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On the Tail Mean–Variance optimal portfolio selection
Insurance: Mathematics and Economics 46 (2010) 547–553
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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
On the Tail Mean–Variance optimal portfolio selection Zinoviy Landsman Department of Statistics, University of Haifa, Mount Carmel, 31905 Haifa, Israel
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Article history: Received June 2009 Received in revised form October 2009 Accepted 5 February 2010 Keywords: Tail condition expectation Tail variance Tail Mean–Variance model Optimal portfolio selection Square root of quadratic functional Elliptical family Quartic equation
abstract In the present paper we propose the Tail Mean–Variance (TMV) approach, based on Tail Condition Expectation (TCE) (or Expected Short Fall) and the recently introduced Tail Variance (TV) as a measure for the optimal portfolio selection. We show that, when the underlying distribution is multivariate normal, the TMV model reduces to a more complicated functional than the quadratic and represents a combination of linear, square root of quadratic and quadratic functionals. We show, however, that under general linear constraints, the solution of the optimization problem still exists and in the case where short selling is possible we provide an analytical closed form solution, which looks more ‘‘robust’’ than the classical MV solution. The results are extended to more general multivariate elliptical distributions of risks. © 2010 Elsevier B.V. All rights reserved.
1. Introduction
constraint (1.2) has the following closed form solution
The classical mean–variance (MV) optimal portfolio selection theory was initiated by Markowitz’s (1952) pioneering paper and further developed by many authors. The subject is so extensively studied that, for example, the review given by Steinbach (2001) contains 208 references and it does not cover all relevant papers. The MV model uses the Mean–Variance risk measure
1 Σ −1 1 x = T −1 + 1 Σ 1 2λ ∗
Σ
−1
µ−
1T Σ −1 µ 1T Σ −1 1
Σ
−1
1
which, in finance literature, is also known as the expected quadratic utility, is the subject of minimization, where L = −R is the loss on the portfolio return R = xT X. Here X = (X1 , . . . , Xn )T is the random vector of returns, xT = (x1 , . . . , xn ) is the vector of generalized weights in the sense that
(see Boyle et al., 1998, Section 6). Additional material and references can be found in books by Markowitz (1987), Elton and Gruberm (1987), Alexander and Sharpe (1989), Zenios (1992), Ziemba and Mulvey (1998) and McNeil et al. (2005). Let us notice that the MV does not pay any special attention to the tail behavior. A growing interest in risk measures based on the tails of distributions is currently being observed among investment and insurance experts. The interest in the tail of the distribution has generated well-known measure, the expected short fall (ES) or the tail conditional expectation (TCE)
1T x = 1,
TCEq (L) = E L|L > VaRq (L) .
MV (L) = E (L) + λV (L),
(1.1)
(1.2)
(1.3)
where 1 is the vector-column of n ones. It is clear that
It is interpreted as the expectation of the worse loss in the sense that this loss exceeds a particular value-at-risk
MV (L) = E (−R) + λV (R) = −µT x + λxT Σ x,
VaRq (L) = inf{x : FL (x) ≥ q},
where µ is the vector of expectations of X, and Σ is its covariance matrix. So the MV model reduces to the problem of quadratic programming, which is quite convenient for technical realization (Luenberger, 1984), and, for the case when short selling is possible, even provides an analytic solution (first noticed by Merton (1972)). So, the problem of minimization of the MV (L) under
where FL (x) is the cdf of loss L. The expression in (1.3) was initially recommended by Artzner et al. (1999) to measure both market and nonmarket risks, presumably for a portfolio of investments. Acerbi and Tasche (2002) have shown the attractive coherence properties of the TCE. Tail conditional expectations for the univariate and multivariate Normal family were developed by Panjer (2002). His results were generalized by Landsman and Valdez (2003) for the larger class of elliptical distributions. Landsman and Valdez (2005) also developed formulae for tail conditional expectations
E-mail address: [email protected]. 0167-6687/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2010.02.001
548
Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
∗ Fig. 2. Changing of ‘‘amplitude’’ wl with increasing of level q from 0 to 1.
Fig. 1. Comparison between TMV-optimal portfolios.
