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Worst-case VaR and robust portfolio optimization with interval random uncertainty set
Expert Systems with Applications 38 (2011) 64–70
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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Worst-case VaR and robust portfolio optimization with interval random uncertainty set Wei Chen a,*, Shaohua Tan b, Dongqing Yang a a b
Key Laboratory of High Confidence Software Technologies (Ministry of Education), School of EECS, Peking University, Beijing 100871, China Department of Machine Intelligence, School of EECS, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Keywords: Interval random uncertainty set Interval random chance-constrained programming Value-at-risk
a b s t r a c t This paper addresses a new uncertainty set – interval random uncertainty set for worst-case value-at-risk and robust portfolio optimization. The form of interval random uncertainty set makes it suitable for capturing the downside and upside deviations of real-world data. These deviation measures capture distributional asymmetry and lead to better optimization results. We also apply our interval random chance-constrained programming to robust worst-case value-at-risk optimization under interval random uncertainty sets in the elements of mean vector and covariance matrix. Numerical experiments with real market data indicate that our approach results in better portfolio performance. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In 1959, Markowitz (1959) published his pioneering work that paved the foundation for modern portfolio analysis. The consideration of estimation risk and model risk has grown in importance. Different criteria for risk measure are proposed, such as variance (Markowitz, 1959), value-at-risk (VaR) (Linsmeier & Pearson, 1996), conditional value-at-risk (CVaR) (Rockafellar & Uryasev, 2000), etc. Among those, VaR remains the most widely accepted measure among practitioners. VaR estimates the maximum potential loss at a certain probability level, i.e., it provides information about the amount of losses that will not be exceeded with certain probability. Mathematically, (1 e)-VaR is defined as the minimum level c such that the probability that the portfolio loss f(x, u) is below c exceeds 1 e. Thus, the VaR can be formulated as
VðxÞ ¼ min; c s:t:
Pðf ðx; uÞ 6 cÞ P 1 e;
ð1Þ
where x = (x1, . . ., xN) is the vector of asset weights, and u = (u1, . . ., uN) is the vector of uncertain portfolio asset returns. When the distribution of returns is Gaussian, with given mean û and covariance matrix C, the VaR can be expressed as
pffiffiffiffiffiffiffiffiffiffiffi ^; VðxÞ ¼ jðeÞ xT Cx xT u
ð2Þ
where j(e) = U1(e), and U1() stands for the inverse cumulative standard normal distribution.pElffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ghaoui, ffi Oks, and Oustry (2003) specified a parameter jðeÞ ¼ ð1 eÞ=e that guaranteed that the * Corresponding author. Tel.: +86 010 62755745. E-mail address: [email protected] (W. Chen). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.06.010
VaR constraint in model (1) would be satisfied probabilistically for all possible distributions for uncertain returns. In practice, VaR suffers from an important drawback: this approach requires a perfect knowledge of the data, in our case the mean and covariance matrix. But data are often prone to errors. Portfolio optimization based on inaccurate point estimates may be highly misleading. For example, the true VaR may be widely worse than the optimal computed VaR. Most recently, researchers have incorporated the uncertainty introduced by estimation errors directly into the portfolio optimization process by robust optimization introduced by Ben-Tal and Nemirovski (1998) and El Ghaoui and Lebret (1997). In this case, the inputs are not classical ones, such as expected returns and covariances, but rather uncertainty sets. Ben-Tal and Nemirovski (2000) summarized two frequently used uncertainty sets: (1) ‘‘Unknown-but-bounded” uncertainty set, such as box uncertainty and ellipsoidal uncertainty (Fabozzi, Kolm, Pachamanova, & Focardi, 2007). (2) ‘‘Random symmetric” uncertainty set, such as ûi = (1 + i)ui, where i are independent random variables symmetrically distributed in the interval [1, 1]. El Ghaoui et al. (2003) used robust optimization to propose a framework for optimization of worst-case VaR based on information about first and second moments of the distribution of returns. Zhu and Fukushima (2005) investigated worst-case CVaR under box uncertainty and ellipsoidal uncertainty. Though these two uncertainty sets are frequently used, they have two serious disadvantages. First, it is difficult to collect all information to determine the precise bounds of ‘‘unknownbut-bounded” uncertainty set in practice. Sometimes only the distributions of the bounds can be found from historical data. In this case, the ‘‘unknown-but-bounded” uncertainty set is actually
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W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
fluctuant instead of stable. Therefore, the variability of bounds cannot be ignored. Second, the assumption of symmetric distribution is also limiting in many applications especially in financial modeling in which distributions are often known to be asymmetric. In our other papers (Chen & Tan, 2009a, 2009b), we proposed a novel uncertainty set: interval random uncertainty, and applied it to robust mean–variance portfolio model. In this paper, we suggest a framework for robust worst-case VaR optimization based on the interval random uncertainty. Roughly speaking, an interval random uncertainty set, namely, interval random variable is an interval with random fluctuant bounds. For example, interval random variables ni are defined as ni ¼ ui h1i ; ui þ h2i . The mean values ui are mean-point of ni. Random variables h1i and h2i are downside and upside deviations of ni around mean values, respectively. Interval random variable ni consider the variability of bounds and asymmetric measures of variability for the distribution of data simultaneously. Hence, it is a good idea to introduce interval random variable as uncertainty set. Then we apply our interval random chance-constrained programming to robust worst-case VaR optimization. Finally, a hybrid-intelligent algorithm is applied to solve the robust worst-case VaR model. Some computational results are discussed that demonstrate the potentially significant economic benefits of investing in portfolios computed using classical models and the model introduced here. The robustness is achieved at relatively high performance and low cost. The rest of this paper is organized as follows. Section 2 presents basic definitions of interval random variable. In Section 3, interval random chance-constrained programming model is discussed. Section 4 presents results of some computational experiments with our robust model. Finally, a few concluding remarks are given in Section 5. 2. Interval random uncertainty set Roughly speaking, an interval random variable is a measurable function from a probability space to a collection of closed intervals. In other words, an interval random variable is a random variable taking interval values. Let I be a collection of closed intervals. For our purpose, we use the following definition of interval random variable (Chen & Tan, 2009a, 2009b). Definition 1. Let ðX; A; PrÞ be a probability space. An interval is a function n : X ! I such that nðxÞ ¼ hrandom variable i nðxÞ; nðxÞ is a measurable function of x. For example, let ðX; A; PrÞ be a probability space. Let X ¼ x1 ; x2 ; . . . ; xm , and l1, l2, . . ., lm be closed intervals in I . Then the function
nðxÞ ¼
8 l1 ; > > >
if x ¼ x1 if x ¼ x2
The assumption of symmetric distribution of traditional uncertainty sets makes no distinction between downside and upside deviations. But in real world, the distributions of returns of assets are often known to be asymmetric. The form of interval random variable makes it suitable for capturing the downside and upside deviations of asset returns. We assume that the uncertainty set for the expected return ui of asset i and covariance dij of asset i and asset j take the form of interval random variables:
Uðui Þ ¼ mi h1i ; mi þ h2i ; h i Uðdij Þ ¼ mij s1ij ; mij þ s2ij ;
where mi are the mean values of returns. Random variables h1i and h2i are downside and upside deviations for the distribution of returns, respectively. mij are the mean values of covariances. Random variables s1ij and s2ij are downside and upside deviations for the distribution of covariances, respectively. Hence, they enable us to capture the asymmetry of asset returns in order to obtain better solutions that satisfy chance constraints. If h1i ¼ h2i or s1ij ¼ s2ij , then the interval random uncertainty sets become symmetric. Thus, the worst-case VaR model (3) can be reformulated as
V ðxÞ ¼
max
fui 2Uðui Þ;dij 2Uðdij Þg
lm ; if x ¼ xm
Definition 2. Let f : I n ! I be a function over the n-dimensional Euclidean space and ni be interval random variable defined on ðXi ; Ai ; Pri Þ; i ¼ 1; 2; . . . ; n, respectively. Then, n = f(n1, n2, . . ., nn) is an interval random variable on ðX1 . . . Xn ; A1 . . . An ; Pr 1 . . . Prn Þ, defined by n(x1, x2, . . ., xn) = f(n1(x1), n2(x2), . . ., nn(xn)), for all (x1, x2, . . ., xn) 2 X1 X2 Xn. In this paper, we define the worst-case VaR under uncertainty set by
max
pffiffiffiffiffiffiffiffiffiffiffi
jðeÞ xT Cx xT u;
j¼1
ð4Þ
i¼1
min fxg
s:t:
V ðxÞ N X
ð5Þ
xi ¼ 1
i¼1
xi P 0: The objective of the robust optimization model (5) is to minimize the worst-case VaR of the portfolio. We expect that the sensitivity of the optimal solution of this mathematical program to parameter fluctuations will be significantly smaller than it would be for its classical counterpart (2). To solve this problem, we will present mathematically meaningful interval random chance-constraint programming model in Section 3. 3. Interval random programming 3.1. Chance of interval random event First, we have the following definition of function F eaðA 6 BÞ (Chen & Tan, 2009a, 2009b) which represents the grade of feasibility of the interval event ‘A to be less than or equal to B’. Definition 3. Let I be the set of all closed intervals on the real line ; B ¼ b; b R. Let A 2 I, B 2 I, A ¼ ½a; a . We define a feasibility function F ea : I I ! ½0; 1 such that
.. > > . > :
fu2UðuÞ;C2UðCÞg
i¼1
The corresponding robust portfolio optimization problem is to solve
is clearly an interval random variable.
