Implications of switching from value at risk to expected shortfall : backtesting the performance of a mean-ES optimal portfolio against the Markovitz portfolio

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Faculty of Economics and Business

Implications of Switching from Value at Risk

to Expected Shortfall

Backtesting the performance of a mean-ES optimal portfolio against the Markovitz portfolio.

by

Alexander Chung Yin Wong

Supervisor:

dr. S.A. Broda

July, 2014

Abstract

The performance of a mean-ES portfolio is backtested against a Markovitz portfolio, using data between 2008 and 2014 from 471 stocks from the S&P 500. There are large differences between the portfolio weights of the two trad-ing strategies. The Markovitz portfolio invests in all stocks, while a mean-ES portfolio invests in only 10 to 20 stocks. The two trading strategies perform equally well, despite the large differences in portfolio weights.

Keywords: Value at Risk, Expected Shortfall, Risk Measures, Coherent Risk Measures, Mean-Variance Portfolio, Markowitz Portfolio, Portfolio Optimization

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3 Data and Methodology 16 3.1 Methodology . . . 16 3.2 Data . . . 17 4 Results 19 4.1 Realized Returns . . . 19 4.2 Regression Results . . . 20 4.3 Discussion . . . 22 5 Conclusion 23 6 Acknowledgement 24 References 25 A Matlab 26 A.1 Markovitz - Mean-Variance Portfolio Optimization Algorithm 26 A.2 mean-ES Portfolio Optimization Algorithm . . . 26

A.3 Backtesting Algorithm . . . 27

A.4 Data Collection Algorithm . . . 29

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1 Introduction

Managing risks is a key process in portfolio management as a trade-off has to be made between the return and the risk of a portfolio. Markowitz (1952) introduced the mean - variance risk management to portfolio optimization. The goal of this strategy is to minimize the variance of the portfolio return subject to a minimal required return.

The development of new risk measures such as Value at Risk (VaR) and Expected Shortfall (ES) propose new methods for portfolio optimization. These risk measures consider only the risk in the tail of the distribution and in particular the potential losses in the tail. The VaR is the minimum loss and the ES is the expected loss incurred in the worst α% cases.

The Markowitz (Mean-Variance) portfolio considers the total risk in the distribution by taking into account the variance associated with potential gains and losses. This is not necessarily desirable as investors can be only interested in minimizing the potential heavy losses instead of the total risk. These investors can follow a different strategy than the Markowitz portfo-lio by minimizing the ES while subject to a minimal required return. The resulting portfolio is called the mean-ES portfolio.

The application of ES in portfolio optimization has been researched by Rockafellar and Uryasev (2000), who developed a non parametric approach in optimizing a mean-ES portfolio. The advantage of this approach is that it is not necessary to know or estimate the underlying distribution of the assets. This approach is also more efficient and therefore easier to compute compared to a parametric approach. Krokhmal, Palmquist, and Uryasev (2002) further developed this approach by adding additional constraints and demonstrated its application with a case study using 100 stocks. However these papers have not researched the performance of a mean-ES portfolio to analyze whether it is a profitable trading strategy. Continuing in the line of these papers this thesis will research the performance of a mean-ES portfolio.

1.1 Goal of Thesis

The goal of this thesis is to evaluate the performance of a mean-ES portfolio compared to the Markowitz portfolio. This will be researched by backtesting the performance of day-to-day portfolio optimization of a mean-ES portfolio and a Markovitz portfolio with the use of a rolling window of 1 year over a 5 year time period using historical data.

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1.2 Data and Software

Computations will be done with MATLABr and stock prices will be obtained from Yahoo!r Finance via a built-in function within MATLABr. Data from the last 5 years of 471 stocks from the S&P 500 will be used to analyze the performance of the portfolios. Regressions to analyze the performance between the two trading strategies will be done with EViewsr.

1.3 Outline Thesis

The theory about risk measures and portfolio optimization techniques will be discussed in section 2. In section 3 the methodology that will be used to analyze the performance of a mean-ES portfolio will be discussed. Section 4 discusses the data that is used. The empirical results can be found in section 5 and section 6 concludes this thesis. Algorithms used in MATLABr can be found in Appendix A.

2 Theory

This section discusses the theory regarding risk measures and in particular VaR and ES. Afterwards portfolio optimization techniques for the Markovitz portfolio and the mean-ES portfolio will be derived and discussed. tech-niques.

2.1 Risk Measures

In order to compare the riskiness between random variables, it is useful to assign a number which quantifies the risk. A risk measure is a function which assigns such a number to random variables (Promislow, 2011).

The concept of a risk measure is useful when deciding how much capital should be set aside in order to cover for potential losses. Real world ap-plications of this concept can be found for example in the banking sector as banks must fulfill the capital requirements set by the Basel Committee (Basel Committee on Banking Supervision, 2013). Another important application of risk measures can be found in the actuarial sector as insurance compa-nies and pension funds must hold a reserve to provide for future obligations (Promislow, 2011).

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2.2 Coherent Risk Measures

It is clear that risk measures are useful for real world applications, but it is necessary to define what properties risk measures should posses in order to be sensible. Artzner, Delbaen, Eber, and Heath (1999) have addressed this issue and introduced the concept of coherent risk measures. They postulated four axioms, which any risk measure ρ should have in order to be coherent. Definition 2.1. Coherent Risk Measure. Let V be a set of random variables. A function ρ : V → R is a coherent risk measure if it is:

(i) monotonous:

X ∈ V, X ≥ 0 ⇒ ρ(X) ≤ 0, (ii) sub-additive:

X, Y, X + Y ∈ V ⇒ ρ(X + Y ) ≤ ρ(X) + ρ(Y ), (iii) positively homogeneous:

X ∈ V, m > 0, mX ∈ V ⇒ ρ(mX) = mρ(X), (iv) translation invariant :

X ∈ V, c ∈ R ⇒ ρ(X + c) = ρ(X) − c.

Let us consider each of these axioms separately to develop a more intu-itive idea behind coherent risk measures. Monotonicity implies that portfolio X should have at least the same amount of risk as portfolio Y , if Y always has higher profits than X. Sub-additivity implies that two portfolios com-bined cannot bear more risk than the sum of the individual risks of the portfolios. This follows from the diversification principle as diversifying re-duces idiosyncratic risk. If you double the size of your portfolio, positive homogeneity implies that the risk is doubled. Multiplying your position in the portfolio X by a factor m will change your risk also by a factor m. Let Y be a deterministic portfolio with a guaranteed return of c. Translation invariance implies that adding Y to your stochastic portfolio X is simply an addition of cash, which lowers the risk.

From sub-additivity and positive homogeneity follows convexity, which is an essential property in portfolio optimization as it allows for unique and well-diversified optimal solution in the risk minimization process (Rockafellar & Uryasev, 2000).

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2.3 Value at Risk

A widely used risk measure is VaR, which was developed in the late 1970s and 1980s. At that time large financial institutions started to develop models to quantify aggregate risks across the institution due to increasing complexity of the business sector(Dowd, 2007). To discuss the VaR method as a risk measure in more detail, it is necessary to introduce more precise definitions and terminology.

The VaR method is a quantile based approach, to see what that means let X be a random variable and α ∈ (0, 1). A number x, such that FX(x) = α

is called an α − quantile of X, where FX(x) is the cumulative distribution

function (CDF) of X. For example the 0.5 - quantile is also known as the median. Multiplying α by 100 results in a percentile. The 0.5 - quantile could also be called the 50th percentile.

If FX(x) is continuous, then FX(α)−1is the unique α−quantile, denoted

by x(α)_{. However if F}

X(x) is not continuous, it is possible that there are

infinitely many quantiles. To cover all cases define x(α) as follows:

Definition 2.2. Value at Risk. Let X be a random variable and α ∈ (0, 1). x(α) = sup {x | P [X ≤ x] ≤ α }, (1)

VaR(α) = −x(α). (2)

Let X represent the future value of the profits or losses of a portfolio for a fixed time horizon T . Then the V aR(α) can be interpreted as follows:

What is the minimum amount lost in the worst α% portfolio outcomes? According to Dowd (2007) the attraction of VaR lies in its straightfor-wardness. The VaR concept is easy to understand and it is a single aggregate risk measure for possible portfolio losses. Losses greater than the value at risk occur with low probability. The VaR concept has two important char-acteristics. It is a widely applicable measure of risk and it also takes into account correlations between risk factors. Risks which offset each other result in a lower VaR and vice versa.