for random risks belonging to the class of exponential dispersion models. The TCE for multivariate gamma models was obtained in Furman and Landsman (2005) and for the multivariate Pareto in Chiragiev and Landsman (2007). Notice that for the case when the variance of L, V (L), exists, TCEq (L) is the best mean square approximation of random L by nonrandom constant c along the tail, in the sense TCEq (L) = arg inf E (L − c )2 |L > VaRq (L) ,
for some
n vector-column µ, a n × n positive-definite matrix Σ = σij i,j=1 , and for some function ψ(t ) called the characteristic generator (see details in Fang et al., 1990). The elliptical class of distributions is popular for the purpose of providing a satisfactory fit in modelling asset or loss returns in the multivariate set up (see Owen and Rabinovitch (1983) and Bingham and Kiesel (2002)). If the density of X exists (Fang et al., 1990, condition (2.19)), it has the form
c
i.e., the TCE provides a risk manager with information about the average of the tail of the loss distribution. In Furman and Landsman (2006) a tail variance risk measure (TV) was proposed TVq (L) = V (L|L > VaRq (L)) = inf E (L − c )2 |L > VaRq (L) ,
c
which estimates the deviance of the loss from this average along such a tail. Furthermore, the explicit expressions for TV in the multivariate normal and more general, multivariate elliptical distribution, were derived. In the following we propose a Tail Mean–Variance (TMV) model instead of MV in the optimal portfolio selection problem. More precisely, instead of (1.1), let us consider for some level q ∈ (0, 1), TMVq (L) = TCEq (L) + λTVq (L) as the subject of minimization. Recall that portfolio return R = −L, and then the problem TMVq (L) → inf E (R|R < VaR1−q (R)) − λV (R|R < VaR1−q (R)) → sup, which means that one is interested in the increase in expectation of only (1 − q) percent of small portfolio returns (not all the returns) and the balanced decrease of their variances. Let us notice that the MV model is a limit case of the TMV model when probability level q decreases to 0. We demonstrate that the volatility of the weights of the optimal solution of the TMV model decreases when q increases from 0 to 1. So the MV model has more volatility in comparison with the TMV model. In this sense the TMV model is essentially more robust then the MV model (see Section 3, Figs. 1 and 2). Let the vector of random variables X = (X1 , X2 , . . . , Xn )T , which can be associated with asset returns, be multivariate elliptically distributed, written as X v En (µ, Σ , ψ). The latter means that the characteristic function of vector X can be expressed as
ϕX (t) = exp(it µ)ψ
1 2
t Σt T
|Σ |
gn
1 2
(x − µ)T Σ −1 (x − µ) ,
(1.5)
for some function gn (·) called the density generator. One may also use the notation X ∼ En (µ, Σ , gn ). From (1.4) it follows that, if A is a m × n matrix of rank m ≤ n, and b is a m dimensional columnvector, then AX + b ∼ Em Aµ + b, AΣ AT , gm ,
(1.6)
i.e., any linear combination of elliptical vectors is another elliptical vector with the same characteristic generator ψ or from the same sequence of density generators g1 , . . . , gn , corresponding to ψ . Using the results of Landsman and Valdez (2003) and Furman and Landsman (2006) results, for portfolio return R = xT X, where xT = (x1 , . . . , xn ) is the vector of real numbers, one can write, TMVq (L)
p = TMVq (−R) = −µT x + λ1,q Var (xT X) + λλ2,q Var (xT X) √ = −µT x + λ1,q xT Σ x + λλ2,q xT Σ x, (1.7)
is equivalent to the problem
T
cn fX (x) = √
(1.4)
where λ1,q , λ2,q are some constant that depends on level q. For a multivariate normal model,
λ1,q =
ϕ(Zq ) , 1 − Φ (Zq )
λ2,q = 1 − λ1,q δq ,
δq = (λ1,q − Zq ),
(1.8)
where ϕ(x) and Φ (x) are density and distribution functions of the standard normal random variable Z , and Zq is its q-quantile. In the case of a more general multivariate elliptical setup, λ1,q , λ2,q look slightly more complicated and can be found in Furman and Landsman (2006). This setup will be considered in more detail in Section 4. So, the problem of the optimal portfolio selection in the Tail MV model is, in fact, equivalent to that of the problem of the minimization of the functional f (x) = −µT x + λ1,q
√
xT Σ x + λλ2,q xT Σ x,
(1.9)
Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
549
l
which is a combination of the linear functional, square root of a quadratic and a quadratic function with balance parameter λ > 0, subject to the linear constraint (1.2). This problem is considered in Section 2, where the analytical closed form solution is evaluated under a system of linear constraints. Section 3 is devoted to the application of the obtained results to stock data returns. A comparison between the TMV-optimal portfolios and efficient frontiers corresponding to different q-levels is provided. In Section 4 the results are extended to the multivariate elliptical model. Special attention was paid to the multivariate generalized Student-t model, which is often regarded to be an appropriate model for stock returns (MacDonald, 1996; Aparicio and Estrada, 2001). The analytic solution of the quartic equation, which is important for the closed form solution of the optimal problem being considered here, is presented in the Appendix.
f0 = cT (BΣ −1 BT )−1 c,
2. Minimization of the combination of linear, root of quadratic and quadratic functionals
1 Proof. Define vector d2 = B− 22 c. Then from system of constraints (2.12) and from (2.14) it follows that xT = (xT1 , dT2 − xT1 D12 )T and then straightforwardly
First we notice that the goal function
√
f (x) = µT x + β( xT Σ x + α xT Σ x),
(2.10)
α, β > 0, which is the main subject of this Section, is convex as a sum of the linear form and the convex functions. The convexity of the square root of the quadratic form follows from the fact that, for any u, v ∈ Rn and t ∈ R,
p p (u + tv)T Σ (u + tv) = vT Σ vt 2 + 2vT Σ ut + uT Σ u
Bx = c,
c 6= 0,
(2.12)
m,n bij i,j=1
where B = ( ) is m × n, m < n, is the rectangular full rank matrix, c is some m × 1 vector, and 0 is a vector-column of m zeros. We advance the approach given in Landsman (2008), where only the combination of linear and square root quadratic functionals were investigated, to the more complicated functional (2.10). Choosing the first n − m variables we have the natural partition of vector xT = (xT1 , xT2 ), x1 = (x1 , . . . , xn−m )T , x2 = (xn−m+1 , . . . , xn )T and the corresponding partition of vectors µT = (µT1 , µT2 ), 1T = (1T1 , 1T2 ), matrix Σ ,
Σ11 Σ= Σ21
Σ12 Σ22
(2.13)
and matrix B = B21
B22 ,
where matrices B21 and B22 are of dimensions m × (m − n) and m × m, respectively. As matrix B is of full rank, then suppose without loss of generality that matrix B22 is nonsingular. Define m × (n − m) and (n − m) × m matrices D12 = DT21
(2.14)
Q = Σ11 − Σ12 D21 − D12 Σ21 + D12 Σ22 D21 = ( )
m qij ni,− j =1
l=
1
and set
T
Q −1 1
x∗ = Σ −1 BT (BΣ −1 BT )−1 c +
w∗ l
(1T Q −1 , −1T Q −1 D12 )T , (2.18)
where w ∗ is the unique rational root of the quartic equation
w − 2kw + f0 + k − 4
3
1
2
4α 2
w 2 − 2kf0 w + k2 f0 = 0,
(2.19)
located in the interval (0, k). The analytic form for w ∗ exists and will be given in the Appendix.
