Robust Portfolio Optimization with Expected

Shortfall

Daniel Isaksson

May 30, 2016

ii
 Abstract
This thesis project studies robust portfolio optimization with Expected Short-
fall applied to a reference portfolio consisting of Swedish linear assets with
stocks and a bond index. Specifically, the classical robust optimization def-
inition, focusing on uncertainties in parameters, is extended to also include
uncertainties in log-return distribution. My contribution to the robust opti-
mization community is to study portfolio optimization with Expected Short-
fall with log-returns modeled by either elliptical distributions or by a normal
copula with asymmetric marginal distributions. The robust optimization
problem is solved with worst-case parameters from box and ellipsoidal un-
certainty sets constructed from historical data and may be used when an
investor has a more conservative view on the market than history suggests.
With elliptically distributed log-returns, the optimization problem is
equivalent to Markowitz mean-variance optimization, connected through the
risk aversion coefficient. The results show that the optimal holding vector
is almost independent of elliptical distribution used to model log-returns,
while Expected Shortfall is strongly dependent on elliptical distribution with
higher Expected Shortfall as a result of fatter distribution tails.
To model the tails of the log-returns asymmetrically, generalized Pareto
distributions are used together with a normal copula to capture multivari-
ate dependence. In this case, the optimization problem is not equivalent
to Markowitz mean-variance optimization and the advantages of using Ex-
pected Shortfall as risk measure are utilized. With the asymmetric log-return
model there is a noticeable difference in optimal holding vector compared
to the elliptical distributed model. Furthermore the Expected Shortfall in-
creases, which follows from better modeled distribution tails.
The general conclusions in this thesis project is that portfolio optimiza-
tion with Expected Shortfall is an important problem being advantageous
over Markowitz mean-variance optimization problem when log-returns are
modeled with asymmetric distributions. The major drawback of portfolio
optimization with Expected Shortfall is that it is a simulation based opti-
mization problem introducing statistical uncertainty, and if the log-returns
are drawn from a copula the simulation process involves more steps which
potentially can make the program slower than drawing from an elliptical
distribution. Thus, portfolio optimization with Expected Shortfall is appro-
priate to employ when trades are made on daily basis.

Keywords: Robust Portfolio Optimization, Risk Management, Expected

Shortfall, Elliptical Distributions, GARCH model, Normal Copula, Hybrid
Generalized Pareto-Empirical-Generalized Pareto Marginals, Markowitz Mean-
Variance Optimization, Contribution Expected Shortfall

iii
Acknowledgements
I would like to thank my colleagues at SAS Institute for all supporting and
encouraging conversations. In particular I would like to thank my supervisor
Jimmy Skoglund for introducing me to the topic of robust portfolio optimiza-
tion, for always being a helping hand and a great source of inspiration. I
would also like to thank my supervisor Professor Henrik Hult at the Royal
Institute of Technology for valuable support throughout this thesis project.

Stockholm, May 30, 2016

Daniel Isaksson

iv
Contents

1 Introduction 1
1.1 Introduction to Portfolio Optimization . . . . . . . . . . . . . 2
1.1.1 Markowitz Mean-Variance Optimization Problem . . . 3
1.1.2 Reference Portfolio and Benchmark Solution . . . . . . 4
1.1.3 Stylized Assumptions in Markowitz Optimization Prob-
lem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Portfolio Optimization with Expected Shortfall 9

2.1 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Expected Shortfall . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Contribution Expected Shortfall . . . . . . . . . . . . 12
2.1.4 Properties of the quantile function . . . . . . . . . . . 14
2.2 Portfolio Optimization Problem Formulation . . . . . . . . . . 14
2.2.1 Remark: Elliptically distributed log-returns . . . . . . 16
2.2.2 General Problem Assumptions . . . . . . . . . . . . . 17
2.3 Rockafellar and Uryasev simplification . . . . . . . . . . . . . 18
2.4 Application on Reference Portfolio . . . . . . . . . . . . . . . 21

3 Portfolio Optimization Under Uncertainty 25

3.1 Model Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.1 Introduction to Robust Optimization . . . . . . . . . . 27
3.1.2 Robust Optimization with Elliptical Distributions . . . 28
3.1.2.1 Solution with Normal Distributed Log-returns 32
3.1.2.2 Solution with Student’s t Distributed Log-
returns . . . . . . . . . . . . . . . . . . . . . 33
3.1.3 Robust Optimization with Asymmetric Distributions . 35
3.1.3.1 The univariate generalized Pareto distribution 35
3.1.3.2 Copula dependence . . . . . . . . . . . . . . 37
3.1.3.3 Sample Preparation with GARCH(1,1) Model 38
3.1.3.4 Solution with Normal Copula and Hybrid GPD-
Empirical-GPD Log-return Marginals . . . . 40
3.2 Statistical Uncertainty . . . . . . . . . . . . . . . . . . . . . . 42

v
 3.2.1 Statistical Uncertainty in the Portfolio Optimization
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4 Analysis and Conclusions 45

4.1 Analysis of Results . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.1 Elliptically distributed log-returns . . . . . . . . . . . 46
4.1.2 Asymmetric Distributed Log-returns . . . . . . . . . . 50
4.1.3 Analysis of Solutions with Euler Risk Decomposition . 53
4.2 Comments on Worst-Case Scenario Based Robust Optimization 57
4.3 More Comments and Further Investigation . . . . . . . . . . . 57

Appendix A Optimization Parameters 61

Appendix B Convergence as Function of Sample Size D 63

Appendix C Reference Solutions 65

C.1 Normal Distributed Log-returns . . . . . . . . . . . . . . . . . 65
C.2 Student’s t Distributed Log-returns . . . . . . . . . . . . . . . 66

Appendix D GARCH and GPD Parameters 69

vi
Chapter 1

Introduction

The goal of this thesis project is to introduce and solve an alternative ro-
bust portfolio optimization problem to Markowitz classical mean-variance
optimization problem. Classical robust optimization focus on parameter un-
certainty for a given distribution and the contribution of this thesis project is
to extend the robust optimization to include uncertainties in log-return dis-
tribution as well. The alternative portfolio optimization problem is solved
with both elliptical distributions and asymmetric log-return distributions,
applied to a reference portfolio consisting of stocks and a bond index from
the Swedish market.
The thesis project is organized as follows. In Chapter 1, I introduce gen-
eral concepts of portfolio optimization and give a brief historical background
on the development of modern portfolio optimization. A reference portfo-
lio is then constructed and the traditional portfolio optimization problem by
Markowitz [12] is solved to obtain a benchmark solution from historical data.
The chapter concludes with arguments on why Markowitz mean-variance op-
timization problem is not particularly good in financial mathematics due to
its narrow area of practical application.
In Chapter 2 I introduce risk measures and based on a risk measure com-
monly used in financial risk management called Expected Shortfall I formu-
late a problem better fit for modern portfolio optimization. With theorems
provided by Rockafellar and Uryasev [15], the portfolio optimization problem
is approximated as a convex linear program that can be solved with stan-
dard optimization algorithms. I then show that Markowitz mean-variance
optimization problem is a special case of the portfolio optimization problem
with Expected Shortfall, connected with a risk aversion coefficient.
Chapter 3 deals with optimization under uncertainties and the concept
of robust optimization is presented, being central when solving optimization
problems with uncertainty in parameters. My contribution to the literature
of robust portfolio optimization is that I first perform a case study under
different elliptical distributions and then study robust portfolio optimization

1
with an asymmetric hybrid generalized Pareto-Empirical-Generalized Pareto
distribution. To analyze the statistical uncertainty in the results, the boot-
strap procedure is applied to calculate standard errors for the holdings and
the Expected Shortfall.
In Chapter 4 I analyze the results obtained throughout the thesis project
and compare them to the benchmark Markowitz solution. I analyze the prop-
erties of the optimization problem and conclude in which areas the optimiza-
tion problem is particularly applicable, being advantageous over Markowitz
mean-variance optimization problem. The thesis ends with remarks on areas
where further investigation can be done.

1.1 Introduction to Portfolio Optimization

Suppose an investor has initial capital V0 at time t = 0 that should be
invested in a market consisting of risky assets. The aim of the investments
is often to, at some future time T = 1, have gained at least as much money
as the investor would have gained if the capital was invested in a risk-free
asset with certain interest rate R0 instead. Since the assets on the market
have unknown future values, the investor cannot be certain of the future
portfolio value V1 and hence the investor is exposed to risk. To limit this
risk, the investor might pose limitations on how much the future portfolio
value is allowed to vary. The more risk averse the investor is, the smaller
the expected value of the future portfolio typically is and hence there is a
trade-off between expected future portfolio value and risk exposure.
Traditionally the log-returns
 
St
Rt = log , (1.1)
St−1

of an asset, St being the asset price at time t, are assumed to have the
Markov property from day to day and are often assumed to be weakly de-
pendent and close to independent and identically distributed. Therefore, by
letting V1 be a function of assets’ log-returns, V1 = f (R1 ) for some multivari-
ate function f , one may construct approximately independent copies of V1
as {f (R−n+1 ), . . . , f (R0 )}. Hence, investors try to predict future portfolio
value by analyzing historical log-returns. An alternative approach which is
equally good is to work directly with the log-returns and maximize the ex-
pected future portfolio log-return since this is equivalent of maximizing the
expected future portfolio value. This approach is used in this thesis project.
Furthermore, since the amount of initial capital V0 is only a scaling factor,
it is common practice to set V0 = 1 so that the solution is expressed as
proportions of the total capital invested rather than expressed as monetary
units.

2
1.1.1 Markowitz Mean-Variance Optimization Problem
Modern portfolio optimization was first introduced by Markowitz [12] in
1952 with what is often referred to as Markowitz mean-variance optimiza-
tion problem. The idea is to maximize the expected return, subject to the
constraint that the variance of the portfolio must be smaller than some pre-
determined tolerance level T . Assume an investor has the possibility to invest
in n assets and let R = (R1 , . . . , Rn ) be a random vector of log-returns at
time t = 1 corresponding to each of the assets as defined by (1.1). The
covariance matrix of returns for the n assets is further given by Σ which
is assumed to be symmetric and positive semi-definite and thus also invert-
ible. Let w = (w1 , . . . , wn ) be a vector of holdings, or weights, of the initial
capital invested in each asset. If the future portfolio log-return is denoted
by X = wT R, then Markowitz mean-variance optimization problem can be
stated

max E[X]
w
Subject to wT Σw ≤ T
Xn
wi = V0 .
i=1

Depending on the value of T the optimal solution will differ and plotting the
expected future portfolio log-return versus its standard deviation for different
values of T gives what is known as the efficient frontier.
Markowitz mean-variance optimization problem can be stated in various
ways and an alternative formulation is the mean-variance trade-off formula-
tion
c T
max wT µ − w Σw
w 2V0
n
X (1.2)
Subject to wi = V0
i=1

where µ is the mean log-return vector and the trade-off parameter c > 0
is a dimensionless constant that should be interpreted as a risk aversion
coefficient. The trade-off problem is convex and has analytical solution

max{1T Σ−1 µ − c, 0}
 
V0 −1
w= Σ µ− 1 .
c 1T Σ−1 1

Additional constraints can easily be added to Markowitz mean-variance op-

timization problem where lower and upper bounds on each weight wi are
often used. A particularly common constraint is to set the lower bounds of
each weight to zero, implying that only long positions in the n assets are

3
allowed. The optimization problem then becomes
c T
max wT µ − w Σw
w 2V0
n
X
Subject to wi = V0 (1.3)
i=1
wi ≥ 0, i = 1, . . . , n.

Since the introduction of Markowitz mean-variance optimization prob-

lem, portfolio optimization has been developed with more complex problem
formulations to improve the optimization further. This thesis project stud-
ies portfolio optimization of one of these alternative optimization problems,
which do not rely on some assumptions that must hold in Markowitz mean-
variance optimization problem. These assumptions are presented in Section
1.1.3.
Before presenting the alternative optimization problem a solution to (1.3)
is calculated by applying the problem to a reference portfolio that will be
used throughout the thesis project. The solution will work as a benchmark
solution to compare future solution with to analyze the sensitivity of portfolio
optimization.

1.1.2 Reference Portfolio and Benchmark Solution

To be able to solve Markowitz mean-variance optimization problem (1.3),
historical log-return data from assets in a reference portfolio is needed so
that estimates of µ and Σ can be calculated. This introduces two questions:

• Which assets are of interest in the reference portfolio?

• From which historical time period should the data be collected?

The answer depends on the investor. In this thesis project the reference
portfolio consists of linear assets being stocks and a bond index from the
Swedish market. The assets included in the reference portfolio are listed in
Table 1.1 and are chosen to reflect some of the largest companies on the
Swedish market, diversified over a large range of business areas. The idea
is to include high and low volatile assets in different business sectors that
are more and less dependent on the current state of economy. An investor
should then have good control of deciding what risk level he is willing to be
exposed to, depending on how the portfolio weights are chosen. The OMRX
Total Bond Index, henceforth abbreviated the Total Bond Index, is included
in the reference portfolio to have a position which can be considered close to
riskless with small expected log-return.
The historical time period used for collecting asset price data is chosen
to be January 2, 2007 until January 22, 2016. The time period includes the

4
global 2007 financial crisis, having its peak between approximately 2007 −
2009 according to the time line in [8], where risky assets typically are more
correlated, and some years afterwards where the market is starting to rise
again and the assets are less correlated.

Table 1.1: Swedish assets included in the reference portfolio and their corre-
sponding business areas.
Asset name Business Area
AstraZeneca Health Care
Ericsson A Technology
Hennes & Mauritz B Retail
ICA Gruppen Retail
Nordea Bank Banks
SAS Travel & Leisure
SSAB A Basic Resources
Swedish Match Personal & Household Goods
TeliaSonera Telecommunications
Volvo Industrial Goods & Services
Total Bond Index Government bonds

The historical data is obtained from the Nasdaq OMX web page and
consists of daily prices for each of the assets in the reference portfolio1 .
Figure 1.1 depicts the historical price developments for the stocks to the left
and the Total Bond Index to the right.

700 6500
AstraZ OMRX Total Bond Index
Ericsson
H&M
600 ICA
Nordea
SAS 6000
SSAB
Swedish Match
500 Telia
Volvo

5500
400

300
5000

200

4500
100

0 4000
2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Figure 1.1: Price developent for the stocks (left) and the bond index (right)
in the reference portfolio during January 2, 2007-January 22, 2016.

From Figure 1.1 one can make several observations. For instance that
1
7 data points for OMRX Total Bond Index was missing in the historical data set and
were linearly interpolated. The data is automatically adjusted for asset splits.

5
some companies managed the financial crisis better than others, for instance
Nordea Bank, and that some assets have historically had a positive trend,
such as Swedish Match, while other assets have had a negative trend, such as
SAS. There are furthermore some assets that have changed very little in value
during the time period of interest, for instance TeliaSonera, while other assets
have been more volatile, for instance ICA Gruppen. All these behaviors were
desired in the construction of the reference portfolio. Ultimately, the assets
included in a portfolio is up to the investor to decide and the reference
portfolio in this thesis project could have been selected differently. The
general conclusions in the thesis project do however not directly depend on
the reference portfolio itself other than the fact that the weight vector and
Expected Shortfall would change if other assets were used.
The optimization problem (1.3) is a quadratic programming problem and
can be solved with standard solving algorithms. The solution is presented in
Table 1.2 when applied to the reference portfolio with risk aversion coefficient
c = 5.33 (turns out later why this particular value was chosen), V0 = 1 and
empirically estimated parameters µ b and Σ.
b The solution was calculated using
the function quadprog in Optimization Toolbox in Matlab version 9.0 on a
Core i5 CPU 2.60 GHz Laptop with 8 GB of RAM using the interior point
method. See Appendix A for numerical values of the empirically estimated
parameters µ b and Σ.
b

Table 1.2: Benchmark solution to Markowitz mean-variance optimization

problem (1.3) applied to the reference portfolio in Table 1.1 with empirically
estimated parameters µb and Σ.b
Asset name Weight
AstraZeneca 0.0001
Ericsson A 0
Hennes & Mauritz B 0.0080
ICA Gruppen 0.0666
Nordea Bank 0
SAS 0
SSAB A 0
Swedish Match 0.1436
TeliaSonera 0
Volvo 0
Total Bond Index 0.7816

As can be seen from Table 1.2, most assets in the portfolio will not be
invested in and about 78% of the initial capital is invested in the Total
Bond Index, about 14% is invested in Swedish Match, approximately 7%
is invested in ICA Gruppen and the remaining 1% is invested in Hennes &
Mauritz B. With the benchmark solution wBM , the expected daily future

6
portfolio log-return is
T
E[wBM T
R] = wBM b = 2.0242 · 10−4
µ (1.4)
T µ
corresponding to a yearly2 expected portfolio log-return of 251 · wBM b ≈
5.08% and the daily portfolio variance is

σp2 = wBM
T
Σw T
b BM = 7.8354 · 10−6 , (1.5)

meaning that the yearly portfolio variance is approximately 0.01%.

1.1.3 Stylized Assumptions in Markowitz Optimization Prob-

lem
Markowitz mean-variance optimization problem (1.3) is theoretically impor-
tant, but when applied to real financial portfolios it suffers from some stylized
assumptions that must hold for the problem formulation to be applicable.
The first stylized assumption is that the log-returns have elliptical distribu-
tion, which is not accurate enough when studying historical data, especially
during times of financial crisis where asset prices become more volatile and
more correlated. Furthermore, elliptical distributions are symmetric which
often is not the case for historical log-returns where either the loss or the
gain tail of the distribution is more prominent than the other tail. Therefore,
parts of the empirical log-return distribution information is lost when mod-
eling with elliptical distributions. The second stylized assumption is that the
portfolio is linear, which is a major limitation. Often highly non-linear assets
such as options or other financial derivatives are included in financial port-
folios, that cannot be taken into consideration by Markowitz mean-variance
optimization problem. With these drawbacks taken into account, alterna-
tive portfolio optimization problems have recently become popular to study.
This thesis project focus on the first drawback discussed but the problem
can easily be extended to include non-linear assets in the reference portfolio
as well.

