤ V₀

(3.23)

As stated previously u is concave and thus (3.23) is a convex problem and the results from Section 3.2.1 can be applied. Hence an optimal solution (w₀^∗, w^∗) fulfills the following conditions:

E[u⁰(w^∗₀R0+ w^∗TR − L)R₀] = λ (3.24) E[u⁰(w₀^∗R₀+ w^∗TR − L)R] = λ111 (3.25)

w^∗₀+ w^∗T111 = V₀ (3.26)

Within the frames of this thesis the exponential utility function has been used with varying values of τ . As stated previously, τ determines the investors’ risk appetite and here τ ∈ (0, 2] has been considered. The exact value of τ in the particular interval is determined through a parameter study, after which suitable τ ’s are obtained. The higher τ the more risk is the investor willing to take. τ = 2 reflects a high risk investor while τ = 0.01 reflects a low risk investor.

Sound and intuitive investment idea and optimization method. Difficulties in choosing an appropriate utility function and fixating τ . Does not capture non-linear dependence when introducing non-linear derivatives in the asset universe. Since the covariance matrix only accounts for linear dependence.

3.2.5 Optimization based on multi-factor models: BarraOne software⁹ MSCI Barra is a leading provider of support tools to investment institutions worldwide, including Nordea Markets. This section briefly outlines a multi-factor model that is used in the software.

The pure factor model Consider factor return f_yof asset y which is based on particular risk exposures, for example market-, fundamental or industry risk exposure. The factor returns f_y are estimated with cross-sectional regression of asset returns, the updating is conducted monthly by MSCI. The chosen assets are modelled using these factors and a covariance matrix Σ is constructed and can be used in various optimization methods, e.g.

standard mean variance analysis. Nordea Markets uses this software when modelling risk and optimizing portfolios to certain clients.

In the BarraOne Global Equity model (BGEM), the factors y consists of a world factor, 12

(31)

55 country factors, 34 industry factors and 8 style factors. With sub indices w, c, i and s respectively. The return of the BGEM model r_n of stock n is modelled according to:

r_n= f_w^P +X

c

X_ncf_c^P +X

i

X_nif_i^P +X

s

X_nsf_s^P + u^P_n (3.27)

Where P denotes the pure factor and X exposure. The model uses regression weights proportional to the square root of market capitalization and an estimation universe based on a MSCI index which is designed to capture the entire spectra of investment oppor- tunities. Each stock has exposure to the world factor and exposure to the belonging country or industry. The style factors have mean zero and standard deviation one. The specific returns u^P_n are assumed to be mutually uncorrelated.

Equation (3.27) contains two collinearities and in order to obtain a unique regression solution, three constraints are introduced. Namely the weighted country factors, industry factors and style factors returns sum to zero. I.e on the form:

X

n

w_nX_ns= 0 (3.28)

This calibrates the model so that the cap weighted world portfolio is style neutral. Factor returns are estimated using weighted and restricted least square regressions, the general solution can be written as:

f_k^P =X

n

Ω^P_nkrn (3.29)

Where Ω^P_nk is the weight of stock n in the pure factor portfolio k. The return of the world portfolio R_w can be rewritten as:

Rw = f_w^P +X

n

wnu^P_n (3.30)

Where w_n is the weight of stock n and u^P_n the corresponding return.

Navigating in Barra The method of obtaining an optimal portfolio is described in Appendix C.

Expanding the perception of risk; captures more than only looking at volatility in the sense of different risk factors. Demanding and expensive software. "Black-box" feel to user (based on user feedback from Nordea Markets). Difficulties when modelling risk for assets with non-linear payoff structure.

(32)

3.2.6 Monte Carlo optimization

In Monte Carlo based optimization the underlying asset is simulated as described in Section 3.3.1 and option prices are calculated with the Black-Scholes model according to Section 4.1.4. When the pricing is done the next and final step is to test different combinations of percentage weights. This is done in order to plot an efficient frontier and choose the appropriate portfolio in line with the clients risk appetite. The weights should always sum up to one and note here that the weights can be negative. This method is used by Nordea Marketes in Copenhagen but due to confidentiality the actual software could not be used in this thesis. However by consulting the supervisors a rough copy for few assets was made.