xT Σ x = xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2 .
(2.20)
The goal-function f (x) = g (x1 ) = µT2 d2 + (µ1 − D12 µ2 )T x1
+β
q
xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2
+ α(
xT1 Q x1
+
2dT2
(Σ21 − Σ22 D21 )x1 +
dT2 Σ22 d2
) ,
is a function of n − m variables, x1 = (x1 , . . . , xn−m )T , and the problem reduces to the problem of finding the unconditional minimum min g (x1 ).
x1 ∈Rn−m
As a corollary of the well-known solution of the quadratic programming problem x0 = arg min xT Σ x =Σ −1 BT (BΣ −1 BT )−1 c,
(2.21)
Bx=c
and xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2 T
≥ x0 Σ x0 = f0 > 0,
x1 ∈ Rn−m ,
(2.22)
as matrix BΣ B > 0 and c 6= 0 (see, for example, Luenberger, 1984, Chapter 14.1). This means that the function q −1 T
xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2 is differentiable for
that function g (x1 ) is strictly convex on Rn−m . Denote by
.
dT dx1
= ( dxd1 · · · dxnd−m ) the vector-row of the first n − m derivatives and let x∗ and x0 be partitioned as follows x∗ = (x∗1T , dT2 − x∗1T D12 )T
(2.23)
and x = ( , − ) Then the vector x1 is the unique solution of the vector-equations (2.15)
In the cited paper it was proved that, together with Σ , matrix Q is positive definite. Denote by
q
(2.17)
Theorem 1. The problem of the minimization of function (2.10) subject to (2.12) has the following solution
0
and (n − m) × (n − m) matrix
1 = D12 µ2 − µ1 ,
,
any x1 ∈ Rn−m . For the same reasons as those given in (2.11) and taking into account the last inequality, one may conclude
1 D21 = B− 22 B21 ,
2αβ
we now state the following theorem:
(2.11)
is a strictly convex function of t (the square root of a quadratic univariate polynomial with a positive leading coefficient √ and negative discriminant is strictly convex) and consequently xT Σ x is strictly convex. We give the explicit closed form solution for the problem of minimization of a function f (x) subject to a system of linear restrictions presented in the matrix form
k=
(2.16)
d dx1
x0T 1
dT2
T x0T 1 D12
∗
g (x1 )
= (µ1 − D12 µ2 ) + β((xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2 )−1/2 + 2α)(Q x1 + (Σ12 − D12 Σ22 )d2 ) = 01 , where 01 is a vector-column of (n − m) zeros, taking into account (2.22). This equation can be rewritten in the form
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Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
(Q x1 + (Σ12 − D12 Σ22 )d2 ) = τ((xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2 )−1/2 + 2α)−1 , where 1
τ = (τ1 , . . . , τn−m )T =
1
(D12 µ2 − µ1 ) = − 1. β β
P4 (0) = k2 f0 > 0, (2.24)
Consider x∗1 in the form x∗1 = x01 + y∗ , where y∗T = (y1 , . . . , yn−m ) is the (n − m) dimension vector. Then, as x0 is a solution of problem (2.21), it follows that Q x01 + (Σ12 − D12 Σ22 )d2 = 01
A brief analysis of quartic equation (2.19) shows that it has a rational root on the interval (0, k). In fact, defining P4 (w) = w 4 − 2kw3 + (f0 + k2 − 4α1 2 )w 2 − 2kf0 w + k2 f0 , one has
(2.25)
∗
and y is the unique solution of the vector-equations
P4 (k) = −
k2
The root is unique in interval (0, k), because the considered optimization problem has a unique solution. The analytic explicit form for this root will be given in the Appendix, where Neumark’s (1965) approach to the solution of quartic equations is implemented. Substituting (2.31) into (2.28) we obtain, taking into account (2.29), (2.24) and (2.27),
y∗ = Q −1 τ((x∗1T Q x∗1 + 2dT2 (Σ21 − Σ22 D21 )x∗1
+ dT2 Σ22 d2 )−1/2 + 2α)−1 .
(2.26)
1 ∗ As τ = 01 (means µ1 = B12 B− 22 µ2 ) it results trivially that y = 01 . n−m −1 Suppose vector τ 6= 0, then as matrix Q = (δij )i,j=1 is nonsin-
gular (positive definite) there exists row δTi = (δi1 , . . . ., δin−m ) of Q −1 such that δTi τ 6= 0. Suppose for convenience, and without loss of generality, that i = 1. Then, using the following partition of matrix Q −1 into the 2 matrices Q1−1 and Q2−1 , Q
−1
Q1−1 Q2−1
=
y∗ = y∗1 (1, rT )T ,
(2.28)
where r = Q2−1 τ/Q1−1 τ.