However there are also disadvantages with VaR as its simplicity comes at a cost. Instead of only knowing what the minimum loss is in the α% worst cases it would be interesting to know what the expected loss is. A

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disadvantage of VaR is that it is a quantile based approach, which ignores the tail of the distribution, which can lead to underestimation of potential losses especially when the distribution has f at tails.

The biggest disadvantage of VaR stems from the fact that it is not a coherent risk measure as it fails the notion of sub-additivity.

Example 2.3. VaR does not satisfy sub-additivity.

Consider two identical portfolios with only two possible outcomes in the next period. Either a profit of $0 with P ($0) = 0.98 or a loss of -$10 with P (−$10) = 0.02. The 97.5% VaR of a single portfolio is $0 as a loss only occurs 2% of the time. However let us consider the 97.5% VaR of a portfolio with equal weights in both portfolios. The possible outcomes for this weighted portfolio are X ∈ {$0, −$5, −$10}. The probabilities of these possible events are:

P (X = $0) = 0.98 ∗ 0.98 = 0.9604,

P (X = −$5) = 2 ∗ 0.98 ∗ 0.02 = 0.0392,

P (X = −$10) = 0.02 ∗ 0.02 = 0.0004.

The 97.5% VaR of two portfolios is therefore $5, which shows that VaR is not sub-additive. If sub-additivity is not satisfied it can prevent firms from diversifying as can be seen from the example above that diversification increases the VaR. It could also lead to firms breaking up two separately incorporated affiliates if capital requirements are not satisfied to reduce the VaR (Artzner et al., 1999).

2.4 Expected Shortfall

The previous section defined the VaR as the minimum loss incurred in the worst α% cases, however the VaR is not a coherent risk measure. Acerbi and Tasche (2002) propose the use of ES instead of VaR. The ES answers the question:

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Acerbi and Tasche (2002) argued that it is more sensible to compute what the expected loss is instead of the minimum loss. Artzner et al. (1999) show that if the distribution function is continuous, that ES is the tail con-ditional expectation.

Definition 2.4. Tail Conditional Expectation (TCE). Let X be a random variable and α ∈ (0, 1). The TCE is then defined as

T CE(α)(X) = −E{X | X ≤ x(α)}. (3) Acerbi and Tasche (2002) conclude however that the TCE is only a co-herent risk measure for continuous distributions. More general distributions can violate the axiom of sub-additivity as P [X ≤ xα] = α does not always hold. They derive a general case for the ES.

Definition 2.5. Expected Shortfall (ES). Let X be a random variable and α ∈ (0, 1). The expected α% shortfall of X is defined as

ES(α)(X) = −1 α EX1_{X ≤ x}(α) − x(α) P [X ≤ x(α)] − α . (4) A different representation of ES requires the introduction of the gener-alized inverse function of F (x)

Definition 2.6. Generalized inverse function of F (x). Let p ∈ (0, α), F←(p) = inf {x | F (x) ≥ p}. (5) With the use of (5) it is possible to rewrite (4) such that

ES(α)(X) = −1 α

Z α

0

F←(p) dp. (6)

From (6) it can be seen that ES(α) _{is continuous in α, while T CE}(α)

and V aR(α) _{are not. Acerbi and Tasche (2002) compared ES to VaR and}

concluded that ES is universal, complete and simple. ES is widely applica-ble, it produces a global assessment of a portfolio’s risk and is also easy to understand. The ES risk measure is also more sensitive to the choice of the distribution in comparison with VaR as it takes into account the entire tail of its distribution, while VaR does not.

Figure 1 is a graphical representation of VaR and ES. The VaR is the minimum amount lost if a loss occurs in the colored area which represents the worst α% cases, while the ES is the expected value of the loss under this area.

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VaRα Probability density Loss Profit ESα = Expectation of the α % largest losses

Figure 1: Value-at-Risk and Expected Shortfall

2.5 Portfolio Optimization

The optimization methods for the mean-ES portfolio and the Markovitz port-folio will be discussed in this subsection.

2.5.1 Mean Variance Portfolio

The Markowitz portfolio optimization minimizes the variance of a portfolio subject to a minimal required return for (Markowitz, 1952).

Theorem 2.1. Derivation of Efficient Frontier. Let there be n risky assets characterized by a return vector µ ∈ Rn _{and a n-by-n covariance matrix Σ.}

To find the efficient frontier the following problem has to be solved: minimize w 1 2w T_Σw subject to wTµ = rp, n X i=1 wi = wT1 = 1.

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Where w is the vector with weights invested in the risky asset and rp the

required portfolio return. This can be solved by using the method of Lagrange multipliers. Leading to the following Lagrangian

Λ(w, λ1, λ2) =

1 2w

T_{Σw − λ}

1(wTµ − rp) − λ2(wT1 − 1). (7)

The first order conditions for optimality are: ∂Λ ∂w = Σw − λ1µ − λ21 = 0, (8) ∂Λ ∂λ1 = −(wTµ − rp) = 0, (9) ∂Λ ∂λ2 = −(wT1 − 1) = 0. (10)

Rewriting (8) and combining (9) and (10) gives: Σw = µ 1 λ1 λ2 , (11) µ 1T w =rp 1 . (12)

Since Σ is a covariance matrix, it is positive definite and therefore in-vertible. Therefore (11) can be written as

w = Σ−1 µ 1 λ1 λ2 (13) ⇒ µ 1T w = µ 1T Σ−1 µ 1 | {z } := A λ₁ λ2 . (14)

A is also positive definite and invertible since Σ is. Implying that (14) can be written as λ₁ λ2 = A−1 µ 1T w = A−1rp 1 . (15)

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The optimal solution for w is:

w∗ = Σ−1 µ 1 A−1rp

1

. (16)

The variance of the optimal portfolio is:

σ∗2_pf = w∗TΣw∗ = rp 1 A−1 µ 1 T Σ−1ΣΣ−1 µ 1 A−1rp 1 = rp 1 A−1 µ 1 T Σ−1 µ 1 | {z } A A−1rp 1 = rp 1 A−1 rp 1 .

The efficient frontier can then be found by optimizing for different rp.

Including a risk-free asset is equivalent to enabling an investor to borrow and lend cash for the risk-free rate. Consider an investor, who invested a fraction α in a risky portfolio with standard deviation σp and return rp and

a fraction 1 − α in the risk-free asset with return rf. The expected return

µ of this investment is α rp + (1 − α) rf and the standard deviation σ is

α σp, because a risk-free asset has no variance. This implies the following

relationship between the return and the standard deviation: σ = α σp ⇒ α = σ σp , µ = α rp+ (1 − α) rf = rf + α (rp− rf) ⇒ µ = rf + rp− rf σp | {z } := Sharpe Ratio σ.

Maximizing this ratio with the following theorem leads to the tangent portfolio.

Theorem 2.2. Derivation of the tangent portfolio. Let there be n risky assets with return vector µT = (µ1, µ2, ..., µn)T and one risk-free asset with mean

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rf, ..., µn− rf)T. The covariance matrix Σ of the returns is unaffected by rf

as it is a constant. Consider the following optimization problem minimize w 1 2 w T Σw subject to wTµ_e= re_p.

Where w is the vector with weights invested in the risky assets. Note that the weights do not have to sum up to 1. The Lagrangian is

Λ(w, λ) = 1 2w

T_{Σw − λ(w}T_µ

e− rpe),

with first order conditions ∂Λ ∂w = Σw − λµe = 0 ⇒ w = Σ −1 µ_eλ, ∂Λ ∂λ = −(w T µ_e− r_pe) = 0 ⇒ µT_ew = r_pe ⇒ µT ew = r e p = µ T eΣ −1 µ_eλ ⇒ λ = (µT eΣ −1 µ_e)−1r_pe ⇒ w = r e p µT eΣ −1 µ_eΣ −1 µ_e. The tangent portfolio can then be found by using

n X i=1 wi = iTw = 1 ⇒ wtan = rpe µT eΣ−1µe iT rpe µT eΣ−1µeΣ −1 µ_eΣ −1 µ_e= 1 iTΣ−1µ_eΣ −1 µ_e. (17)

Figure 2 summarizes theorems 2.1 and 2.2. The tangent portfolio has the highest Sharpe ratio and is tangent to the efficient frontier, which only contains risk assets.