This method is often regarded by the industry as superior because it yields better price es- timates. This method also captures the non-linear dependency when including non-linear derivatives e.g. options in the asset universe. However the method requires additional resources (larger Central Processing Unit (CPU) usage) resulting in longer computation time. Increasing the asset universe extends the CPU usage exponentially.

3.3 Simulations and data generation

Both Section 3.3.1 and 3.3.2 describes means of generating additional data sets in order to obtain robust optimizations and more reliable results. The concept presented in Section 3.3.1 is also used in Section 3.2.6.

3.3.1 Monte Carlo simulation¹⁰

The Monte Carlo simulation is in this setting used to forecast the underlying assets price many times over and to create a new larger sample. It is also used to value the option i.e. Monte Carlo valuation. The new sample consists of the historical part (in sample) and the new simulated part (out of sample) which is used in order to get a more robust optimization. To simulate asset prices and generate possible trails of the underlying assets price a mathematical model is used, which in this case is the Geometric Brownian Motion (GBM) model. The basic formula to simulate non-normal GBM structure is presented in (3.31), which is the discrete rate of change of the stock price S.

∆S

S = µ∆t + σ(p)

√

∆t (3.31)

where µ is the mean of the log returns, σ is the standard deviation of the log returns, t is the time and (p) is the random non-normal distributed term from z_t where z_t = [x1, ..., xt] and xt∼ N (0, 1):

(p) = z + 1

6(z²− 1)τ + 1

24(z³− 3z)κ − 1

36(2z³− 5z)τ² (3.32) 14

(33)

Then Cholesky decomposition of the covariance matrix (of log returns) is done in order to convert the random samples into correlated values.

This method is used in order to use current values as a starting point and generate thousands of possible outcomes of the underlying assets price which produce a distribution of outcomes. In order to use this data and retrieve the mean path of all the possible outcomes (sample average) the weak law of large numbers of the daily log returns is used:

R_n= 1

n(R₁+ ... + R_n) (3.33)

where R_i, i ∈ (1, n) is the daily log return and n is the number of simulation. This is done for all the days covering the whole simulation. The following holds for the weak law of large numbers:

R_n→ µ^P when n → ∞ (3.34)

This means that the sample average R_n converges in probability towards the expected value µ. The expected value in this case is the mean path of the portfolio return. In each of the paths the payoff of the option is calculated with Black-Scholes, the average of these payoffs is then the value of the option. The option price have then been computed by Monte Carlo valuation.

3.3.2 The Bootstrap method¹¹

The purpose of using bootstrapping on a chosen data set is to generate new synthetic data through re-sampling of the existing data and estimate some statistic measure of interest. Bootstrapping in this case is based on random sampling with replacement from the original data in order to generate 100 new samples. The new samples are created by drawing randomly with replacement. The original data is a time series of the daily log returns of a particular asset. Since bootstrapping is done on the daily log returns the drift (trend) in the data set is unchanged.

When the 100 new samples are retrieved the optimization can be done for each of the samples and the mean value of all the weights and ratios from the optimizations can be computed. This result represent a more robust optimization since 100 samples have been used, instead of only the original sample. The implementation of bootstrapping data samples was done in all cases except for in-and-out of sample and Monte Carlo optimization (due to limited computation power).

(34)

(35)

Chapter 4

Asset classes, portfolios and market scenarios

The following chapter defines the asset classes, portfolios and market scenarios used in the project. This chapter could be seen as the foundation of which this thesis has been written on.

4.1 Asset classes and portfolios

4.1.1 Asset classes

On the very top level one will find specific asset classes that make up the universe of financial instruments available to investors. An asset class could intuitive define instruments but for the sake of clarity, a specific structure has been selected. A structure that Nordea Market approves of. A simplified top-down tree is sketched below.