(2.29)
Substituting (2.28) into the first equation of (2.26), we get straightforwardly y∗1 = Q1−1 τ((x∗1T Q x∗1 + 2dT2 (Σ21 − Σ22 D21 )x∗1
+ dT2 Σ22 d2 )−1/2 + 2α)−1
−1
1 f0 + y1 (1, rT )Q (1, rT )T ∗2
+ 2α
−1
1 f0 +
y ∗2 l2
1 (Q1−1 τ)2 β 2
+ 2α
.
(2.30)
The latter was done by taking into account (2.25), (2.20), the right hand side of (2.22) and equality
(1, rT )Q (1, rT )T =
1
(Q1−1 τ)2
τ T Q −1 τ,
obtained in Landsman (2008, Eq.(29)). After substituting
w = ∗
l
y1 , ∗
β Q1−1 τ
(2.31)
the Eq. (2.30) reduces to
√
1
f0 + w ∗
=
l
βw∗
−1
βw∗ l
1,
(Q2−1 τ)T
!T
Q1 τ −1
=
w∗ l
Q −1 1.
This finishes the proof of the Theorem, taking into account that from (2.23) and (2.21) x∗ = ((x01 + y∗ )T , (d2 − D21 x01 − D21 y∗ )T )T
= x0 + (y∗T , −y∗T D12 )T .
(2.27)
where Q1−1 is simply the first row of Q −1 (i.e., Q1−1 = δT1 ) and Q2−1 consists of other (n − m − 1) rows of Q −1 , we have from (2.26)
= Q1−1 τ r
y = Q1 τ ∗
3. Application to stock data returns
,
= Q1−1 τ q
< 0.
4α 2
− 2α,
w ∗ > 0.
The latter is equivalent to the quartic equation (2.19) together with conditions βwl ∗ − 2α > 0, w ∗ > 0. This means that w ∗ is a root of Eq. (2.19) in interval (0, k), recalling (2.17).
Consider a portfolio of 10 stocks from NASDAQ/Computers (ADOBE Sys. Inc., Top Image System (TISA), Hauppauge Digit (HAUP), Immerson Corp (IMMR), Logithch Int. Sa. (LOGI), NVIDIA Corp., O2micro Intl Lt (OIIM), Universal Displ (PANL), SCM Microsystem (SCMM), Stratasys Inc (SSYS)) for the year 2007, and denote by X = (X1 , . . . , Xn )T with n = 10 stock weekly returns. The vector of means and the covariance matrix of weekly returns are given in Tables 1 and 2. Substituting B, c and µ in (2.18) with λ
1T , 1 and −µ, respectively, and setting β = λ1,q , α = λ λ2,q , 1,q we obtain the explicit solution of the problem of minimization of functional (1.9) under constraint (1.2). For the investigated data we found l = 0.430761 and the TMV-optimal solutions are reported in Table 3 for q = 0.6 and q = 0.95, respectively. In this table we also present the classical MV-solution, that corresponds to the case q = 0. In Fig. 1, we give the graphic illustration of these solutions, where cases q = 0.6 and q = 0.95 are presented by dashed and solid lines, respectively. For comparison, we plotted the limit cases, MV solution (q = 0) and minimum variance solution (q = 1), denoted by big-dotted and small-dotted lines, respectively. From the explicit solution of minimization problem (2.18) it follows that the ‘‘amplitude’’ of changing the corrected term ∗ ∗ depends on value wl . In Fig. 2 one can see that wl decreases when q increases from 0 to 1. The maximal value of wl corresponds to the case q = 0 (classical MV-optimal portfolio) and for the case ∗ q = 1 (minimum variance-optimal portfolio), value wl vanishes. Now, using the formula for optimal solution (2.18) and taking into account (1.7) we can write the equation for the efficient frontier in the (σq2,x , µq,x ) system of coordinates, where µq,x = ∗
TCEq (Rx ), σq2,x = TVq (Rx ), and Rx = xT X is a return on portfolio
xT X. Recall x0 = Σ −1 BT (BΣ −1 BT )−1 c and denote by u = (1T Q −1 , −1T Q −1 D12 )T , r0 = µT x0 , r1 = µT u, a = xT0 Σ x0 , b = uT Σ u. Then the efficient frontier can be given by
q λ1,q 2 σq2,x − aλ2,q + σ . λ2,q q,x bλ2,q
µq ,x = r 0 + p
r1
In Fig. 3 we present the graphs of efficient frontiers for the TMV model with q = 0.8 and q = 0.9, respectively. For comparison we present also the frontier of the MV model (q = 0).
Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
σ11 = V (X1 ), and fZ ∗ (z ) is the density of some spherical random
Table 1 Portfolio mean returns. Stock Mean
−0.0016
ADBE
Stock Mean
551
TISA 0.0092
HAUP 0.0053
IMMR −0.0148
−0.0059
LOGI
NVDA
OIIM
−0.0093
−0.0080
PANL −0.0085
SCMM −0.0023
−0.0121
SSYS
variable Z ∗ , the distribution of which is called the distribution associated with the elliptical family. Finally, r (z ) =
F Z ∗ (z ) F Z (z )
(see details in Landsman, 2006; Furman and Landsman, 2006). D
Notice that for a multivariate normal model, Z ∗ = Z , and then ¯ (z ), r (z ) = 1, and λ1,q , λ2,q coincide h(Z , Z ∗ ) = h(z ) = ϕ(z )/Φ with those given in (1.8). Suppose that the vector of returns X = (X1 , . . . , Xn )T has the multivariate generalized Student-t distribution (GST), which is the natural generalization of the multivariate normal family. GST has a density of the form (1.5) with density generator gn (u) = 1 + u/kn,p
−p
,
(4.32)
where the power parameter p > n/2 and the normalized constant
kn,p =
p − (n + 2)/2, p > (n + 2)/2 1/2, n/2 < p ≤ (n + 2)/2.
We write X ∼ tn µ, Σ ;p, kn,p (see details in Landsman and Valdez, 2003; Landsman, 2004, 2006). Taking, for example, p = (n + m) /2, where m is a number of the degrees of freedom, and kn,p = m/2, we get the traditional form of the multivariate Student t distribution with the density
Fig. 3. Efficient frontiers for TMV and MV models.
4. Elliptical underlying model Let the vector of asset returns X = (X1 , X2 , . . . , Xn ) now have the multivariate elliptical distribution, X ∼ En (µ, Σ , gn ), where gn is a density generator. As mentioned in the Introduction, property (1.6) guarantees that TMVq (L) = TMVq (−R) is represented by the form of (1.7), i.e., in fact, it is still a combination of linear, root of quadratic and quadratic functionals. So, one may use Theorem 1 to evaluate explicitly the TMV-optimal portfolio in the elliptical context. Here constants λ1,q , λ2,q have a more complicated form T
λ1,q = h(Z , Z ∗ ) λ2,q = r zq + hZ ,Z ∗ zq zq − hZ ,Z ∗ zq , where h(Z , Z ∗ ) =
fZ ∗ (z ) F Z (z )
is the elliptical distorted hazard function, FZ (z ) is the distribution √ function of standardized random variable Z = (X1 − µ1 )/ σ11 ,
fX (x) =
Γ ((n + m)/2) √ (π m)n/2 Γ (m/2) |6| −(n+m)/2 (x − µ)T 6−1 (x − µ) × 1+ .
(4.33)
m
Notice that for X ∼ tn µ, Σ ;p, kn,p , the univariate marginal Xi ∼
t1 µi , σii ;p1 , k1,p , where p1 = p − (n − 1)/2 and for p1 > 1.5, the covariance exists. The normal distribution is the right limit case, p1 → ∞, of the GST distribution. In Fig. 4 one can see that when p1 increases from 1.5 to ∞, w∗ , the ‘‘amplitude’’ of the corrected term of the explicit solution l (2.18), decreases from ∞ and approaches the normal case. The ∗ limit value of wl corresponds to the case p1 = ∞, the normal underlying distribution. In Fig. 5 we present a graphic illustration of TMV-optimal solutions corresponding to q = 0.6 and GST underlying distributions with different power parameters. Here cases p1 = 1.501, 1.7 are presented by dashed and solid lines, respectively.
Table 2 Portfolio covariance return. ADBE
TISA
HAUP
IMMR
LOGI
ADBE TISA HAUP IMMR LOGI
0.0012088 0.0001915 0.0001390 0.0005170 0.0005389
0.0001915 0.0030343 −0.0003295 0.0008143 0.0001490
0.0001390 −0.0003295 0.0050096 0.0005269 0.0005666
0.0005170 0.0008143 0.0005269 0.0053998 0.0003593
0.0005389 0.0001490 0.0005666 0.0003593 0.0018776
NVDA
OIIM
PANL
SCMM
SYSS
ADBE TISA HAUP IMMR LOGI
0.0006330 0.0000345 0.0007979 0.0007018 0.0010585
0.0004924 −0.0001516 0.0001053 0.0012359 0.0006486
0.0008320 0.0004867 0.0004440 −0.0000263 0.0011565
−0.0002446
0.0008472
0.0000974 0.0006469 −0.0000338 0.0004212
−0.0002585
NVDA
OIIM
PANL
SCMM
SSYS
NVDA OIIM PANL SCMM SSYS
0.0024724 0.0007340 0.0007735 0.0001867 0.0003532
0.0007340 0.0033428 0.0007140 −0.0006152 0.0007267
0.0007735 0.0007140 0.0034164 −0.0004335 0.0016879
0.0001867 −0.0006152 −0.0004335 0.0034052 −0.0007061
0.0003532 0.0007267 0.0016879 −0.0007061 0.0045641
0.0011251 0.0007976 0.0006734
552
Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
Table 3 TMV-optimal portfolios. Stock TMVq=0.95 TMVq=0.6 MV
ADBE 0.2741 0.2099 −0.1621
TISA 0.1159 0.0544 −0.3014
HAUP 0.0232 −0.0197 −0.268
IMMR 0.0296 0.0724 0.3207
LOGI 0.0334 0.0317 0.0214
Stock TMVq=0.95 TMVq=0.6 MV
NVDA 0.079 0.1247 0.3899
OIIM 0.1289 0.1272 0.1172
PANL 0.026 0.0624 0.2737
SCMM 0.2091 0.2271 0.3313
SSYS 0.081 0.1099 0.2773
At the same time, the Mean–Variance risk measure uses all the information about the distribution of risks and does not pay any special attention to their tail behavior. There is a growing interest in risk measures based on the tails of distributions among insurance and investment experts. This results in the multipurpose use of the Tail Condition Expectation, TCEq (X ) = E (X |X > VaRq (X )), and the recently introduced Tail Variance, TVq (X ) = E (X |X > VaRq (X )). In the present paper we proposed the Tail Mean–Variance model, based on TMVq (R) = TCEq (−R) + λTVq (R), as a subject for optimal portfolio selection. We have shown that under multivariate normal underlying distribution the TMVq (R) reduces to a functional that is more complicated than quadratic, TMVq (R) = −µT x+λ1,q
√
xT 6x + λλ2,q xT 6x.
