PORTFOLIO VALUE-AT-RISK OPTIMIZATION Текст научной статьи по специальности «Математика»

Wschodnioeuropejskie Czasopismo Naukowe (East European Scientific Journal) #9(13)/2016

Valentyn Khokhlov, CFA

PhD, International Marketing Manager Global Spirits, Kiev, Ukraine

PORTFOLIO VALUE-AT-RISK OPTIMIZATION

Abstract

Portfolio optimization with regards to the variance-covariance Value-at-Risk (VaR), which is one of the most popular risk measures, is generally a non-linear problem. We show that minimum VaR portfolios reside on the Markowitz efficient frontier and derive the risk aversion factor that corresponds to the VaR-optimal portfolio with a given probability. Since the Markowith problem is can be solved using quadratic programming, which allows developing a quadratic programming model and algorithm for the VaR optimization. The algorithm is tested on the Dow Jones constituents in 2013-2015 and the optimal portfolios with regards to the variance-covariance VaR are the same as the ones found with the non-linear optimization algorithm. As expected, the VaR-optimal portfolios are located on the efficient frontier close to the minimum-variance portfolio. We tried three different distributions for estimating VaR — normal, Student's t with 3 degrees of freedom, Laplace — but the choice of distribution does not materially affect the optimal portfolio composition.

Introduction

An optimal allocation of assets in the portfolio is one of the key tasks for asset managers. Starting with classic works of Markowitz [1], Sharpe [2], and Mer-ton [3], the widely used approach is to use the mean-variance optimization (MVO), for which the variance is used as the measure of risk. The MVO problem can be solved using the quadratic programming, and a significant amount of tools has been developed to address it. However, using the variance as the measure of risk implies that investors do not differentiate between the downside risk and the upside risk, whereas in practice it's the former that actually poses concerns.

Nowadays, the Value-at-Risk (VaR) appears to be a more relevant risk measure, since it catches only the downside risk. The popularity of VaR boosted after it had been promoted in the Basel II2 and Basel III3 Accords. Unfortunately, one cannot easily switch from the mean-variance to the mean-VaR optimization, and even VaR minimization is a significant problem itself. Only a few researchers have addressed these issues so far.

One approach was developed by Benati and Rizzi [4], who suggest using mixed integer linear model for empirical mean-VaR optimization. Their model requires the historical sample of realized returns and, as such, is not suitable for optimizing the forward-looking VaR. The structural limitation of the model results in inability to use it for VaR minimization. Another approach was developed by Goh [5] who consider the return as a random variable and work with the forward-looking VaR-based measures, such as WVaR and PVaR. It was possible to develop generic quadratic models for them, however they are much more complex that the classic MVO model and existing portfolio optimization tools can't be utilized to solve the WVaR or PVaR problems. Finally, Deng [6] address a related but not similar problem of VaR-adjusted Sharpe ratio optimization. Thus, the unsolved aspect of the problem is portfolio optimization with

regards to the forward-looking VaR using its classic definition under the variance-covariance approach.

The purpose of this paper is to develop a mathematical model and create an algorithm for the portfolio VaR minimization using the variance-covariance approach to measuring VaR. We start with defining the portfolio VaR and introduce the model, then show that the minimum VaR portfolio is MV-optimal and, therefore, MVO can be used to find it, so we can develop a quadratic programming algorithm to solve this problem, and finally show how it works on a numerical example.

Mathematical Model for VaR Optimization

The variance-covariance approach to measuring VaR allows estimating a forward-looking risk measure by assuming that the portfolio and its components returns follow a certain elliptical distribution. In that case, VaR per $1 of the portfolio can be expressed using the asset weights:

(rp +фа°р ) = -(w'r + фа>/ w'Zw ) , (1)

2 See Basel II Accord (article 178) available at http://www.bis.org/publ/bcbs107.pdf

3 See Basel III Accord available at http://www.bis.org/publ/bcbs189.pdf

v„ =-

where V is the VaR @ tt of the portfolio with the initial value of $1,

V , CT are the expected return and the standard deviation of the portfolio,

= O 1 (a) is the inverse standard

CDF value at a ,

W is the vector of asset weights in the portfolio,

r is the vector of the expected returns for the portfolio assets,

^ is the covariance matrix for the portfolio

assets.