V ðxÞ ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N N N X uX X jðeÞt xi xj dij xi ui :
ð3Þ
where U(u) is the uncertainty set for return u and U(C) is the uncertainty set for covariance matrix C.
F eaðA 6 BÞ ¼
8 1; > > > > < 1;
6 b; a
mðAÞ ¼ mðBÞ; mðBÞmðAÞ > b; > ; mðAÞ < mðBÞ and a > wðBÞþwðAÞ > > : 0; otherwise;
ð6Þ
where m(A) and w(A) are the mid-point and half-width of interval A, a aþa bþb a bb mðAÞ ¼ 2 ; wðAÞ ¼ 2 . Similarly, mðBÞ ¼ 2 ; wðBÞ ¼ 2 . At least one of A and B must be interval. If one of them is a real number, wðBÞ ¼ 0. for example B, then mðBÞ ¼ b ¼ b; Now, let us consider the chance of an interval random event. Like the chance of fuzzy random event (Liu, 2001a), the chance for an interval random event is also a function. Generally, we have the following definition of chance of interval random event (Chen & Tan, 2009a, 2009b).
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W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
Definition 4. Let n = (n1, n2, . . ., nn) be a n-dimensional interval random vector and f : I n ! I be functions. Then the chance of interval random event characterized by f(n) 6 0, is a function Ch from (0, 1] to [0, 1] such that for any a 2 (0, 1]. We have
Chff ðnÞ 6 0gðaÞ ¼ sup fbjPrfx 2 XjF eaff ðnðxÞÞ 6 0g P bg P ag: Here, we define two critical values: optimistic value and pessimistic value (Chen & Tan, 2009a, 2009b). Definition 5. For any given decision x
fopt ¼ maxfrjChff ðx; nÞ P rgðaÞ P bg is called the (a, b)-optimistic value to the return function f(x, n), where a 2 (0, 1], b 2 [0, 1].
3.2. Interval random simulation Here, we recall two simulation algorithms (Chen & Tan, 2009a, 2009b) to calculate the (c, d)-optimistic and (c, d)-pessimistic value to the function f(n). According to Definition 5, the (c, d)-optimistic value is the max imal value f such that Ch f ðnÞ P f ðaÞ P b holds. It is obvious that the (a, b)-optimistic value f must be achieved at the equality case
Prfx 2 XjF ea f ðnðxÞÞ P f P bg ¼ a:
ð9Þ
We sample x1, x2, . . ., xN from X according to the probability meaf can be tasure Pr. Let N0 be the integer part of aN. Then the value 0 ken as the N th largest element in the sequence f 1 ; f 2 ; . . . ; f N with f n ¼ supffn jF eaff ðnðxÞÞ P fn g P bg for n = 1, 2, . . ., N. Proposition 1.
Definition 6. For any given decision x
f n ¼ mðf ðnðxn ÞÞÞ wðf ðnðxn ÞÞÞ b:
g pes ¼ minfrjChff ðx; nÞ 6 rgðaÞ P bg is called the (a, b)-pessimistic value to the return function f(x, n), where a 2 (0, 1], b 2 [0, 1]. Thus, model (5) can be reformulated to the following optimization model with chance constraint:
pffiffiffiffiffiffiffiffi
jðeÞ g pes fopt
min fxg
N X
s:t:
ð7Þ
xi ¼ 1
i¼1
where fopt is the (1 e, b1)-optimistic value to the portfolio return Pn function Uðui Þ. And gpes is the (1 e, b2)-pessimistic i¼1 ui xi ; ui 2 P P j¼N value to the function i¼N i¼1 j¼1 xi xi dij ; dij 2 Uðdij Þ.
( N X
fui 2Uðui Þg
)
u i xi
i¼1
8 ( ) N < X ¼ max f jCh Uðui Þxi P f : i¼1
ð1eÞ
g pes ¼ max
( j¼N i¼N X X
fdij 2Uðdij Þg
i¼1
9 = P b1 ; ;
xi xi dij
j¼1
ð1eÞ
9 =
P b2 ; ;
4. Computational results
Uðui Þ ¼ mi h1i ; mi þ h2i ; h i Uðdij Þ ¼ mij s1ij ; mij þ s2ij : where e, b1 and b2 are given levels (typically, e = 1%, or 5%,b1 and b2 = 90%, or 95%). We have that model (7) is equivalent to
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
min
jðeÞ minfgg maxff g
s:t:
Ch
) Uðui Þxi P f
i¼1
Ch
( j¼N i¼N X X i¼1
N X
j¼1
xi ¼ 1
i¼1
xi P 0:
P b1 ð1eÞ
A group of intelligent algorithms can be employed to find the optimal solutions of the interval random programming model (8), such as NN, GA, etc. In fact, a small change makes the hybrid NN and GA algorithm proposed by Liu (2001a, 2001b) applicable to interval random programming. A detailed descriptions of this algorithm can be found in Liu (2001a, 2001b). For interval random chance-constrained programming, the simulation methods introduced above will substitute for the original ones in the algorithm. The uncertain functions are defined as follows:
ð1eÞ
where
( N X
f n ¼ mðf ðnðxn ÞÞÞ þ wðf ðnðxn ÞÞÞ b:
8 9 ( ) N < = X U 1 : x ! sup f jCh Uðui Þxi P f P b1 ; : ; i¼1 ð1eÞ 8 9 ( ) j¼N < = i¼N X X U 2 : x ! inf gjCh xi xi Uðdij Þ 6 g P b2 : : ; i¼1 j¼1
)
8 ( ) j¼N i¼N X < X ¼ min gjCh xi xi Uðdij Þ 6 g : i¼1 j¼1
fxg
Similarly, we can find the pessimistic value f can be taken as the N0 th smallest element in the sequence f 1 ; f 2 ; . . . ; f N with f n ¼ infffn jF eaff ðnðxÞÞ 6 fn g P bg for n = 1, 2, . . ., N. It is obvious that f n must be achieved at the equality case
3.3. Hybrid-intelligent algorithm
xi P 0;
fopt ¼ min
The proof of Proposition 1 can be found in Chen and Tan (2009a, 2009b).