Asset class Equity

Derivatives

Linear Non-linear

Direct exposure

Fixed income

Derivatives

Linear Non-linear

Direct exposure

Commodities

Derivatives

Linear Non-linear

Direct exposure

Currencies

Derivatives

Linear Non-linear

Direct exposure

In the scope for this thesis, equity intends common stock, fixed income bonds of different types, commodities oil or metals and currencies the exchange rate between two currency pairs. In order for the investor to position him or herself to one of the assets, direct or in-direct exposure are to chose from.

(36)

CHAPTER 4. ASSET CLASSES, PORTFOLIOS AND MARKET SCENARIOS

Direct exposure means that the actual asset is purchased or sold. In the case of stocks or bonds, the investor goes out to the market and buys (or short sells) the desired amount of stock or bond. This is the most common way to gain exposure to a particular market (or say country, industry, company etc) and hence what one would expect to find in an customers’ investment policy.

However a less capital intensive (and fairly modern) method is to use the asset in question as an underlying asset in a derivatives position (contract). In this setting two main topics are often discussed: linear and non-linear derivatives. The former includes futures, forwards and swaps and the latter options that could either be "standard" (vanilla) or exotic ones, to name a few.

These derivatives are linear or non-linear in terms of their payoff structure. With a linear payoff (so called delta one derivatives) the value of the derivative contract change according to the underlying asset. For example a SEK10 movement in Ericsson B would directly result in a positive or negative change in the futures contract of Ericsson B (with some maturity T ). A non-linear payoff however changes with space and time. In the case of a vanilla option, the strike K, maturity T , risk-free rate r and volatility σ affects the price of the option. The complexity of pricing non-linear derivatives escalates rather quickly and this brief text could be seen as a light introduction, however the interested reader is advised to read the excellent book by John C. Hull, Options, Futures, and Other Derivatives first published in 1997.

4.1.2 Portfolios

With the structure given in the previous section seen as a foundation, the portfolios are the extended foundation. For the sake of structure, three different portfolio classes have been defined. Namely: non-complex portfolios, standard portfolios and complex portfolios. The distinction is made with respect to the complexity of the assets and what kind of position (long/short) that is allowed. No constraints on how much of each asset the portfolio can contain.

Non-complex portfolio, restricted to assets with linear payoffs

• Equity, fixed income, commodities and currencies - direct exposure

• No short positions allowed

• Buy and hold strategy with yearly client meeting and potential reallocation Standard portfolio, restricted to assets with linear payoffs

• Equity, fixed income, commodities and currencies - direct exposure and linear derivatives

• Short positions allowed

17

(37)

• Buy and hold strategy with yearly client meeting and potential reallocation Complex portfolio no constrain on payoff characteristics

• Equity, fixed income, commodities and currencies - direct exposure, linear- and non-linear derivatives

• Short positions allowed

• Shorter time frame with client meetings held quarterly with potential reallocation 4.1.3 Selected cases

Along with the three different portfolios (non-complex, standard and complex) two different cases are considered to each portfolio – each representing a typical client. Furthermore the cases are tested during the different market scenarios defined in Section 4.2.

4.1.4 Asset pricing¹²

When pricing derivatives, in particular call options (of European character) Black-Scholes formula has been used. It is a common pricing model which provides the market with a fairly accurate start price C₀. Here the formula is simply stated, omitting the actual derivation:

C0= S0Φ(d1) − B0KΦ(d2) (4.1) Where d₁ and d₂ are given respectively by:

d1 = log(S0/(B0K)) σ√

T +σ√

T

2 , (4.2)

d₂ = d₁− σ√

T . (4.3)

Behind the model are seven assumptions worth mentioning, namely:

1. Constant volatility

2. Efficient markets, i.e prices always reflect the available information on the market 3. No dividends paid, this can however be accounted for using different approaches 4. Known interest rates that are constant

5. Lognormally distributed returns

6. European-style options i.e the option cannot be exercised prematurely 7. Perfect liquidity

(38)

4.2 Market scenarios

Moreover it makes sense to take an additional parameter into account when building portfolios and optimizing, namely a view on the market. Three different markets are here considered: stagnating market, bull market with positive drift and bear market with negative drift.