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σ µ rf Tangent portfolio Efficient frontier

Figure 2: Efficient frontier and Tangent portfolio The vector µT _{= (µ}

1, µ2, ..., µn)T and the covariance matrix Σ can be

estimated by taking the day-to-day returns for T consecutive business days for all n risky assets:

r_it= p t+1 i − pti pt i , ˆ µi = 1 T T X t=1 rt_i, ˆ σij = 1 T − 1 T X t=1 (r_it− ˆµi)(rjt− ˆµj), i, j = 1, 2, ..., n. (18)

Where ˆσij is the sample covariance between asset i and j and an element

of Σ on the ith row and the jth column. Note that ˆσii = ˆσi2 is the sample

variance of asset i.

It is also possible to add short selling constraints (wi ≥ 0, i = 1, 2, ..., n).

Although an analytical solution cannot be derived, when such constraints are added. To solve the optimization problem quadratic programming algorithms are needed.

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2.5.2 Expected Shortfall Portfolio

The approach for optimizing the ES portfolio follows Krokhmal et al. (2002) and Rockafellar and Uryasev (2000). Let f (x, y) represent the loss associated with x ∈ Rn, a portfolio to be chosen from a set of available portfolios (possibly subject to constraints) X. The loss function f (x, y) has a underlying distribution in R induced by y, the vector y ∈ Rn _represents

uncertainties such as market prices with probability density function p(y). A negative loss can also occur, which is a gain. The α% ES is then:

ζα = VaRα,

ESα(x) = (1 − α)−1

Z

ζα≤f (x,y)

f (x, y) p(y) dy. (19) The key to the approach of Krokhmal et al. (2002) and Rockafellar and Urya-sev (2000) is that ESα and ζα can be characterized by a function Fα(x, ζ):

Fα(x, ζ) = ζ + (1 − α)−1

Z

y∈Rn

[f (x, y) − ζ]+p(y) dy, (20) where [a]+ _{= max{a, 0}. Rockafellar and Uryasev (2000) show that}

min

x∈XESα(x) =(x,ζ)∈X×Rmin Fα(x, ζ) (21)

are equivalent. The solution (x∗, ζ∗) of Fα(x, ζ) minimize the ESα(x) and

ζ∗ gives us the VaRα. Furthermore Rockafellar and Uryasev (2000) show

that if the constraints are such that X is a convex set, the optimization problem is an instance of convex programming. This key property ensures that there exists no other local minimum that differs from a global minimum (Rockafellar, 1970).

The integral Fα(x) can be approximated by sampling from the

proba-bility density p(y) and generating a collection of vectors y₁, y₂, ..., y_J such that (20) can be approximated by

ˆ Fα(x, ζ) = ζ + (1 − α)−1 J X j=1 πj [ f (x, yj) − ζ ] +_, (22)

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where πj represents the probability of scenario yj. To further linearize

(22) additional linear constraints are added:

ζ + (1 − α)−1 J X j=1 πj zj, (23) zj ≥ f (x, yj) − ζ, zj ≥ 0, j = 1, 2, ..., J. (24)

For the one period portfolio optimization the loss function over the pe-riod is

f (x, y; x0, y0) = −y

T _{x + y}T

0 x0, (25)

where the portfolio xT

0 = (x01, x02, ..., x0n)T represents the initial

portfo-lio positions (i.e. number of shares) and yT₀ = (y01, y02, ..., y0n)T is the vector

with initial prices. The expected loss is defined as

E[f (x, y)] = E[−yT x + yT₀ x0] = E[−yT x] + yT0 x0, (26)

E[−yT x] =

n

X

i=1

−E[yi] xi. (27)

The portfolio can be subjected to capital requirements such that

ESα(x) ≤ ω yT0 x0, (28)

where ω is a percentage of the initial portfolio value yT

0 x0 allowed for

risk exposure. This can be seen as an upper bound for the risk measure. This corresponds with the linear constraints (23) and (24). Short selling constraints can be added such that

xi ≥ 0, i = 1, 2, ..., n. (29)

The transaction costs can also be taken into account. This thesis as-sumes linear costs, where ci is the percentage of transaction cost for the total

value bought or sold of asset i. The initial value of the portfolio should be enough to pay for the transaction costs resulting in the following balancing constraint n X i=1 y0ix0i = n X i=1 ciy0i|x0i− xi| + n X i=1 y0ixi. (30)

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This constraint can be rewritten in the following set of linear constraints: n X i=1 y0ix0i = n X i=1 ciy0i(u+i + u − i ) + n X i=1 y0ixi, xi− x0i = u+i + u − i , i = 1, 2, ..., n, u+_i ≥ 0, u−_i ≥ 0, i = 1, 2, ..., n. (31)

The complete optimization problem for the mean-ES portfolio is: min x,ζ n X i=1 −E[yi] xi, subject to ζ + (1 − α)−1 J X j=1 πj zj ≤ ω n X i=1 y0ix0i, zj ≥ n X i=1 (−yijxi+ y0ix0i) − ζ, zj ≥ 0, j = 1, 2, ..., J, n X i=1 y0ix0i= n X i=1 ciy0i(u+i + u − i ) + n X i=1 y0ixi, xi− x0i= u+i + u − i , u+_i ≥ 0, u−_i ≥ 0, xi ≥ 0, i = 1, 2, ..., n. (32)

Solving this problem results in the optimal vector x∗ and the corre-sponding ESα and VaRα. The efficient return - ES frontier can be found by

optimizing for different levels of ω.

This approach approximates (20) by a weighted sum over all scenarios. Following Rockafellar and Uryasev (2000) historical returns are used for the scenario generation. The trading strategy comprises of a day-to-day portfolio optimization. The scenario set for random variable yi consists of J elements

where element yij is constructed as follows

yij = y0i ptj+1 i ptj i , j = 1, 2, ..., J, (33) where ptj

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ex-pected price for asset i for the next day is E[yi] = J X j=1 πjyij = 1 J J X j=1 yij, (34)

assuming that each scenario is equally likely with probability 1_J.

3 Data and Methodology

This section describes the methodology used to test the performance of a Markovitz portfolio against a mean-ES portfolio. It also describes the data that will be used in this thesis.

3.1 Methodology

This section describes the methodology used to test the performance of a Markovitz portfolio against a mean-ES portfolio. The performance will be measured with the use of backtesting over a rolling window.

Starting from 30/5/2009 the portfolios will be optimized for 1/6/2009 with the use of 250 consecutive trading days prior to 30/5/2009 (J = 250). Afterwards the portfolios will be optimized for 2/6/2009 with the use of 250 consecutive trading days prior to 1/6/2009, this day-to-day optimization process will continue until 30/5/2014.

Short selling constraints will be imposed implying that the analytical solution (17) cannot be used. A quadratic programming algorithm will be used to optimize the Markovitz portfolio. The script of this algorithm can be found in Appendix A.1.

The optimization process of the mean-ES portfolio follows (32), (33) and (34). Short selling constraints will also imposed in this case. For simplicity capital requirements (ω = 1) and transaction costs (ci = 0, ∀i) will be

disre-garded. The mean-ES portfolio is linear in its goal function and constraints implying that this optimization problem can be solved with the use of linear programming techniques like the simplex algorithm (Dantzig, 1951). The script can be found in Appendix A.2.

The day-to-day optimization will generate five years of day-to-day re-turns for both portfolios. This will be repeated for various minimal required returns (ranging from 0.0068% to 0.05%) and for different α’s of ESα (0.9,

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0.95, 0.975 and 0.99). The backtesting algorithm can be found in Appendix A.3.

To test which portfolio performed better, the difference of the day-to-day returns of the two portfolios will be regressed on a constant resulting in the following regression

rES_t − rM V

t = γ + t, t = 1, 2, ..., T, (35)

where rES_t & rM V_t is the day-to-day return of the mean-ES portfolio and the Markovitz portfolio respectively. HAC standard errors will be used to account for heteroscedasticity, which is a common property of financial time series (Heij, De Boer, Franses, Kloek, & Van Dijk, 2004, p. 532). If γ is positive and significant, then the mean-ES portfolio outperformed the Markovitz portfolio and vice versa. This regression will done in EViewsr and repeated for various minimal required returns and ES confidence levels.