2
There are about 251 trading days in a year on the Swedish market.

7
8
Chapter 2

Portfolio Optimization with

Expected Shortfall

With the introduction to portfolio optimization in Chapter 1 I have so far

concluded that Markowitz mean-variance optimization problem (1.3) is not
sufficient to employ in portfolio optimization unless the log-returns are el-
liptically distributed, but due to its historical importance it may be used
as a benchmark problem. In this chapter, I introduce the concept of risk
measures and construct an alternative optimization problem based on a risk
measure better fit for modern portfolio optimization. The aim is to construct
an appropriate portfolio optimization problem that does not rely on the styl-
ized assumptions that must hold for Markowitz mean-variance optimization
problem to be considered good.

2.1 Risk Measures

Financial risk can be measured in several different ways, but it is often de-
sirable to measure risk in monetary units, meaning that the risk is expressed
as buffer capital that needs to be added to a portfolio to protect it from un-
desired outcomes. In Markowitz mean-variance optimization problem, the
risk is measured as the variance of the future portfolio return. However,
variance is not considered to be a very good risk measure in finance since
it is defined as the expected squared deviation from the mean value and
thus does not make difference between positive deviations, portfolio gain,
and negative deviations, portfolio loss. Furthermore, standard deviation can
only be considered accurate enough to be translated into monetary risk if the
future value of the portfolio value is approximately normal distributed. This
is often a too strict assumption that simplifies the real portfolio log-return
distribution too much. Instead, it is often desirable to utilize a risk measure
that make difference between good and bad deviations from the expected fu-
ture portfolio value. In this section I first present basic risk measure theory

9
and later specify two risk measures widely used in financial risk management.
Generally, let ρ(X) be a function measuring the risk of a stochastic vari-
able X. Different risk measures have different properties and below is a list
of such mathematical properties that are considered to be useful or desirable,
together with brief explanations on how they can be interpreted.

1. Translation invariance. ρ(cR0 + X) = −c + ρ(X) for c ∈ R. This

means that adding the amount c with risk-free interest rate R0 to a
portfolio reduces the risk equally much.

2. Monotonicity. If X2 < X1 , then ρ(X1 ) ≤ ρ(X2 ). This means that if

we know for certain that a portfolio X1 is greater than a portfolio X2
at a future time, then the first portfolio is considered to be less risky.

3. Convexity. ρ(λX1 + (1 − λ)X2 ) ≤ λρ(X1 ) + (1 − λ)ρ(X2 ), for any real

λ ∈ [0, 1]. The risk measure rewards diversification, which means that
it takes into account that it often is wise to spread out the investment
in several risky positions, rather than investing everything in one.

4. Normalization. ρ(0) = 0. This means that it is acceptable to not

invest in risky assets at all, so that an empty portfolio is riskless.

5. Positive homogeneity. ρ(λX) = λρ(X) for λ ≥ 0. This means that

for instance investing twice as much in one position is twice as risky.

6. Subadditivity. ρ(X1 + X2 ) ≤ ρ(X1 ) + ρ(X2 ). This property should

also be interpreted as that the risk measure rewards diversification. A
company consisting of two business units is interpreted as less risky
compared to the two units considered as separate companies.

The reader is refered to the book [5, Ch. 6] by Hult, Lindskog, Hammarlid
and Rehn for a more thorough presentation on general risk measure theory
and more comments on the above properties, as well as more on why variance
is considered a bad risk measure in finance.
A risk measure with the properties translation invariance and monotonic-
ity is said to be a monetary measure of risk, and a risk measure considered to
replace variance in Markowitz mean-variance optimization problem should
satisfy at least these two properties. A risk measure that in addition to
translation invariance and monotonicity also satisfies convexity is said to be
a convex risk measure. The convex risk measure family is thus a subset of
the monetary risk measure family. Finally, a third risk measure family is the
coherent risk measure, where ρ(X) satisfies the properties translation invari-
ance, monotonicity, positive homogeneity and subadditivity. It is easy to see
that a risk measure satisfying positive homogeneity also satisfies normaliza-
tion by setting λ = 0. It can further be shown that positive homogeneity
and convexity together implies subadditivity but not the reverse. Hence, a

10
coherent risk measure is also a convex risk measure but the opposite does not
generally hold, so the coherent risk measure family is a subset of the convex
risk measure family and thus also a subset of the monetary risk measure fam-
ily. When choosing an appropriate risk measure for a portfolio optimization
problem replacing Markowitz mean-variance optimization problem one may
therefore consider both convex and coherent risk measures to be at least as
good as monetary risk measures. Below, two risk measures that are com-
monly used in risk management are presented that are considered good in
financial mathematics.

2.1.1 Value-at-Risk
The first risk measure presented is the Value-at-Risk, abbreviated VaR.
Value-at-Risk satisfies translation invariance, monotonicity and positive ho-
mogeneity and is hence a monetary risk measure. The Value-at-Risk at level
p ∈ (0, 1) of a stochastic variable X is defined as

V aRp (X) = min{m ∈ R : P (mR0 + X < 0) ≤ p}.

If X is assumed to have right continuous and increasing cumulative distri-

bution function F (x), which will be the case in this thesis project, it follows
that
V aRp (X) = min{m ∈ R : P (−X > mRo ) ≤ p}
= min{m ∈ R : P (L ≤ m) ≥ 1 − p} = FL−1 (1 − p)

where L = −X/R0 should be interpreted as the discounted loss. Hence, the

Value-at-Risk of the stochastic variable X is the (1 − p)-level quantile of the
associated discounted loss L.
A direct advantage with Value-at-Risk over traditional variance is that it
can be used when variance is not a relevant risk measure, for instance when
the expected value of a distribution does not represent a fair picture of the
distributional appearance. Furthermore, since Value-at-Risk computes the
(1−p)-level quantile of the discounted loss it accounts only for large losses but
not large gains and hence Value-at-Risk makes difference between negative
and positive deviations from the expected future portfolio value. However,
since Value-at-Risk only look at a particular level (1 − p) of the loss quantile,
traders can take advantage of this to "hide" risky investments by making
the losses more extreme so that the potential loss ends up further out in the
loss distribution tail so that it is not discovered by Value-at-Risk. With this
strategy, very risky portfolios could seem acceptable that otherwise would
not have been if the risk was visible for risk managers. This might result in
that companies are exposed to extremely large loss scenarios, although with
relatively small probability, but with the possible outcome that the company
suffers a huge loss and possibly bankruptcy.

11
2.1.2 Expected Shortfall
A better risk measure than Value-at-Risk in the sense that it takes into
account all losses located in the whole (1 − p)-level quantile tail of the loss
distribution is Expected Shortfall. For a stochastic variable X, the Expected
Shortfall, abbreviated ES, at level p ∈ (0, 1) is defined as
Z p Z 1
1 1
ESp (X) = V aRu (X)du = V aRu (L)du, (2.1)
p 0 1−q q

where q = 1 − p. Expected Shortfall appears under several different names

in literature and with minor changes it is goes under the names Condi-
tional Value-at-Risk (CVaR), Average Value-at-Risk (AVaR), Tail Value-at-
Risk (TVaR), Expected Tail Loss (ETL) and Tail Conditional Expectation
(TCE). Similarly to Value-at-Risk, if X has right continuous and increasing
distribution function F (x) then the Expected Shortfall has representation

1 p −1
Z
ESp (X) = F (1 − u)du.
p 0 L

Since Expected Shortfall is defined through Value-at-Risk, it inherits the

properties of Value-at-Risk, being translation invariance, monotonicity and
positive homogeneity. It can further be shown that Expected Shortfall also
satisfies subadditivity and is hence a coherent risk measure.

2.1.3 Contribution Expected Shortfall

In addition to the total portfolio risk, the investor might be interested in
how much each asset in the reference portfolio contributes to the total risk
when allocating the capital. The contributions go under the names Marginal
Risk, Contribution Risk and Euler Allocations/Decomposition and originate
from Euler’s decomposition theorem. The theorem states that a function f
of n variables x1 , . . . , xn with continuous partial derivatives is homogenous
of degree k if and only if the equation
n
X ∂f
xi = kf
∂xi
i=1

holds for all x1 , . . . , xn ∈ D being an open domain. There exists several

different approaches for decomposing the total portfolio risk but a natural
decomposition given a homogenous risk measure would based on Euler’s
decomposition theorem be
n
X ∂ρ(X)
ρ(X) = wi . (2.2)
∂wi
i=1

12
This is natural since exactly the total risk is decomposed, meaning that
the sum of all contributions is exactly the total risk. Expected Shortfall is
(positive) homogenous of degree 1 and a decomposition of the total portfolio
Expected Shortfall is
n
X ∂ESp (X)
ESp (X) = Ui
∂Ui
i=1

where U = (U1 , . . . , Un )T is a vector of sensitivities defined by

n
X ∂Fj (S(t))
Ui = wj ,
∂Si (t)
j=1

wj being the asset weight and Fj (S(t)) is a financial instrument. The deriva-
tive ∂ESp (X)/∂Ui should then be interpreted as the Euler Allocation for
asset i.
Skoglund and Chen show in [18] that if the log-returns are assumed to
have multivariate normal distribution, the Euler Allocations can be calcu-
lated analytically as
∂ESp (X) ∂σp
= λ(Φ−1 (q)) . (2.3)
∂U ∂U
Here λ(x) is the hazard function for the normal distribution, defined as
φ(x)
λ(x) = . (2.4)
1 − Φ(x)
and
∂σp
= (UT ΣU)−1/2 ΣU.
∂U
If the log-returns are modeled by their empirical distribution, the deriva-
tive ∂ESp (X)/∂U does not exist since the portfolio distribution is discrete.
In this case, we may use the results of Tasche [21] and approximate the
derivative as PD
E[Li,. |Ld > Ls ]
DESp (X) (wi ) = d=1
PD (2.5)
d=1 1(Ld >Ls )

where {Li,d }n,D

i=1,d=1 are the loss components, n being the number of assets
and D the amount of available loss scenarios, Li,. = (Li,1 , . . . , Li,D )0 , Ls is
the loss sample point corresponding to the p-level Value-at-Risk and
(
1, Ld > Ls
1(Ld >Ls ) =
0, otherwise.

In this thesis project the investment horizon is one day and the risk-free
interest rate can be approximated to zero, which implies that Li = −Ri ,

13
i.e. the log-returns with switched sign. With linear assets, the portfolio loss
scenarios are then obtained by multiplying the weight vector with the loss
vector, L = wT L = −wT R. Furthermore, by ordering the portfolio losses
such that L1 > L2 > · · · > LD then the portfolio Value-at-Risk at level p,
defined as the p-level loss quantile, is

V aRp (X) = Ls = L[Dp]+1 (2.6)

where [Dp] is the integer part of Dp.

2.1.4 Properties of the quantile function

Since the quantile function turns out to play a central role in both Value-
at-Risk and Expected Shortfall, I here state two properties of the quantile
function that will be used later.

Proposition 2.1 If g : R → R is a nondecreasing function and left contin-

−1
uous, then for any random variable Z it holds that Fg(Z) (p) = g(FZ−1 (p)) for
all p ∈ (0, 1).

Proposition 2.2 For any random variable X with continuous and strictly
−1
increasing density function FX , F−X (p) = −FX−1 (1 − p) for all p ∈ (0, 1).

I refer the reader to Hult et al.[5, pp. 170-172] for complete proofs.

2.2 Portfolio Optimization Problem Formulation

In optimization, we are generally interested in finding the global extreme
point to a function under some constraining requirements. Convex optimiza-
tion problems is a particularly nice family of optimization problems where
there exists a unique optimal solution being the global extreme point. Since
Expected Shortfall is a coherent risk measure, it is also convex and hence
an appealing risk measure to employ in financial optimization problems, op-
posite to Value-at-Risk which does not satisfy the convexity property and
may therefore produce several local extreme points. With this in mind, it
becomes natural to use Expected Shortfall rather than Value-at-Risk as risk
measure in a portfolio optimization problem. Additionally, since Expected
Shortfall does not demand elliptically distributed log-returns or linear port-
folios, it does not rely on the stylized assumptions that must be assumed in
Markowitz mean-variance optimization problem. In this section I formulate
the portfolio optimization problem based on minimizing Expected Shortfall
that will be the subject of study in the remaining part of this thesis project.
Simply minimizing the Expected Shortfall in a portfolio optimization
problem is trivial without constraining equations and in a market consisting
of only risky assets the solution would be to not invest at all. Therefore,

14
a reasonable initial constraint would be that the entire initial capital V0
must be invested in the market. Additionally, it is common to constrain the
asset allocations to long positions, i.e. wi ≥ 0, i = 1, . . . , n. The portfolio
optimization problem can then be stated mathematically as

min ESp (wT R)

w
n
X
Subject to wi = V 0
i=1
wi ≥ 0, i = 1, . . . , n,

where n is the number of assets available in the portfolio.

Often, investors are not interested in only minimizing the risk, but they
want some reward for exposing their capital to risk. Therefore, investors add
the constraint that the future expected portfolio (log)-return should be larger
some threshold (log-)return θ. The optimization problem then becomes

min ESp (wT R)

w
Subject to wT µ ≥ θ
Xn (2.7)
wi = V 0
i=1
wi ≥ 0, i = 1, . . . , n.

The above problem formulation will be the foundation of this thesis project.
Next I remark on a special case when (2.7) have the same solution as the
benchmark solution to Markowitz mean-variance optimization problem and
then I state a few general model assumptions that is used when later solving
(2.7) numerically. In Section 2.3 I show that the portfolio optimization
problem can be approximated and solved by ordinary linear programming.
In section 2.1.3 it was said that a natural decomposition of the total
portfolio risk is obtained by calculating the Euler Allocations since then
the risk contributions sum up to exactly the total risk. But this property
can be achieved for any risk decomposition method by simply normalizing.
However, Tasche proves in [20] that Euler decomposition is the only risk
decomposition method that is consistent with (local) portfolio optimization
which motivates the benefits of using Euler decomposition in this thesis
project.
In portfolio optimization the Euler Allocations have a second field of
application in addition to measuring the risk contribution from a specific
asset to the total portfolio risk. The famous Sharpe ratio, introduced by
Sharpe [17], relates the expected portfolio (log)-return to the risk as

wT µ
S= (2.8)
ρ

15
and the larger Sharpe ratio the better is the investment. Similarly to the
decomposition of the portfolio risk a natural decomposition of the Sharpe
ratio is to define
wi µ i
Si∗ = ∂ρ
(2.9)
wi ∂w i

as the marginal Sharpe ratio of asset i, i = 1, . . . , n. An asset should then

be considered a good investment if the marginal Sharpe ratio is large. With
the marginal Sharpe ratio as argument for investing in an asset or not, it can
be seen that the only information an investor requires for (local) portfolio
optimization is the expected log-return and Euler Allocation. This observa-
tion will be used later when attempting to analyze the optimal solutions to
the portfolio optimization problem with Expected Shortfall.

2.2.1 Remark: Elliptically distributed log-returns

The following remark points out a special case where portfolio optimization
problem (2.7) can be simplified to a modified version of Markowitz mean-
variance problem (1.3). This is important since the solution to (2.7) should
in this case be the same as the benchmark solution in Table 1.2.
Consider a scenario where the vector of log-returns R is assumed to be
elliptically distributed with mean µ and covariance matrix Σ. Then R can
be represented as
d
R = µ + W AZ

where A is a matrix such that AAT = Σ and Z ∈ N (0, Id ), Id being the

d-dimensional identity matrix, is independent of W ≥ 0. The portfolio log-
return is then
√
wT R = wT µ + wT ΣwW Z1 (2.10)

where I have used a standard property for spherically distributed random

variables and Z1 is univariate standard normal distributed. Minimizing the
Expected Shortfall then yields
Z p
T 1 −1
min ESp (w R) = min − Fw T R (u)du
w w p 0
√ 1
Z p
T −1
= min −w µ − wT Σw FW Z1 (u)du
w p 0

where I have used both Proposition 2.1 and Proposition 2.2. Now, since
minimizing −f (w) corresponds to maximizing f (w), portfolio optimization

16
 problem (2.7) with elliptically distributed log-returns can be written
c √ T
max wT µ − w Σw
w 2V0
Subject to wT µ ≥ θ
Xn
wi = V 0
i=1
wi ≥ 0, i = 1, . . . , n,

where Z p
2V0 −1
c=− FW Z1 (u)du. (2.11)
p 0

Note the similarity between the above special case and Markowitz mean-
variance optimization problem (1.3). Optimization problem (2.7) can in the
special case considered hence be rewritten as an optimization problem with
solution equivalent to that of the standard quadratic optimization problem.

2.2.2 General Problem Assumptions

This section presents some assumptions that need to be specified for the
portfolio optimization problem (2.7) to be numerically solvable. The as-
sumptions stated are used when solving the portfolio optimization problem
unless otherwise explicitly stated.

Assumption 1 The investor’s initial capital V0 = 1. This was already assumed when
calculating the benchmark solution and is used to simplify the analysis
work of the solutions since the asset weights wi can be seen as propor-
tions of the whole initial capital rather than monetary weights. The
solution is then easily interpreted for arbitrary initial capital V0 .

Assumption 2 The portfolio lives for one day. This means that the goal is to invest
optimally for one day ahead, which is convenient since the historical
data consists of daily log-returns calculated using (1.1).

Assumption 3 p = 0.01 (q = 0.99). Banks often use the p-value p = 0.005 but other
common levels are p = 0.01 and p = 0.05. The smaller value p, the
more unlikely it is to be exposed to a loss larger than the calculated
Expected Shortfall. p = 0.01 means that with a portfolio life of one
day, one would expect the loss to be larger than the Expected Shortfall
in 1 out of every 100 days. Plugging in p = 0.01 in (2.11) yields with
Z1 being standard normal distributed and W = 1
0.01
2φ(Φ−1 (0.01))
Z
2
c=− Φ−1 (u)du = = 5.33, (2.12)
0.01 0 0.01

17
 where φ(x) denotes the probability density function of the standard
normal distribution. This explains why the risk aversion coefficient
c = 5.33 was used earlier in the benchmark solution.

Assumption 4 The threshold for acceptable expected daily portfolio log-return is θ =

2.0242·10−4 corresponding to 5.08% yearly log-return. This is the same
expected portfolio log-return as for the benchmark solution calculated
in (1.4). The value of θ is ultimately up to the investor to decide, being
a trade-off between return and risk, where larger θ could expose the
investor to larger risk. The impact on the optimal solution of varying
θ is analyzed in Section 4.1.