4.2.1 Stagnating market

A market with no distinguishable trend i.e. constant drift, in which there is little or no growth is said to be stagnating. This kind of situation often occurs between economic cyclicals.

4.2.2 Bull market – positive drift

A bull market is a positive market with positive drift where there are more buyers than sellers and asset prices are rising. This state of the economy often occurs in a positive cyclical and investors are willing to relocate capital from risk-less assets to risky ones, hence the stock market is usually booming in a bull market.

4.2.3 Bear market – negative drift

A bear market is a market with negative drift in direct opposite to a bull market. A market in which prices are falling and investors are selling off risky assets such as stocks (more sellers than buyers). This scenario occurs in a negative cyclical such as a recession or a crisis.

19

(39)

(40)

Chapter 5

Empirical study

The following chapter contains the assets and parameters from the times series used when executing the optimizations.

5.1 Assets and time series

Most assets are directly accessible on the financial exchange markets while some, such as bonds and currencies are traded over the counter between two parties (buyer and seller).

In order to invest in some assets, exchange traded funds – commonly referd to as ETFs – have been considered. A long safe asset is represented by the 10 year generic Swedish Governmental bond with the highest credit rating possible. American High Yield bonds (high credit risk) are represented by an index by iTraxx. Gold and USDSEK assets are represented by SEK denoted ETFs. The stock index OMXS30 resembles the 30 largest companies on the Stockholm Stock Exchange. Two European corporate bond indices maintained by Bloomberg represet investment grade bonds (low credit risk) and high yield bonds (high credit risk).

In Table 5.1 the ten assets considered are presented along with their associated asset class and alphabetic abbreviation. The time series of these assets have been fetched into the constructed software. The time series originates from the Bloomberg terminal database. In order to gain greater reliability in the results, new data sets have been generated using a bootstrapping technique on the existing data, see Section 3.3.2.

20

(41)

CHAPTER 5. EMPIRICAL STUDY

Table 5.1: Considered assets

Asset Asset class Abbreviation

10 Year Swedish Government Bond Fixed income GB

American High Yield Bond iTraxx index Fixed income AH

Gold ETF Commodity GO

USDSEK ETF Currency US

OMXS30 Index Equity OM

European Investment Grade Bond Bloomberg Index Fixed income EI European High Yield Bond Bloomberg Index Fixed income EH

H&M B Stock Equity HS

Volvo B Stock Equity VS

Volvo B European ATM Call Option Equity derivative VO

5.1.1 Non-complex and standard portfolios

Case 1 and Case 3 - Portfolio consisting of assets GB, AH, GO and US Stagnating market

This artificial client can invest in the following four assets: the 10 year Swedish governmental bond, the American high yield bond, gold and USDSEK.

In Table 5.2 the annual (250 trading days) log returns and standard deviation based on the sample of 1598 days are presented.

Table 5.2: Historical parameters - Case 1 and Case 3 µ (Annual) σ (Annual)

GB 0.0133 0.0470

AH 0.0113 0.0839

GO 0.0179 0.1695

US 0.0275 0.1157

Below follows the associated covariance matrix Σ of the log returns. The matrix Σ is positive semi definite since all real eigenvalues are positive, this is necessary since Σ has to be inverted when optimizing in for example the mean variance framework.

Σ = 10⁻³·







0.1150 0.0009 −0.0117 0.0018 0.0009 0.0088 0.0066 −0.0003

−0.0117 0.0066 0.0535 −0.0003 0.0018 −0.0003 −0.0003 0.0281







In Figure 5.1 the historical asset prices over the time considered can be seen.

(42)

Figure 5.1: Historical prices of assets GB, AH, GO and US - Case 1 and Case 3 Case 2 and Case 4 - Portfolio consisting of assets OM, GB, EI and EH Bull market - positive drift

This client can invest in the following four assets: the OMXS30 index, 10 year Swedish governmental bond, European investment grade bond index and European high yield bond index.

Table 5.3: Historical parameters - Case 2 and Case 4 (bull market) µ (Annual) σ (Annual)

OM 0.0578 0.1949

GB 0.0133 0.0470

EI 0.0449 0.0239

EH 0.0711 0.0377

Below follows the associated covariance matrix Σ of the log returns. As in the previous case the matrix Σ is positive semi definite since all real eigenvalues are positive.