3.2 Data

The dataset consists of stocks of the S&P 500 index, though not all stocks currently in the index have six consecutive years of data and are therefore dropped from the dataset leaving 471 stocks in total to optimize from. The data is obtained from Yahoo!r Finance with the use of MATLABr. The data collection algorithm can be found in Appendix A.4.

Table 5 in Appendix B provides a table with descriptive statistics of all 471 stocks. The daily return of stocks has a mean near zero for all stocks, however the standard deviation can vary substantially between stocks. Min-imum and maxMin-imum daily returns can be found in columns three and four.

Figure 3 represents the time series of an average stock by averaging over all stocks through time. The mean is concentrated around zero and the figure shows that volatility is clustered, which is an indication for heteroscedasticity.

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2008 2009 2010 2011 2012 2013 2014 2015 −0.2 −0.1 0 0.1 Year Return (%)

Figure 3: Time series of an average stock

The efficient portfolios for the Markovitz trading strategy and mean-ES trading strategy can be found in Figure 4. This figure is made by using all returns from 30/5/2008 till 30/5/2014. The dots represent individual stock returns and standard deviations. The efficient frontier of the mean-ES portfolio lies under the Markovitz portfolio. This is expected as the Markovitz strategy minimizes the standard deviation.

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0 0.04 0.08 0.12 0.16 −0.002 −0.001 0 0.001 0.002 0.003 0.004 0.005 Standard Deviation (%) Rate of return (%) Stocks MV portfolio mean-ES portfolio

Figure 4: 99%-ES and Mean-Variance Efficient Portfolios

4 Results

This section discusses the results found from the backtesting algorithm and also the regression results from (35) for various minimum required returns and confidence levels for ESα.

4.1 Realized Returns

In Table 1 the geometric average realized daily return can be found for the Markovitz portfolio and the mean-ES portfolio under several confidence levels and both subjected to minimum daily returns ranging from 0.0068% to 0.5%. The portfolio with the highest return for a given minimum required daily return is marked in bold. The geometric mean is used in this table to account for compounding in order to compare the historical performances between the Markovitz and mean-ES portfolios. For lower minimum required returns the

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mean-ES portfolio outperforms the Markovitz portfolio, although for higher minimum required returns the performance is the opposite. To bring matters in perspective let us compare these realized returns with a simple strategy, where one buys the S&P 500 (technically the SPDR S&P500) and holds it for 5 years. The total realized return of this simple strategy is 37.36% implying a 0.0252% return per trading day. From Table 1 it follows that most of the portfolios outperform this strategy except when the minimum required return is 0.2% or 0.3%. Min. req. return (%) Markovitz return (%) mean-ESα return (%) 0.9 0.95 0.975 0.99 0.0068 0.0386 0.0440 0.0429 0.0449 0.0498 0.0133 0.0416 0.0433 0.0424 0.0447 0.0446 0.0198 0.0437 0.0419 0.0415 0.0440 0.0445 0.0261 0.0436 0.0415 0.0405 0.0449 0.0438 0.0323 0.0402 0.0403 0.0400 0.0447 0.0432 0.0383 0.0391 0.0401 0.0391 0.0431 0.0424 0.0442 0.0447 0.0398 0.0385 0.0410 0.0411 0.0500 0.0447 0.0394 0.0379 0.0401 0.0404 0.1 0.0405 0.0361 0.0358 0.0381 0.0370 0.2 0.0240 0.0203 0.0191 0.0192 0.0182 0.3 0.0242 0.0225 0.0221 0.0226 0.0258 0.4 0.0310 0.0265 0.0291 0.0295 0.0303 0.5 0.0346 0.0438 0.0441 0.0409 0.0417

Table 1: Geometric average daily returns in percentages. Trading strategies are optimized for several minimum required returns and confidence levels of α.

4.2 Regression Results

The regressions results from (35) are needed to see whether one trading strategy consistently outperforms the other. These results can be found in Table 2 and 3. In Table 2 the returns of the mean-ES0.9 and the

mean-ES0.95 portfolios are compared to the Markovitz portfolio subject to various

minimum required returns. In all regressions in Table 2 the constant γ is highly insignificant implicating that neither portfolios perform better than the other. In Table 3 the same is done for the mean-ES0.975 and the

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mean-ES0.99 portfolios, the constant γ is also highly insignificant for all regressions

in this table. Min. req. return (%)

0.90 0.95

γ S.E. P-value γ S.E. P-value

0.0068 0.00008 0.00022 0.72561 0.00005 0.00023 0.80730 0.0133 -0.00001 0.00022 0.96070 -0.00003 0.00023 0.90360 0.0198 -0.00009 0.00022 0.67380 -0.00010 0.00023 0.66500 0.0261 -0.00009 0.00020 0.64800 -0.00011 0.00021 0.58120 0.0323 -0.00004 0.00020 0.84060 -0.00004 0.00020 0.82930 0.0383 -0.00002 0.00019 0.93526 -0.00004 0.00020 0.84850 0.0442 -0.00016 0.00020 0.42550 -0.00019 0.00021 0.35420 0.0500 -0.00017 0.00020 0.40310 -0.00020 0.00021 0.32040 0.1 -0.00016 0.00021 0.46060 -0.00016 0.00022 0.45720 0.2 -0.00020 0.00024 0.40210 -0.00026 0.00025 0.29680 0.3 -0.00014 0.00024 0.55110 -0.00016 0.00025 0.51830 0.4 -0.00030 0.00026 0.26420 -0.00021 0.00027 0.43390 0.5 0.00001 0.00033 0.98360 0.00001 0.00034 0.98270 Table 2: Regression results for ESα vs. Markovitz for α = (0.90, 0.95).

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Min. req. return (%)

0.975 0.99

γ S.E. P-value γ S.E. P-value

0.0068 0.00011 0.00024 0.65030 0.00012 0.00025 0.64760 0.0133 0.00003 0.00025 0.89990 0.00003 0.00026 0.89830 0.0198 -0.00003 0.00024 0.89420 -0.00002 0.00025 0.95090 0.0261 0.00000 0.00022 0.98930 -0.00002 0.00024 0.92120 0.0323 0.00008 0.00022 0.73080 0.00004 0.00023 0.84790 0.0383 0.00007 0.00022 0.74950 0.00006 0.00023 0.80820 0.0442 -0.00012 0.00022 0.57550 -0.00011 0.00023 0.61730 0.0500 -0.00015 0.00022 0.50180 -0.00013 0.00022 0.54940 0.1 -0.00010 0.00022 0.66200 -0.00012 0.00022 0.57550 0.2 -0.00025 0.00025 0.32710 -0.00029 0.00025 0.24130 0.3 -0.00014 0.00025 0.57800 -0.00001 0.00025 0.97030 0.4 -0.00019 0.00027 0.47610 -0.00016 0.00028 0.57230 0.5 -0.00006 0.00034 0.85470 -0.00003 0.00034 0.93400 Table 3: Regression results for ESα vs. Markovitz for α = (0.975, 0.99).

Number of observations is 1260 for each regression.

4.3 Discussion

The empirical results imply that both trading strategies perform equally well, however several factors can affect these results.

Firstly, the dataset can be too small as it only consists of 6 years of data of 471 stocks. Increasing the backtesting period and therefore the number of observations can increase the precision of the results. Although a longer backtesting period increases the difficulty of finding sufficiently long time series of stock prices. This can result in infeasible solutions for the mean-ES optimization algorithm as the number of stocks to choose from decreases. The portfolio weights of the Markovitz portfolio is very different compared to the mean-ES portfolio. The Markovitz portfolio invests small fractions in all stocks to minimize the variance. While the mean-ES portfolio invests in less than 5% of the stocks available as can be seen in Table 4. Despite this large difference in portfolio weights the performance is not significantly different between the two trading strategies.

Secondly, the investment period can affect the results, increasing the investment period from daily to monthly could for example lead to different

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results between the performance of mean-ES and Markovitz. However a longer investment period requires more data and as discussed in the first criticism this could lead to infeasible solutions due to the decrease of number of stocks to invest in.

Thirdly, the mean-ES optimization uses a non-parametric approach, it is possible that a parametric approach can lead to different results. However this is beyond the scope of this thesis to test whether this is indeed the case. This thesis has neglected transaction costs, which have a negative impact on portfolios returns. If a trading strategy involves high turnover ratios it can diminish returns especially when transaction costs are high. However daily turnovers for day-to-day trading strategies are low, because the day-to-day portfolio weight changes are very small.