)
xi xi Uðdij Þ 6 g
P b2 ð1eÞ
ð8Þ
We test the viability of the proposed parametric approaches to VaR optimization using real market data. This experiment investigates whether asymmetric interval random uncertainty set is useful in real-world situations with imperfect information. We compare the performance of asymmetric robust worst-case VaR (ARVaR) model (8) taking asymmetric interval random uncertainty set to the following alternative approaches: symmetric robust worst-case VaR (SRVaR) taking symmetric interval random uncertainty set, Interval robust worst-case VaR (IRVaR) taking interval uncertainty set introduced in Tutuncu and Koenig (2004) and normal VaR (NVaR) model (1). We consider a portfolio of 24 small cap stocks from different industry categories of the S& P 600 index, and use 2000 daily historical returns from April, 1998 to June, 2006. The entire data sequence is divided into time periods of length T = 200 days. In all there are p = 10 time periods. For each period p, first, we consider moving windows of n = 20 days and compute mean returns r 0it of
W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
asset i and covariance d0 ijt of asset i and j in each such window, i = 1, . . ., 24, t = 1, . . ., T n + 1 (there are T n + 1 windows in each period). Then, we compute the following uncertainty sets for period p = 1, . . ., 10: (1) ARVaR. As defined in Section 2, the asymmetric interval random uncertainty sets take the following forms in period p:
h1i ; mi
h2i
; i
Uðui Þ ¼ mi þ h 1 Uðdij Þ ¼ mij sij ; mij þ s2ij : Then, the uncertainty set U(ui) can be formulated as
mi ¼
Tnþ1 X 1 r0 T n þ 1 t¼1 it
for i = 1, 2, h1i and h2i follow normal distri . . ., n,in period p. Suppose 1 2 bution N li ; n1i and N li ; n2i , which is given by
1 s
1
li ¼
1 n1i ¼ s 1 q
2
li ¼ n2i
1 ¼ q
Tnþ1 X t¼1
max mi r 0it ; 0 ; 8 2 1 mi r 0it li ; mi r0it > 0; : 0; mi r0it < 0;
Tnþ1 X < t¼1 Tnþ1 X t¼1
max r 0it mi ; 0 ; 8 2 2 r 0it mi li ; r 0it mi > 0; : 0; r 0it mi < 0;
Tnþ1 X < t¼1
where s is the number of r0it where mi r 0it > 0. And q is the number of r 0it where r0it mi > 0. In the same way, we can calculate the uncertainty set U(dij). (2) SRVaR. The downside and upside deviations are equivalent for symmetric interval random uncertainty. Thus, the symmetric interval random uncertainty sets take the following forms in period p:
Uðui Þ ¼ ½mi hi ; mip þ hi ; Uðdij Þ ¼ mij sij ; mij þ sij : Then, the uncertainty set U(ui) can be formulated as
mi ¼
Tnþ1 X 1 r0 T n þ 1 t¼1 it
for i = 1, 2, . . ., n, in period p. Suppose hi follows a normal distribution N(li, ni), which is given by
li ¼
1 T nþ1
Tnþ1 X
r 0 mi ; it
t¼1
Tnþ1 X 1 r 0 mi li 2 : ni ¼ it T n þ 1 t¼1
In the same way, we can calculate the uncertainty set U(dij). The VaR model for asymmetric and symmetric sets will use the ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuncertainty same model (8) with jðeÞ ¼ ð1 eÞ=e. (3) IRVaR. In Tutuncu and Koenig (2004), the uncertainty set for the expected return vector u and the covariance matrix Q take the form of intervals:
UðuÞ ¼ u : uL 6 u 6 uU ; n o UðQ Þ ¼ Q : Q L 6 Q 6 Q U ; Q P 0 : Here, uL1, uU1, QL1, QU1 take the 5% and 95% percentile values for mean returns r0it and covariance d0 ijt. uL2, uU2, QL2, QU2 take the 10% and 90% percentile values for mean returns r 0it and covariance d0ijt .
67
Thus, the worst-case VaR under interval uncertainty set is formulated as:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jðeÞ xT Q U x xT uL
min fxg
N X
s:t:
ð10Þ
xi ¼ 1
i¼1
xi P 0; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where jðeÞ ¼ ð1 eÞ=e. (4) NVaR. It is supposed that stock returns follow the normal distribution in period p. When the distribution is Gaussian, with given mean u and covariance matrix C, the NVaR portfolio model can be expressed as
min fxg
s:t:
pffiffiffiffiffiffiffiffiffiffiffi
jðeÞ xT Cx xT u N X
ð11Þ
xi ¼ 1
i¼1
xi P 0; where j(e) = U1(e). Once all the parameters are set, the robust portfolio X pasy taking asymmetric interval random uncertainty set (respectively, robust portfolio X psym taking symmetric interval random uncertainty set, robust portfolio X p5—95% taking 5–95% percentile interval, robust portfolio X p10—90% taking 10–90% percentile interval, and portfolio X pN computed by NVaR) for period p is computed by solving the robust interval random chance-constrained portfolio selection model (8) (respectively, model (8), model (10), model (10), model (11)). The portfolio X pasy ; X psym ; X p5—95% ; X p10—90% ; and X pN are held constant for the period p and then rebalanced to the portfolio ðpþ1Þ ðpþ1Þ ðpþ1Þ ðpþ1Þ X ðpþ1Þ for period p + 1. asy ; X sym ; X 5—95% ; X 10—90% ; and X N Let us compute the realized in-sample VaR values of portfolio X pasy (respectively, X psym ; X p5—95% ; X p10—90% ; X pN ) for the original training set in period p, as well as the realized out-of-sample VaR values of the test set in period p + 1. We note that the results for VaR represent the worst portfolio loss, and it is desirable to have low values for the VaR. Negative VaR value means that the portfolio loss is a gain. To compute the in-sample VaR values, we apply the following equations:
ARVaRpin ¼ Rp X pasy ð1 eÞ; SRVaRpin ¼ Rp X psym ð1 eÞ; p
IRVaRin5—95% ¼ Rp X p5—95% ð1 eÞ; p
IRVaRin10—90% ¼ Rp X p10—90% ð1 eÞ; NVaRpin
¼ Rp X pN ð1 eÞ;
Q 0 where p = 1, . . ., 10, and Rp ¼ ðp1ÞT
ARVaRpout ¼ Rp X ðp1Þ ð1 eÞ; asy SRVaRpout ¼ Rp X ðp1Þ sym ð1 eÞ; p
ðp1Þ
5—95% ¼ Rp X 5—95% ð1 eÞ; IRVaRout
p
ðp1Þ
10—90% ¼ Rp X 10—90% ð1 eÞ; IRVaRout
NVaRpout
¼ R
p
ðp1Þ XN
ð1 eÞ;
Q 0 where p = 2, . . ., 10, and Rp ¼ ðp1ÞT
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W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
Table 1 Realized portfolio VaR for in sample experiments with window n = 20. Period
In-sample VaR value ARVaR
1 2 3 4 5 6 7 8 9 10 Average
SRVaR
IRVaR5–95%
IRVaR10–90%
NVaR
e = 5%
1%
5%
1%
5%
1%
5%
1%
5%
1%
0.957 3.098 1.313 1.536 1.565 1.130 2.127 1.087 1.198 1.107 1.507
1.061 2.749 1.282 1.557 1.606 1.080 1.997 1.075 1.286 1.175 1.486
0.963 1.303 1.205 1.442 1.487 0.974 2.367 1.057 1.281 1.206 1.324
0.995 1.289 1.273 1.695 1.690 1.150 2.038 1.178 1.246 1.170 1.371
0.895 1.061 1.045 1.427 1.405 0.917 1.426 1.116 1.047 0.998 1.129
0.873 1.160 1.042 1.461 1.384 0.915 1.497 1.183 1.109 1.033 1.165
0.850 1.204 1.083 1.481 1.428 0.903 1.420 1.108 1.177 1.027 1.163
0.886 1.104 1.094 1.543 1.353 0.975 1.500 1.145 1.089 1.044 1.172
0.892 1.530 1.053 1.387 1.401 0.918 1.516 1.137 1.141 1.096 1.102
0.928 1.413 1.088 1.391 1.452 0.952 1.452 1.165 1.199 1.144 1.217
Table 2 Realized portfolio VaR for out of sample experiments with window n = 20. Period
Out-of-sample VaR value ARVaR
2 3 4 5 6 7 8 9 10 Average
SRVaR
IRVaR5–95%
IRVaR10–90%
NVaR
e = 5%
1%
5%
1%
5%
1%
5%
1%
5%
1%
1.922 1.138 1.307 1.335 0.909 1.747 0.988 1.191 1.025 1.279
2.466 1.025 1.349 1.408 0.851 1.846 1.047 1.140 1.072 1.355
1.301 1.361 1.315 1.255 0.795 1.505 0.887 1.106 1.009 1.165
1.184 0.961 1.470 1.333 0.919 1.717 0.979 1.204 1.108 1.207
1.258 0.977 1.255 1.273 0.786 1.556 1.070 1.034 1.048 1.134
1.375 1.054 1.264 1.373 0.876 1.595 1.193 1.165 1.103 1.221
1.413 1.163 1.319 1.312 0.869 1.541 1.053 1.169 1.059 1.205
1.632 1.105 1.367 1.434 0.772 1.547 1.137 1.154 1.074 1.246
1.172 1.053 1.304 1.421 0.893 1.608 1.104 1.157 1.039 1.189
1.570 1.154 1.355 1.413 0.885 1.596 1.158 1.143 1.071 1.259
Table 3 Realized portfolio VaR for in sample experiments with window n = 30. Period
In-sample VaR value ARVaR
1 2 3 4 5 6 7 8 9 10 Average
SRVaR
IRVaR5–95%
IRVaR10–90%
NVaR
e = 5%
1%
5%
1%
5%
1%
5%
1%
5%
1%
0.964 2.547 1.292 1.529 1.568 1.027 2.016 1.086 1.159 1.304 1.444
1.069 2.467 1.383 1.552 1.611 1.029 1.995 1.144 1.248 1.218 1.470
0.975 1.232 1.306 1.456 1.713 0.924 2.166 1.109 1.223 1.091 1.315
0.966 1.213 1.246 1.614 1.636 0.939 1.988 1.099 1.224 1.091 1.301
0.890 1.080 1.070 1.361 1.256 0.932 1.440 1.088 1.098 1.028 1.119
0.880 1.224 0.993 1.483 1.396 0.938 1.516 1.184 1.064 1.035 1.170
0.874 1.038 1.071 1.398 1.398 0.944 1.442 1.134 1.042 1.016 1.131
0.889 1.148 1.076 1.422 1.409 0.893 1.475 1.143 1.111 1.086 1.164
0.861 1.318 1.093 1.447 1.374 1.013 1.477 1.165 1.145 1.095 1.194
0.889 1.507 1.103 1.454 1.490 1.034 1.570 1.174 1.155 1.116 1.248