When analyzing the Expected Shortfall of a portfolio it is common to

look at the relative Expected Shortfall defined as the ratio

ES
ESrel = .
Portfolio market value
With Assumption 1 the portfolio market value is 1 implying that in this
thesis project the relative Expected Shortfall equals the Expected Short-
fall calculated when solving the portfolio optimization problem (2.7) which
simplifies the analysis.

2.3 Rockafellar and Uryasev simplification

Since the definition of Expected Shortfall involves an integral of the Value-
at-Risk, being the quantile function of the loss distribution, it is difficult
to work directly with and optimize (2.1). Rockafellar and Uryasev provide
in [15] two theorems that simplify problem formulation (2.7) much. This
section presents this simplification.
The key is to characterize ESp (X) and VaRp (X) with the auxiliary func-
tion Hp (w, α) on W, Rm defined by
Z
1
Hp (w, α) = α + [L(w, R) − α]+ f (R)dR (2.13)
1−q
R∈Rm

where (
+ x, x>0
[x] =
0, x≤0

and f (R) is the probability density function of the log-return vector R. The
theorems regarding the characterization are central in this thesis project and
are stated below.

18
Theorem 2.3 As a function of α, Hp (w, α) is convex and continuously dif-
ferentiable. The ESp of the loss associated with any w ∈ W can be determined
from the formula
ESp (X) = min Hp (w, α). (2.14)
α∈R
In this formula, the set consisting of the values α for which the minimum is
attained, namely
Ap (w) = argmin Hp (w, α),
α∈R
is a nonempty and closed bounded interval (perhaps reducing to a single
point), and the VaRp of the loss is given by
V aRp (X) = left endpoint of Ap (w). (2.15)
In particular, one always has
V aRp (X) = argmin Hp (w, α), ESp (X) = Hp (w, V aRp (w)).
α∈R

The theorem relies on the assumption that the cumulative loss distribution
function FL is continuous.
Proof. Hp (w, α) is convex from its definition (2.13) and has derivative
∂ 1 1
Hp (w, α) = 1 + (FL (w, α) − 1) = (FL (w, α) − q) .
∂α 1−q 1−q
Therefore, the values of α that minimize Hp (w, α), i.e. the set Ap (w), are
those for which FL (w, α) = q. These values form a non-empty and closed
interval since FL (w, α) is continuous and nondecreasing with limits 0 as
α → −∞ and 1 as α → ∞. This proves (2.15). Furthermore,
Z
1
min Hp (w, α) = Hp (w, αq (w)) = αq (w) + [L(w, R) − αq (w)]+ f (R)dR,
α∈R 1−q
R∈Rm

and the integral is simplified as

Z Z
[L(w, R) − αq (w)]+ f (R)dR = [L(w, R) − αq (w)]f (R)dR
R∈Rm L(w,R)≥αq (w)
Z Z
= L(w, α)f (R)dR − αq (w) f (R)dR
L(w,α)≥αq (w) L(w,α)≥αq (w)

= (1 − q)ESp (X) − αq (w) (1 − FL (w, αq )) ,

where in the last equality I have used the definition of ESp . Moreover,
FL (w, αq ) = q thus yielding
min Hp (w, α) = ESp (X),
α∈R

which proves equation (2.14) and completes the proof. 

19
Theorem 2.4 Minimizing the ESp of the loss associated with all w ∈ W
is equivalent to minimizing Hp (w, α) over all (w, α) ∈ W × R, in the sense
that
min ESp (X) = min Hp (w, α),
w∈W (w,α)∈W×R

where, moreover, a pair (w∗ , α∗ ) achieves the second minimum if and only
if w∗ achieves the first minimum and α∗ ∈ Ap (w∗ ). In particular, therefore,
in circumstances where the interval Ap (w∗ ) reduces to a single point (as is
typical), the minimization of H(w, α) over (w, α) ∈ W × R produces a pair
(w∗ , α∗ ), not necessarily unique, such that w∗ minimizes ESp and α∗ gives
the corresponding VaRp .
Furthermore, Hp (w, α) is convex with respect to (w, α), and ESp is con-
vex with respect to w, when L(w, R) is convex with respect to w, in which
case, if the constraints are such that W is convex, the joint minimization is
an instance of convex programming.

Proof. The initial claims in the first section follows directly from Theorem
1 and by realizing that minimization of Hp (w, α) with respect to (w, α) ∈
W × R can be carried out by first minimizing over α ∈ R for fixed w and
then minimizing over w ∈ W.
Proving the claim in the second section starts by the observation that
Hp (w, α) is convex with respect to (w, α) when [L(w, R) − α]+ is convex.
Since a decomposition of two convex functions is convex, this is true when
L(w, α) is convex with respect to w. The convexity of ESp follows from
the fact that minimizing an extended real-valued convex function of two
variables with respect to one of these variables results in a convex function
of the remaining variable or by recalling that Expected Shortfall is a coherent
risk measure and thus satisfies the convexity property. 

From Theorem 2.3 and 2.4 it follows that instead of minimizing Expected
Shortfall directly through its definition (2.1) one can equivalently minimize
Hp (w, α) defined by (2.13). With convex loss function L(w, R) this is partic-
ularly nice since then the optimization problem becomes a convex program.
By sampling D samples from the probability density function f (R) of
the return vector, Rockafellar and Uryasev then argues that the integral in
(2.13) can be approximated with the sum

D
1 X
Hp (w, α) ≈ H̃p (w, α) = α + [L(w, Rd ) − α]+
(1 − q)D
d=1

which is convex and piecewise linear in α.

Finally, by introducing auxiliary variables zd , d = 1, . . . , D the original

20
problem formulation (2.7) can be approximated as
D
1 X
min α + zd
w,α (1 − q)D
d=1
T
Subject to w µ ≥ θ
zd ≥ 0, d = 1, . . . , D
(2.16)
− L(w, Rd ) + α + zd ≥ 0, d = 1, . . . , D
Xn
wi = V0
i=1
wi ≥ 0, i = 1, . . . , n.
where n is as always the number of assets available in the reference portfolio.
The above problem formulation is a standard result in portfolio optimization
with Expected Shortfall and is for instance presented by Skoglund and Chen
in [19, pp. 156-157]. The problem is a convex linear program with stan-
dardized numerical solving algorithms such as the Simplex or interior point
method [3, Ch. 5,10], when L(w, R) is convex. This is for instance true for
the reference portfolio used in this thesis project, where L(w, R) = −wT R
is linear.

2.4 Application on Reference Portfolio

In this section portfolio optimization problem (2.16) is solved applied to the
reference portfolio in the special case where the solution theoretically should
equal the benchmark solution. To arrive at this scenario, the parameters
in (2.16) must be cleverly chosen to coincide with the parameters in the
benchmark problem formulation (1.3).
Since the benchmark solution relies on the assumption of multivariate
normal distributed log-returns, this must be assumed in this case as well.
To compute loss scenarios L(w, Rd ), I should hence simulate log-return
vectors R from a multivariate normal distribution with parameters µ and
Σ estimated from the historical data presented graphically in Figure 1.1.
Also, q = 0.99 since p = 0.01. For the sum to be approximately equal
to the integral in (2.13), I should choose D big. In this application I use
D = 15, 0001 . The solution presented in Table 2.1 was calculated using the
function linprog in Optimization Toolbox in Matlab version 9.0 on a Core
i5 CPU 2.60 GHz Laptop with 8 GB of RAM with the empirically esti-
mated parameters µ b and Σ.b The last row corresponds to the total portfolio
Expected Shortfall the investor is exposed to if investing according to the
optimal solution.
1
In Appendix B I evaluate how large D must be for the simulated solutions to converge
and have a narrow confidence interval.

21
Table 2.1: Solution to (2.16) with log-returns simulated from multivariate
normal distribution with empirically estimated parameters µb and Σ.
b
Asset name Weight
AstraZeneca 0
Ericsson A 0
Hennes & Mauritz B 0.0066
ICA Gruppen 0.0693
Nordea Bank 0
SAS 0
SSAB A 0
Swedish Match 0.1422
TeliaSonera 0
Volvo 0
Total Bond Index 0.7819
Expected Shortfall 0.0072

By comparing the optimal solution in Table 2.1 to the benchmark solution

in Table 1.1 it is seen that the solutions are almost the same. The small
differences in the weights exist due to numerical simulations of log-returns
and from the approximation of the integral to a sum.
Skoglund and Chen present under the assumption of normal distributed
log-returns two equations in [19, pp. 28-29] to transform the portfolio vari-
ance to Value-at-Risk according to

V aRp (X) = Φ−1 (q)σp (2.17)

where Φ−1 (x) is the normal quantile function and portfolio variance to Ex-
pected Shortfall as
 
V aRp (X)
ESp (X) = σp λ . (2.18)
σp

where λ(x) is again the hazard function for the normal distribution defined in
(2.4). Hence, by recalling the portfolio variance for the benchmark solution
calculated in (1.5), the benchmark Value-at-Risk becomes
T
V aR0.01 (wBM R) = 0.0065

and the benchmark Expected Shortfall is

T
ES0.01 (wBM R) = 0.0075.

The Expected Shortfall by investing according to Table 2.1 is close to the

benchmark Expected Shortfall. Therefore I can verify the claim that in the

22
special case of multivariate normal distributed log-returns, portfolio opti-
mization problem (2.7) is indeed identical to Markowitz mean-variance opti-
mization problem (1.3). Furthermore, since the two optimization problems
are connected only by the risk aversion coefficient c, calculated as (2.11)
for all elliptical distributions, it is clear that Markowitz mean-variance opti-
mization problem is a special case of portfolio optimization with Expected
Shortfall for all elliptical distributions. Since portfolio optimization with Ex-
pected Shortfall does not rely on the stylized assumptions that Markowitz
mean-variance optimization problem requires, (2.7) is a more general prob-
lem formulation that can be applied to more general investment situations
with possibility to include non-linear assets or model the log-returns with
asymmetric distributions for instance. In this thesis project I focus on the
modeling of asset log-returns with non-elliptical distributions.

23
24
Chapter 3

Portfolio Optimization Under

Uncertainty

Recall that the future portfolio log-return is a linear function of portfolio

weights wi and random log-returns Ri , i = 1, . . . , n. Since the set of histor-
ical log-returns is finite, there will be uncertainty in estimated parameters
such as the expected log-return vector and covariance matrix. Parameter
uncertainty might in turn cause large errors in the final portfolio decision
deviating much from the true optimal asset allocations. Parameter uncer-
tainty can be thought of as a sub class to the larger family of model uncer-
tainty, with may also include uncertainty in distribution model. Distribution
uncertainty originates from the fact that the true distribution of empirical
log-returns is unknown and must be approximated. The approximation can
often not be made perfect and there exists uncertainty in the distribution
approximation.

As mentioned in Section 2.2.1, when log-returns are assumed to have el-

liptical distribution, portfolio optimization problem (2.7) has solution that
is equal to that of Markowitz mean-variance optimization problem. In this
special case, there is no need to simulate log-return outcomes since the solu-
tion only depends on the expected log-return vector and covariance matrix.
When the log-returns are assumed to be other than elliptically distributeds
problem (2.7) does not have analytical solution and simulations of log-returns
is necessary. The simulations introduce another source of uncertainty called
statistical uncertainty. By the nature of simulations, one cannot be certain
that the outcomes reflect a fair picture of the distribution they were drawn
from, which is why statistical uncertainty is a problem. The remaining part
of this thesis project will study portfolio optimization with Expected Short-
fall under both model and statistical uncertainty and present approaches on
how to handle optimization under those uncertainties.

25
3.1 Model Uncertainty
Markowitz mean-variance optimization problem assumes the expected log-
return vector µ and covariance matrix Σ to be known and the same holds
for the portfolio optimization problem (2.7). However in real applications
the parameters are often estimated from historical market data, as in this
thesis project. Since there is only a limited amount of historical data avail-
able on the market, the expected log-return vector has to be estimated by
the empirical mean vector and similarly the covariance matrix must also be
approximated. One cannot be certain that the approximated parameters
equals the true market parameters and hence there exists model uncertainty
in the parameters when solving the portfolio optimization problem yielding
uncertainty in the optimal solution.
A long established problem in portfolio optimization is referred to as the
problem of error maximization, discussed by Scherer in [16, pp. 185-186].
The problem origins from that the optimization algorithm tends to select
assets with the best properties, in this case high log-return and low variance
and correlation, and not select assets with the worst properties. These are
the assets where estimation errors in µ and Σ are likely to be largest, with
strong dependence on outliers in the data. Hence the optimal solution will
have strong dependence on parameter uncertainty, where positive estimation
error leads to over-weighted assets and negative estimation error leads to
under-weighted assets.
In addition to parameter uncertainty, the multivariate distribution of the
empirical log-returns can generally only be approximated, leading to uncer-
tainty in distribution in the model. Different distributions are more or less
successful in modeling historical data and where some are better at model-
ing the tails of the empirical distribution, others are better at capturing the
behavior in the center of the empirical distribution. A perfect distribution
fit on the entire empirical distribution is generally not possible to find and
hence uncertainty in distribution must be regarded as a relevant factor in
the model.
In the 1990s, two approaches were developed to tackle model uncertainty.
One approach is called robust statistics, which involves removing or down-
weighting what is thought of as being outliers in the empirical data set. The
second approach is the concept of robust optimization, which will be consid-
ered in this thesis project. Robust optimization can intuitively be thought of
as attempting to optimize the worst-case scenario given a confidence region.
Traditionally, parameter uncertainty in Markowitz mean-variance optimiza-
tion problem has received a lot of attention from the robust optimization
community but less has been said about distribution uncertainty. This the-
sis project contributes to the robust optimization community by studying
worst-case scenario based robust optimization of portfolio optimization prob-
lem (2.16) under different distribution models. First I will perform a case

26
study on robust optimization under elliptical distributions and then move
on to study robust optimization under asymmetric log-return distributions.
A third possibility for model uncertainty is that the model itself is wrong.
For instance, there might exist liquidity risk in assets, invalidating the trans-
lation invariance property of the risk measure, included in the portfolio which
is not covered by the model, or it could be that the covariance matrix is de-
pendent on time and state of economy. These types of model uncertainties
can be harder to evaluate directly without changing the entire problem for-
mulation and will not be considered in this thesis project.

3.1.1 Introduction to Robust Optimization

This section introduces robust optimization and specifies what is meant by
worst-case scenario based robust optimization. Generally, let f (w; X, a) and
gk (w; X, b), k = 1, . . . , m be functions of the weight vector w, where X is
a stochastic variable with cumulative distribution function FX (x) and a, b
are parameters and consider the optimization problem

min f (w; X, a)
w
Subject to gk (w; X, b) ≤ gk,0 , k = 1, . . . , m,

where gk,0 k = 1, . . . , m are constants. If f (w; X, a) is a convex function and

the constraining equations span a convex region, there exists a unique global
optimal solution that can be found using standard optimization algorithms.
Now suppose that there is some uncertainty in the parameters a, b and in
the cumulative distribution function FX (x) and it is only known that a ∈ A,
b ∈ B and FX (x) ∈ F for some sets A, B and family of distributions F. The
problem formulation then becomes

min f (w; X, a)
w
Subject to gk (w; X, b) ≤ gk,0 , k = 1, . . . , m
a ∈ A, b ∈ B, FX (x) ∈ F.

Since ordinary optimization problems require parameters to be known, this

problem cannot be solved with traditional optimization methods. Instead,
the concept of robust optimization has been developed. The general objec-
tive of robust optimization is to compute solutions that ensure feasibility
independent of the parameters and distribution included in the uncertainty
sets. Note that there is a key difference between robust optimization and sen-
sitivity analysis, which is typically applied after the optimization to study
the change in objective function under small variations in the underlying
data. With robust optimization, the goal is instead to compute solutions
when the uncertainty is included in the problem formulation, prior to opti-
mization. Similarly to ordinary optimization, if f (w; X, a) and gk (w; X, b)

27
are convex functions and A, B, F are convex sets, the robust optimization
problem is particularly nice since when an extreme point is found, we know
it is the global optimal solution.
There are different approaches on how to solve a robust optimization
problem, and one of the most commonly used approaches in portfolio opti-
mization is the worst-case scenario based robust optimization approach in-
troduced by Tütüncü and König in [22]. They argue it is a good idea to solve
the robust optimization problem by first finding the worst-case scenarios for
the parameters given the uncertainty sets and then consider the resulting
optimization problem with these worst-case scenario parameters and solve
it with ordinary optimization algorithms. For instance, if the worst-case
scenario is attained for the smallest a in the uncertainty set A and for the
largest b in the uncertainty set B then the worst-case scenario based robust
optimization problem would be

min f (w; X, a)
w,a∈A

Subject to max gk (w; X, b) ≤ gk,0 , k = 1, . . . , m

b∈B

under the additional constraint that X is drawn from the worst-case distribu-
tion, measured in some way, that is included in the distribution uncertainty
set F.
Since the worst-case scenario based robust optimization approach is widely
used in robust portfolio optimization problems, it is used in this thesis project
as well. Section 4.2 discuss in greater detail the effect of interpreting robust
optimization as finding the worst-case scenario and then optimize the result-
ing worst-case scenario optimization problem.