Markovitz mean-ES Min. req.

return (%) average number of stocks

0.0068 471 20.14 0.0133 471 19.98 0.0198 471 19.91 0.0261 471 19.73 0.0323 471 19.56 0.0383 471 19.43 0.0442 471 19.24 0.05 471 19.02 0.1 471 17.70 0.2 471 15.47 0.3 471 13.50 0.4 471 11.70 0.5 471 9.90

Table 4: Average number of stocks Markovitz and mean-ES portfolios in-vested in.

5 Conclusion

The development of risk measures like VaR and ES has provided new methods for portfolio optimization. This thesis focused primarily on ES as it is a

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coherent risk measure, whereas VaR is not a coherent risk measure due to the lack of sub-additivity.

The goal of this thesis was to test the performance of the mean-ES port-folio against the Markovitz portport-folio. The mean-ES portport-folio minimizes the ESα, while subject to a minimum return. The Markovitz portfolio minimizes

the variance of its portfolio, while subject to a minimum return.

These trading strategies were backtested over a period of 5 years with the use of a rolling window with a length of one year. The trading strategies were optimized day by day for various minimum required returns and confidence levels of ESα with the use of linear and quadratic programming techniques.

The empirical results found in this thesis indicate that the Markovitz portfolio and the mean-ES portfolio perform equally well for all confidence levels and minimum required returns. The Markovitz portfolio invests in all stocks to minimize the variance, while the mean-ES portfolio invests in 10 to 20 stocks. These large differences in portfolio weights do not lead to a significant performance difference between the two trading strategies.

It is however possible that the results are affected by several factors. Firstly, the dataset can be too small or short. Secondly, it is possible that varying the length of the investment period can lead to different results. Thirdly, a non-parametric approach is used to optimize the mean-ES portfo-lio, which can lead to different results compared to a parametric approach.

To further research the performance of the mean-ES portfolio, this thesis proposes the use of a larger variety of assets as only stocks were considered in this thesis. It is also interesting to research how the performance of mean-ES portfolios is for various lengths of investment periods. A parametric approach could also be researched and used to test the performance of the mean-ES portfolio.

6 Acknowledgement

I would like to thank my thesis supervisor dr. S.A. Broda for providing me the optimization algorithms for the Markovitz portfolio and the mean-ES portfolio and also for his help and comments on my thesis.

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References

Acerbi, C., & Tasche, D. (2002). Expected shortfall: a natural coherent alternative to value at risk. Economic notes, 31 (2), 379–388.

Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical finance, 9 (3), 203–228.

Basel Committee on Banking Supervision. (2013). Fundamental review of the trading book: A revised market risk framework. Consultative doc-ument . (Basel: Bank for International Settlements)

Dantzig, G. B. (1951). Application of the simplex method to a transportation problem. Activity analysis of production and allocation, 13 , 359–373. Dowd, K. (2007). Measuring market risk. John Wiley & Sons.

Heij, C., De Boer, P., Franses, P. H., Kloek, T., & Van Dijk, H. K. (2004). Econometric methods with applications in business and economics. Ox-ford University Press.

Krokhmal, P., Palmquist, J., & Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk , 4 , 43–68.

Markowitz, H. (1952). Portfolio selection. The journal of finance, 7 (1), 77–91.

Promislow, S. D. (2011). Fundamentals of actuarial mathematics. John Wiley & Sons.

Rockafellar, R. T. (1970). Convex analysis, volume 28 of princeton mathe-matics series. Princeton university press.

Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk , 2 , 21–42.

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A

Matlab

A.1 Markovitz - Mean-Variance Portfolio

Optimiza-tion Algorithm

function [all_weights] = markovitz_opt(... min_return_vec, return_matrix)

[~, num_stocks] = size(return_matrix); num_min_return = length(min_return_vec);

mean_returns = mean(return_matrix); %A of Ax <= b cov_matrix = cov(return_matrix); %H

Aeq = ones(1, num_stocks); beq = 1;

lb = zeros(num_stocks, 1);

all_weights = zeros(num_stocks, num_min_return); f = zeros(num_stocks, 1);

options = optimset(’Algorithm’,...

’interior-point-convex’, ’Display’, ’off’); for i = 1:num_min_return,

min_return = min_return_vec(i); %b of Ax <= b [w] = quadprog(cov_matrix, f, -mean_returns,...

-min_return,Aeq, beq, lb, [], [], options); all_weights(:, i) = w;

end end

A.2 mean-ES Portfolio Optimization Algorithm

function [W, ES]=mean_ES_portopt(MinRetvec, Rets, Beta) if nargin < 3,

Beta = 0.99; end

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L = J+N; LM = length(MinRetvec); mu = mean(Rets); f = [1 zeros(1,N) ones(1,J)/J/(1-Beta)]’; A = [ 0 mu zeros(1,J) zeros(L,1) eye(L)

ones(J,1) Rets eye(J)];

A = -A;

ES = zeros(LM,1); W = zeros(N,LM);

OPTIONS = optimset(’Algorithm’, ’simplex’, ’Display’, ’off’); for loop = 1:LM %repeat for all values in MinRetvec

MinRet = MinRetvec(loop);

b = [MinRet zeros(1,L) zeros(1,J)]’; b = -b;

Aeq = [0 ones(1,N) zeros(1,J)]; beq = 1; [axu,fval] = linprog(f,A,b,Aeq,beq,[],[],[],OPTIONS); W(:,loop) = max(axu(2:N+1),0); ES(loop) = fval; end end

A.3 Backtesting Algorithm

load sp500_returns.mat

min_return_vec = linspace(0.001, 0.005, 5); beta = 0.99;

num_min_returns = length(min_return_vec); rolling_window = 249;

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for j = 1:num_min_returns, pct_complete = 0.01;

fprintf(’Minimal required return: %d/%d\n’,... j, num_min_returns)

fprintf(’Computing... 0%% Completed\n’) mean_es_weights_time_series =...

zeros(num_stocks, num_obs - rolling_window - 1); es_time_series = zeros(num_obs - rolling_window - 1); markovitz_weights_time_series =...

zeros(num_stocks, num_obs - rolling_window - 1); for i = 1:(num_obs - rolling_window - 1),

if i > (num_obs - rolling_window - 1)*pct_complete fprintf(’Computing... %d%% Completed\n’,... round(100 * pct_complete)) pct_complete = pct_complete + 0.01; end returns_rolling_window =... returns( i:(i+rolling_window), :); try markovitz_weights_time_series(:,i) = markovitz_opt(min_return_vec(j),... returns_rolling_window); [w, es] = mean_ES_portopt(... min_return_vec(j),... returns_rolling_window, beta); mean_es_weights_time_series(:,i) = w; es_time_series(i, 1) = es; catch continue end end

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results_markovitz(j).min_return = min_return_vec(j); results_markovitz(j).markovitz_time_series =... markovitz_weights_time_series; results_es(j).min_return = min_return_vec(j); results_es(j).es_time_series = es_time_series(:, 1); results_es(j).mean_es_time_series =... mean_es_weights_time_series; end

save results_medium_returns.mat results_es results_markovitz

A.4 Data Collection Algorithm

data_collector.m clear

%Import array with tickers. filename = ’ticker_list.xlsx’; full_list = importdata(filename); ticker_list = full_list(2:end); num_tickers = length(ticker_list); %Collect data from Yahoo! Finance c = yahoo; j = 1; pct_complete = 0.1; fprintf(’Collecting data...\n’) for i = 1:num_tickers, if i > num_tickers * pct_complete

fprintf(’Collecting data... %d%% Completed\n’,... pct_complete * 100)

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pct_complete = pct_complete + 0.1; end

try

raw_data = fetch(c, ticker_list(i), ’Close’,... ’5/30/2008’, ’5/30/2014’); catch continue end if length(raw_data(:,1)) == 1258 data(j).name = ticker_list(i); data(j).dates = raw_data(:,1); data(j).close_prices = raw_data(:,2); j = j + 1; end end