Table 4 Realized portfolio VaR for out of sample experiments with window n = 30. Period
Out-of-sample VaR value ARVaR
2 3 4 5 6 7 8 9 10 Average
SRVaR
IRVaR5–95%
IRVaR10–90%
NVaR
e = 5%
1%
5%
1%
5%
1%
5%
1%
5%
1%
1.611 1.278 1.350 1.374 0.944 1.539 0.932 1.117 1.026 1.236
1.977 1.107 1.341 1.575 0.913 1.683 0.915 1.188 1.079 1.308
1.830 0.952 1.220 1.305 0.859 1.540 0.983 1.150 1.045 1.204
2.033 1.163 1.222 1.373 0.847 1.563 1.024 1.165 1.066 1.272
1.159 1.006 1.254 1.252 0.828 1.473 1.101 1.035 1.058 1.124
1.430 1.132 1.344 1.332 0.863 1.550 1.184 1.176 1.083 1.232
1.237 1.032 1.204 1.324 0.858 1.629 1.126 1.093 1.055 1.168
1.347 1.103 1.367 1.335 0.937 1.646 1.174 1.139 1.096 1.237
1.463 1.095 1.288 1.375 0.841 1.592 1.086 1.128 1.029 1.205
1.275 1.096 1.328 1.401 0.865 1.712 1.159 1.186 1.105 1.235
W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
with window n = 20. We also consider the moving window n = 30 and recompute all parameters. Table 3 shows the in-sample VaR values with window n = 30. Table 4 plots the out-of-sample VaR values with window n = 30. The average values of realized VaR for the ARVaR method are lower than the average values of
69
other methods. It is apparent that ARVaR outperforms SRVaR, IRVaR and NVaR in terms of realized VaR. In real world, transaction cost is another important concern for portfolio managers. When choosing investment strategy, it cannot be ignored. Here, we compare the costs of implementing the
Fig. 1. Relative cost with e = 0.05 and window n = 20 at each investment period.
Fig. 2. Relative cost with e = 0.01 and window n = 20 at each investment period.
Fig. 3. Relative cost with e = 0.05 and window n = 30 at each investment period.
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W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
Fig. 4. Relative cost with e = 0.01 and window n = 30 at each investment period.
asymmetric measure strategy with those of implementing other two strategies. We calculate the transaction cost by kXp X(p1)k1. Fig. 1 shows the ratios of the costs,
. p p ðp1Þ ðp1Þ X asy X asy X sym X sym ; 1 1 . p p ðp1Þ ðp1Þ X asy X asy X 5—95% X 5—95% ; 1 1 . p p ðp1Þ ðp1Þ X asy X asy X 10—90% X 10—90% ; 1 1 . p p ðp1Þ ðp1Þ X asy X asy X N X N ; 1
1
for e = 0.05 and window n = 20. The average ratios of the costs are 1.101, 1.022, 0.874, and 1.200, respectively. As the robust asymmetric strategy becomes less conservative, it pays more in transaction costs. Fig. 2 plots the same quantity for e = 0.01 and now the average ratios of the costs are 0.967, 0.878, 0.782, 1.128, respectively. The transaction costs of asymmetric strategy were approximately 3%, 12%, 22% less than those of robust symmetric strategy and interval (5–95%, 10–90%) strategy, respectively. We also consider the moving window n = 30. Fig. 3 shows the relative costs for e = 0.05 and window n = 30, and the average ratios of the costs are 0.839, 0.795, 0.843, 0.984, respectively. Fig. 4 plots the same quantity for e = 0.01, and now the average ratios of the costs are 0.999, 0.934, 0.929, 1.093, respectively. Therefore, the robust asymmetric strategy incurs relatively less transaction costs than other two robust strategies. 5. Conclusion Building on recent research in robust portfolio, this paper introduces a novel uncertainty set: interval random uncertainty, for worst-case VaR and robust portfolio optimization. It can consider the variability of bounds and asymmetric measures of variability for the distribution of returns simultaneously. We present a robust worst-case VaR optimization model under interval random uncertainty in the elements of mean vector and covariance matrix, and reformulate this model to an mathematically meaningful one by using our interval random chance-constrained programming. A method for generating the uncertainty set from historical data and an hybrid-intelligent algorithm for solving this model are dis-
cussed. The numerical experiments presented in this paper suggest that the behavior of portfolios can be improved significantly by using the robust worst-case VaR model under interval random uncertainty set. And the robustness is achieved at relatively high performance and low cost. Acknowledgement This work is supported by the National Key Technology R;D Program of China(No. 2009BAK63B08) and the National High Technology Research and Development Program of China(’863’ Program)(No. 2009AA01Z150). References Ben-Tal, A., & Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23, 769–805. Ben-Tal, A., & Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88, 411–424. Chen, W., & Tan, S. H. (2009a). Robust portfolio selection using interval random programming. In: Proceedings of the 18th IEEE international conference on fuzzy systems. Chen, W., & Tan, S. H. (2009b). Interval random dependent-chance programming and its application to portfolio selection. In: Proceedings of the 18th IEEE international conference on fuzzy systems. El Ghaoui, L., & Lebret, H. (1997). Robust solutions to least-squares problems with uncertain data. SIAM Journal on Matrix Analysis and Applications, 18, 1035–1064. El Ghaoui, L., Oks, M., & Oustry, F. (2003). Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Operations Research, 51, 543–556. Fabozzi, F. J., Kolm, P. N., Pachamanova, D. A., & Focardi, S. M. (2007). Robust portfolio optimization. The Journal of Portfolio Management, 33, 40–48. Linsmeier, T. J., & Pearson, N. D. (1996). Risk measurement: An introduction to valueat-risk. Unpublished manuscript. Liu, B. (2001a). Fuzzy random chance-constrained programming. IEEE Transactions on Fuzzy Systems, 9, 713–720. Liu, B. (2001b). Fuzzy random dependent-chance programming. IEEE Transactions on Fuzzy Systems, 9, 721–726. Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investments. New York: Wiley. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21–41. Tutuncu, R. H., & Koenig, M. (2004). Robust asset allocation. Annals of Operations Research, 132, 157–187. Zhu, S. S., & Fukushima, M. (2005). Worst-case conditional value-at-risk with application to robust portfolio management. Unpublished manuscript.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Worst-case VaR and robust portfolio optimization with interval random uncertainty set Wei Chen a,*, Shaohua Tan b, Dongqing Yang a a b
Key Laboratory of High Confidence Software Technologies (Ministry of Education), School of EECS, Peking University, Beijing 100871, China Department of Machine Intelligence, School of EECS, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Keywords: Interval random uncertainty set Interval random chance-constrained programming Value-at-risk
a b s t r a c t This paper addresses a new uncertainty set – interval random uncertainty set for worst-case value-at-risk and robust portfolio optimization. The form of interval random uncertainty set makes it suitable for capturing the downside and upside deviations of real-world data. These deviation measures capture distributional asymmetry and lead to better optimization results. We also apply our interval random chance-constrained programming to robust worst-case value-at-risk optimization under interval random uncertainty sets in the elements of mean vector and covariance matrix. Numerical experiments with real market data indicate that our approach results in better portfolio performance. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction In 1959, Markowitz (1959) published his pioneering work that paved the foundation for modern portfolio analysis. The consideration of estimation risk and model risk has grown in importance. Different criteria for risk measure are proposed, such as variance (Markowitz, 1959), value-at-risk (VaR) (Linsmeier & Pearson, 1996), conditional value-at-risk (CVaR) (Rockafellar & Uryasev, 2000), etc. Among those, VaR remains the most widely accepted measure among practitioners. VaR estimates the maximum potential loss at a certain probability level, i.e., it provides information about the amount of losses that will not be exceeded with certain probability. Mathematically, (1 e)-VaR is defined as the minimum level c such that the probability that the portfolio loss f(x, u) is below c exceeds 1 e. Thus, the VaR can be formulated as