3.1.2 Robust Optimization with Elliptical Distributions

Before presenting the worst-case scenario based robust version of (2.16) I
mention some work that historically has been done on robust optimization
of Markowitz mean-variance optimization problem. Lobo and Boyd discuss
in [11] a robust variant of Markowitz mean-variance problem with uncertain
covariance matrix Σ. The problem is presented as

min max wT Σw
w Σ∈S
Subject to wT µ ≥ Rmin
Xn
wi = 1
i=1
wi ≥ wmin

and aims at finding the portfolio weights that minimize the total portfolio
variance with the worst-case covariance matrix. The authors assume that

28
the investor is ambiguous about the covariance matrix and have several pos-
sible candidates. The robust problem is discussed with box and ellipsoidal
constraints on the covariance matrix, mathematically as

Sbox = Σ ∈ Sn+ |Σij,low ≤ Σij ≤ Σij,high , i, j = 1, . . . , n



Sellipsoid = Σ ∈ Sn+ |(s − b

s)T Q(s − b

s) ≤ 1 ,

where s is a vector representation of the upper triangular elements of Σ, b

s is
the corresponding empirically estimated vector and Q is the second moment
of the Wishart distribution for the empirically estimated covariance matrix.
A similar robust optimization problem is also presented where instead the
box and ellipsoidal uncertainty sets constrain the mean log-return vector µ,
mathematically written as

Mbox = {µ ∈ Rn |µi,low ≤ µi ≤ µi,high , i = 1, . . . , n}

b)T S −1 (µ − µ
Mellipsoid = µ ∈ Rn |(µ − µ

b) ≤ 1 ,

where S is a scaled version of Σ and µb is the empirically estimated expected

log-return vector. Both the box and ellipsoidal uncertainty set are convex
sets so they are attractive from an optimization perspective. Ye, Parpas and
Rustem [23] takes the problem formulation by Lobo and Boyd further and
propose a problem where both µ and Σ are uncertain so that the optimization
problem becomes

min max wT Σw
w Σ∈S
Subject to min wT µ ≥ Rmin
µ∈M
Xn
wi = 1
i=1
wi ≥ wmin

and the objective is to find the portfolio weights that minimize the risk given
worst-case scenarios in both expected log-return and covariance.
The worst-case scenario based robust version of problem (2.16) is now
easy to formulate in a similar manner. Let the covariance matrix Σ b and the
mean log-return vector µ b be estimates from historical data of the uncertain
parameters Σ and µ and assume we know that the true parameters are
somewhere in the uncertainty sets M, S. The robust version of portfolio

29
optimization problem (2.16) with linear loss function is then given by
D
1 X
min α + zd
w,α (1 − q)D
d=1
T
Subject to min w µ ≥ θ
µ∈M

zd ≥ 0, d = 1, . . . , D
(3.1)
min max wT Rd + α + zd ≥ 0, d = 1, . . . , D
µ∈M Σ∈S
n
X
wi = V0
i=1
wi ≥ 0, i = 1, . . . , n

and the log-return vectors Rd , d = 1, . . . , D are simulated from some dis-

tribution that is interpreted as the worst one of all distributions included in
the distribution uncertainty set F.
To solve (3.1) numerically, the uncertainty sets M, S must be specified so
that the worst-case scenarios can be found, and the distribution uncertainty
set F must be defined. I begin by constructing the uncertainty sets M, S.
To ensure that the robust optimization problem remains convex, I let
M, S be boxes or ellipsoids as discussed by Lobo and Boyd. One approach
to construct a box uncertainty set for µ is to add or subtract some term ±|b
µi |
to each empirically estimated element µ bi . This yields the box uncertainty
set
Mbox = {µ ∈ Rn : µ = µ b ± |b
µ|}
and the worst-case scenario expected log-return vector would then be
(box)
b − |b
µmin = µ µ|. (3.2)

In this thesis project I have set  = 0.20. A reasonable sanity check is

(box)
to investigate if µmin lies within the approximate 95% confidence interval
calculated from the historical data to make sure that it is not unrealistically
far from the empirically estimated parameter. With the data used in this
(box)
thesis project and  = 0.20, µmin does indeed lie within the 95% confidence
interval.
To calculate the worst-case scenario covariance matrix requires a bit more
thought process. It is not possible to employ the same strategy as when
calculating a box uncertainty set for µ since this could result in a non-
invertible matrix which is not desired. There are several methods to define
the worst-case covariance matrix and the general idea is that one wants to
increase the eigenvalues and rotate the eigenvectors by some angle so that
the resulting matrix is scaled and oriented worse than the original covariance
matrix. One approach that does this implicitly is to study the historical price

30
development for the assets in the reference portfolio and locate a time period
where the assets seem the most correlated. Typically this occurs during times
of financial crisis. Looking at the asset price development in Figure 1.1, it
seems as the assets in the reference portfolio are the most correlated during
2007 − 2009 where most asset prices have a negative trend, which fits well
with the time line for the financial crisis presented in [8]. Therefore, the
worst-case scenario covariance matrix originating from a box uncertainty set
is defined with basis on the financial crisis and taken as the covariance matrix
estimated from historical data from the first trading day of 2007 until the
last trading day of 2009, i.e.

Σ(box)
max = Σ between January 2, 2007 – December 30, 2009.
b (3.3)

Since an empirically estimated covariance matrix is always positive semi-

definite, this approach ensures an invertible worst-case scenario covariance
matrix. Throughout the remaining part of the thesis project I refer to the
(box) (box)
worst-case scenario parameters µmin and Σmax given by (3.2) and (3.3)
respectively as "box uncertainty parameters". See Appendix A for numerical
values.
With ellipsoidal uncertainty sets the elements depend on each other which
makes it harder to find the worst-case scenarios µmin and Σmax . The value
of one element restricts the range of values the other elements can take so
it is not possible to find the worst-case scenario for each element separately
without taking into account the other element values. One approach to
find the worst-case scenario expected log-return vector originating from an
ellipsoidal uncertainty set is however to solve the optimization problem

(ellipsoidal)
min µ−µb
µmin = µ (3.4)
Subject to (µ − µb)T S −1 (µ − µ
b) ≤ 1.

In this thesis project, the above optimization problem is solved with S −1 =

100Σ−1 to limit the resulting worst-case scenario parameter µmin to not
deviate too much from µ b. In a similar way, the worst-case scenario covariance
matrix originating from an ellipsoidal uncertainty set can be obtained by
solving the optimization problem

max s−b
s

 s
s)T Q(s − b

Subject to (s − b s) ≤ 1
Σ(ellipsoidal)
max = (n−1)bs (3.5)

 s ≤ χ2
 0.975
(ellipsoidal)

) ≥ 0.

diag(Σmax

The two last constraints are posed to ensure that Σmax is not too far from
Σ
b and so that all variances are positive. In this thesis project the second

31
moment of the Wishart distribution, Q, is estimated by simulating 10, 000
covariance matrices from the Wishart distribution and then calculate the
covariance matrix for all pairwise elements in the simulated covariance ma-
trices. The solution to (3.5) might not be positive semi-definite which is a
requirement for the portfolio optimization problem to be solvable. In that
case Rebonato and Jäckel present in [14] a general methodology to find the
closest symmetric and positive semi-definite matrix given a non positive semi-
definite matrix that can be used to obtain a feasible covariance matrix. The
method involves spectral decomposition of the matrix and setting negative
eigenvalues to zero. Throughout the remaining part of the thesis project I
(ellipsoidal) (ellipsoidal)
refer to the worst-case parameters µmin and Σmax given by (3.4)
and (3.5) respectively as "ellipsoidal uncertainty parameters". See Appendix
A for numerical values.
Regarding the distribution uncertainty set F, the thesis project first con-
siders elliptical distributions with the historically popular normal distribu-
tion and Student’s t distribution with different degrees of freedom, i.e.

Felliptical = {N (µ, Σ), t(µ, Σ, ν)} , (3.6)

and later on considers asymmetric log-return distributions with the general-

ized Pareto distribution.
For the elliptical distributions in Felliptical , it is easy to determine the
worst-case scenario distribution. Since the Student’s t distribution has heav-
ier tails than the normal distribution, the normal distribution is ruled out
directly as the worst-case scenario distribution, having lower probability of
encountering extremely large losses. For the Student’s t distribution, the
variance is undefined for ν < 2, infinite for ν = 2 and for increasing ν > 2 the
variance is decreasing. Hence, the smaller the degrees of freedom, the heav-
ier the tails of the distribution and the greater is the variance and Expected
Shortfall. The worst-case scenario distribution that ensures that variance
exists is hence Student’s t distribution with lim ν ≈ 2.1 degrees of freedom.
ν&2
Since not much work has been done by the robust optimization commu-
nity on uncertainty in distribution, it would be interesting to analyze the
behavior of the solution to (3.1) under different distributions. Therefore,
instead of simply solving (3.1) with the worst-case distribution in Felliptical ,
this thesis project contributes to the robust portfolio optimization commu-
nity with a case study by solving the problem for each of the distributions
in Felliptical with both box and ellipsoidal uncertainty parameters. This is
done in the two following sub sections.

3.1.2.1 Solution with Normal Distributed Log-returns

This section presents solutions to the robust portfolio optimization prob-
lem (3.1) applied to the reference portfolio with log-return vectors Rd , d =

32
1, . . . , D drawn from the multivariate N (µmin , Σmax ) distribution. The left
column in Table 3.1 presents the solution with D = 15, 000 simulated log-
return vector samples and box uncertainty parameters. The right column
of Table 3.1 presents the solution with ellipsoidal uncertainty parameters.
The last row in each column corresponds to the Expected Shortfall when
investing according to the corresponding column. Since the robust portfolio
optimization problem is still convex and linear, the problem can be solved by
the same algorithm as in Section 2.4 for the non-robust problem formulation.

Table 3.1: Solutions to (3.1) applied to the reference portfolio with

N (µmin , Σmax ) distributed log-returns. Left column with box uncertainty
parameters. Right column with ellipsoidal uncertainty parameters.
Box uncertainty Ellipsoidal uncertainty
Asset name Weight Weight
AstraZeneca 0 0
Ericsson A 0 0
Hennes & Mauritz B 0 0.0024
ICA Gruppen 0.1509 0.0051
Nordea Bank 0 0
SAS 0 0
SSAB A 0 0
Swedish Match 0.3485 0.0032
TeliaSonera 0 0
Volvo 0 0
Total Bond Index 0.5006 0.9892
Expected Shortfall 0.0208 0.0051

Note that since the log-return vector is assumed to have multivariate

normal distribution, the problem is equivalent to a Markowitz mean-variance
optimization problem according to the remark in Section 2.2.1. Hence, in-
stead of simulating log-returns and introducing statistical uncertainty to the
solution, it would be possible to solve (1.3) with the same worst-case pa-
rameters and risk aversion coefficient c = 5.33 calculated in (2.12) and avoid
statistical uncertainty. Those solutions are presented in Appendix C.1 as
reference solutions to validate the accuracy of the simulated solutions.

3.1.2.2 Solution with Student’s t Distributed Log-returns

This section presents the robust solutions to (3.1) applied to the refer-
ence portfolio with 15, 000 log-return vectors simulated from multivariate
Student’s t distributions with different degrees of freedom ν. Maximum-
Likelihood estimation of the degrees of freedom to the historical log-return
data in the reference portfolio gives νb = 3.58. Hence low degrees of freedom
when simulating log-returns from a Student’s t distribution is reasonable. It

33
is also interesting to study larger degrees of freedom as well to observe if the
robust solution converges to the robust solution with normal distributed log-
returns. Therefore I decide to solve (3.1) with ν = {2.1, 3.58, 10, 20} in this
thesis project. With box uncertainty parameters, the solutions for varying
degrees of freedom are presented in Table 3.2. With ellipsoidal uncertainty
parameters the corresponding solutions are presented in Table 3.3.

Table 3.2: Robust solutions to (3.1) with multivariate Student’s t distributed

log-returns, box uncertainty parameters and varying degrees of freedom.
ν = 2.1 ν = 3.58 ν = 10 ν = 20
Asset name Weight Weight Weight Weight
AstraZeneca 0 0 0 0
Ericsson A 0 0 0 0
Hennes & Mauritz B 0 0 0 0
ICA Gruppen 0.1494 0.1414 0.1454 0.1512
Nordea Bank 0 0 0 0
SAS 0 0 0 0
SSAB A 0 0 0 0
Swedish Match 0.3495 0.3551 0.3523 0.3482
TeliaSonera 0 0 0 0
Volvo 0 0 0 0
Total Bond Index 0.5011 0.5035 0.5023 0.5005
Expected Shortfall 0.0902 0.0427 0.0260 0.0231

Table 3.3: Robust solutions to (3.1) with multivariate Student’s t distributed

log-returns, ellipsoidal uncertainty parameters and varying degrees of free-
dom.
ν = 2.1 ν = 3.58 ν = 10 ν = 20
Asset name Weight Weight Weight Weight
AstraZeneca 0 0.0004 0 0
Ericsson A 0 0 0 0
Hennes & Mauritz B 0.0012 0.0041 0 0.0003
ICA Gruppen 0.0010 0 0.0006 0.0006
Nordea Bank 0.0007 0 0.0011 0.0024
SAS 0 0 0 0
SSAB A 0 0 0 0
Swedish Match 0.0091 0.0075 0.0024 0.0056
TeliaSonera 0 0 0 0
Volvo 0 0 0 0
Total Bond Index 0.9880 0.9884 0.9919 0.9911
Expected Shortfall 0.0237 0.0113 0.0063 0.0056

34
 Similarly to the case with normal distributed log-returns, to avoid intro-
ducing statistical uncertainty in the solutions, it is possible to solve Markowitz
mean-variance optimization problem (1.3) with the same worst-case parame-
ters and risk aversion coefficient c calculated using (2.11) with the Student’s
t quantile function. Those solutions are presented in Appendix C.2 as refer-
ence solutions to validate the accuracy of the simulated solutions.

3.1.3 Robust Optimization with Asymmetric Distributions

A well known problem when fitting distributions to empirical data with
Maximum-Likelihood estimation is that most focus is on fitting the center of
the distribution and less focus is on the tails. This is because by definition
more observations are located in the center than in the tails of the empirical
distribution. This behavior is unfortunate when calculating risk measures
such as Expected Shortfall which depends heavily on the loss tail of the
modeled distribution. One attempt to solve this problem is to model the
distributional tails and center separately.

3.1.3.1 The univariate generalized Pareto distribution

Modeling the tails and center of a univariate empirical distribution sepa-
rately results in an asymmetric log-return distribution and there is no longer
a simple analytical relationship between Markowitz mean-variance optimiza-
tion problem and the portfolio optimization problem with Expected Short-
fall. This is of course of interest because then problem formulation (2.7) is
a unique portfolio optimization problem not identical to a Markowitz mean-
variance optimization problem.
A distribution that is popular to use for modeling the tails of an empirical
log-return distribution is the generalized Pareto distribution (GPD). More
correctly, given independent and identically distributed residuals r1 , . . . , rn
with unknown distribution function F and a high threshold uh , the excesses
rk − uh often turns out to be well modeled by the generalized Pareto dis-
tribution. For a shape parameter γ > 0 and a scale parameter β > 0, the
generalized Pareto distribution is given by
 1
γx − γ

Gγ,β (x) = 1 − 1 + , x≥0
β

and for γ = 0 it is given by

x
−β
Gγ,β (x) = 1 − e , x ≥ 0.

The choice of a suitably high threshold uh is crucial, but often hard,

for the generalized Pareto distribution to be a good fit to the excesses. If
uh is too large then few observations can be used for parameter estimation

35
yielding poor estimates with large variation. If instead uh is too small then
more observations can be used to estimate the parameters but modeling
the excesses with the generalized Pareto distribution becomes questionable.
One approach to estimate the parameters is to pick some uh1 far out in the
empirical distribution, say the 90% quantile of the empirical distribution,
and estimate the parameters with Maximum Likelihood estimation. Then a
new threshold uh2 is chosen farther out in the empirical distribution, say the
91% quantile, and the parameters are estimated once again with Maximum
Likelihood estimation. The procedure is repeated and the threshold uh is
chosen as the first candidate for which the parameters to the generalized
Pareto distribution are stable from that point on. This method relies on the
assumption that the parameters will converge before we are not too far out
in the distribution tail. With data originating from the historical log-returns
in this thesis project the method could not be used because the parameters
did not converge until very far out in the distribution tail. Instead, the
threshold were chosen as the 90% empirical quantile, i.e. uh = Fn−1 (0.90),
for each asset in the reference portfolio1 . In Section 4.3 I mention another
strategy that could be used to choose better values of uh .
Note that the generalized Pareto distribution models excesses above a
high threshold uh which means that we are looking at the upper tail of the
log-return distribution, but Expected Shortfall only depends on the lower
tail of the log-return distribution. Luckily, if the lower tail of the log-return
distribution consists solely of negative values below some negative threshold
ul one may simply look at the absolute values and the "excess" of an ob-
servation in the lower tail is then the positive distance from the observation
to ul and one can model the lower tail of the log-return distribution with a
generalized Pareto distribution as well. Levine makes use of this strategy in
[9] when modeling the lower tail of the total monthly return distribution for
the A-rated 7- to 10-year corporate bond component of Citi’s U.S. Broad In-
vestment Grade Bond Index from January 1980 to August 2008. Similarly to
the upper threshold uh , the lower thresholds are chosen as the 10% empirical
quantile, i.e. ul = Fn−1 (0.10), for each asset in the reference portfolio.
Since generalized Pareto distributions can only be used to model the tails
of a distribution a third distribution must be used as a bridge to connect the
two tails with the center of the distribution. In this thesis project, this is
done with the empirical distribution.
A mathematical expression for the entire underlying distribution will
be derived by considering two scenarios and then combining them. Consider
first the scenario where the excesses above some threshold uh of a distribution
consisting of independent and identically distributed random variables is
modeled with a generalized Pareto distribution Ghγ,β (x − uh ). For x ≥ uh

1
Skoglund and Nyström show in [13] that Maximum Likelihood estimation of general-
ized Pareto distribution parameters is not sensitive of the choice of threshold value.

36
the cumulative distribution function of the underlying distribution is given
by

P (X ≤ x) = P (X ≤ uh ) + P (uh ≤ X ≤ x)
= P (X ≤ uh ) + (1 − P (X ≤ uh ))Ghγ,β (x − uh ).

The probability P (X ≤ uh ) is estimated by the empirical distribution func-

tion as P (X ≤ uh ) ≈ Fn (uh ). For x < uh the remaining underlying dis-
tribution is modeled with the empirical distribution Fn (x). The cumulative
distribution function for the entire underlying distribution is then modeled
as (
Fn (x), x < uh
F (x) =
Fn (uh ) + (1 − Fn (uh ))Ghγ,β (x − uh ), x ≥ uh .
Now consider the reversed scenario where the "excesses" of the independent
and identically distributed random variables below some threshold ul is mod-
eled by a generalized Pareto distribution Glγ,β (ul − x) and for x > ul the
remaining underlying distribution is modeled with the empirical distribu-
tion Fn (x). The cumulative distribution function for the entire underlying
distribution is in this case modeled as
(  
Fn (ul ) 1 − Glγ,β (ul − x) , x ≤ ul
F (x) =
Fn (x), x > ul .