B

Descriptive Statistics

Table 5: Descriptive Statistics for S&P 500 Stocks

Ticker Mean Standard Deviation Minimum Maximum A 0.00001 0.02392 -0.12188 0.12377 AA 0.00126 0.03291 -0.18839 0.19124 AAPL -0.00057 0.02176 -0.12208 0.21832 ACE -0.00012 0.02223 -0.17186 0.24623 ACN -0.00030 0.01804 -0.14065 0.15546 ACT -0.00117 0.01735 -0.10888 0.11212 ADBE 0.00004 0.02427 -0.12555 0.23510 ADI -0.00007 0.01967 -0.13529 0.15291 ADM 0.00018 0.02309 -0.14776 0.16856 ADP -0.00030 0.01472 -0.10576 0.08884 ADS -0.00067 0.02401 -0.12682 0.17069 ADSK 0.00021 0.02742 -0.12856 0.18520 AEE 0.00023 0.01637 -0.13641 0.21029 AEP -0.00004 0.01471 -0.11672 0.09482 AES 0.00062 0.02855 -0.22052 0.18652 AET -0.00001 0.02521 -0.15076 0.16066 AFL 0.00066 0.03563 -0.23241 0.58384 AGN -0.00051 0.02007 -0.13229 0.15081 AIV 0.00087 0.03858 -0.21603 0.37255 AIZ 0.00040 0.02836 -0.17898 0.33215 AKAM 0.00030 0.03239 -0.19378 0.33890 ALL 0.00028 0.02739 -0.17822 0.26869 ALTR 0.00003 0.02303 -0.10869 0.15542 AMAT 0.00025 0.02320 -0.12489 0.14417 AMGN -0.00048 0.01776 -0.12218 0.10404 AMP -0.00002 0.03343 -0.21758 0.25291 AMT -0.00024 0.02014 -0.16819 0.11667 AMZN -0.00054 0.02645 -0.21133 0.14655 AN -0.00044 0.02884 -0.18132 0.19349

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Table 5: (continued)

Ticker Mean Standard Deviation Minimum Maximum AON -0.00029 0.01645 -0.10266 0.15998 APA 0.00057 0.02579 -0.17572 0.22304 APC 0.00025 0.03063 -0.19094 0.24299 APD 0.00009 0.01987 -0.12781 0.14041 APH -0.00021 0.02312 -0.13972 0.15844 ARG -0.00011 0.02349 -0.28593 0.17987 ATI 0.00108 0.03703 -0.20462 0.23704 AVB 0.00018 0.02847 -0.15308 0.20664 AVP 0.00100 0.02592 -0.16883 0.28000 AVY 0.00026 0.02249 -0.10822 0.17195 AXP -0.00003 0.02914 -0.17115 0.21352 AZO -0.00082 0.01609 -0.09912 0.13698 BA -0.00011 0.02052 -0.13392 0.08587 BAC 0.00158 0.04626 -0.26073 0.40784 BAX -0.00002 0.01514 -0.07791 0.15294 BBBY -0.00018 0.02211 -0.19527 0.20435 BBT 0.00034 0.03041 -0.19102 0.30484 BBY 0.00080 0.03038 -0.15210 0.40030 BCR -0.00023 0.01355 -0.06510 0.13185 BDX -0.00013 0.01308 -0.08784 0.09215 BHI 0.00058 0.02949 -0.21248 0.28336 BIIB -0.00082 0.02305 -0.13170 0.39492 BK 0.00067 0.03190 -0.19876 0.37283 BLK 0.00017 0.02725 -0.16361 0.18377 BMS -0.00015 0.01707 -0.11111 0.12420 BMY -0.00039 0.01574 -0.08526 0.09370 BRCM 0.00026 0.02573 -0.10443 0.17845 BSX 0.00037 0.02646 -0.20716 0.19526 BTU 0.00173 0.03837 -0.16220 0.27015 BXP 0.00030 0.02985 -0.19310 0.20027 CA 0.00014 0.01970 -0.10411 0.14691 CAG -0.00012 0.01336 -0.08417 0.09322 CAH 0.00004 0.02002 -0.09134 0.37714 CAM 0.00033 0.03029 -0.17999 0.17078 CAT 0.00014 0.02369 -0.12833 0.12910 CB -0.00017 0.01961 -0.14390 0.14362 CBG 0.00099 0.04848 -0.39024 0.41615 CBS -0.00013 0.03307 -0.20999 0.26194 CCE -0.00027 0.02406 -0.24725 0.44480 CCI -0.00012 0.02360 -0.15295 0.21021 CCL 0.00032 0.02554 -0.11600 0.15811 CELG -0.00036 0.02252 -0.15705 0.18339 CF 0.00018 0.03423 -0.16451 0.52828 CHK 0.00107 0.03644 -0.19602 0.38054 CHRW 0.00025 0.02009 -0.08704 0.18543 CI -0.00012 0.02883 -0.19055 0.27407 CINF 0.00003 0.02269 -0.15435 0.25106 CLX -0.00022 0.01210 -0.08209 0.06597 CMA 0.00040 0.03376 -0.17143 0.23022 CMCSA -0.00032 0.02186 -0.19693 0.16936 CMG -0.00081 0.02736 -0.13864 0.27409 CMI -0.00003 0.03131 -0.18079 0.21147 CMS -0.00031 0.01513 -0.09655 0.11237 CNP -0.00009 0.01695 -0.11759 0.13208 CNX 0.00127 0.03913 -0.16399 0.28674 COF 0.00040 0.03856 -0.20905 0.33408 COH 0.00031 0.02792 -0.16976 0.22806 COL 0.00001 0.01869 -0.11114 0.09707 COP 0.00034 0.02202 -0.14243 0.26756 COST -0.00020 0.01539 -0.09708 0.13511 COV -0.00010 0.01712 -0.12751 0.10201 CPB -0.00013 0.01254 -0.07850 0.08975 CSC 0.00011 0.02347 -0.15642 0.18172 CSCO 0.00028 0.02148 -0.13756 0.19347 CTAS -0.00033 0.01793 -0.09713 0.14120 CTL 0.00014 0.01931 -0.15035 0.29191 CTXS -0.00005 0.02623 -0.16481 0.18729 CVC 0.00071 0.02964 -0.15689 0.39484 CVS -0.00023 0.01849 -0.12221 0.25216 CVX 0.00004 0.01894 -0.17256 0.14271 D -0.00018 0.01321 -0.09514 0.08161

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Ticker Mean Standard Deviation Minimum Maximum DAL -0.00044 0.04011 -0.20981 0.26795 DD -0.00003 0.02059 -0.10287 0.12781 DE 0.00023 0.02469 -0.13564 0.16566 DFS -0.00030 0.03221 -0.21458 0.17647 DGX 0.00001 0.01584 -0.09770 0.10262 DHI 0.00027 0.03700 -0.27536 0.26511 DIS -0.00041 0.02003 -0.13772 0.10772 DISCA -0.00042 0.02431 -0.20127 0.32513 DNB 0.00006 0.01692 -0.11834 0.16390 DNR 0.00112 0.03643 -0.24210 0.29391 DO 0.00097 0.02555 -0.19213 0.22501 DOV -0.00008 0.02199 -0.10154 0.19216 DOW 0.00022 0.02823 -0.15563 0.26240 DPS -0.00039 0.01808 -0.13206 0.14953 DRI 0.00002 0.02347 -0.16643 0.14195 DTV -0.00052 0.01976 -0.15560 0.13512 DUK -0.00053 0.02132 -0.66109 0.11749 DVN 0.00062 0.02540 -0.17673 0.20032 EA 0.00064 0.02832 -0.17378 0.21730 EBAY -0.00006 0.02409 -0.11694 0.16116 ECL -0.00045 0.01664 -0.10248 0.12118 ED -0.00013 0.01134 -0.08402 0.07146 EFX -0.00025 0.01783 -0.09803 0.14898 EIX 0.00010 0.01587 -0.11559 0.08724 EMC -0.00006 0.02079 -0.12068 0.13069 EMR 0.00013 0.02102 -0.13344 0.17672 EQR 0.00027 0.03228 -0.19061 0.27589 EQT 0.00005 0.02558 -0.19192 0.17742 ESS 0.00006 0.02597 -0.12946 0.24543 ESV 0.00061 0.02861 -0.19248 0.24843 ETR 0.00043 0.01513 -0.12433 0.08451 EXC 0.00073 0.01772 -0.14677 0.10861 EXPD 0.00027 0.02205 -0.15232 0.11807 EXPE -0.00024 0.03167 -0.22190 0.37712 F -0.00000 0.03435 -0.22791 0.33333 FDO -0.00045 0.02079 -0.17446 0.14891 FDX -0.00006 0.02198 -0.08394 0.16936 FE 0.00072 0.01770 -0.15031 0.11375 FFIV -0.00042 0.02938 -0.13479 0.27146 FLIR 0.00034 0.02287 -0.14127 0.16020 FMC 0.00055 0.03784 -0.18396 1.02296 FOXA -0.00009 0.02701 -0.17197 0.18523 FRX -0.00045 0.01950 -0.21584 0.16699 FSLR 0.00194 0.04425 -0.31283 0.33927 FTR 0.00073 0.02295 -0.13914 0.15294 GAS -0.00017 0.01420 -0.07889 0.10929 GCI 0.00071 0.03697 -0.28267 0.31532 GD -0.00001 0.01742 -0.10497 0.09102 GE 0.00036 0.02338 -0.16460 0.12539 GHC 0.00017 0.02122 -0.13097 0.14406 GLW 0.00048 0.02516 -0.12394 0.14370 GME 0.00061 0.02953 -0.14920 0.24814 GPC -0.00032 0.01602 -0.09199 0.09944 GPS -0.00026 0.02370 -0.21405 0.21176 GRMN 0.00022 0.02653 -0.19251 0.28048 GS 0.00048 0.02896 -0.20929 0.23395 GT 0.00069 0.03807 -0.16567 0.24368 GWW -0.00053 0.01761 -0.10258 0.10890 HAL 0.00025 0.03002 -0.19061 0.20633 HAR -0.00000 0.03375 -0.17047 0.28364 HAS -0.00009 0.01864 -0.11265 0.10647 HCBK 0.00068 0.02398 -0.13557 0.16295 HCN 0.00009 0.02302 -0.13211 0.16925 HCP 0.00033 0.03019 -0.18554 0.19809 HD -0.00054 0.01865 -0.12332 0.08958 HES 0.00065 0.03054 -0.14308 0.23695 HOG 0.00010 0.03030 -0.17006 0.18436 HON -0.00010 0.01965 -0.10512 0.10385 HOT 0.00017 0.03163 -0.15160 0.17338 HP 0.00015 0.03244 -0.20951 0.23808 HPQ 0.00050 0.02356 -0.14602 0.25042