We have further shown that under general linear constraints the solution of the optimization problem still exists and in the case where short selling is possible we provide the analytical closed form solution, which coincides with the classical MV when q → 0. The results were extended for more general multivariate elliptical distributions of risks. Appendix. Analytic solution of the quartic equation (2.19)
Fig. 4. Changing of ‘‘amplitude’’ wl with increasing power parameter of univariate GST p1 from 1.5 to ∞; q = 0.6, λ = 5. ∗
The solution of a quartic equation is actually based on the solution of some cubic equation, called the resolvent. Neumark (1965) suggests a special form of the resolvent, which in the case of (2.19) looks like
1 w 3 − 2 f0 + k2 − 2 w2 4α 2 k2 f0 1 + f0 + k2 − 2 w + 2 = 0. 4α α
(6.35)
A brief analysis of Eq. (6.35) shows that it has only a negative rational root. In fact, when w ≥ 0, polynomial P3 (w) = w 3 − k2 f
k2 f
2bw 2 + b2 w + α 20 = w(w − b)2 + α 20 > 0, b = (f0 + k2 − 4α1 2 ). For the analytic solution of this cubic equation we use the k4 f 2
approach suggested by Nickalls (1993) and, denoting D = α 40 + 4k2 f0 (f + k2 − 4α1 2 )3 , the discriminant of Eq. (6.35), we obtain the 27α 2 0 rational root,
Fig. 5. TMV-optimal solutions corresponding q distributions with different power parameters.
= 0.6 and GST underlying
For comparison, we plotted the solution under normal distribution of returns (p1 = ∞) and MV-solution (small-dotted line and bigdotted line, respectively). 5. Conclusions In classical optimal portfolio selection theory, the Mean–Variance (MV) risk measure of the portfolio return R MV (R) = E (−R) + λV (R) = −µ x+λx 6x, T
T
(5.34)
was minimized subject to certain linear constraints. Here R = xT X, X = (X1 , . . . , Xn )T is a vector of random variables with expectations EX =µ, covariance matrix cov(X) = 6; λ > 0 is the balancing coefficient. The corresponding model reduces to the quadratic programming problem (see (5.34)), which is quite convenient for the technical realization, and, for the case where short selling is possible, even provides the analytic solution.
w R = x0 q√ q√ 1 3 3 D − y − D + y D ≥ 0, 0 0 , 2 + 2 1 1 y0 f0 + k2 − cos arccos − , 3 4α 2 3 hR
D < 0, k2 f
2 where x0 = 32 (f0 + k2 − 4α1 2 ), hR = 27 (f0 +k2 − 4α1 2 )3 , y0 = hR + α20 . The solution of the quartic equation (2.19), located in interval (0, k), can be derived using wR . First, recall that in our case wR is always negative. Then k2 − wR ≥ 0 and
G = −k +
p
g = −k −
p
k2 − wR ≥ 0,
(6.36)
k2 − wR ≤ −2k.
Denote h1 =
1 2
((f0 + k2 −
(6.37) 1 4α 2
) − wR ), h2 = k √
f0 −h1 k2 −wR
,H =
h1 + h2 , h = h1 − h2 . Then the quartic equation (2.19) has two real solutions.
w1,2
p 1 1 (−G ∓ G2 − 4H ), if G2 − H ≥ 0 4 = 2 p 1 (−g ∓ g 2 − 4h), if 1 G2 − H < 0. 2
4
Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
For case 14 G2 − H ≥ 0, we have, taking into account (6.36), that w1 < 0 and the resulting solution of our quartic equation 1
r
w∗ = w2 = − G + 2
For the case w2 > k and
w∗ = w1 =
1 2 G 4
1 2
1 4
G2 − H .
− H < 0 we have, taking into account (6.37),
(−g −
p
g 2 − 4h).