By combining the two scenarios, the cumulative distribution function for the
entire underlying distribution is given by
  
F (ul ) 1 − Gl (ul − x) , x ≤ ul
 n

 γ,β
F (x) = Fn (x), ul < x < uh (3.7)

F (uh ) + (1 − F (uh ))Gh (x − uh ), x ≥ uh .

n n γ,β

where the shape and scale parameters γ and β may differ between the two
generalized Pareto distribution functions Glγ,β (x) and Ghγ,β (x). The cumula-
tive distribution function (3.7) is asymmetric and will be referred to in this
thesis project as the hybrid GPD-Empirical-GPD distribution.

3.1.3.2 Copula dependence

The generalized Pareto distribution is univariate so for each asset in the refer-
ence portfolio two unique generalized Pareto distributions and the empirical
distribution must be modeled to obtain unique univariate cumulative distri-
bution functions. To capture the dependence between the different assets a
copula may be used. A copula can be explained as a function that couples a
joint distribution function to univariate marginal distribution functions and
it is from the joint distribution function where the multivariate dependence

37
between the univariate random variables is inherited. A copula is constructed
by combining probability and quantile transforms. If X is a random variable
with continuous distribution function F then the probability transform states
that F (X) has uniform distribution on (0, 1), i.e. F (X) is U (0, 1). If U is a
random variable with uniform distribution on (0, 1) and G is some distribu-
tion function then the quantile transform states that G−1 (U ) has distribution
function G, i.e. P (G−1 (U ) ≤ x) = G(x). Using the probability transform it
follows that for a random vector U = (U1 , . . . , Un ) whose components have
uniform distribution on (0, 1) a random vector X = (X1 , . . . , Xn ) whose
components have marginal distributions F1 , . . . , Fn can be constructed as
X = (F1−1 (U1 ), . . . , Fn−1 (Un )). (3.8)
Now, to describe the dependence between the components of X one may
instead describe the dependence between the components of U and use (3.8).
The distribution function C(u1 , . . . , un ) is called a copula and using the
quantile transform it follows that
C(F1 (x1 ), . . . , Fn (xn )) = P (U1 ≤ F1 (x1 ), . . . , Un ≤ Fn (xn )
= P (F1−1 (U1 ) ≤ x1 , . . . , Fn−1 (Un ) ≤ xn )
= F (x1 , . . . , xn )
where F is the joint multivariate distribution of X.
Since the components of U are transformed to components in X using
the marginal distributions of X it is important to realize that the measure of
dependence between the random variables must be invariant to this transfor-
mation. Ordinary linear correlation is not invariant to this transformation
which makes it necessary to find such a measure. One measure that has
the desired property is the rank correlation Kendall’s tau. Kendall’s tau is
defined on the random vector (X1 , X2 ) as
τ (X1 , X2 ) = P (X1 − X10 )(X2 − X20 ) > 0 − P (X1 − X10 )(X2 − X20 ) < 0

where (X10 , X20 ) is an independent copy of (X1 , X2 ). Lindskog, McNeil and

Schmock show in [10] that if (X1 , X2 ) have an elliptical distribution then the
relationship
2
τ (X1 , X2 ) = arcsin(ρ) (3.9)
π
holds under some technical conditions. The relationship makes it easy to
convert the linear correlation matrix C into a transformation invariant rank
correlation matrix Cτ that can be used when simulating from an elliptical
copula.

3.1.3.3 Sample Preparation with GARCH(1,1) Model

The validity of fitting a generalized Pareto distribution to excesses in a dis-
tribution tail relies heavily on the assumption of independent and identically

38
distributed random variables. Hence, to model the tails of the empirical dis-
tributions to each asset in the reference portfolio with generalized Pareto
distribution it is of great importance that the data has the right properties.
When initially defining log-returns in (1.1) it was mentioned that log-returns
are often assumed to be weakly dependent and close to independent and
identically distributed. In financial time series analysis of risk factors (here
the log-returns), stylized facts are effects that are commonly observed in the
data. Josefsson summarizes the stylized facts in [7] as

1. Return series are not independent and identically distributed although

they show little serial correlation.

2. Series of absolute or squared returns show significant serial correlation.

3. Conditional expected returns are close to zero.

4. Volatility appears to vary over time.

5. Return series are leptokurtic2 or heavy tailed.

6. Extreme returns appear in clusters.

The stylized facts are problematic since they imply that the empirical log-
returns cannot be considered independent and identically distributed and
hence the excesses in the distribution tails cannot be modeled directly with
generalized Pareto distributions. Some sample preparation has to be done,
typically by filtering the data through a time series model. A popular model
used for filtering financial log-return data is the Generalized Autoregressive
Conditional Heteroscedasticity, GARCH, model.
The general GARCH(p,q) process is defined in the following way. Let
{Zt } be independent and identically distributed N (0, 1). Then {Xt } is called
a GARCH(p,q) process if

Xt = µ + σt Zt , t∈Z

where {σt } is the nonnegative process

p
X q
X
σt2 = α0 + 2
αi Xt−i + 2
βj σt−j ,
i=1 j=1

and α0 > 0, αj , βj ≥ 0, j = 1, 2, . . . . For more on GARCH(p,q) processes

and other time series models I refer to [1] by Brockwell and Davis. Note
from the GARCH model that if Xt is interpreted as the log-return at time
t then the model is a linear combination of squared historical log-returns
2
A leptokurtic distribution has fatter tails and higher peak than the curvature found
in a normal distribution.

39
which fits stylized fact 2. Also, the volatility is dependent on time and on
previous volatilities, which fits stylized facts 4 and 6. This is an intuitive
motivation on why GARCH models often fits historical log-return time series
well. Furthermore, by applying Hölder’s inequality when calculating the
kurtosis of σt Zt ,
E[(σt Zt )4 ] E[σt4 ]
k(σt Zt ) = = k(Z t ) ≥ k(Zt )
E[(σt Zt )2 ]2 E[σt2 ]2
it is seen that it is greater than the kurtosis of Zt . Hence, even though
the residuals are assumed to be normal distributed, the GARCH(p,q) model
takes into account that log-returns often have fatter tails than the normal
distribution. This fits well with stylized fact 5.
There are methods for fitting general GARCH(p,q) models to time series
data which require us to find p, q, α0 , . . . , αp , β1 , . . . , βq . In this thesis project
I settle with fitting GARCH(1,1) models to the historical log-returns, which
makes the process easier. GARCH(1,1) models are used since they are often
good enough to filter financial data and the parameters can for instance be
approximated by Maximum Likelihood estimation.
When the parameters have been estimated the standardized residuals
{Zt } are obtained as
Rt − µ b
Zt =
σ
bt
and if the GARCH filtration is successful then the stylized facts should be ac-
counted for, meaning that the standardized residuals should be independent
and identically distributed.
In Appendix D I analyze the standardized residuals to the filtered empir-
ical log-returns and conclude that the distribution tails are asymmetric and
heavier than for the standard normal distribution assumed in the definition
of the GARCH process. In practice this is not a problem since the standard-
ized residuals can be modeled by any suitable distribution. This motivates
the idea of modeling the tails with generalized Pareto distributions.

3.1.3.4 Solution with Normal Copula and Hybrid GPD-Empirical-

GPD Log-return Marginals
In this section I solve the robust portfolio optimization problem (3.1) with
worst-case scenario parameters originating from both box and ellipsoidal
uncertainty sets and log-returns simulated from a normal copula with hy-
brid GPD-Empirical-GPD marginal distributions. Skoglund and Wei note
in [19, p. 132] that the choice of copula has a second-order effect when
modeling market risk factors such as stock log-returns, the most important
factor being to model stochastic volatility and volatility clustering which is
accounted for in the GARCH(p,q) model. Hence choosing the normal copula
for dependency structure is not the most important decision in the modeling

40
procedure and if a Student’s t copula were to be chosen instead, it would
outperform the normal copula model only by a small margin. In Appendix
D, Table D.1, I present the parameters for the GARCH(1,1) models used for
filtering the historical log-returns to standardized residuals. With help of
Figure D.1 I argue on basis of the stylized facts for financial time series why
the GARCH(1,1) models have filtered the empirical log-return data well into
standardized independent and identically distributed residuals. Secondly I
use Figure D.2 to describe why the standardized residuals are not well mod-
eled with the standard normal distribution which motivates the use of hybrid
GPD-Empirical-GPD distributions instead. In Table D.2 I present values of
estimated thresholds and parameters for the generalized Pareto distributions
when fitted to the standardized residuals and in Figure D.3 the estimated
generalized Pareto distribution functions are plotted to show that the models
approximate the true data well.
In Section 3.1.2.1 and Section 3.1.2.2 it was easy to simulate log-returns
by generating random variables from the multivariate normal and Student’s t
distribution. With hybrid GPD-Empirical-GPD marginal distributions and
dependence between the standardized residuals from a normal copula the
simulation of log-returns is not as straight forward. Below I provide an
algorithm that can be used to simulate log-returns in this case.

Algorithm 1 Simulation of log-returns from normal copula with hybrid

GPD-Empirical-GPD marginal distributions
1: Simulate dependent random variables x1 , . . . , xn from the multivariate
normal distribution N (0, Cτ ), where Cτ is the Kendall’s tau rank corre-
lation matrix with each element calculated with (3.9).
2: Transform x1 , . . . , xn to U (0, 1) distributed random variables as ui =
Φ(xi ) where Φ(x) is the standard normal cumulative distribution func-
tion.
−1
3: The standardized residuals Zi = Fi (ui ), i = 1, . . . , n, where the quan-
tile function Fi−1 is the inverse function of (3.7) for asset i with param-
eters found in Table D.2.
4: The simulated log-returns Ri = σt+1 i Z +µ i
bi , i = 1, . . . , n, where σt+1
i
is the forecasted volatility obtained from the GARCH(1,1) model with
parameters in Table D.1 and µ bi is the empirically estimated mean log-
return value for asset i.

Simulation from a copula can often be written in a similar algorithm

as the one above, specified for the particular task it should be used for.
Cherubini, Luciano and Vecchiato write in [2, Ch. 6] several algorithms to
generate samples from other types of copulas, for instance the Student’s t
copula but also archimedean copulas such as the Clayton, Gumbel and Frank
copula.
To generate the log-returns needed for the optimization problem the

41
above algorithm is repeated 15, 000 times to obtain the appropriate sam-
ple size D. Note that log-returns with worst-case scenario parameters are
as easy to generate by simply replacing Cτ with a corresponding worst-case
scenario rank correlation matrix in step 1 and replace the forcasted standard
i
deviation σt+1 and the empirically estimated µi with corresponding worst-
case scenario forcasted standard deviation and worst-case scenario expected
log-return in step 4. To solve the robust portfolio optimization problem
(3.1) the simulated log-returns can then be directly inserted in the same
optimization solving algorithm as when solving with elliptically distributed
log-returns since the optimization problem itself is unchanged. Table 3.4
presents the optimal weight vector and portfolio Expected Shortfall when
solving (3.1) with both box and ellipsoidal uncertainty parameters.

Table 3.4: Solutions to (3.1) applied to the reference portfolio with log-
returns modeled by a normal copula with hybrid GPD-Empirical-GPD
marginals. Left column with box uncertainty parameters and right column
with ellipsoidal uncertainty parameters.
Box uncertainty Ellipsoidal uncertainty
Asset name Weight Weight
AstraZeneca 0 0
Ericsson A 0 0
Hennes & Mauritz B 0.0123 0.0037
ICA Gruppen 0.1855 0
Nordea Bank 0 0
SAS 0 0
SSAB A 0 0
Swedish Match 0.3201 0.0080
TeliaSonera 0 0
Volvo 0 0
Total Bond Index 0.4822 0.9884
Expected Shortfall 0.0233 0.0056

3.2 Statistical Uncertainty

Portfolio optimization problem (2.7), or equivalently problem (2.16), has in
the case of elliptically distributed log-returns with known parameters µ and
Σ analytical solution that corresponds to the solution to Markowitz mean-
variance optimization problem. Under other model assumptions the problem
might however not have analytical solution. This applies for instance to
log-returns simulated from the normal copula with hybrid GPD-Empirical-
GPD marginals, and an optimal solution must be numerically calculated
from simulations. Simulating D log-return vectors R several times would
then produce different solutions since statistical uncertainty is present. A

42
popular strategy to evaluate the sensitivity of a solution in the presence of
statistical uncertainty is the bootstrap strategy, discussed in [6] by James,
Witten, Hastie and Tibshirani.
The idea behind bootstrapping is to estimate the accuracy of some esti-
mated quantity κ when it is not possible or practical to create new samples
from the original population. This could for instance be the case when the
original distribution is unknown, because it is too expensive or too time
consuming or because it is in other ways impossible to generate new data
samples. With the reference portfolio in this thesis project it is impossible to
obtain more historical daily log-return data for the time period of interest and
simulating log-returns from the normal copula with hybrid GPD-Empirical-
GPD marginal distributions is quite time consuming. Therefore the boot-
strap procedure is appropriate to use to estimate standard errors for the
portfolio weights. In the (non-parametric) bootstrap procedure one obtains
new distinct artificial data sets by repeatedly sampling with replacement
from the original data set. That is, from the original data set X we repeat-
(1) (B)
edly draw and replace elements to construct new data sets X̃ , . . . , X̃ ,
(b)
for B being big, where in each artificial data set X̃ a specific element in
X may occur multiple or no times. From each artificial data set the desired
quantity is calculated so that we have B samples κ(1) , . . . , κ(B) originating
from the bootstrapped data sets. An estimate of the standard error can then
be calculated as
v !2
B B
u
u 1 X 1 X 0
SEB (κ) = t κ(b) − κ(b ) . (3.10)
B−1 B 0
b=1 b =1

Small standard error implies that the original data set X is large enough to
produce a precise solution.

3.2.1 Statistical Uncertainty in the Portfolio Optimization

Problem
In this section I apply the bootstrap strategy when solving (3.1) with log-
returns simulated from the normal copula with hybrid GPD-Empirical-GPD
marginal distributions so that standard errors for each of the weights in Table
3.4 can be estimated. In this case what is referred to as the original data set
X would be a set R consisting of D log-return vectors R simulated from the
¯
normal copula. Drawing with replacement one element from X would further
correspond to drawing with replacement one log-return vector R from R and
¯
the desired quantity κ that should be calculated for each new artificial data
set R̃(b) is the optimal weight vector w(b) and portfolio Expexted Shortfall
¯
(b)
ESp (X). This results in B samples of optimal weight vectors w(1) , . . . , w(B)
(1) (B)
and portfolio Expected Shortfall ESp (X), . . . , ESp (X) and the standard
errors can then be calculated by using (3.10).

43
 In this application, I let B = 1000. With D = 15, 000 simulated log-
returns in the original data set R this means that I should draw with re-
¯
placement 15, 000 samples of log-return vectors, calculate the optimal solu-
tion to (3.1) and repeat the procedure 1000 times. The standard errors for
each asset weight wi can then be calculated with (3.10). Table 3.5 summa-
rizes each of the standard errors when conducting the described procedure
for both box and ellipsoidal uncertainty parameters so that standard errors
are estimated for every weight and the Expected Shortfall in Table 3.4.

Table 3.5: Standard errors calculated with the bootstrap procedure when
solving (3.1) with original log-returns simulated from the normal copula with
hybrid GPD-Empirical-GPD marginal distributions. Left column with box
uncertainty parameters. Right column with ellipsoidal uncertainty parame-
ters.
Box uncertainty Ellipsoidal uncertainty
Asset name Standard error Standard error
AstraZeneca 0 0.0019
Ericsson A 0 0
Hennes & Mauritz B 0.0148 0.0016
ICA Gruppen 0.0111 0.0024
Nordea Bank 0 0.0004
SAS 0 0
SSAB A 0 0
Swedish Match 0.0078 0.0037
TeliaSonera 0 0.0001
Volvo 0 0.0001
Total Bond Index 0.0092 0.0018
Expected Shortfall 6.6754 · 10−4 1.7177 · 10−4

A short analysis of the results is presented in Section 4.1.2 together with

a comparison of the asymmetric solutions in Table 3.4 with the solutions in
Table 3.1.

44
Chapter 4

Analysis and Conclusions

In this chapter the results found in the previous chapter are analyzed. I
also give comments on the interpretation of robust optimization as worst-
case scenario based optimization. The thesis project ends with comments on
alternative approaches that could have been made and comments on areas
of further investigation is mentioned.

4.1 Analysis of Results

In this section the results obtained throughout the thesis project are analyzed
and compared to gain knowledge of how the different model assumptions
investigated affect the optimal solution to the robust portfolio optimization
problem with Expected Shortfall.
An initial observation valid for all solutions is that even though 11 assets
are included in the reference portfolio, only a few of them are actually in-
vested in. This could initially be seen as counter-intuitive since the resulting
portfolio would be more diversified if the initial capital were spread across
more assets and the investor could become exposed to less risk. The key
to understand this behavior is however to recall that with highly correlated
assets the idea of a well diversified portfolio lose its meaning. As an example,
consider a portfolio consisting of two assets with correlation 1 but different
expected log-returns. In that case the risk would not decrease if half the cap-
ital were allocated in each of the assets since they are perfectly correlated.
It would hence be better to invest everything in the asset with the higher
expected log-return. The same principle holds for the reference portfolio in
this thesis project and diversifying the initial capital more than suggested by
the optimal solution would decrease the expected portfolio log-return more
than the decrease in Expected Shortfall and make the solution sub-optimal.
If a more distributed portfolio allocation is preferred by the investor then
additional lower and upper bounds on the weights could be added to the
portfolio optimization problem formulation. As an example, pension funds

45
often have restrictions on the portfolio weights in different asset classes and
industries.