Σ = 10⁻³·







0.1519 −0.0119 −0.0043 0.0092

−0.0119 0.0088 0.0017 −0.0017

−0.0043 0.0017 0.0023 0.0010 0.0092 −0.0017 0.0010 0.0057







22

(43)

Figure 5.2: Historical prices of assets OM, GB, EI and EH - Case 2 and Case 4 (bull market)

Bear market - negative drift

This investor is eligable to invest in the same assets as in the previously described bull market scenario.

Table 5.4: Historical parameters - Case 2 and Case 4 (bear market) µ (Annual) σ (Annual)

OM -0.0579 0.3036

GB 0.0210 0.0609

EI -0.0010 0.0328

EH 0.0226 0.0988

Below follows the associated covariance matrix Σ of the log returns. As in the previous case the matrix Σ is positive semi definite since all real eigenvalues are positive.

Σ = 10⁻³·







0.3687 −0.0025 −0.0019 −0.0085

−0.0025 0.0149 0.0003 0.0010

−0.0019 0.0003 0.0043 0.0112

−0.0085 0.0010 0.0112 0.0391







(44)

Figure 5.3: Historical prices of assets OM, GB, EI and EH - Case 2 and Case 4 (bear market)

24

(45)

5.1.2 Complex portfolios

Case 5 - Portfolio consisting of assets VS and VO Bull market - positive drift

In this case the investor can invest in two different assets with varying complexity.

Namely Volvo B stock and option exposure through a rolling strategy. The rolling option strategy contains of at the money (ATM) call options with strike price K = S_t, maturity T = 3 months and are priced with market volatility. Note the price dynamics of the option in Figure 5.4 and 5.5.

In Table 5.5 the monthly (20 days) log returns and standard deviation based on the sample of 64 days are presented.

Table 5.5: Historical parameters - Case 5 (bull market) µ (Monthly) σ (Monthly)

VS 0.0532 0.0990

VO 0.0919 0.3355

Below follows the associated positive semi definite covariance matrix Σ of the log returns.

Σ =0.0005 0.0009 0.0009 0.0056

In Figure 5.4 the historical option prices along with the implied volatility over the time considered can be seen.

Figure 5.4: Implied volatility and option prices

(46)

Figure 5.5: Historical prices of assets VS and VO - Case 5 (bull market) Case 6 - Portfolio consisting of assets VS, HS and VO

Bull market - positive drift

This investor can invest in three assets: Volvo B stock, H&M stock and Volvo B option with the rolling option strategy described in Case 5.

In Table 5.6 the monthly (20 days) log returns and standard deviation based on the sample of 64 days are presented.

Table 5.6: Historical parameters - Case 6 (bull market) µ (Monthly) σ (Monthly)

VS 0.0532 0.0990

HS 0.0025 0.0881

VO 0.0919 0.3355

Below follows the associated positive semi definite covariance matrix Σ of the log returns.

Σ =





0.0005 0.0003 0.0009 0.0003 0.0004 0.0005 0.0009 0.0050 0.0056





Market volatility is used when pricing the options (as in Case 5, Figure 5.4). In Figure 5.6 the historical asset prices over the time considered can be seen.

26

(47)

Figure 5.6: Historical prices of assets VS, HS and VO - Case 6 (bull market)

(48)

(49)

Chapter 6

Results and analysis

In the following chapter the results from the optimizations are presented. Weights and ratios are presented in Table 6.1 - 6.18. The ratios marked inblueare the best performing ones, according to Section 2.2. All portfolio value plots have been normalized to a start value of 100 SEK in order to easily benchmark the performance of the portfolios against each other.

6.1 Non-complex portfolios

6.1.1 Case 1 - Portfolio consisting of assets GB, AH, GO and US Stagnating market

In sample optimization

In Table 6.1 the results of the optimization are given, where w_GB is the weight percentage invested in the governmental bond, w_AH is the weight percentage invested in the high yield bond, w_GO is the weight percentage invested in the commodity and w_{U S} is the weight percentage invested in the foreign exchange. The ratios are defined on an annual basis as stated in Section 2.2.