Here we assume that the portfolio return is distributed like the asset returns, so they have the same standard CDF (i.e. the CDF with the location parameter of 0 and the scale parameter of 1) that we denote as

O.

The mathematical model for finding the minimum VaR portfolio, therefore, is formulated as fol-

Wschodnioeuropejskie Czasopismo Naukowe (East European Scientific Journal) #9(13)/2016 lows: find a vector of asset weights w* that maximizes the objective function

UVaR = w'r + фа%/w'Sw ^ max,

'VaR so that n

Ew = i

i=i

w,.

minimum VaR portfolio I Г

(* * \ Гр, aP j i

is not efficient,

V„

For

ГР +^ + фа^р

(5),

j, using (1) we can express

VaR as

* \ * * j = -va -S< V*,

dU d

(2)

(3)

< w < wmax for every i = 1,...,n,

(4)

Where n is the amount

of assets we can choose for the portfolio, mm max

' are the vectors of lower and

upper bounds on asset weights.

The optimization problem (2)-(4) is a nonlinear programming problem, and this type of problems doesn't have a simple algorithm. However, we know that finding a mean-variance efficient portfolio is a quadratic programming problem that is rather easy to solve under the Markowitz framework.

Proposition 1. The minimum VaR portfolio resides on the mean-variance efficient frontier for

a <0.5.

To prove this proposition let's assume that the

which contradicts to our initial statement that Va is

so we can find a portfolio with either a higher return and the same standard deviation or a lower standard

deviation and the same return. Let's suppose there's a

(* *

Tp +8, Op

its

the minimum VaR. The same is true for any portfolio

(* * \

r*, Op — 8 ), which we can easily prove considering the fact that < 0 for any a <0.5.

Using the Proposition 1, we can conclude that solving the Markowitz problem

U = rp — Aop ^ max, (5)

will for a certain A result in finding the minimum VaR portfolio. Therefore, we need to find that

i *

value (we denote it as A ) for which solving the problem (5) is equivalent to solving the problem (2). Proposition 2. The solution of the Markowitzh

problem for A = — 2o p corresponds to the

minimum VaR portfolio.

To prove this proposition, we can compare the gradients of (2) and (5) and ensure that both objective functions reach the extremes at the same point for all components, i.e. all partial derivatives are zero at the same w.

д

д U

VaR

д

dw

д

dw

w'r - X—w'Sw = r' - 2Xw'£.

dw

For

(2),

w'r + фа —(w'Sw j05 = r' + фа (w'Sw j-05 w'S = r' + ф w'SS

0.5

Фа

dw dw

dw

a,

. Since both gradients are identical when —2A = ^ a/op , both objective functions reach the

extreme values at the same w.

Therefore, in order to find the minimum VaR portfolio we can use the MVO approach, but instead of the constant risk aversion A we should use a specific value A =— §aj2op that depends on the

portfolio variance and is unknown a priori. However, the iterative structure of the MVO algorithm ensures

i *

convergence to the required A . Optimization Algorithm

The VaR optimization algorithm we introduce here is based on the MVO algorithm developed by Sharpe [7] and is modified in order to use the Proposition 2. The algorithm uses the following inputs:

• = O 1 (a) is the inverse standard

CDF value at a, where a is the probability with which the VaR is calculated,

(0)

• w is the initial vector of asset weights in the portfolio,

• r is the vector of the expected returns for the portfolio assets,

• ^ is the covariance matrix for the portfolio assets,

min max

• W , W are the vectors of lower and upper bounds on asset weights.

The algorithm has multiple iterations and converges to the optimal solution. Each iteration k = 0,1,2,... consists of the following steps:

1. The variable risk aversion value is calculated as per Proposition 2:

Фс

W: MU гш = max \MUl

w,

(k )

< w

2d,

îVw( k >Lw( k )

■p 2 V w( A (

2. The marginal utilities (partial derivatives) of each asset are calculated:

lsub : MUiSub

= min I

MU

w(k ) > wmin

MUi = /

k )]Tw(k )a

j=1

j

y

3. Two assets are selected to modify weights: the weight of the asset iadd with the highest marginal utility is increased and the weights of the asset isub with the lowest marginal utility is decreased:

If we can't find either ¿add or ¿sub the algorithm terminates.