4.1.1 Elliptically distributed log-returns

Here the different solutions with elliptically distributed log-returns will be
compared and analyzed. More specifically, the benchmark solution in Ta-
ble 1.2 and the solutions in Table 2.1 and Table 3.1 through Table (3.3)
are investigated. Since portfolio optimization with Expected Shortfall is
directly related to Markowitz mean-variance optimization with elliptically
distributed log-returns I may compare the robust solutions to (2.16) to the
benchmark solution to analyze the effect of model uncertainty in portfolio
optimization.
The first observation that I make is that with box uncertainty parame-
ters the robust solutions resembles the benchmark solution more than with
ellipsoidal uncertainty parameters. This is seen by observing that the solu-
tions in the left columns of Table 3.1 and Table 3.2 have similar structure as
the benchmark solution in Table 1.2 while the solutions in the right columns
of Table 3.1 and Table 3.3 differs more from the benchmark solution. With
box uncertainty parameters the initial capital is invested in the same as-
sets as in the benchmark solution and the only difference is that the robust
solutions are a bit more diversified. With ellipsoidal uncertainty sets the
wast majority of the initial capital is invested in the Total Bond Index and
approximately one percent of the capital is diversified across a few other as-
sets. Note that this result dependents on how much µmin and Σmax deviate
from the empirically estimated parameters where larger deviations lead to
larger solution deviation from the benchmark solution. I can therefore con-
clude from the optimal solutions that the ellipsoidal uncertainty parameters
deviate more from the empirically estimated parameters than the box un-
certainty parameters, which is confirmed by comparing the numerical values
in Appendix A. The results from robust optimization with box uncertainty
parameters can be seen as a stress test of the optimization problem where
the investor has a more conservative view on the market than the historical
data suggests, while robust optimization with ellipsoidal uncertainty param-
eters could be seen as exposing the portfolio optimization problem to a large
financial crisis. A conclusion independent of the particular parameters used
in this thesis project is that portfolio optimization with Expected Shortfall is
sensitive to parameter uncertainty. This can be concluded since the robust
optimal solutions deviate notably from the benchmark solution regardless
of which worst-case scenario parameters that are used. If the optimization
problem would had been insensitive to parameter uncertainty then the robust
solutions would have been close to each other and close to the benchmark
solution. It is therefore of great importance to take into consideration uncer-
tainties in parameters when solving a portfolio optimization problem. Quite

46
small variations in parameters can have huge impact on the optimal solution
and the risk exposure.
With µmin and Σmax held fixed, the optimal robust solutions seem to be
almost independent of the underlying log-return distribution. This can for
instance be seen by comparing the left (right) column of Table 3.1 with Table
3.2 (3.3) or equivalently by comparing the reference solutions in Appendix C.
The reference solutions suggest that the optimal solution is almost indepen-
dent of the multivariate elliptical distribution used for modeling log-returns.
The multivariate distribution model only have significant impact on the Ex-
pected Shortfall where Student’s t distributed log-returns yield larger risk
than normal distributed log-returns. These two observations should come
as no surprise. The constraint wT µ ≥ θ is independent of the log-return
distribution and the optimal weight vector is insensitive of the log-return
distribution. On the contrary, the risk is directly dependent on the distri-
bution model since fatter tails implies a greater probability of encountering
extremely negative (and positive) log-returns which increases the Expected
Shortfall. The Student’s t distribution with low degrees of freedom have fat-
ter tails than the normal distribution and therefore Student’s t distributed
log-returns imply larger Expected Shortfall. Additionally, as the degrees of
freedom increases the Expected Shortfall should converge to the Expected
Shortfall for normal distributed log-returns. This behavior is verified by solv-
ing (3.1) with Student’s t distributed log-returns and plot Expected Shortfall
as function of the degrees of freedom. Figure 4.1 depicts this study with a
reference level of Expected Shortfall for normal distributed log-returns in-
cluded to observe the convergence of Expected Shortfall.

Expected Shortfall as function of degrees of freedom

Student's t distribution
0.1 Normal distribution

0.09

0.08

0.07
Expected Shortfall

0.06

0.05

0.04

0.03

0.02

0.01

0
5 10 15 20 25 30 35 40
Degrees of freedom

Figure 4.1: Expected Shortfall with Student’s t distributed log-returns is

converging for increasing degrees of freedom to Expected Shortfall with nor-
mal distributed log-returns.

47
 Until now the threshold for acceptable expected daily log-return has been
held constant to θ = 2.0242 · 10−4 (5.08% expected yearly log-return). Fig-
ure 4.2 presents robust solutions to (3.1) with box uncertainty parameters
and both multivariate normal and Student’s t distributed log-returns for
varying θ. As can be seen, the solutions are almost identical with small dif-
ferences originating from statistical uncertainty and small differences in the
risk aversion coefficient. Hence the choice of distribution has little impact
on the optimal weights regardless of the level of θ. As long as θ is feasible1
the optimal solution is affected very little by different elliptical distributions
and portfolio optimization with Expected Shortfall is insensitive to which
elliptical distribution model that is used.

Multivariate normal distributed log-returns Multivariate Student's t distributed log-returns with ν = 2.1
100 100

90 90

80 80
Optimal asset weight w i in percent

Optimal asset weight w i in percent

70 70
AstraZeneca AstraZeneca
Ericsson Ericsson
60 H&M 60 H&M
ICA ICA
Nordea Nordea
50 SAS 50 SAS
SSAB SSAB
Swedish Match Swedish Match
40 Telia 40 Telia
Volvo Volvo
OMRX Total Bond Index OMRX Total Bond Index
30 30

20 20

10 10

0 0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Threshold for yearly portfolio log-return (251 · θ) Threshold for yearly portfolio log-return (251 · θ)

Figure 4.2: Left: Optimal robust solutions to (3.1) with box uncertainty
parameters and multivariate normal distributed log-returns as function of
θ. Right: Corresponding solutions with multivariate Student’s t distributed
log-returns and 2.1 degrees of freedom.

Previously I concluded that Expected Shortfall depends on the log-return

distribution and the heavier tails of the distribution the larger Expected
Shortfall. It is further known that Expected Shortfall increases when θ in-
creases as a result of a less diversified portfolio. It would be interesting to
analyze how Expected Shortfall depends on both these factors simultaneously
and if some structure can be found in the dependence. Figure 4.3 visual-
ize Expected Shortfall when solving (3.1) for different elliptical distributions
with box uncertainty parameters.

1
If θ is too large then the constraint wT µ ≥ θ cannot be satisfied and the portfolio
optimization problem has no solution.

48
 Expected Shortfall vs. threshold for expected yearly portfolio log-return
0.2
Normal distribution
0.18 Student's t distribution, ν = 2.1
Student's t distribution, ν = 3.58
Student's t distribution, ν = 10
Student's t distribution, ν = 20
0.16

0.14
Expected Shortfall

0.12

0.1

0.08

0.06

0.04

0.02

0
0.01 0.02 0.03 0.04 0.05 0.06 0.07
Threshold for expected yearly portfolio log-return (251 · θ)

Figure 4.3: Expected Shortfall for problem (3.1) for different elliptically
distributed log-returns as function of the threshold for expected portfolio
log-return θ.

Two interesting observations can be made. The first is that given a log-
return distribution, Expected Shortfall increases marginally when the thresh-
old for acceptable expected yearly portfolio log-return is less than 3% and
afterwards it increases more pronounced. This behavior should be specific
for the reference portfolio used in this thesis project and for another reference
portfolio the behavior could differ. Looking at Figure 4.2 one notice that an
expected yearly portfolio log-return of 3% seems to be a breakpoint in the
optimal weight vector as well. It is stable and almost unchanged for expected
yearly portfolio log-returns less than 3% but then changes continuously for
larger values. This explains the behavior of Expected Shortfall. I conclude
that for the specific reference portfolio used in this thesis project the investor
can increase the acceptable expected yearly portfolio log-return up to 3% in
the robust portfolio optimization problem with box uncertainty parameters
without increasing the Expected Shortfall since the optimal asset allocations
are unchanged. With acceptable expected yearly portfolio log-return greater
than 3% the investor must accept that the Expected Shortfall will increase
more rapidly. The second interesting observation from Figure (4.3) is that
there seems to be some structure in the ratio ES Student’s t /ES Normal and the
figure looks like a folding fan. Since the optimal weight vector is almost un-
changed between different elliptical distributions, the dependence between
ES Student’s t and ES Normal for a given threshold θ must be possible to explain
from how Expected Shortfall is calculated analytically for elliptical distribu-
tions. Recalling that the portfolio log-return under elliptical distributions

49
can be written as (2.10), the portfolio Expected Shortfall is
√
ESp (wT R) = −wT µ + wT ΣwESp (W Z1 ), (4.1)

where W Z1 determines which standard elliptical distribution is used. Now,

since w is almost unchanged between different elliptical distributions and the
parameters µ and Σ are not dependent on the distribution, it is differences
in Expected Shortfall for standard elliptical distributions, ESp (W Z1 ), that
determines the ratio ES Student’s t /ES Normal . The ratio should therefore be
approximately proportional to ESpStandard Student’s t /ESpStandard Normal where
the numerator depends on the degrees of freedom.

4.1.2 Asymmetric Distributed Log-returns

I continue the analysis of the results by comparing the robust solutions be-
tween asymmetric and elliptically distributed log-returns. This allows me to
conclude what impact the implementation of portfolio optimization problem
(2.7) in favor of Markowitz mean-variance optimization problem (1.3) has,
since it can handle asymmetric distributions. If an asymmetric distribution
model have visible impact on the optimal solution then I can conclude that
problem (2.7) is important since it changes the investment strategy. Some-
thing that was not achieved by altering between elliptical distributions.
First of all I give a short comment on the GARCH(1,1) filtering and
generalized Pareto distribution fit to the log-return distribution tails. When
modeling historical log-returns, the stylized facts in Section 3.1.3.3 should
be accounted for in a good model. Modeling the log-returns directly with
an elliptical distribution may take into account the heavy tailed behavior
of the log-returns if the multivariate Student’s t distribution is used, but
for instance volatility clustering and significant serial autocorrelation is not
accounted for explicitly. By filtering the log-returns with a GARCH(1,1)
model, the heavy tailed behavior is captured, as well as volatility clustering
and serial correlation. Hence, more stylized facts are taken into considera-
tion which should result in a better model. A GARCH(1,1) model together
with generalized Pareto distribution models in the tails to account for asym-
metry is thus a much more accurate model than an elliptical distribution
model. The optimization results when simulating log-returns from a normal
copula with hybrid GPD-Empirical-GPD marginals should therefore be more
reliable than the results when simulating from elliptical distributions.
Since the log-returns are simulated from a normal copula, the robust so-
lutions in Table 3.4 will be compared to the solutions in Table 3.1 where
the log-returns have multivariate normal distribution. Comparing the ta-
bles shows that with both box and ellipsoidal uncertainty parameters the
Expected Shortfall is greater when simulating log-returns from the normal
copula. The result is expected. As has been discussed before, the multi-
variate normal distribution has not fat enough tails to be good at model-

50
ing historical log-return tails and therefore the Expected Shortfall calculated
with multivariate normal distribution will be smaller than the true Expected
Shortfall. Modeling the tails with generalized Pareto distributions solves this
problem and the tails are better modeled with fatter tails which results in
larger Expected Shortfall. In this sense, the Expected Shortfall in Table 3.4
should be thought of as closer to reality than the Expected Shortfall in Table
3.1.
It is not only the Expected Shortfall that changes when simulating log-
returns from a normal copula with asymmetric marginals instead of simu-
lating from a multivariate normal distribution. The optimal asset weights
change notably as well, seen by comparing the solutions in Table 3.1 and
Table 3.4. The general structure of the investments is very similar between
the tables, but quite large differences can be seen in the amount of capi-
tal invested when comparing individual assets. This behavior was not seen
when changing between elliptical distributions. It can for instance be seen
that with box uncertainty parameters a switch to simulating with the normal
copula results in an investment increase of approximately 4% in ICA Grup-
pen and 1% in Hennes & Mauritz B and a decrease of approximately 3% in
Swedish Match and 2% in the Total Bond Index. These can be significant
changes. For instance, if $1 million is invested then there is a shift of ad-
ditional $40, 000 invested in ICA Gruppen. The optimal solutions with box
uncertainty parameters differ more than with ellipsoidal uncertainty param-
eters but this is likely due to that with ellipsoidal uncertainty parameters the
optimal solution suggests that almost all initial capital should be invested
in one asset and this decision cannot be changed much without violating
the constraint wT µ ≥ θ. In Section 4.1.3 I make an attempt to explain the
differences in optimal solution by considering changes in each asset’s risk
contribution to the entire portfolio’s Expected Shortfall.
With elliptically distributed log-returns the optimal solutions to (3.1)
could be compared to the analytical solutions in Appendix C where (1.3) is
solved with risk aversion coefficient calculated by (2.11) to find an appropri-
ate sample size. In that case it was concluded that with D = 15, 000 log-
return samples the simulated solutions are close to the analytical solutions.
With hybrid GPD-Empirical-GPD distributed log-returns the solutions can-
not be compared with analytical solutions. Instead, standard errors to the
optimal solutions in Table 3.4 were calculated using the bootstrap method
with results presented in Table 3.5. I begin by analyzing the solutions when
box uncertainty parameters were used. By comparing the magnitudes of the
standard errors with the magnitudes of the asset weights I see that most
uncertainty lies in the Hennes & Mauritz B investment. The standard error
is larger than the investment and the true optimal weight might differ quite
much from the one suggested and it is not possible to reject at a 95% con-
fidence level that the true optimal investment is zero. In this sense, larger
sample size D of log-returns is desired to lower the standard error of the

51
optimal weight on that particular asset. On the other hand, the portfo-
lio Expected Shortfall has very small standard error so the uncertainty in
the Hennes & Mauritz B investment seems to have little influence on the
uncertainty in Expected Shortfall. In this sense larger sample size seems
unnecessary. I conclude that solving (3.1) with 15, 000 log-return samples
simulated from a normal copula with hybrid GPD-Empirical-GPD marginal
distributions yields acceptable accuracy, but increasing the sample size a bit
further could increase the accuracy in the Hennes & Mauritz B investment
if this is important to the investor.
When it comes to the results in the right columns of Table 3.4 and 3.5
where ellipsoidal uncertainty parameters have been used, the standard errors
points out that it is not possible to reject that the true optimal weights for
Hennes & Mauritz B and Swedish Match are zero. This means that the
optimal solution with ellipsoidal uncertainty parameters could be to invest
everything in the Total Bond Index. The result is easily interpreted. With
large volatility and correlations and small expected log-returns on the risky
assets in a portfolio, a conservative investor would, if he is forced to invest,
invest all his capital in the least risky asset, being the Total Bond Index. The
standard error for the portfolio Expected Shortfall is furthermore small and
I conclude that with ellipsoidal uncertainty parameters, 15, 000 log-return
samples simulated from a normal copula with hybrid GPD-Empirical-GPD
marginal distributions is large enough to obtain accurate solutions regardless
of which parameters are used.
In Section 4.1.1 I concluded that portfolio optimization with Expected
Shortfall is sensitive to uncertainties in the parameters µ and Σ. Now, af-
ter analyzing the optimal robust solutions with different types of log-return
distribution models, I will extend the general conclusion and say something
about sensitivity to distribution. It has already been seen that the optimal
weight vector is quite insensitive to different elliptical log-return distribu-
tions. However, changing from an elliptical to an asymmetric distribution
changes the optimal weight vector significantly. I therefore conclude that
portfolio optimization with Expected Shortfall is sensitive to distribution
uncertainty - if both elliptical and asymmetric distributions are included in
the distribution uncertainty set F. This is an interesting conclusions since
the robust optimization community has not put much focus on distribution
uncertainty. I can thus conclude that problem (2.7) indeed is important since
not only parameter uncertainty but also distribution uncertainty affects the
optimal solution.
The major drawback with simulating log-returns from a normal copula
with hybrid GPD-Empirical-GPD marginal distributions is that the investor
is required to use an algorithm such as Algorithm 1. The algorithm is po-
tentially more time consuming than simulating log-returns from a simple
multivariate elliptical distribution since the data must be treated in four
steps rather than just one. With many risk factors the additional steps 2-

52
4 in Algorithm 1 can potentially take relatively long time. Therefore, this
approach requires careful implementation for high frequency trading opti-
mization but works perfectly fine in this application when trades are made
once a day and can be used in favor of the simpler multivariate distribution
approach. The reasoning can be taken further and it can be concluded that
if it is of great importance to model the log-returns well, then portfolio opti-
mization with Expected Shortfall is appropriate to use since the log-returns
can be modeled by copulas with different marginal distributions. The cost
is that the optimization problem becomes computationally heavier, is more
time consuming and introduces statistical uncertainty since simulations has
to be made. If a fast optimization procedure is important then Markowitz
mean-variance optimization problem might be better to consider. An ex-
ception is if the reference portfolio includes non-linear assets where portfolio
optimization with Expected Shortfall must be used or the assets must be
approximated by linear functions.

4.1.3 Analysis of Solutions with Euler Risk Decomposition

Up to this point I have motivated the appearance of the optimal weight
vectors on basis of the expected log-return and volatility of each asset. Posi-
tive expected log-return and small volatility should make an asset attractive,
but with ellipsoidal uncertainty parameters, the optimal weight vector sug-
gests to invest a small part of the capital in assets with negative expected
log-return. In an attempt to explain this behavior, I will analyze the Euler
Allocations defined in section 2.1.3 together with the expected log-return
vector, having the marginal Sharpe ratio (2.9) in mind.
In this thesis project, all financial instruments are linear assets, i.e.
Fj (S(t)) = Sj (t) meaning that the sensitivities U = w and the Euler Allo-
cation defined in (2.3) for normal distributed log-returns is simplified to

∂ESp (X) ∂ESp (X)

= = λ(Φ−1 (q))(wT Σw)−1/2 Σw
∂U ∂w
Table 4.1 presents robust Euler Allocations in the case where w is solved
(box)
with box uncertainty parameters. The table also includes µmin on yearly
basis in percentage to make the analysis more convenient. For ellipsoidal
uncertainty parameters, the corresponding Euler Allocations are presented
in Table 4.2. Recall that the optimal weight vectors are given in Table 3.1.