Table 6.1: Results non-complex – Case 1

MV CVaR Utility (τ = 2) Utility (τ = 0.1) Barra EWP ERC

wGB 0.6514 0.6555 0.0121 0.5524 0.5247 0.2500 0.4455

wAH 0.2241 0.2328 0.0113 0.1947 0.2006 0.2500 0.2498

wGO 0.0622 0.0573 0.0606 0.0821 0.1573 0.2500 0.1236

wU S 0.0624 0.0544 0.9160 0.1708 0.1174 0.2500 0.1811

Risk 0.0823 0.0788 0.1524 0.0986 0.1235 0.1465 0.1174

Return 0.0151 0.0150 0.0186 0.0157 0.0164 0.0172 0.0162

Sharpe ratio 0.1835 0.1901 0.1222 0.1592 0.1331 0.1172 0.1383

Drawdown 0.2520 0.2426 0.4021 0.2932 0.3524 0.4018 0.3386

Calmar ratio 0.0599 0.0618 0.0462 0.0535 0.0465 0.0428 0.0478

(50)

CHAPTER 6. RESULTS AND ANALYSIS

Figure 6.1 represent the performance of the back tracked optimal portfolios from Table 6.1. Note how the utility based optimization dramatically changes with varying τ . The portfolio value for all portfolios increases drastically in the beginning of period and then it decreases drastically. The period ends with a small increase in the portfolio value for all portfolios hence the returns are positive in Table 6.1.

Figure 6.1: Optimal portfolios – Case 1

In sample optimization using bootstrapped data samples

In Table 6.2 the mean value of the bootstrapped optimization results are presented. By using the bootstrap technique more robust results are obtained since the optimization is done 100 times using 100 different samples.

Table 6.2: Results non-complex with bootstrapping – Case 1

MV CVaR Utility (τ = 2) Utility (τ = 0.1) Barra EWP ERC

wGB 0.6357 0.6363 0.0121 0.5524 0.5247 0.2500 0.4431

wAH 0.2035 0.2033 0.0113 0.1947 0.2006 0.2500 0.2504

wGO 0.0526 0.0512 0.0606 0.0821 0.1573 0.2500 0.1245

wU S 0.1082 0.1091 0.9160 0.1708 0.1174 0.2500 0.1820

Risk 0.0771 0.0763 0.1515 0.0980 0.1223 0.1451 0.1165

Return 0.0202 0.0209 0.0216 0.0205 0.0207 0.0205 0.0202

Sharpe ratio 0.2296 0.2384 0.1373 0.1830 0.1508 0.1328 0.1573

Drawdown 0.1734 0.1703 0.3346 0.2219 0.2756 0.3227 0.2632

Calmar ratio 0.1706 0.1751 0.1451 0.1525 0.1454 0.1435 0.1472

29

A performance investigation and evaluation of selected portfolio optimization methods with varying assets and market scenarios

A performance investigation and evaluation of selected portfolio optimization methods with varying assets and market scenarios

GUSTAF CALLERT

FILIP HALÉN DAHLSTRÖM

A performance investigation and evaluation of selected portfolio optimization methods with varying assets and market scenarios

G U S T A F C A L L E R T F I L I P H A L É N D A H L S T R Ö M

Acknowledgements

Contents

List of Figures

List of Tables

Chapter 1

Introduction

1.1 Background

1.2 Purpose

1.3 Problem formulation

1.4 The indented reader

1.5 Method

Chapter 2

Evaluation of optimization methods

2.1 Performance evaluation by asset allocation models

2.2 Performance evaluation by ratios

Chapter 3

Optimization methods and simulations

3.1 Risk measures

3.2 Portfolio optimization

3.3 Simulations and data generation

Chapter 4

Asset classes, portfolios and market scenarios

4.1 Asset classes and portfolios

4.2 Market scenarios

Chapter 5

Empirical study

5.1 Assets and time series

Chapter 6

Results and analysis

6.1 Non-complex portfolios