4. The effect of the substitution is calculated as the difference in the marginal utilities:

AMU = MUf - MU,

'add 'sub

5. If the difference is less than a pre-specified threshold (e.g. 10-8) the algorithm terminates.

6. The feasible change in the selected asset weights is calculated as:

Aw = min

MU: - MU ]

ladd lsub

2 («

ladd

+ a

lsub

laddlsub

k )

w

ladd

-w

(k )

ladd

w(k )

lsub

-w

'sub

7. The weights of two selected assets are modified for the next iteration:

w

( k+1)

w

(k )

+ Aw if l = l .

add,

W(k) -Aw ifl = l'sub,

w

(k )

otherwise.

As the original Sharpe algorithm, our algorithm converges to a certain mean-variance optimal portfolio, subject to the lower and upper boundaries on the

asset weights. Step 1 of our algorithm, additionally, ensures compliance with the Proposition 2.

Numerical Example

Our numerical example is based on the Dow Jones components in 2013-2015 (30 U.S. companies stocks). We use the daily variance-covariance VaR, which are based on the expected value and its standard deviation of daily stock returns. We use the sample mean and the sample standard deviation to estimate those statistics (see Table 1).

Table 1. Sample statistics of the selected securities in 2013-2015

Ticker Company Name Return Sample Log-return Sample

Mean St.dev Mean St.dev

MMM 3M Company 0.0786% 1.0021% 0.0736% 1.0036%

IBM IBM -0.0265% 1.2005% -0.0338% 1.2092%

GS Goldman Sachs 0.0593% 1.2908% 0.0509% 1.2921%

UNH UnitedHealth Group 0.1183% 1.3729% 0.1088% 1.3719%

HD Home Depot 0.1152% 1.1409% 0.1087% 1.1375%

BA Bank of America 0.1038% 1.3085% 0.0952% 1.3080%

MCD McDonalds 0.0563% 0.9423% 0.0519% 0.9373%

JNJ Johnson & Johnson 0.0662% 0.9189% 0.0620% 0.9186%

TRV The Travelers Companies 0.0735% 0.9468% 0.0690% 0.9482%

CVX Chevron -0.0013% 1.2863% -0.0096% 1.2859%

UTX United Technologies 0.0355% 1.0657% 0.0298% 1.0683%

DIS The Walt Disney Company 0.1113% 1.2297% 0.1037% 1.2326%

T AT&T 0.0282% 0.9495% 0.0237% 0.9504%

XOM Exxon Mobil 0.0041% 1.1202% -0.0021% 1.1205%

PG Procter & Gamble 0.0375% 0.9289% 0.0332% 0.9300%

V Visa 0.1069% 1.3580% 0.0977% 1.3539%

CAT Caterpillar -0.0165% 1.3448% -0.0256% 1.3480%

WMT Wal-Mart 0.0010% 1.0011% -0.0041% 1.0089%

DD E. I. du Pont de Nemours 0.0783% 1.2910% 0.0700% 1.2828%

JPM JP Morgan 0.0723% 1.2478% 0.0645% 1.2483%

AXP American Express 0.0374% 1.2092% 0.0301% 1.2106%

MRK Merck 0.0542% 1.2109% 0.0469% 1.2087%

VZ Verzion Communications 0.0316% 1.0040% 0.0265% 1.0034%

NKE Nike 0.1308% 1.3488% 0.1218% 1.3298%

MSFT Microsoft 0.1195% 1.5322% 0.1078% 1.5320%

KO Coca-Cola 0.0389% 0.9474% 0.0344% 0.9479%

PFE Pfizer 0.0527% 1.1048% 0.0466% 1.1041%

INTC Intel 0.0911% 1.4159% 0.0811% 1.4113%

GE General Electric 0.0721% 1.1550% 0.0655% 1.1466%

CSCO Cisco 0.0635% 1.3868% 0.0539% 1.3834%

(Source: Yahoo! Finance)