53
Table 4.1: Euler Allocations calculated under the assumption of normal
distributed log-returns with box uncertainty parameters and the expected
yearly log-return in percent.
Multivariate normal distribution - Box uncertainty
(box)
Asset name ∂ESp (X)/∂wi µmin (% per year)
AstraZeneca 0.0139 3.3057
Ericsson A 0.0243 −7.7406
Hennes & Mauritz B 0.0201 4.6686
ICA Gruppen 0.0355 6.0308
Nordea Bank 0.0293 0.4732
SAS 0.0283 −30.1588
SSAB A 0.0314 −27.2710
Swedish Match 0.0461 7.2086
TeliaSonera 0.0196 −4.8182
Volvo 0.0306 −2.3136
Total Bond Index −0.0004 3.3135

Table 4.2: Euler Allocations calculated under the assumption of normal dis-
tributed log-returns with ellipsoidal uncertainty parameters and the expected
yearly log-return in percent.
Multivariate normal distribution - Ellipsoidal uncertainty
(ellipsoidal)
Asset name ∂ESp (X)/∂wi µmin (% per year)
AstraZeneca −0.0035 −11.4241
Ericsson A −0.0160 −42.7518
Hennes & Mauritz B −0.0131 −20.5889
ICA Gruppen −0.0072 −12.5113
Nordea Bank −0.0251 −39.4785
SAS −0.0193 −80.5988
SSAB A −0.0293 −73.9089
Swedish Match −0.0021 −6.9987
TeliaSonera −0.0121 −29.2233
Volvo −0.0260 −46.4973
Total Bond Index 0.0053 5.2738

Now recall that the optimal solution with box uncertainty parameters
according to Table 3.1 is to invest approximately 50% of the initial capital
in the Total Bond Index, approximately 35% in Swedish Match and the
remaining 15% in ICA Gruppen. Looking at the yearly expected log-return
one might wonder why Hennes & Mauritz B is not invested in, having larger
expected log-return than the Total Bond Index. The answer lies in the Euler
Allocations, where it is seen that the Total Bond Index works as a hedge
against all other assets in the reference portfolio, hence decreasing the total

54
portfolio Expected Shortfall.
With ellipsoidal uncertainty parameters, the optimal solution is accord-
ing to Table 3.1 to invest approximately 99% in the Total Bond Index and
the remaining 1% split between ICA Gruppen, Swedish Match and Hennes
& Mauritz B, all three having negative expected log-return. Investing in
assets having negative expected log-return could seem counter intuitive but
is explained by fact that the assets act as hedges against investments in the
Total Bond Index, as seen in Table 4.2. Hence, by allowing the expected
total portfolio log-return to decrease a little, still being larger than θ, by
investing 1% in assets with negative expected log-return the investor is able
to decrease the portfolio’s Expected Shortfall.
Equation (2.5) and (2.6) can be used to approximate the Euler Alloca-
tions when the log-returns are simulated from the normal copula with hybrid
GPD-Empirical-GPD marginal distributions. With p = 0.01 and sample size
D = 15, 000 it follows that V aRp (X) = L151 , the 151th largest simulated
loss. The Euler Allocation for asset i is then the mean of the 150 largest
losses for that asset. With box uncertainty parameters the Euler Allocations
are presented in Table 4.3 and with ellipsoidal uncertainty parameters the
Euler Allocations are presented in Table 4.4. Recall that the optimal weight
vectors are presented in Table 3.4.

Table 4.3: Euler Allocations calculated for log-returns simulated from a nor-
mal copula with hybrid GPD-Empirical-GPD marginal distributions with
box uncertainty parameters and the expected yearly log-return in percent.
Normal copula - Box uncertainty
(box)
Asset name DESp (X) (wi ) µmin (% per year)
AstraZeneca 0.0097 3.3057
Ericsson A 0.0164 −7.7406
Hennes & Mauritz B 0.0177 4.6686
ICA Gruppen 0.0360 6.0308
Nordea Bank 0.0246 0.4732
SAS 0.0185 −30.1588
SSAB A 0.0281 −27.2710
Swedish Match 0.0518 7.2086
TeliaSonera 0.0122 −4.8182
Volvo 0.0213 −2.3136
Total Bond Index −0.0003 3.3135

55
Table 4.4: Euler Allocations calculated for log-returns simulated from a nor-
mal copula with hybrid GPD-Empirical-GPD marginal distributions with
ellipsoidal uncertainty parameters and the expected yearly log-return in per-
cent.
Normal copula - Ellipsoidal uncertainty
(ellipsoidal)
Asset name DESp (X) (wi ) µmin (% per year)
AstraZeneca −0.0008 −11.4241
Ericsson A −0.0102 −42.7518
Hennes & Mauritz B −0.0097 −20.5889
ICA Gruppen −0.0040 −12.5113
Nordea Bank −0.0203 −39.4785
SAS −0.0147 −80.5988
SSAB A −0.0211 −73.9089
Swedish Match −0.0013 −6.9987
TeliaSonera −0.0089 −29.2233
Volvo −0.0219 −46.4973
Total Bond Index 0.0057 5.2738

Similar conclusions that were drawn from analyzing the Euler Allocations
with multivariate normal distributed log-returns can be drawn from Table
4.3 and Table 4.4 as well. More interesting is instead to see if the changes in
Euler Allocations can motivate the differences in optimal solutions between
Table 3.1 and Table (3.4). With ellipsoidal uncertainty parameters it is
hard to tell whether the differences in solutions depend on the change of
underlying distribution or being an artifact of statistical uncertainty so I
focus on the solutions with box uncertainty parameters. As was previously
noted, the investment in Swedish Match decreases by 3% when going from
the multivariate normal model to the normal copula model. Furthermore,
the Euler Allocation increases compared to the multivariate normal model.
This seems reasonable that if there is an increase in Euler Allocation, i.e.
an increase in the risk contribution, the consequence is that less capital is
invested in that asset for a risk averse investor. The same principle holds for
the investment changes for Hennes & Mauritz B and the Total Bond Index
as well. Surprisingly, the investment in ICA Gruppen increases by 4% when
switching to the normal copula even though the risk contribution increases
for that asset. This behavior goes against the intuitive conclusion that an
increase in risk contribution makes an asset less attractive.
I conclude that combining Euler Allocations and expected log-return,
having in mind the marginal Sharpe ratio, can be used as a great tool when
analyzing the structure behind optimal portfolio solutions but should be used
together with other methods to understand the entire structure.

56
4.2 Comments on Worst-Case Scenario Based Ro-
bust Optimization
This thesis project has, as commonly done in optimization, interpreted ro-
bust optimization as worst-case scenario based optimization, presented in
Section 3.1.1. This section discuss the effect of this interpretation on the
optimization problem.
Firstly, worst-case scenario based robust optimization should be seen as
a conservative optimization approach. When the worst-case scenarios from
the uncertainty sets M, S are used in the optimization problem, it means
that the investor has a more negative view on the market than what the
empirically estimated parameters suggests. The investor expects the log-
returns of the assets in his portfolio to be worse than it has been historically
and believes that the assets are more correlated than seen in the covariance
matrix. In short, the investor has a conservative view on the market and
optimizes his asset allocations according to his view on the market.
One negative aspect with worst-case scenario based robust optimization
is that the approach is very sensitive to outliers in the historical data. As an
example, with a 95% confidence interval used as uncertainty set, the worst-
case scenario is more extreme when outliers are present in the historical data
than what the worst-case scenario would be with no outliers. This results in
an even more conservative view on the market for the investor. One approach
to solve this problem is to search the data for outliers and remove them prior
to constructing the 95% confidence interval.
A positive aspect with the worst-case scenario based robust optimization
approach is that the solution is quite stable if re-solving the problem period-
ically as more historical data is available. This is positive in the sense that
the investor does not need to re-balance his asset allocations very often and
trading fees can be held low for the investor.
An alternative approach to worst-case scenario based robust optimization
could be to grid the uncertainty sets and solve the optimization problem for
each combination of realizations in each uncertainty set and then analyze how
the solution changes as function of parameter location within the uncertainty
sets. This approach would however be very time consuming and perhaps
computationally impossible to solve with many assets in the portfolio and
fine grids of the uncertainty sets.

4.3 More Comments and Further Investigation

The thesis project ends by a few remarks on some alternative approaches that
could have been used and discuss areas where further investigation could be
of interest.
(box)
In this thesis project the worst-case box uncertainty log-return µmin

57
is constructed as in (3.2) by subtracting 20% of each element in the vec-
tor. The result is then compared to the lower limit of the 95% confidence
interval to see whether the worst-case scenario is reasonable or not. An al-
(box)
ternative approach is to define µmin directly as the lower limit of the 95%
confidence interval. With the particular historical data in this thesis project
(box)
this approach resulted in that all elements in µmin were negative. This
yields a robust optimization problem without solutions since the constraint
wT µmin ≥ θ cannot be satisfied with long positions. In light of this, this
alternative approach was not used in the thesis project but could be more
intuitive to use in other applications or with other reference portfolios.
All solutions in this thesis project are based on the same reference portfo-
lio with assets listed in Table 1.1. It would however be interesting to analyze
how different reference portfolios consisting of other types of financial as-
sets affect the optimal solution. For instance, how is the optimal solution
affected if the amount of financial assets included in the reference portfolio
increases? It seems reasonable that larger reference portfolios would decrease
the Expected Shortfall since the optimization algorithm has more assets to
choose from. However, as has been seen in all solutions throughout this
thesis project, the optimal solution often turns out to be to investment in
a small sub group of the possible assets. Therefore, including more assets
in the reference portfolio does not necessarily decrease the risk and could
at worst result in a less time efficient program that can be crucial for the
investor. What can be said is that including more assets in the reference
portfolio does at least not increase the financial risk. The perhaps most
interesting area of further investigation would be to extend the reference
portfolio to include other types of financial assets such as foreign curren-
cies or non-linear financial derivatives such as options. This would introduce
possibilities of more complex hedging opportunities than possible with linear
assets. It is furthermore interesting because non-linear portfolios makes full
use of the risk measure Expected Shortfall since Markowitz mean-variance
optimization problem cannot handle non-linear financial assets. In that case
the non-linear assets have to be approximated by linear functions. Thirdly
it would be interesting from a practical point of view to include non-linear
assets since it is common that investors have non-linear assets in their port-
folios.
In this thesis project, the p-value has been held constant to 1% but can
of course be altered as well. Decreasing the p-value should however only
increase the Expected Shortfall or the investor must decrease the threshold
for acceptable expected log-return to keep the risk constant. Hence the
investor has to find a p-level where he feels confident with both the expected
portfolio log-return and the risk he is exposed to.
Another interesting area of further investigation is to study the effect of
different copulas on the optimal solution. In this thesis project, the normal

58
copula was used together with hybrid GPD-Empirical-GPD marginal dis-
tributions but the dependency structure between different log-returns could
just as well had been modeled by another copula. However, as was men-
tioned earlier, according to Skoglund and Nyström the choice of copula has
a second-order effect on the model accuracy when modeling market risk fac-
tors. If a Student’s t copula were used instead, the results had probably only
been marginally improved.
The last area that I will mention where further investigation could be
made is in the decision of the upper and lower thresholds for the generalized
Pareto distributions. In this thesis project the thresholds were chosen as the
90% and 10% empirical quantiles respectively but no investigation was made
to establish whether these thresholds were good representatives of where the
tails begin. A better and more consistent approach to find the thresholds is
to use a parametric estimation method such as the Hill’s tail-index estimator
[4]. This could improve the results a bit further but also make the program
more time consuming.
Different approaches taken in a portfolio optimization problem, whether
it is about which problem formulation to use, which models to use or which
parameters to use, is ultimately up to the investor to decide and a trade-
off between model accuracy and time efficiency/computational possibility
always has to be considered.

59
60
 Appendix A

Optimization Parameters

When solving the portfolio optimization problem (2.16) the parameters µ and
Σ are needed for instance in the process of simulating log-returns. Since the
portfolio lives for one day, µ is the expected daily log-return and Σ is the co-
variance matrix for daily log-returns. In the thesis project I begin by solving
the portfolio optimization problem with empirically estimated parameters
from historical data and then solve the robust portfolio optimization prob-
lem (3.1) after manipulating the empirical parameters to obtain two different
kinds of worst-case scenario parameters. The worst-case scenario box uncer-
tainty parameters are defined by (3.2) and (3.3) and the worst-case scenario
ellipsoidal uncertainty parameters are defined by (3.4) and (3.5). In this
Appendix I clarify the parameters by writing out their explicit numerical
values.
The numerical values of the empirical parameters estimated for the time
period January 2, 2007 until January 22, 2016 from daily log-return data to
the assets in the reference portfolio are

−4
1.6501 T
 
µ
b = 10 1.6463 −2.5699 2.3250 3.0034 0.2357 −10.0129 −9.0541 3.5899 −1.5997 −0.7681
 2.2076 0.8564 0.7462 0.5434 0.8873 0.9597 0.9827 0.6127 0.7489 0.8089 −0.0201
 0.8564 6.7548 1.4816 0.8375 2.3592 1.8613 2.5425 0.8313 1.3073 2.5546 −0.0688
 0.7462
 1.4816 2.6549 0.8594 1.9553 1.7679 2.1897 0.7987 1.2000 2.0837 −0.0606
 0.5434
 0.8375 0.8594 3.2819 1.2290 1.2987 1.4293 0.4836 0.8364 1.3421 −0.0424
 0.8873 2.3592 1.9553 1.2290 4.9191 2.7372 3.5812 1.0444 1.8956 3.3907 −0.1052
b = 10−4  0.9597
 
Σ  1.8613 1.7679 1.2987 2.7372 13.5689 3.0979 0.7071 1.5030 2.9935 −0.0821 .

 0.9827
 2.5425 2.1897 1.4293 3.5812 3.0979 8.3021 1.0916 1.9876 4.4259 −0.1217 

 0.6127
 0.8313 0.7987 0.4836 1.0444 0.7071 1.0916 2.4675 0.7539 1.0733 −0.0271 

 0.7489 1.3073 1.2000 0.8364 1.8956 1.5030 1.9876 0.7539 2.6576 1.9369 −0.0538
−0.1094
 
0.8089 2.5546 2.0837 1.3421 3.3907 2.9935 4.4259 1.0733 1.9369 5.8128
−0.0201 −0.0688 −0.0606 −0.0424 −0.1052 −0.0821 −0.1217 −0.0271 −0.0538 −0.1094 0.0195

The numerical values of the box uncertainty parameters defined by (3.2) and
(3.3) are

61
 (box) −4
1.3201 T
 
µmin = 10 1.3170 −3.0839 1.8600 2.4027 0.1885 −12.0155 −10.8649 2.8719 −1.9196 −0.9217
 3.3115 1.2960 0.9321 0.8164 1.0944 1.7779 1.2788 0.8737 0.9871 0.9052 −0.0278
 1.2960 8.4445 2.3581 1.4760 4.2318 3.2969 4.3982 1.5645 2.1447 4.3579 −0.0883
 0.9321
 2.3581 3.9946 1.3956 3.1897 3.1002 3.6072 1.2078 1.7751 3.4598 −0.0772
 0.8164
 1.4760 1.3956 5.1444 1.9984 2.4270 2.4435 0.7163 1.3759 2.4078 −0.0562
 1.0944 4.2318 3.1897 1.9984 8.9431 5.1257 6.1330 1.8146 3.0073 6.0763 −0.1361
(box) −4  
Σmax = 10  1.7779
 3.2969 3.1002 2.4270 5.1257 18.4093 5.4142 1.4777 2.9255 5.3976 −0.1051.
 1.2788
 4.3982 3.6072 2.4435 6.1330 5.4142 13.4753 1.8367 2.9929 7.6285 −0.1678
 0.8737
 1.5645 1.2078 0.7163 1.8146 1.4777 1.8367 3.7902 1.1495 1.7437 −0.0398
 0.9871 2.1447 1.7751 1.3759 3.0073 2.9255 2.9929 1.1495 4.6483 3.0786 −0.0602
−0.1439
 
0.9052 4.3579 3.4598 2.4078 6.0763 5.3976 7.6285 1.7437 3.0786 9.6670
−0.0278 −0.0883 −0.0772 −0.0562 −0.1361 −0.1051 −0.1678 −0.0398 −0.0602 −0.1439 0.0234

The numerical values of the ellipsoidal uncertainty parameters defined by

(3.4) and (3.5) are

(ellipsoidal) −4
2.1011 T
 
µmin = 10 −4.5514 −17.0326 −8.2027 −4.9846 −15.7285 −32.1111 −29.4458 −2.7883 −11.6428 −18.5248
 4.5494 1.7649 1.5377 1.1198 1.8286 1.9776 2.0250 1.2627 1.5434 1.6669 −0.0413
 1.7649 13.9200 3.0532 1.7258 4.8617 3.8357 5.2395 1.7130 2.6940 5.2644 −0.1417
 1.5377
 3.0532 5.4712 1.7710 4.0294 3.6433 4.5124 1.6459 2.4729 4.2941 −0.1248
 1.1198
 1.7258 1.7710 6.7631 2.5327 2.6763 2.9455 0.9966 1.7237 2.7658 −0.0873
 1.8286 4.8617 4.0294 2.5327 10.1372 5.6408 7.3801 2.1522 3.9064 6.9874 −0.2168
(ellipsoidal) −4  
Σmax = 10  1.9776
 3.8357 3.6433 2.6763 5.6408 27.9625 6.3842 1.4572 3.0972 6.1689 −0.1693.
 2.0250
 5.2395 4.5124 2.9455 7.3801 6.3842 17.1087 2.2496 4.0960 9.1208 −0.2508
 1.2627
 1.7130 1.6459 0.9966 2.1522 1.4572 2.2496 5.0849 1.5537 2.2118 −0.0558
 1.5434 2.6940 2.4729 1.7237 3.9064 3.0972 4.0960 1.5537 5.4767 3.9915 −0.1109
−0.2254
 
1.6669 5.2644 4.2941 2.7658 6.9874 6.1689 9.1208 2.2118 3.9915 11.9787
−0.0413 −0.1417 −0.1248 −0.0873 −0.2168 −0.1693 −0.2508 −0.0558 −0.1109 −0.2254 0.0402

By comparing the different parameters it is easy to see that the worst-case

scenario parameters indeed are worse than the empirically estimated param-
eters. The worst-case scenario expected log-return vectors are more negative
than the empirically estimated expected log-return vector and the worst-
case scenario covariance matrices have larger pairwise covariances than the
empirically estimated covariance matrix. It is also easy to observe that the el-
lipsoidal uncertainty parameters are more extreme than the box uncertainty
parameters. Hence, if the box uncertainty parameters are seen as stress tests
of the empirically estimated parameters, then the ellipsoidal uncertainty pa-
rameters could be seen as parameters from a large global financial crisis.