VðxÞ ¼ min; c s:t:
Pðf ðx; uÞ 6 cÞ P 1 e;
ð1Þ
where x = (x1, . . ., xN) is the vector of asset weights, and u = (u1, . . ., uN) is the vector of uncertain portfolio asset returns. When the distribution of returns is Gaussian, with given mean û and covariance matrix C, the VaR can be expressed as
pffiffiffiffiffiffiffiffiffiffiffi ^; VðxÞ ¼ jðeÞ xT Cx xT u
ð2Þ
where j(e) = U1(e), and U1() stands for the inverse cumulative standard normal distribution.pElffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ghaoui, ffi Oks, and Oustry (2003) specified a parameter jðeÞ ¼ ð1 eÞ=e that guaranteed that the * Corresponding author. Tel.: +86 010 62755745. E-mail address: [email protected] (W. Chen). 0957-4174/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2010.06.010
VaR constraint in model (1) would be satisfied probabilistically for all possible distributions for uncertain returns. In practice, VaR suffers from an important drawback: this approach requires a perfect knowledge of the data, in our case the mean and covariance matrix. But data are often prone to errors. Portfolio optimization based on inaccurate point estimates may be highly misleading. For example, the true VaR may be widely worse than the optimal computed VaR. Most recently, researchers have incorporated the uncertainty introduced by estimation errors directly into the portfolio optimization process by robust optimization introduced by Ben-Tal and Nemirovski (1998) and El Ghaoui and Lebret (1997). In this case, the inputs are not classical ones, such as expected returns and covariances, but rather uncertainty sets. Ben-Tal and Nemirovski (2000) summarized two frequently used uncertainty sets: (1) ‘‘Unknown-but-bounded” uncertainty set, such as box uncertainty and ellipsoidal uncertainty (Fabozzi, Kolm, Pachamanova, & Focardi, 2007). (2) ‘‘Random symmetric” uncertainty set, such as ûi = (1 + i)ui, where i are independent random variables symmetrically distributed in the interval [1, 1]. El Ghaoui et al. (2003) used robust optimization to propose a framework for optimization of worst-case VaR based on information about first and second moments of the distribution of returns. Zhu and Fukushima (2005) investigated worst-case CVaR under box uncertainty and ellipsoidal uncertainty. Though these two uncertainty sets are frequently used, they have two serious disadvantages. First, it is difficult to collect all information to determine the precise bounds of ‘‘unknownbut-bounded” uncertainty set in practice. Sometimes only the distributions of the bounds can be found from historical data. In this case, the ‘‘unknown-but-bounded” uncertainty set is actually
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W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
fluctuant instead of stable. Therefore, the variability of bounds cannot be ignored. Second, the assumption of symmetric distribution is also limiting in many applications especially in financial modeling in which distributions are often known to be asymmetric. In our other papers (Chen & Tan, 2009a, 2009b), we proposed a novel uncertainty set: interval random uncertainty, and applied it to robust mean–variance portfolio model. In this paper, we suggest a framework for robust worst-case VaR optimization based on the interval random uncertainty. Roughly speaking, an interval random uncertainty set, namely, interval random variable is an interval with random fluctuant bounds. For example, interval random variables ni are defined as ni ¼ ui h1i ; ui þ h2i . The mean values ui are mean-point of ni. Random variables h1i and h2i are downside and upside deviations of ni around mean values, respectively. Interval random variable ni consider the variability of bounds and asymmetric measures of variability for the distribution of data simultaneously. Hence, it is a good idea to introduce interval random variable as uncertainty set. Then we apply our interval random chance-constrained programming to robust worst-case VaR optimization. Finally, a hybrid-intelligent algorithm is applied to solve the robust worst-case VaR model. Some computational results are discussed that demonstrate the potentially significant economic benefits of investing in portfolios computed using classical models and the model introduced here. The robustness is achieved at relatively high performance and low cost. The rest of this paper is organized as follows. Section 2 presents basic definitions of interval random variable. In Section 3, interval random chance-constrained programming model is discussed. Section 4 presents results of some computational experiments with our robust model. Finally, a few concluding remarks are given in Section 5. 2. Interval random uncertainty set Roughly speaking, an interval random variable is a measurable function from a probability space to a collection of closed intervals. In other words, an interval random variable is a random variable taking interval values. Let I be a collection of closed intervals. For our purpose, we use the following definition of interval random variable (Chen & Tan, 2009a, 2009b). Definition 1. Let ðX; A; PrÞ be a probability space. An interval is a function n : X ! I such that nðxÞ ¼ hrandom variable i nðxÞ; nðxÞ is a measurable function of x. For example, let ðX; A; PrÞ be a probability space. Let X ¼ x1 ; x2 ; . . . ; xm , and l1, l2, . . ., lm be closed intervals in I . Then the function
nðxÞ ¼
8 l1 ; > > >
if x ¼ x1 if x ¼ x2
The assumption of symmetric distribution of traditional uncertainty sets makes no distinction between downside and upside deviations. But in real world, the distributions of returns of assets are often known to be asymmetric. The form of interval random variable makes it suitable for capturing the downside and upside deviations of asset returns. We assume that the uncertainty set for the expected return ui of asset i and covariance dij of asset i and asset j take the form of interval random variables:
Uðui Þ ¼ mi h1i ; mi þ h2i ; h i Uðdij Þ ¼ mij s1ij ; mij þ s2ij ;
where mi are the mean values of returns. Random variables h1i and h2i are downside and upside deviations for the distribution of returns, respectively. mij are the mean values of covariances. Random variables s1ij and s2ij are downside and upside deviations for the distribution of covariances, respectively. Hence, they enable us to capture the asymmetry of asset returns in order to obtain better solutions that satisfy chance constraints. If h1i ¼ h2i or s1ij ¼ s2ij , then the interval random uncertainty sets become symmetric. Thus, the worst-case VaR model (3) can be reformulated as
V ðxÞ ¼
max
fui 2Uðui Þ;dij 2Uðdij Þg
lm ; if x ¼ xm
Definition 2. Let f : I n ! I be a function over the n-dimensional Euclidean space and ni be interval random variable defined on ðXi ; Ai ; Pri Þ; i ¼ 1; 2; . . . ; n, respectively. Then, n = f(n1, n2, . . ., nn) is an interval random variable on ðX1 . . . Xn ; A1 . . . An ; Pr 1 . . . Prn Þ, defined by n(x1, x2, . . ., xn) = f(n1(x1), n2(x2), . . ., nn(xn)), for all (x1, x2, . . ., xn) 2 X1 X2 Xn. In this paper, we define the worst-case VaR under uncertainty set by
max
pffiffiffiffiffiffiffiffiffiffiffi
jðeÞ xT Cx xT u;
j¼1
ð4Þ
i¼1
min fxg
s:t:
V ðxÞ N X
ð5Þ
xi ¼ 1
i¼1
xi P 0: The objective of the robust optimization model (5) is to minimize the worst-case VaR of the portfolio. We expect that the sensitivity of the optimal solution of this mathematical program to parameter fluctuations will be significantly smaller than it would be for its classical counterpart (2). To solve this problem, we will present mathematically meaningful interval random chance-constraint programming model in Section 3. 3. Interval random programming 3.1. Chance of interval random event First, we have the following definition of function F eaðA 6 BÞ (Chen & Tan, 2009a, 2009b) which represents the grade of feasibility of the interval event ‘A to be less than or equal to B’. Definition 3. Let I be the set of all closed intervals on the real line ; B ¼ b; b R. Let A 2 I, B 2 I, A ¼ ½a; a . We define a feasibility function F ea : I I ! ½0; 1 such that
.. > > . > :
fu2UðuÞ;C2UðCÞg
i¼1
The corresponding robust portfolio optimization problem is to solve
is clearly an interval random variable.