References Aparicio, F.M., Estrada, J., 2001. Empirical distribution of stock returns: European security markets, 1990–1995. The European Journal of Finance 7, 1–21. Acerbi, C., Tasche, D., 2002. On the coherence of expected shortfall. Journal of Banking & Finance 26 (7), 1487–1503. Alexander, A.J., Sharpe, 1989. Fundamentals of Investment. Prentice-Hall, Englewood Cliffs, NJ. Artzner, P., Delbaen, F., Eber, J.M., Heath, D., 1999. Coherent measures of risk. Mathematical Finance 9, 203–228. Bingham, N.H., Kiesel, R., 2002. Semi-parametric modelling in finance: Theoretical foundations. Quantitative Finance 2, 241–250. Boyle, P.P, Cox, S.H., et al., 1998. Financial economics: With applications to investments, insurance and pensions. Actuarial Foundation, Schaumburg. Fang, K.T., Kotz, S., Ng, K.W., 1990. Symmetric Multivariate and Related Distributions. Chapman & Hall, London. Chiragiev, A., Landsman, Z., 2007. Multivariate Pareto portfolios: TCE-based capital allocation and divided differences. Scandinavian Actuarial Journal 4, 261–280. Elton, E.J., Gruberm, M.J., 1987. Portfolio Theory and Investment Analysis. Wiley, New York. Furman, E., Landsman, Z., 2005. Risk capital decomposition for a multivariate dependent gamma portfolio. Insurance: Mathematics and Economics 37 (3), 635–649.
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Furman, E., Landsman, Z., 2006. Tail variance premium with applications for elliptical portfolio of risks. ASTIN Bulletin 36 (2), 433–462. Landsman, Z., 2004. On the generalization of Esscher and variance premiums modified for the elliptical family of distributions. Insurance: Mathematics and Economics 35, 563–579. Landsman, Z., 2006. On the generalization of Stein’s Lemma for elliptical class of distributions. Statistics & Probability Letters 76, 1012–1016. Landsman, Z., 2008. Minimization of the root of a quadratic functional under a system of affine equality constraints with application to portfolio management. Journal of Computational and Applied Mathematics 216 (2), 319–327. Landsman, Z., Valdez, E., 2003. Tail conditional expectations for elliptical distribution. North American Actuarial Journal 7 (4), 55–71. Landsman, Z., Valdez, E., 2005. Tail conditional expectation for exponential dispersion models. ASTIN Bulletin 35 (1), 189–209. Luenberger, D.G., 1984. The Linear and Nonlinear Programming. Addison-Wesley, CA. MacDonald, J.B., 1996. Probability distributions for financial models. Handbook of Statistics 14, 427–461. Markowitz, H.M., 1952. Portfolio selection. Journal of Finance 7, 77–91. Markowitz, H.M., 1987. Mean–Variance Analysis in Portfolio Choice and Capital Markets. Basil Blackwell, Oxford. UK. Merton, R.C., 1972. An analytic derivation of the efficient portfolio frontier. Journal of Financial Quantitative Analysis 7, 1851–1872. McNeil, A.J., Frey, R., Embrechts, P., 2005. Quantitative Risk Management. Princeton University Press, Princeton. Neumark, S., 1995. Solution of Cubic and Quartic equations. Pergamon Press, Oxford. Nickalls, R.W.D., 1993. The Mathematical Gazette 3, 354–359. Owen, J., Rabinovitch, R., 1983. On the class of elliptical distributions and their applications to the theory of portfolio choice. Journal of Finance 38 (3), 745–752. Panjer, H.H., 2002. Measurement of risk, solvency requirements, and allocation of capital within financial conglomerates. Institute of Insurance and Pension Research, University of Waterloo Research. Steinbach, M.C., 2001. Markowitz revisited: Mean-variance models in financial portfolio analysis. SIAM Review 43 (1), 31–85. Zenios, S.A., 1992. Financial Optimization. Cambridge University Press, Cambridge,UK. Ziemba, W.T., Mulvey, J.M., 1998. Worldwide Asset and Liability Modeling. Cambridge University Press, Cambridge, UK.
Contents lists available at ScienceDirect
Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime
On the Tail Mean–Variance optimal portfolio selection Zinoviy Landsman Department of Statistics, University of Haifa, Mount Carmel, 31905 Haifa, Israel
article
info
Article history: Received June 2009 Received in revised form October 2009 Accepted 5 February 2010 Keywords: Tail condition expectation Tail variance Tail Mean–Variance model Optimal portfolio selection Square root of quadratic functional Elliptical family Quartic equation
abstract In the present paper we propose the Tail Mean–Variance (TMV) approach, based on Tail Condition Expectation (TCE) (or Expected Short Fall) and the recently introduced Tail Variance (TV) as a measure for the optimal portfolio selection. We show that, when the underlying distribution is multivariate normal, the TMV model reduces to a more complicated functional than the quadratic and represents a combination of linear, square root of quadratic and quadratic functionals. We show, however, that under general linear constraints, the solution of the optimization problem still exists and in the case where short selling is possible we provide an analytical closed form solution, which looks more ‘‘robust’’ than the classical MV solution. The results are extended to more general multivariate elliptical distributions of risks. © 2010 Elsevier B.V. All rights reserved.
1. Introduction
constraint (1.2) has the following closed form solution
The classical mean–variance (MV) optimal portfolio selection theory was initiated by Markowitz’s (1952) pioneering paper and further developed by many authors. The subject is so extensively studied that, for example, the review given by Steinbach (2001) contains 208 references and it does not cover all relevant papers. The MV model uses the Mean–Variance risk measure
1 Σ −1 1 x = T −1 + 1 Σ 1 2λ ∗
Σ
−1
µ−
1T Σ −1 µ 1T Σ −1 1
Σ
−1
1
which, in finance literature, is also known as the expected quadratic utility, is the subject of minimization, where L = −R is the loss on the portfolio return R = xT X. Here X = (X1 , . . . , Xn )T is the random vector of returns, xT = (x1 , . . . , xn ) is the vector of generalized weights in the sense that
(see Boyle et al., 1998, Section 6). Additional material and references can be found in books by Markowitz (1987), Elton and Gruberm (1987), Alexander and Sharpe (1989), Zenios (1992), Ziemba and Mulvey (1998) and McNeil et al. (2005). Let us notice that the MV does not pay any special attention to the tail behavior. A growing interest in risk measures based on the tails of distributions is currently being observed among investment and insurance experts. The interest in the tail of the distribution has generated well-known measure, the expected short fall (ES) or the tail conditional expectation (TCE)
1T x = 1,
TCEq (L) = E L|L > VaRq (L) .