62
Appendix B

Convergence as Function of
Sample Size D

This chapter studies how large the sample size D of simulated log-returns
must be for the solutions to the portfolio optimization problem (2.16) to
be precise and have small statistical uncertainty. While holding all other
parameters constant, let us first increase the sample size until the solution
weight vector no longer have visible oscillations and then find the sample
size D∗ where further increments no longer improve the solution accuracy
much. Figure B.1 presents the solution weight vector to the problem (2.16)
as function of sample size D. The simulated log-returns come from a mul-
tivariate normal distribution with empirically estimated parameters µ b and
Σ.
b

Asset weights development as function of sample size D

100
AstraZeneca
Ericsson
90
H&M
ICA
Nordea
80
SAS
SSAB
Optimal asset weight w i in percent

Swedish Match
70
Telia
Volvo
OMRX Total Bond Index
60

50

40

30

20

10

0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Sample size D

Figure B.1: The optimal weight vector as function of the sample size D of
simulated log-returns.

63
 From the figure it seems as the solution converges quite fast, perhaps
as early as for D∗ = 1000 samples. However, if convergence of Expexted
Shortfall as function of sample size is investigated the behavior is different.
This behavior is presented in Figure B.2 and depicts approximate 95% confi-
dence intervals for Expected Shortfall when solving (2.16) 100 times each for
increasing sample size. With multivariate normal distributed log-returns the
Expected Shortfall seems to have converged with narrow confidence interval
for approximately D∗ = 10, 000 and for Student’s t distributed log-returns
with 2.1 degrees of freedom the threshold is approximately D∗ = 15, 000
samples. Both observations are far above the initial one of D∗ = 1000. For
an investor, accuracy in both optimal weight vector and Expected Shortfall
is important and the sample size must be chosen sufficiently large for both
factors to be accurate estimates of the true values. Therefore, when solving
the portfolio optimization problem I should use a step size of at least 10, 000
when simulating log-returns from the multivariate normal distribution and
15, 000 when simulating from the Student’s t distribution. For simplicity I
use D = 15, 000 in both cases throughout this thesis project.

Figure B.2: 95% confidence intervals for Expected Shortfall as function of

the sample size D for (left) multivariate normal and (right) multivariate
Student’s t distributed log-returns with 2.1 degrees of freedom.

64
Appendix C

Reference Solutions

This Appendix presents reference solutions to the robust portfolio opti-

mization problem (3.1) when instead Markowitz mean-variance optimization
problem (1.3) is solved. By doing this, statistical uncertainty is avoided. The
two optimization problems are connected through the risk aversion coefficient
c which for elliptically distributed log-returns is calculated as (2.11). The ref-
erence portfolio Expected Shortfall in the last row of each table is calculated
using (2.17) and (2.18).

C.1 Normal Distributed Log-returns

Under the assumption of multivariate normal distributed log-returns, the
risk aversion coefficient c in Markowitz mean-variance optimization problem
is already calculated in (2.12) to be c = 5.33. With the same worst-case
scenario parameters µmin and Σmax used to obtain the two solutions in Ta-
ble 3.1, the corresponding reference solutions to Markowitz mean-variance
problem (1.3) are presented in Table C.1.

65
Table C.1: Solutions to (1.3) with multivariate normal distributed log-
returns. Left column with box uncertainty parameters and right column
with ellipsoidal uncertainty parameters.
Box uncertainty Ellipsoidal uncertainty
Asset name Weight Weight
AstraZeneca 0 0.0015
Ericsson A 0 0.0004
Hennes & Mauritz B 0 0.0006
ICA Gruppen 0.1479 0.0015
Nordea Bank 0 0.0004
SAS 0 0
SSAB A 0 0.0002
Swedish Match 0.3506 0.0029
TeliaSonera 0 0.0004
Volvo 0 0.0004
Total Bond Index 0.5015 0.9917
Expected Shortfall 0.0213 0.0052

C.2 Student’s t Distributed Log-returns

Under the assumption of multivariate Student’s t distributed log-returns, the

risk aversion coefficient c is defined by (2.11), and the Student’s t quantile
function depends on the degrees of freedom. Table C.2 presents the different
values of the risk aversion coefficient for ν = 2.1, 3.58, 10, 20.

Table C.2: The risk aversion coefficient calculated numerically using (2.11)
with the Student’s t quantile function for different degrees of freedom.
Degrees of freedom ν 2.1 3.58 10 20
Risk aversion coefficient c 25.2278 11.5312 6.7265 5.9538

Note that as the degrees of freedom increases, the risk aversion coefficient de-
creases. This behavior is expected and c should converge to the risk aversion
coefficient for normal distributed log-returns, i.e. converge to c = 5.33.
With the same box uncertainty parameters that were used to obtain the
solutions in Table 3.2, the corresponding reference solutions to Markowitz
mean-variance optimization problem are presented in Table C.3. Table C.4
presents in turn the reference solutions to the simulated solutions with ellip-
soidal uncertainty parameters in Table 3.3.

66
Table C.3: Solutions to (1.3) with multivariate Student’s t distributed log-
returns with different degrees of freedom and box uncertainty parameters.
ν = 2.1 ν = 3.58 ν = 10 ν = 20
Asset name Weight Weight Weight Weight
AstraZeneca 0 0 0 0
Ericsson A 0 0 0 0
Hennes & Mauritz B 0 0 0 0
ICA Gruppen 0.1479 0.1479 0.1479 0.1479
Nordea Bank 0 0 0 0
SAS 0 0 0 0
SSAB A 0 0 0 0
Swedish Match 0.3506 0.3506 0.3506 0.3506
TeliaSonera 0 0 0 0
Volvo 0 0 0 0
Total Bond Index 0.5015 0.5015 0.5015 0.5015
Expected Shortfall 0.0914 0.0447 0.0265 0.0235

Table C.4: Solutions to (1.3) with multivariate Student’s t distributed log-

returns with different degrees of freedom and ellipsoidal uncertainty param-
eters.
ν = 2.1 ν = 3.58 ν = 10 ν = 20
Asset name Weight Weight Weight Weight
AstraZeneca 0.0001 0.0012 0.0014 0.0015
Ericsson A 0 0.0003 0.0004 0.0004
Hennes & Mauritz B 0.0027 0.0007 0.0006 0.0006
ICA Gruppen 0.0035 0.0020 0.0016 0.0016
Nordea Bank 0.0001 0.0004 0.0004 0.0004
SAS 0 0 0 0
SSAB A 0 0.0001 0.0002 0.0002
Swedish Match 0.0044 0.0036 0.0030 0.0029
TeliaSonera 0 0.0003 0.0003 0.0003
Volvo 0 0.0004 0.0004 0.0004
Total Bond Index 0.9892 0.9910 0.9916 0.9916
Expected Shortfall 0.0238 0.0112 0.0065 0.0058

67
68
Appendix D

GARCH and GPD Parameters

The generalized Pareto distribution Gγ,β (x) is given by

 1
γx − γ

Gγ,β (x) = 1 − 1 + , x≥0
β

for some shape parameter γ > 0 and scale parameter β > 0 and

x
−β
Gγ,β (x) = 1 − e , x≥0

if γ = 0.

Modeling excesses in the tail of a distribution over a suitably high thresh-

old u with a generalized Pareto distribution is possible if the underlying data
is independent and identically distributed. The log-returns in the reference
portfolio can only be assumed to be weakly dependent and close to inde-
pendent and identically distributed and typically suffers from the stylized
facts presented in Section 3.1.3.3. To obtain independent and identically dis-
tributed data the historical log-returns to each asset in the reference portfolio
are filtered with fitted GARCH(1,1) models, also defined in Section 3.1.3.3.
The GARCH(1,1) parameters were numerically estimated with the Maxi-
mum Likelihood method using the function estimate in Matlab and are
presented in Table D.1 together with other statistical quantities of interest.

69
Table D.1: Maximum Likelihood estimated parameters to the GARCH(1,1)
models used for filtering historical log-returns to standardized residuals.
Asset name Parameter estimate Standard error t statistic
AstraZeneca α0 = 7.28056 · 10−6 5.95667 · 10−7 12.2225
α1 = 0.0931352 0.00721758 12.9039
β1 = 0.875473 0.00857937 102.044
Ericsson A α0 = 0.000272561 2.47196 · 10−5 11.0261
α1 = 0.165719 0.0223635 7.41023
β1 = 0.411872 0.0546948 7.53037
Hennes & Mauritz B α0 = 1.24182 · 10−5 1.52487 · 10−6 8.14376
α1 = 0.0763801 0.00970769 7.868
β1 = 0.874378 0.0140866 62.0714
ICA Gruppen α0 = 1.56923 · 10−5 2.08814 · 10−6 7.51496
α1 = 0.168988 0.0114856 14.713
β1 = 0.794041 0.0133304 59.566
Nordea Bank α0 = 4.66648 · 10−6 1.23123 · 10−6 3.7901
α1 = 0.0646265 0.00657526 9.82874
β1 = 0.923464 0.00792685 116.498
SAS α0 = 4.38621 · 10−5 5.27758 · 10−6 8.31103
α1 = 0.118331 0.0067498 17.5311
β1 = 0.858769 0.00767209 111.934
SSAB A α0 = 6.72565 · 10−6 8.38453 · 10−7 8.0215
α1 = 0.041997 0.00414664 10.1279
β1 = 0.948355 0.00437622 216.707
Swedish Match α0 = 1.22517 · 10−5 1.64289 · 10−6 7.45739
α1 = 0.103236 0.0101418 10.1793
β1 = 0.847721 0.0149104 56.8544
TeliaSonera α0 = 4.23416 · 10−6 6.64369 · 10−7 6.37321
α1 = 0.079082 0.0052433 15.0825
β1 = 0.909637 0.00517679 175.715
Volvo α0 = 7.43774 · −6 1.28612 · −6 5.78307
α1 = 0.0647348 0.00877252 7.37927
β1 = 0.921605 0.00983831 93.6752
Total Bond Index α0 = 2 · 10−7 9.32652 · 10−8 2.14442
α1 = 0.0865343 0.0102208 8.46645
β1 = 0.813981 0.00897679 90.6761

Figure D.1 depicts realizations and histograms of the standardized resid-

uals in the two upper plots and sample autocorrelation functions of squared
log-returns and squared standardized residuals in the two lower plots for each
asset in the reference portfolio. According to stylized fact 2 for financial risk
factor time series the squared log-returns should show sign of autocorrelation
which can be observed for every asset in the lower left plots. After filtering
the log-returns the goal is to have squared residuals without autocorrelation.
This seems to hold for the standardized residuals by observing the lower right
plot for every asset and hence it is concluded that the GARCH(1,1) filtra-
tion has been successful. Looking now at the histograms of the standardized
residuals, the distributions are leptokurtic, or heavy tailed, and hence the
standard normal distribution is not appropriate to use as model distribution.

70
71
Figure D.1: Each sub figure consists of realizations and histograms of the standardized residuals and sample autocorrelation
functions of squared log-returns and squared standardized residuals.
 The same conclusion can be drawn by observing Figure D.2 which shows
Quantile-Quantile plots for the empirical residual quantile function plotted
against the standard normal quantile function. A good fit should produce
a straight line and a reverted S-shaped curve indicates that the tails are
heavier than for the standard normal distribution. As can be seen, reverted
S-shapes appear in every plot indicating that the standardized residuals have
distributions with fatter tails than the standard normal distribution. Notice
also that the S-shapes are asymmetric, meaning that the two tails of the
residual distribution are of different size and length. Therefore, an elliptical
distribution with fat tails, for instance the Student’s t distribution, would not
capture the entire empirical residual distribution since elliptical distributions
are symmetric. Thus generalized Pareto distributions are appropriate to
model the tails of the residual distributions since this can give an asymmetric
distribution.

72
 QQ Plot of Standardized Residuals to AstraZeneca versus Standard Normal QQ Plot of Standardized Residuals to Ericsson A versus Standard Normal QQ Plot of Standardized Residuals to Hennes & Mauritz B versus Standard Normal QQ Plot of Standardized Residuals to ICA Gruppen versus Standard Normal
8 25 6 6

6 20
4
4

4 15
2
2
2 10
0

0 5 0

-2
-2 0
-2

Quantiles of Input Sample

Quantiles of Input Sample
Quantiles of Input Sample
Quantiles of Input Sample
-4
-4 -5

-4
-6
-6 -10

-8 -15 -8 -6
-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4
Standard Normal Quantiles Standard Normal Quantiles Standard Normal Quantiles Standard Normal Quantiles

QQ Plot of Standardized Residuals to Nordea Bank versus Standard Normal QQ Plot of Standardized Residuals to SAS versus Standard Normal QQ Plot of Standardized Residuals to SSAB A versus Standard Normal QQ Plot of Standardized Residuals to Swedish Match versus Standard Normal
6 8 8 6

5
6
6
4
4
4
3 4
2
2
2
2

1 0 0

0
0
-2
-2

Quantiles of Input Sample

Quantiles of Input Sample
Quantiles of Input Sample
-1 -2 Quantiles of Input Sample
-4

73
-2
-4
-4
-6
-3

-4 -8 -6 -6
-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4
Standard Normal Quantiles Standard Normal Quantiles Standard Normal Quantiles Standard Normal Quantiles

QQ Plot of Standardized Residuals to TeliaSonera versus Standard Normal QQ Plot of Standardized Residuals to Volvo versus Standard Normal QQ Plot of Standardized Residuals to OMRX Total Bond Index versus Standard Normal
6 8 8

4 6
6

2 4
4

0 2

-2 0

Quantiles of Input Sample

Quantiles of Input Sample
Quantiles of Input Sample

-4 -2

-2
-6 -4

-8 -4 -6
-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4
Standard Normal Quantiles Standard Normal Quantiles Standard Normal Quantiles

Figure D.2: Each sub figure consists of Quantile-Quantile plots of the empirical quantile function for the standardized residuals
versus the standard normal quantile function.
 Table D.2 presents Maximum Likelihood estimated parameters to the
modeled generalized Pareto distributions with tail thresholds chosen as uh =
Fn−1 (0.90) and ul = Fn−1 (0.10) respectively for each asset in the reference
portfolio.

Table D.2: Maximum Likelihood estimated generalized Pareto distribution

parameters to standardized residuals calculated for each asset in the reference
portfolio. The thresholds are chosen as uh = Fn−1 (0.90) and ul = Fn−1 (0.10).
Asset name Threshold γ
b β
b
AstraZeneca ul = −1.0957 0.1348 0.5897
uh = 1.0984 −0.0127 0.7010
Ericsson A ul = −0.9369 0.1829 0.5844
uh = 0.9762 0.2569 0.4286
Hennes & Mauritz B ul = −1.1497 0.1112 0.5477
uh = 1.1377 0.0701 0.6366
ICA Gruppen ul = −1.1299 −0.0201 0.6120
uh = 1.2474 0.0527 0.6210
Nordea Bank ul = −1.1740 −0.0745 0.5647
uh = 1.2112 0.0104 0.6627
SAS ul = −1.0576 0.1897 0.5085
uh = 1.1622 0.0712 0.6935
SSAB A ul = −1.1388 0.0557 0.5270
uh = 1.2375 0.1244 0.5508
Swedish Match ul = −1.1020 0.1509 0.5094
uh = 1.1974 −0.0139 0.6881
TeliaSonera ul = −1.0573 0.2619 0.4897
uh = 1.1462 0.0709 0.6137
Volvo ul = −1.1655 −0.0123 0.5449
uh = 1.2166 0.0637 0.5970
Total Bond Index ul = −1.2251 0.0695 0.5012
uh = 1.1286 0.2358 0.4493

The thresholds ul and uh and parameters γ b and βb are used in Section

3.1.3.4 when solving the robust optimization problem (3.1). They are used
in Algorithm 1 step 3 when calculating Fi−1 being the inverse of the hybrid
GPD-Empirical-GPD distribution function (3.7).
Figure D.3 displays the generalized Pareto distributions with parameters
in Table D.2 together with the empirical cumulative distribution for the
standardized residuals. The estimated tail distributions are found to fit the
empirical residual data well in the tails and the hybrid GPD-Empirical-GPD
distributions will reflect the entire residual distribution much more accurate
than the standard normal or Student’s t distribution.

74
 Cumulative distribution function for standardized residuals to AstraZeneca Cumulative distribution function for standardized residuals to Ericsson A Cumulative distribution function for standardized residuals to Hennes & Mauritz B Cumulative distribution function for standardized residuals to ICA Gruppen
1 1 1 1

0.9 0.9 0.9 0.9

0.8 0.8 0.8 0.8

0.7 0.7 0.7 0.7

0.6 0.6 0.6 0.6

0.5 0.5 0.5 0.5

F(x)
F(x)
F(x)
F(x)
0.4 0.4 0.4 0.4

0.3 0.3 0.3 0.3

0.2 0.2 0.2 0.2

0.1 0.1 0.1 0.1

0 0 0 0
-8 -6 -4 -2 0 2 4 6 8 -15 -10 -5 0 5 10 15 20 25 -8 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
Standardized residuals Standardized residuals Standardized residuals Standardized residuals

Cumulative distribution function for standardized residuals to Nordea Bank Cumulative distribution function for standardized residuals to SAS Cumulative distribution function for standardized residuals to SSAB A Cumulative distribution function for standardized residuals to Swedish Match
1 1 1 1

0.9 0.9 0.9 0.9

0.8 0.8 0.8 0.8

0.7 0.7 0.7 0.7

0.6 0.6 0.6 0.6

0.5 0.5 0.5 0.5

F(x)
F(x)
F(x)
F(x)
0.4 0.4 0.4 0.4

0.3 0.3 0.3 0.3

75
0.2 0.2 0.2 0.2

0.1 0.1 0.1 0.1

0 0 0 0
-4 -3 -2 -1 0 1 2 3 4 5 6 -8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6
Standardized residuals Standardized residuals Standardized residuals Standardized residuals

Cumulative distribution function for standardized residuals to TeliaSonera Cumulative distribution function for standardized residuals to Volvo Cumulative distribution function for standardized residuals to OMRX Total Bond Index
1 1 1

0.9 0.9 0.9

0.8 0.8 0.8

0.7 0.7 0.7

0.6 0.6 0.6

0.5 0.5 0.5

F(x)
F(x)
F(x)

0.4 0.4 0.4

0.3 0.3 0.3

0.2 0.2 0.2

0.1 0.1 0.1

0 0 0
-8 -6 -4 -2 0 2 4 6 -4 -2 0 2 4 6 8 -6 -4 -2 0 2 4 6 8
Standardized residuals Standardized residuals Standardized residuals

Figure D.3: Empirical cumulative distribution functions for the standardized residuals for each asset in the reference portfolio
together with estimated generalized Pareto distributions in the distribution tails.
76
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