V ðxÞ ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N N N X uX X jðeÞt xi xj dij xi ui :
ð3Þ
where U(u) is the uncertainty set for return u and U(C) is the uncertainty set for covariance matrix C.
F eaðA 6 BÞ ¼
8 1; > > > > < 1;
6 b; a
mðAÞ ¼ mðBÞ; mðBÞmðAÞ > b; > ; mðAÞ < mðBÞ and a > wðBÞþwðAÞ > > : 0; otherwise;
ð6Þ
where m(A) and w(A) are the mid-point and half-width of interval A, a aþa bþb a bb mðAÞ ¼ 2 ; wðAÞ ¼ 2 . Similarly, mðBÞ ¼ 2 ; wðBÞ ¼ 2 . At least one of A and B must be interval. If one of them is a real number, wðBÞ ¼ 0. for example B, then mðBÞ ¼ b ¼ b; Now, let us consider the chance of an interval random event. Like the chance of fuzzy random event (Liu, 2001a), the chance for an interval random event is also a function. Generally, we have the following definition of chance of interval random event (Chen & Tan, 2009a, 2009b).
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W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
Definition 4. Let n = (n1, n2, . . ., nn) be a n-dimensional interval random vector and f : I n ! I be functions. Then the chance of interval random event characterized by f(n) 6 0, is a function Ch from (0, 1] to [0, 1] such that for any a 2 (0, 1]. We have
Chff ðnÞ 6 0gðaÞ ¼ sup fbjPrfx 2 XjF eaff ðnðxÞÞ 6 0g P bg P ag: Here, we define two critical values: optimistic value and pessimistic value (Chen & Tan, 2009a, 2009b). Definition 5. For any given decision x
fopt ¼ maxfrjChff ðx; nÞ P rgðaÞ P bg is called the (a, b)-optimistic value to the return function f(x, n), where a 2 (0, 1], b 2 [0, 1].
3.2. Interval random simulation Here, we recall two simulation algorithms (Chen & Tan, 2009a, 2009b) to calculate the (c, d)-optimistic and (c, d)-pessimistic value to the function f(n). According to Definition 5, the (c, d)-optimistic value is the max imal value f such that Ch f ðnÞ P f ðaÞ P b holds. It is obvious that the (a, b)-optimistic value f must be achieved at the equality case
Prfx 2 XjF ea f ðnðxÞÞ P f P bg ¼ a:
ð9Þ
We sample x1, x2, . . ., xN from X according to the probability meaf can be tasure Pr. Let N0 be the integer part of aN. Then the value 0 ken as the N th largest element in the sequence f 1 ; f 2 ; . . . ; f N with f n ¼ supffn jF eaff ðnðxÞÞ P fn g P bg for n = 1, 2, . . ., N. Proposition 1.
Definition 6. For any given decision x
f n ¼ mðf ðnðxn ÞÞÞ wðf ðnðxn ÞÞÞ b:
g pes ¼ minfrjChff ðx; nÞ 6 rgðaÞ P bg is called the (a, b)-pessimistic value to the return function f(x, n), where a 2 (0, 1], b 2 [0, 1]. Thus, model (5) can be reformulated to the following optimization model with chance constraint:
pffiffiffiffiffiffiffiffi
jðeÞ g pes fopt
min fxg
N X
s:t:
ð7Þ
xi ¼ 1
i¼1
where fopt is the (1 e, b1)-optimistic value to the portfolio return Pn function Uðui Þ. And gpes is the (1 e, b2)-pessimistic i¼1 ui xi ; ui 2 P P j¼N value to the function i¼N i¼1 j¼1 xi xi dij ; dij 2 Uðdij Þ.
( N X
fui 2Uðui Þg
)
u i xi
i¼1
8 ( ) N < X ¼ max f jCh Uðui Þxi P f : i¼1
ð1eÞ
g pes ¼ max
( j¼N i¼N X X
fdij 2Uðdij Þg
i¼1
9 = P b1 ; ;
xi xi dij
j¼1
ð1eÞ
9 =
P b2 ; ;
4. Computational results
Uðui Þ ¼ mi h1i ; mi þ h2i ; h i Uðdij Þ ¼ mij s1ij ; mij þ s2ij : where e, b1 and b2 are given levels (typically, e = 1%, or 5%,b1 and b2 = 90%, or 95%). We have that model (7) is equivalent to
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
min
jðeÞ minfgg maxff g
s:t:
Ch
) Uðui Þxi P f
i¼1
Ch
( j¼N i¼N X X i¼1
N X
j¼1
xi ¼ 1
i¼1
xi P 0:
P b1 ð1eÞ
A group of intelligent algorithms can be employed to find the optimal solutions of the interval random programming model (8), such as NN, GA, etc. In fact, a small change makes the hybrid NN and GA algorithm proposed by Liu (2001a, 2001b) applicable to interval random programming. A detailed descriptions of this algorithm can be found in Liu (2001a, 2001b). For interval random chance-constrained programming, the simulation methods introduced above will substitute for the original ones in the algorithm. The uncertain functions are defined as follows:
ð1eÞ
where
( N X
f n ¼ mðf ðnðxn ÞÞÞ þ wðf ðnðxn ÞÞÞ b:
8 9 ( ) N < = X U 1 : x ! sup f jCh Uðui Þxi P f P b1 ; : ; i¼1 ð1eÞ 8 9 ( ) j¼N < = i¼N X X U 2 : x ! inf gjCh xi xi Uðdij Þ 6 g P b2 : : ; i¼1 j¼1
)
8 ( ) j¼N i¼N X < X ¼ min gjCh xi xi Uðdij Þ 6 g : i¼1 j¼1
fxg
Similarly, we can find the pessimistic value f can be taken as the N0 th smallest element in the sequence f 1 ; f 2 ; . . . ; f N with f n ¼ infffn jF eaff ðnðxÞÞ 6 fn g P bg for n = 1, 2, . . ., N. It is obvious that f n must be achieved at the equality case
3.3. Hybrid-intelligent algorithm
xi P 0;
fopt ¼ min
The proof of Proposition 1 can be found in Chen and Tan (2009a, 2009b).