We run our optimization algorithm using log- mization are summarized in the Table 2, where "Mod-

return sample statistics as inputs and use three differ- el VaR" column shows the variance-covariance VaR

ent distributions — normal, Student's t with 3 degrees estimate (the one used for the optimization model) and

of freedom and Laplace — to calculate è„ . While "Actual VaR" column shows the historical VaR calcu-

a lated as the percentile of the actual sample of the opte choice of distabuti°n agmficandy mfluemx« to timal portfolio daily returns in 2013-2015. VaR estimates, it doesn't materially affect composition of the optimal portfolios. The results of our opti-

_Table 2. VaR-optimal portfolio results for selected distributions

Distribution Probability Return St.deviation Model VaR Actual VaR

5% 0.0539% 0.6882% 0.8812% 1.1211%

Student's t (df 2% 0.0509% 0.6863% 1.3288% 1.4946%

= 3) 1% 0.0493% 0.6856% 1.7481% 1.6313%

0.5% 0.0473% 0.6849% 2.2624% 2.0634%

5% 0.0523% 0.6872% 1.0665% 1.1066%

Laplace 2% 0.0502% 0.6860% 1.5112% 1.4914%

1% 0.0489% 0.6855% 1.8472% 1.6313%

0.5% 0.0476% 0.6850% 2.1830% 2.0638%

5% 0.0523% 0.6871% 1.0779% 1.1058%

Normal 2% 0.0507% 0.6863% 1.3587% 1.4942%

1% 0.0500% 0.6859% 1.5457% 1.6319%

0.5% 0.0495% 0.6857% 1.7167% 2.0687%

(Source: author's calculations)

Figure 1 shows the location of the VaR-optimal portfolios and the Markowitz efficient frontier for the selected securities. We can see that VaR-optimal port-

folios are located on the efficient frontier close to the minimum-variance portfolio.

10% 15%

Standard deviation annualized

F¿gure 1. VaR-opt¿mal portfoUos and the €//¿€¿6^ /^^¿er

Figure 2 shows a zoomed out picture of the VaR-optimal region of the efficient frontier. Different VaR probabilities (0.5% to 5%) result in very close portfo-

lios, and both Student's t and Laplace distribution also reside quite close to each other.

11.4% 11.5% 11.6%

Standard deviation annualized

F¿gure 2. The effl€¿ent front¿er reg¿on w¿th VaR-opt¿mal portfoUos

The composition of the optimal portfolios for the Student's t distribution with 3 degrees of freedom and the Laplace distribution are shown in Table 3. These portfolios are essentially the same as VaR-optimal portfolios derived with a non-linear optimization algo-

rithm used by the Microsoft Excel™ Solver. That confirms both that our algorithm results in the same outcomes as the non-linear optimization and that the Propositions 1 and 2 are correct.