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Ticker Mean Standard Deviation Minimum Maximum HRB 0.00011 0.02348 -0.15749 0.14353 HRL 0.00011 0.02875 -0.07359 0.99691 HRS 0.00012 0.02117 -0.10822 0.18309 HSP 0.00008 0.01948 -0.08409 0.26601 HST 0.00060 0.03922 -0.21250 0.27027 HSY -0.00050 0.01402 -0.08065 0.10369 HUM -0.00027 0.02583 -0.12104 0.31495 IBM -0.00012 0.01493 -0.10326 0.09026 ICE 0.00028 0.03211 -0.28977 0.21532 IFF -0.00043 0.01702 -0.13879 0.11412 IGT 0.00116 0.03091 -0.14556 0.24915 INTC 0.00008 0.01950 -0.10594 0.11175 INTU -0.00050 0.01847 -0.13072 0.12437 IP 0.00010 0.03078 -0.17955 0.22730 IPG 0.00009 0.03217 -0.25993 0.26820 IR 0.00016 0.02705 -0.13540 0.28546 IRM 0.00022 0.02199 -0.12129 0.18826 ISRG 0.00023 0.02758 -0.21164 0.20326 ITW -0.00014 0.01896 -0.11541 0.10659 IVZ 0.00039 0.03399 -0.21906 0.24231 JBL 0.00032 0.03408 -0.26731 0.25846 JCI 0.00011 0.02632 -0.12051 0.14273 JEC 0.00080 0.02988 -0.17766 0.17736 JNJ -0.00021 0.01118 -0.10897 0.08301 JNPR 0.00048 0.02838 -0.15055 0.26399 JOY 0.00088 0.03573 -0.15580 0.23940 JPM 0.00037 0.03280 -0.20062 0.26147 JWN -0.00004 0.02818 -0.19980 0.15590 K -0.00011 0.01230 -0.07232 0.08430 KEY 0.00119 0.04455 -0.35171 0.50000 KLAC 0.00005 0.02387 -0.10983 0.12723 KMB -0.00030 0.01209 -0.07937 0.08630 KMX -0.00015 0.02782 -0.15854 0.14286 KR -0.00024 0.01594 -0.09235 0.13512 KSS 0.00011 0.02187 -0.12891 0.13616 KSU -0.00010 0.02871 -0.16634 0.21646 L 0.00037 0.02374 -0.19120 0.22066 LB -0.00037 0.02636 -0.17969 0.22985 LEG -0.00015 0.02144 -0.10326 0.13996 LEN 0.00028 0.04183 -0.24046 0.25789 LH -0.00012 0.01400 -0.09884 0.12351 LLL 0.00004 0.01574 -0.09091 0.09845 LLTC 0.00002 0.01855 -0.10354 0.17682 LLY -0.00002 0.01569 -0.12549 0.14088 LM 0.00073 0.03688 -0.23286 0.32895 LMT -0.00013 0.01627 -0.09517 0.10726 LNC 0.00145 0.05401 -0.30396 0.66347 LOW -0.00021 0.02152 -0.14162 0.11529 LRCX 0.00006 0.02625 -0.13187 0.13837 LUK 0.00098 0.03129 -0.17113 0.26154 LUV -0.00019 0.02387 -0.14577 0.22625 M -0.00013 0.03135 -0.15385 0.21247 MAC 0.00086 0.04075 -0.24113 0.36650 MAR -0.00010 0.02497 -0.13172 0.14090 MAS 0.00043 0.03226 -0.15438 0.17706 MAT -0.00024 0.01967 -0.13191 0.19244 MCD -0.00028 0.01244 -0.08584 0.08671 MCHP 0.00002 0.01948 -0.09224 0.15074 MCK -0.00062 0.01817 -0.13819 0.20564 MCO -0.00015 0.02834 -0.13725 0.15295 MDT 0.00002 0.01686 -0.09338 0.15253 MET 0.00085 0.03880 -0.21875 0.36556 MHFI -0.00014 0.02476 -0.19252 0.15984 MHK 0.00001 0.02841 -0.22816 0.13248 MKC -0.00036 0.01220 -0.07530 0.10509 MMC -0.00025 0.01784 -0.12611 0.13968 MMM -0.00028 0.01586 -0.08990 0.08577 MO -0.00032 0.01334 -0.14071 0.14187 MON 0.00029 0.02286 -0.15997 0.19303 MRK -0.00011 0.01762 -0.11230 0.12767 MSFT -0.00006 0.01921 -0.15686 0.13267

(35)