)
xi xi Uðdij Þ 6 g
P b2 ð1eÞ
ð8Þ
We test the viability of the proposed parametric approaches to VaR optimization using real market data. This experiment investigates whether asymmetric interval random uncertainty set is useful in real-world situations with imperfect information. We compare the performance of asymmetric robust worst-case VaR (ARVaR) model (8) taking asymmetric interval random uncertainty set to the following alternative approaches: symmetric robust worst-case VaR (SRVaR) taking symmetric interval random uncertainty set, Interval robust worst-case VaR (IRVaR) taking interval uncertainty set introduced in Tutuncu and Koenig (2004) and normal VaR (NVaR) model (1). We consider a portfolio of 24 small cap stocks from different industry categories of the S& P 600 index, and use 2000 daily historical returns from April, 1998 to June, 2006. The entire data sequence is divided into time periods of length T = 200 days. In all there are p = 10 time periods. For each period p, first, we consider moving windows of n = 20 days and compute mean returns r 0it of
W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
asset i and covariance d0 ijt of asset i and j in each such window, i = 1, . . ., 24, t = 1, . . ., T n + 1 (there are T n + 1 windows in each period). Then, we compute the following uncertainty sets for period p = 1, . . ., 10: (1) ARVaR. As defined in Section 2, the asymmetric interval random uncertainty sets take the following forms in period p:
h1i ; mi
h2i
; i
Uðui Þ ¼ mi þ h 1 Uðdij Þ ¼ mij sij ; mij þ s2ij : Then, the uncertainty set U(ui) can be formulated as
mi ¼
Tnþ1 X 1 r0 T n þ 1 t¼1 it
for i = 1, 2, h1i and h2i follow normal distri . . ., n,in period p. Suppose 1 2 bution N li ; n1i and N li ; n2i , which is given by
1 s
1
li ¼
1 n1i ¼ s 1 q
2
li ¼ n2i
1 ¼ q
Tnþ1 X t¼1
max mi r 0it ; 0 ; 8 2 1 mi r 0it li ; mi r0it > 0; : 0; mi r0it < 0;
Tnþ1 X < t¼1 Tnþ1 X t¼1
max r 0it mi ; 0 ; 8 2 2 r 0it mi li ; r 0it mi > 0; : 0; r 0it mi < 0;
Tnþ1 X < t¼1
where s is the number of r0it where mi r 0it > 0. And q is the number of r 0it where r0it mi > 0. In the same way, we can calculate the uncertainty set U(dij). (2) SRVaR. The downside and upside deviations are equivalent for symmetric interval random uncertainty. Thus, the symmetric interval random uncertainty sets take the following forms in period p:
Uðui Þ ¼ ½mi hi ; mip þ hi ; Uðdij Þ ¼ mij sij ; mij þ sij : Then, the uncertainty set U(ui) can be formulated as
mi ¼
Tnþ1 X 1 r0 T n þ 1 t¼1 it
for i = 1, 2, . . ., n, in period p. Suppose hi follows a normal distribution N(li, ni), which is given by
li ¼
1 T nþ1
Tnþ1 X
r 0 mi ; it
t¼1
Tnþ1 X 1 r 0 mi li 2 : ni ¼ it T n þ 1 t¼1
In the same way, we can calculate the uncertainty set U(dij). The VaR model for asymmetric and symmetric sets will use the ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiuncertainty same model (8) with jðeÞ ¼ ð1 eÞ=e. (3) IRVaR. In Tutuncu and Koenig (2004), the uncertainty set for the expected return vector u and the covariance matrix Q take the form of intervals:
UðuÞ ¼ u : uL 6 u 6 uU ; n o UðQ Þ ¼ Q : Q L 6 Q 6 Q U ; Q P 0 : Here, uL1, uU1, QL1, QU1 take the 5% and 95% percentile values for mean returns r0it and covariance d0 ijt. uL2, uU2, QL2, QU2 take the 10% and 90% percentile values for mean returns r 0it and covariance d0ijt .
67
Thus, the worst-case VaR under interval uncertainty set is formulated as:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jðeÞ xT Q U x xT uL
min fxg
N X
s:t:
ð10Þ
xi ¼ 1
i¼1
xi P 0; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where jðeÞ ¼ ð1 eÞ=e. (4) NVaR. It is supposed that stock returns follow the normal distribution in period p. When the distribution is Gaussian, with given mean u and covariance matrix C, the NVaR portfolio model can be expressed as
min fxg
s:t:
pffiffiffiffiffiffiffiffiffiffiffi
jðeÞ xT Cx xT u N X
ð11Þ
xi ¼ 1
i¼1
xi P 0; where j(e) = U1(e). Once all the parameters are set, the robust portfolio X pasy taking asymmetric interval random uncertainty set (respectively, robust portfolio X psym taking symmetric interval random uncertainty set, robust portfolio X p5—95% taking 5–95% percentile interval, robust portfolio X p10—90% taking 10–90% percentile interval, and portfolio X pN computed by NVaR) for period p is computed by solving the robust interval random chance-constrained portfolio selection model (8) (respectively, model (8), model (10), model (10), model (11)). The portfolio X pasy ; X psym ; X p5—95% ; X p10—90% ; and X pN are held constant for the period p and then rebalanced to the portfolio ðpþ1Þ ðpþ1Þ ðpþ1Þ ðpþ1Þ X ðpþ1Þ for period p + 1. asy ; X sym ; X 5—95% ; X 10—90% ; and X N Let us compute the realized in-sample VaR values of portfolio X pasy (respectively, X psym ; X p5—95% ; X p10—90% ; X pN ) for the original training set in period p, as well as the realized out-of-sample VaR values of the test set in period p + 1. We note that the results for VaR represent the worst portfolio loss, and it is desirable to have low values for the VaR. Negative VaR value means that the portfolio loss is a gain. To compute the in-sample VaR values, we apply the following equations:
ARVaRpin ¼ Rp X pasy ð1 eÞ; SRVaRpin ¼ Rp X psym ð1 eÞ; p
IRVaRin5—95% ¼ Rp X p5—95% ð1 eÞ; p
IRVaRin10—90% ¼ Rp X p10—90% ð1 eÞ; NVaRpin
¼ Rp X pN ð1 eÞ;
Q 0 where p = 1, . . ., 10, and Rp ¼ ðp1ÞT
ARVaRpout ¼ Rp X ðp1Þ ð1 eÞ; asy SRVaRpout ¼ Rp X ðp1Þ sym ð1 eÞ; p
ðp1Þ
5—95% ¼ Rp X 5—95% ð1 eÞ; IRVaRout
p
ðp1Þ
10—90% ¼ Rp X 10—90% ð1 eÞ; IRVaRout
NVaRpout
¼ R
p
ðp1Þ XN
ð1 eÞ;
Q 0 where p = 2, . . ., 10, and Rp ¼ ðp1ÞT
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W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
Table 1 Realized portfolio VaR for in sample experiments with window n = 20. Period
In-sample VaR value ARVaR
1 2 3 4 5 6 7 8 9 10 Average
SRVaR
IRVaR5–95%
IRVaR10–90%
NVaR
e = 5%
1%
5%
1%
5%
1%
5%
1%
5%
1%
0.957 3.098 1.313 1.536 1.565 1.130 2.127 1.087 1.198 1.107 1.507
1.061 2.749 1.282 1.557 1.606 1.080 1.997 1.075 1.286 1.175 1.486
0.963 1.303 1.205 1.442 1.487 0.974 2.367 1.057 1.281 1.206 1.324
0.995 1.289 1.273 1.695 1.690 1.150 2.038 1.178 1.246 1.170 1.371
0.895 1.061 1.045 1.427 1.405 0.917 1.426 1.116 1.047 0.998 1.129
0.873 1.160 1.042 1.461 1.384 0.915 1.497 1.183 1.109 1.033 1.165
0.850 1.204 1.083 1.481 1.428 0.903 1.420 1.108 1.177 1.027 1.163
0.886 1.104 1.094 1.543 1.353 0.975 1.500 1.145 1.089 1.044 1.172
0.892 1.530 1.053 1.387 1.401 0.918 1.516 1.137 1.141 1.096 1.102
0.928 1.413 1.088 1.391 1.452 0.952 1.452 1.165 1.199 1.144 1.217
Table 2 Realized portfolio VaR for out of sample experiments with window n = 20. Period
Out-of-sample VaR value ARVaR
2 3 4 5 6 7 8 9 10 Average
SRVaR
IRVaR5–95%
IRVaR10–90%
NVaR
e = 5%
1%
5%
1%
5%
1%
5%
1%
5%
1%
1.922 1.138 1.307 1.335 0.909 1.747 0.988 1.191 1.025 1.279
2.466 1.025 1.349 1.408 0.851 1.846 1.047 1.140 1.072 1.355
1.301 1.361 1.315 1.255 0.795 1.505 0.887 1.106 1.009 1.165
1.184 0.961 1.470 1.333 0.919 1.717 0.979 1.204 1.108 1.207
1.258 0.977 1.255 1.273 0.786 1.556 1.070 1.034 1.048 1.134
1.375 1.054 1.264 1.373 0.876 1.595 1.193 1.165 1.103 1.221
1.413 1.163 1.319 1.312 0.869 1.541 1.053 1.169 1.059 1.205