Table 3. VaR-optimal portfolios composition

Ticker Student's t distribution (df = 3) Laplace distribution

5% 2% 1% 0.5% 5% 2% 1% 0.5%

MMM 1.98% 1.80% 1.67% 1.40% 1.90% 1.76% 1.61% 1.44%

IBM 0.00% 0.00% 0.02% 0.56% 0.00% 0.00% 0.15% 0.49%

GS 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

UNH 4.47% 3.89% 3.60% 3.30% 4.17% 3.75% 3.53% 3.34%

HD 4.11% 3.09% 2.60% 2.21% 3.59% 2.85% 2.51% 2.26%

BA 0.39% 0.07% 0.00% 0.00% 0.23% 0.00% 0.00% 0.00%

MCD 16.40% 16.58% 16.64% 16.53% 16.49% 16.62% 16.62% 16.54%

JNJ 5.99% 5.69% 5.52% 5.30% 5.83% 5.61% 5.47% 5.33%

TRV 7.63% 7.25% 7.06% 6.80% 7.43% 7.16% 7.00% 6.83%

CVX 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

UTX 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

DIS 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

T 10.69% 11.19% 11.43% 11.45% 10.95% 11.32% 11.44% 11.45%

XOM 0.00% 0.00% 0.00% 0.17% 0.00% 0.00% 0.00% 0.12%

PG 9.43% 9.65% 9.77% 9.89% 9.54% 9.70% 9.81% 9.88%

V 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

CAT 0.00% 0.00% 0.09% 0.48% 0.00% 0.00% 0.19% 0.43%

WMT 8.67% 10.26% 11.02% 11.60% 9.48% 10.64% 11.15% 11.53%

DD 4.13% 4.08% 4.04% 3.91% 4.10% 4.07% 4.01% 3.93%

JPM 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

AXP 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

MRK 0.68% 0.81% 0.87% 0.91% 0.75% 0.84% 0.88% 0.91%

VZ 6.90% 7.11% 7.19% 7.23% 7.01% 7.15% 7.20% 7.22%

NKE 2.59% 1.91% 1.57% 1.31% 2.24% 1.74% 1.51% 1.34%

MSFT 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

KO 12.29% 12.51% 12.62% 12.70% 12.40% 12.57% 12.64% 12.69%

PFE 2.12% 2.59% 2.83% 3.03% 2.36% 2.71% 2.88% 3.01%

INTC 1.48% 1.24% 1.10% 0.90% 1.35% 1.18% 1.06% 0.93%

GE 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

CSCO 0.07% 0.28% 0.37% 0.34% 0.18% 0.33% 0.36% 0.34%

Conclusions

VaR optimization is a non-linear problem with regards to the portfolio asset weights. However, the propositions 1 and 2 show that the minimum VaR portfolios reside on the Markowitz efficient frontier and derive the risk aversion factor that corresponds to the optimal portfolio with particular VaR probability. Hence the MVO models and algorithms can be used for VaR optimization.

In this paper we introduce a quadratic algorithm for VaR optimization, which is based on the classic MVO algorithm developed in Sharpe (1987). Unlike existing VaR-optimization algorithms, which optimize portfolios with regards to the historical VaR and require full sample of securities returns, our algorithm

(Source: author's calculations) only the sample statistics (the expected return and the covariance matrix).

The VaR optimization algorithm developed in this paper results in the same portfolios as a non-linear optimization algorithm used in Microsoft Excel™ Solver. Thus, it confirms that the VaR-optimal portfolios reside on the efficient frontier. All those portfolios are quite close to the minimum-variance portfolio, and the choice of distribution (Students't, Laplace or normal) does not materially affects the portfolio composition. Our tests show that the Laplace distribution leads to a better match between the variance-covariance and the historical VaR estimates. It should be noted, however, that the optimal portfolio with regards to the variance-covariance VaR is not the same as the optimal portfolio with regards to the historical VaR.

optimizes the variance-covariance VaR and requires

References

1. Markowitz, Harry. 1952. "Portfolio Selection." The Journal of Finance. 7 (1): 77-91.

2. Sharpe, William F. 1963. "A Simplified Model for Portfolio Analysis." Management Science. 9 (2): 277-293.

3. Merton, Robert C. 1972. "An Analytic Derivation of the Efficient Portfolio Frontier." The Journal of Financial and Quantitative Analysis. 7 (4): 1851-1872.

4. Benati, Stefano, Romeo Rizzi. 2007. "A mixed integer linear programming formulation of the optimal mean/Value-at-Risk portfolio problem." European Journal of Operational Research. 176: 423-434.

u

5. Goh, Joel Weiqiang, Kian Guan Lim, Melvyn Sim, Weina Zhang. 2012. "Portfolio value-at-risk optimization for asymmetrically distributed asset returns." European Journal of Operational Research. 221 (2): 397-406.

6. Deng, Geng, Tim Dulaney, Craig McCann, Olivia Wang. 2013. "Robust portfolio optimization with Value-at-Risk-adjusted Sharpe ratios." Journal of Asset Management. 14 (5): 293-305.

7. Sharpe, William F. 1987. "An Algorithm for Portfolio Improvement." In: Advances in Mathematical Programming and Financial Planning, K.D. Lawrence, J.B. Guerard, Jr., and Gary D. Reeves (editors). JAI Press, 1987: 155-170.