Ticker Mean Standard Deviation Minimum Maximum MTB 0.00013 0.02650 -0.17395 0.18497 MU -0.00004 0.04008 -0.18991 0.22485 MUR 0.00061 0.02643 -0.14691 0.20690 MWV -0.00000 0.02467 -0.11136 0.20508 MYL -0.00061 0.02288 -0.17011 0.20685 NBR 0.00092 0.03516 -0.13913 0.26125 NDAQ 0.00036 0.02870 -0.14871 0.15317 NE 0.00088 0.02927 -0.21536 0.26174 NEE -0.00012 0.01587 -0.12237 0.13310 NEM 0.00086 0.02751 -0.20111 0.16404 NFX 0.00092 0.03356 -0.15348 0.22436 NI -0.00035 0.01607 -0.09187 0.10640 NOC -0.00017 0.01704 -0.09834 0.15550 NOV 0.00058 0.03416 -0.19629 0.27212 NRG 0.00045 0.02632 -0.22680 0.21065 NSC -0.00004 0.02133 -0.09933 0.14831 NTAP 0.00007 0.02631 -0.15038 0.16336 NTRS 0.00051 0.02682 -0.23613 0.23169 NU -0.00026 0.01433 -0.09712 0.08877 NUE 0.00066 0.02859 -0.21736 0.24726 NVDA 0.00070 0.03306 -0.15404 0.44355 NWL 0.00008 0.02587 -0.17033 0.37578 OI 0.00082 0.03030 -0.24313 0.20981 OMC -0.00008 0.01848 -0.11853 0.10286 ORCL -0.00021 0.01981 -0.11559 0.13194 ORLY -0.00097 0.01898 -0.15530 0.16741 OXY 0.00028 0.02600 -0.15332 0.22689 PAYX 0.00001 0.01562 -0.09613 0.08609 PBCT 0.00028 0.01917 -0.15169 0.18529 PBI 0.00043 0.02262 -0.16863 0.18857 PCAR 0.00021 0.02547 -0.15855 0.14038 PCG 0.00001 0.01441 -0.11025 0.10303 PCL 0.00033 0.02494 -0.19671 0.17421 PCLN -0.00113 0.02698 -0.17999 0.20892 PCP -0.00023 0.02263 -0.15602 0.12860 PDCO 0.00007 0.01790 -0.11760 0.12400 PEG 0.00023 0.01698 -0.14622 0.11615 PEP -0.00009 0.01234 -0.07877 0.13548 PETM -0.00038 0.02064 -0.11602 0.13063 PFE -0.00015 0.01620 -0.09233 0.11502 PFG 0.00105 0.04433 -0.28958 0.42035 PG -0.00006 0.01252 -0.09268 0.08574 PGR 0.00007 0.02100 -0.19338 0.14966 PH -0.00000 0.02271 -0.12021 0.12372 PHM 0.00037 0.03697 -0.18721 0.22688 PKI -0.00007 0.02195 -0.16600 0.22573 PLD 0.00105 0.04120 -0.20848 0.43019 PLL -0.00025 0.02180 -0.12182 0.20510 PM -0.00023 0.01526 -0.11669 0.10385 PNR -0.00022 0.02196 -0.13083 0.12001 PNW -0.00023 0.01387 -0.08535 0.09247 POM 0.00013 0.01713 -0.14836 0.15993 PPG -0.00057 0.02023 -0.12918 0.12992 PPL 0.00038 0.01638 -0.12892 0.15186 PRGO -0.00068 0.01996 -0.10612 0.26809 PRU 0.00082 0.04231 -0.27668 0.32836 PSA -0.00009 0.02674 -0.16863 0.22959 PVH -0.00034 0.02700 -0.16811 0.16098 PWR 0.00037 0.02890 -0.19023 0.25852 PX -0.00005 0.01854 -0.12939 0.14416 PXD -0.00016 0.03348 -0.15948 0.28536 QCOM -0.00012 0.02062 -0.14515 0.16601 R 0.00025 0.02701 -0.11735 0.21897 RDC 0.00074 0.03203 -0.20086 0.24017 REGN -0.00124 0.03372 -0.22237 0.26587 RHI -0.00012 0.02412 -0.13268 0.11348 RHT -0.00012 0.02648 -0.16344 0.13268 RIG 0.00125 0.02888 -0.16576 0.24108 RL -0.00021 0.02489 -0.11732 0.16701 ROK -0.00016 0.02555 -0.14157 0.14649 ROP -0.00029 0.02110 -0.20340 0.13285

(36)

Ticker Mean Standard Deviation Minimum Maximum RRC 0.00025 0.03113 -0.16610 0.21507 RSG 0.00012 0.01850 -0.14280 0.15291 RTN -0.00016 0.01563 -0.10851 0.09319 SBUX -0.00065 0.02317 -0.15526 0.12327 SCG -0.00008 0.01325 -0.10385 0.09842 SCHW 0.00030 0.02761 -0.16423 0.17878 SE -0.00009 0.01880 -0.15436 0.12316 SEE 0.00008 0.02392 -0.13946 0.28423 SHW -0.00069 0.01832 -0.13237 0.14979 SIAL -0.00017 0.01870 -0.10689 0.16995 SJM -0.00034 0.01435 -0.13167 0.09638 SLB 0.00032 0.02640 -0.12979 0.22555 SNA -0.00020 0.02092 -0.10614 0.09575 SNDK -0.00009 0.03840 -0.28107 0.46283 SO -0.00006 0.01123 -0.09964 0.05677 SPG 0.00017 0.03191 -0.20224 0.25000 SPLS 0.00080 0.02522 -0.13464 0.18160 SRCL -0.00032 0.01560 -0.09539 0.11980 SRE -0.00024 0.01602 -0.13047 0.19340 STI 0.00106 0.04170 -0.23409 0.37307 STJ -0.00012 0.01956 -0.15025 0.14491 STX -0.00000 0.03496 -0.21790 0.31579 STZ -0.00065 0.02237 -0.27131 0.21044 SWK -0.00013 0.02286 -0.11162 0.16625 SWN 0.00052 0.03322 -0.26828 0.41446 SWY 0.00017 0.02122 -0.12364 0.16201 SYK -0.00004 0.01681 -0.12660 0.12338 SYMC 0.00026 0.02353 -0.11898 0.21475 SYY -0.00002 0.01458 -0.12029 0.10249 T 0.00019 0.01523 -0.14001 0.08367 TAP 0.00005 0.01647 -0.09615 0.12972 TDC 0.00000 0.02450 -0.09415 0.22536 TE 0.00026 0.01730 -0.13745 0.11612 TEG 0.00008 0.01826 -0.10281 0.36387 TEL 0.00004 0.02437 -0.12705 0.13892 TGT 0.00016 0.02027 -0.15113 0.14286 TIF -0.00012 0.02637 -0.14221 0.12839 TJX 0.00001 0.03086 -0.09701 0.97278 TMK 0.00025 0.02946 -0.14673 0.50314 TMO -0.00026 0.01961 -0.14456 0.10440 TROW 0.00019 0.02899 -0.15374 0.21860 TRV -0.00018 0.02161 -0.20354 0.22222 TSN -0.00021 0.02591 -0.14563 0.30920 TSO 0.00009 0.03583 -0.22736 0.24853 TSS 0.00003 0.01859 -0.11633 0.14202 TWX -0.00060 0.02538 -0.53758 0.15134 TXN -0.00005 0.01990 -0.09343 0.17125 TXT 0.00104 0.03890 -0.32817 0.46315 UNH -0.00025 0.02464 -0.25791 0.22905 UNM 0.00029 0.03263 -0.18135 0.42112 UNP -0.00036 0.02122 -0.08300 0.16291 UPS -0.00012 0.01603 -0.08555 0.09291 URBN 0.00037 0.02846 -0.15414 0.25901 USB 0.00027 0.02952 -0.18594 0.22198 UTX -0.00018 0.01732 -0.12008 0.09603 V -0.00037 0.02181 -0.13041 0.15799 VAR -0.00017 0.01976 -0.13193 0.13880 VIAB -0.00030 0.02370 -0.20673 0.21697 VLO 0.00041 0.03124 -0.15239 0.25000 VMC 0.00053 0.02733 -0.14521 0.11819 VNO 0.00043 0.03148 -0.18154 0.24100 VRSN 0.00016 0.02481 -0.14138 0.18304 VRTX -0.00010 0.03119 -0.38236 0.19402 VTR 0.00020 0.02928 -0.16157 0.25095 VZ -0.00006 0.01525 -0.12765 0.08777 WAG -0.00029 0.01826 -0.14263 0.08872 WAT -0.00012 0.02035 -0.13986 0.13006 WDC -0.00012 0.02973 -0.17316 0.16267 WFC 0.00024 0.03588 -0.24679 0.31272 WHR -0.00001 0.02942 -0.16097 0.16737 WIN 0.00041 0.01943 -0.14153 0.11596

(37)

Ticker Mean Standard Deviation Minimum Maximum WLP -0.00021 0.02135 -0.15550 0.13720 WM 0.00001 0.01535 -0.15235 0.08660 WMB 0.00025 0.02824 -0.20836 0.23117 WMT -0.00011 0.01253 -0.09969 0.08771 WU 0.00055 0.02510 -0.17373 0.40848 WYN -0.00010 0.03828 -0.26138 0.43960 WYNN 0.00018 0.03691 -0.23013 0.20081 X 0.00212 0.03997 -0.19642 0.21535 XEL -0.00017 0.01215 -0.07159 0.07609 XLNX -0.00016 0.01998 -0.09818 0.09987 XOM 0.00007 0.01728 -0.14669 0.16215 XRAY 0.00005 0.01724 -0.09686 0.10075 XRX 0.00039 0.02586 -0.15186 0.23041 YHOO 0.00013 0.02490 -0.10707 0.26368 YUM -0.00027 0.01875 -0.10291 0.11017 ZION 0.00119 0.04311 -0.21604 0.30208 ZMH -0.00007 0.01836 -0.11026 0.15596

Implications of switching from value at risk to expected shortfall : backtesting the performance of a mean-ES optimal portfolio against the Markovitz portfolio