1.632 1.105 1.367 1.434 0.772 1.547 1.137 1.154 1.074 1.246
1.172 1.053 1.304 1.421 0.893 1.608 1.104 1.157 1.039 1.189
1.570 1.154 1.355 1.413 0.885 1.596 1.158 1.143 1.071 1.259
Table 3 Realized portfolio VaR for in sample experiments with window n = 30. Period
In-sample VaR value ARVaR
1 2 3 4 5 6 7 8 9 10 Average
SRVaR
IRVaR5–95%
IRVaR10–90%
NVaR
e = 5%
1%
5%
1%
5%
1%
5%
1%
5%
1%
0.964 2.547 1.292 1.529 1.568 1.027 2.016 1.086 1.159 1.304 1.444
1.069 2.467 1.383 1.552 1.611 1.029 1.995 1.144 1.248 1.218 1.470
0.975 1.232 1.306 1.456 1.713 0.924 2.166 1.109 1.223 1.091 1.315
0.966 1.213 1.246 1.614 1.636 0.939 1.988 1.099 1.224 1.091 1.301
0.890 1.080 1.070 1.361 1.256 0.932 1.440 1.088 1.098 1.028 1.119
0.880 1.224 0.993 1.483 1.396 0.938 1.516 1.184 1.064 1.035 1.170
0.874 1.038 1.071 1.398 1.398 0.944 1.442 1.134 1.042 1.016 1.131
0.889 1.148 1.076 1.422 1.409 0.893 1.475 1.143 1.111 1.086 1.164
0.861 1.318 1.093 1.447 1.374 1.013 1.477 1.165 1.145 1.095 1.194
0.889 1.507 1.103 1.454 1.490 1.034 1.570 1.174 1.155 1.116 1.248
Table 4 Realized portfolio VaR for out of sample experiments with window n = 30. Period
Out-of-sample VaR value ARVaR
2 3 4 5 6 7 8 9 10 Average
SRVaR
IRVaR5–95%
IRVaR10–90%
NVaR
e = 5%
1%
5%
1%
5%
1%
5%
1%
5%
1%
1.611 1.278 1.350 1.374 0.944 1.539 0.932 1.117 1.026 1.236
1.977 1.107 1.341 1.575 0.913 1.683 0.915 1.188 1.079 1.308
1.830 0.952 1.220 1.305 0.859 1.540 0.983 1.150 1.045 1.204
2.033 1.163 1.222 1.373 0.847 1.563 1.024 1.165 1.066 1.272
1.159 1.006 1.254 1.252 0.828 1.473 1.101 1.035 1.058 1.124
1.430 1.132 1.344 1.332 0.863 1.550 1.184 1.176 1.083 1.232
1.237 1.032 1.204 1.324 0.858 1.629 1.126 1.093 1.055 1.168
1.347 1.103 1.367 1.335 0.937 1.646 1.174 1.139 1.096 1.237
1.463 1.095 1.288 1.375 0.841 1.592 1.086 1.128 1.029 1.205
1.275 1.096 1.328 1.401 0.865 1.712 1.159 1.186 1.105 1.235
W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
with window n = 20. We also consider the moving window n = 30 and recompute all parameters. Table 3 shows the in-sample VaR values with window n = 30. Table 4 plots the out-of-sample VaR values with window n = 30. The average values of realized VaR for the ARVaR method are lower than the average values of
69
other methods. It is apparent that ARVaR outperforms SRVaR, IRVaR and NVaR in terms of realized VaR. In real world, transaction cost is another important concern for portfolio managers. When choosing investment strategy, it cannot be ignored. Here, we compare the costs of implementing the
Fig. 1. Relative cost with e = 0.05 and window n = 20 at each investment period.
Fig. 2. Relative cost with e = 0.01 and window n = 20 at each investment period.
Fig. 3. Relative cost with e = 0.05 and window n = 30 at each investment period.
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W. Chen et al. / Expert Systems with Applications 38 (2011) 64–70
Fig. 4. Relative cost with e = 0.01 and window n = 30 at each investment period.
asymmetric measure strategy with those of implementing other two strategies. We calculate the transaction cost by kXp X(p1)k1. Fig. 1 shows the ratios of the costs,
. p p ðp1Þ ðp1Þ X asy X asy X sym X sym ; 1 1 . p p ðp1Þ ðp1Þ X asy X asy X 5—95% X 5—95% ; 1 1 . p p ðp1Þ ðp1Þ X asy X asy X 10—90% X 10—90% ; 1 1 . p p ðp1Þ ðp1Þ X asy X asy X N X N ; 1
1
for e = 0.05 and window n = 20. The average ratios of the costs are 1.101, 1.022, 0.874, and 1.200, respectively. As the robust asymmetric strategy becomes less conservative, it pays more in transaction costs. Fig. 2 plots the same quantity for e = 0.01 and now the average ratios of the costs are 0.967, 0.878, 0.782, 1.128, respectively. The transaction costs of asymmetric strategy were approximately 3%, 12%, 22% less than those of robust symmetric strategy and interval (5–95%, 10–90%) strategy, respectively. We also consider the moving window n = 30. Fig. 3 shows the relative costs for e = 0.05 and window n = 30, and the average ratios of the costs are 0.839, 0.795, 0.843, 0.984, respectively. Fig. 4 plots the same quantity for e = 0.01, and now the average ratios of the costs are 0.999, 0.934, 0.929, 1.093, respectively. Therefore, the robust asymmetric strategy incurs relatively less transaction costs than other two robust strategies. 5. Conclusion Building on recent research in robust portfolio, this paper introduces a novel uncertainty set: interval random uncertainty, for worst-case VaR and robust portfolio optimization. It can consider the variability of bounds and asymmetric measures of variability for the distribution of returns simultaneously. We present a robust worst-case VaR optimization model under interval random uncertainty in the elements of mean vector and covariance matrix, and reformulate this model to an mathematically meaningful one by using our interval random chance-constrained programming. A method for generating the uncertainty set from historical data and an hybrid-intelligent algorithm for solving this model are dis-
cussed. The numerical experiments presented in this paper suggest that the behavior of portfolios can be improved significantly by using the robust worst-case VaR model under interval random uncertainty set. And the robustness is achieved at relatively high performance and low cost. Acknowledgement This work is supported by the National Key Technology R;D Program of China(No. 2009BAK63B08) and the National High Technology Research and Development Program of China(’863’ Program)(No. 2009AA01Z150). References Ben-Tal, A., & Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23, 769–805. Ben-Tal, A., & Nemirovski, A. (2000). Robust solutions of linear programming problems contaminated with uncertain data. Mathematical Programming, 88, 411–424. Chen, W., & Tan, S. H. (2009a). Robust portfolio selection using interval random programming. In: Proceedings of the 18th IEEE international conference on fuzzy systems. Chen, W., & Tan, S. H. (2009b). Interval random dependent-chance programming and its application to portfolio selection. In: Proceedings of the 18th IEEE international conference on fuzzy systems. El Ghaoui, L., & Lebret, H. (1997). Robust solutions to least-squares problems with uncertain data. SIAM Journal on Matrix Analysis and Applications, 18, 1035–1064. El Ghaoui, L., Oks, M., & Oustry, F. (2003). Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Operations Research, 51, 543–556. Fabozzi, F. J., Kolm, P. N., Pachamanova, D. A., & Focardi, S. M. (2007). Robust portfolio optimization. The Journal of Portfolio Management, 33, 40–48. Linsmeier, T. J., & Pearson, N. D. (1996). Risk measurement: An introduction to valueat-risk. Unpublished manuscript. Liu, B. (2001a). Fuzzy random chance-constrained programming. IEEE Transactions on Fuzzy Systems, 9, 713–720. Liu, B. (2001b). Fuzzy random dependent-chance programming. IEEE Transactions on Fuzzy Systems, 9, 721–726. Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investments. New York: Wiley. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21–41. Tutuncu, R. H., & Koenig, M. (2004). Robust asset allocation. Annals of Operations Research, 132, 157–187. Zhu, S. S., & Fukushima, M. (2005). Worst-case conditional value-at-risk with application to robust portfolio management. Unpublished manuscript.