MV (L) = E (L) + λV (L),
(1.1)
(1.2)
(1.3)
where 1 is the vector-column of n ones. It is clear that
It is interpreted as the expectation of the worse loss in the sense that this loss exceeds a particular value-at-risk
MV (L) = E (−R) + λV (R) = −µT x + λxT Σ x,
VaRq (L) = inf{x : FL (x) ≥ q},
where µ is the vector of expectations of X, and Σ is its covariance matrix. So the MV model reduces to the problem of quadratic programming, which is quite convenient for technical realization (Luenberger, 1984), and, for the case when short selling is possible, even provides an analytic solution (first noticed by Merton (1972)). So, the problem of minimization of the MV (L) under
where FL (x) is the cdf of loss L. The expression in (1.3) was initially recommended by Artzner et al. (1999) to measure both market and nonmarket risks, presumably for a portfolio of investments. Acerbi and Tasche (2002) have shown the attractive coherence properties of the TCE. Tail conditional expectations for the univariate and multivariate Normal family were developed by Panjer (2002). His results were generalized by Landsman and Valdez (2003) for the larger class of elliptical distributions. Landsman and Valdez (2005) also developed formulae for tail conditional expectations
E-mail address: [email protected]. 0167-6687/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.insmatheco.2010.02.001
548
Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
∗ Fig. 2. Changing of ‘‘amplitude’’ wl with increasing of level q from 0 to 1.
Fig. 1. Comparison between TMV-optimal portfolios.
for random risks belonging to the class of exponential dispersion models. The TCE for multivariate gamma models was obtained in Furman and Landsman (2005) and for the multivariate Pareto in Chiragiev and Landsman (2007). Notice that for the case when the variance of L, V (L), exists, TCEq (L) is the best mean square approximation of random L by nonrandom constant c along the tail, in the sense TCEq (L) = arg inf E (L − c )2 |L > VaRq (L) ,
for some
n vector-column µ, a n × n positive-definite matrix Σ = σij i,j=1 , and for some function ψ(t ) called the characteristic generator (see details in Fang et al., 1990). The elliptical class of distributions is popular for the purpose of providing a satisfactory fit in modelling asset or loss returns in the multivariate set up (see Owen and Rabinovitch (1983) and Bingham and Kiesel (2002)). If the density of X exists (Fang et al., 1990, condition (2.19)), it has the form
c
i.e., the TCE provides a risk manager with information about the average of the tail of the loss distribution. In Furman and Landsman (2006) a tail variance risk measure (TV) was proposed TVq (L) = V (L|L > VaRq (L)) = inf E (L − c )2 |L > VaRq (L) ,
c
which estimates the deviance of the loss from this average along such a tail. Furthermore, the explicit expressions for TV in the multivariate normal and more general, multivariate elliptical distribution, were derived. In the following we propose a Tail Mean–Variance (TMV) model instead of MV in the optimal portfolio selection problem. More precisely, instead of (1.1), let us consider for some level q ∈ (0, 1), TMVq (L) = TCEq (L) + λTVq (L) as the subject of minimization. Recall that portfolio return R = −L, and then the problem TMVq (L) → inf E (R|R < VaR1−q (R)) − λV (R|R < VaR1−q (R)) → sup, which means that one is interested in the increase in expectation of only (1 − q) percent of small portfolio returns (not all the returns) and the balanced decrease of their variances. Let us notice that the MV model is a limit case of the TMV model when probability level q decreases to 0. We demonstrate that the volatility of the weights of the optimal solution of the TMV model decreases when q increases from 0 to 1. So the MV model has more volatility in comparison with the TMV model. In this sense the TMV model is essentially more robust then the MV model (see Section 3, Figs. 1 and 2). Let the vector of random variables X = (X1 , X2 , . . . , Xn )T , which can be associated with asset returns, be multivariate elliptically distributed, written as X v En (µ, Σ , ψ). The latter means that the characteristic function of vector X can be expressed as
ϕX (t) = exp(it µ)ψ
1 2
t Σt T
|Σ |
gn
1 2
(x − µ)T Σ −1 (x − µ) ,
(1.5)
for some function gn (·) called the density generator. One may also use the notation X ∼ En (µ, Σ , gn ). From (1.4) it follows that, if A is a m × n matrix of rank m ≤ n, and b is a m dimensional columnvector, then AX + b ∼ Em Aµ + b, AΣ AT , gm ,
(1.6)
i.e., any linear combination of elliptical vectors is another elliptical vector with the same characteristic generator ψ or from the same sequence of density generators g1 , . . . , gn , corresponding to ψ . Using the results of Landsman and Valdez (2003) and Furman and Landsman (2006) results, for portfolio return R = xT X, where xT = (x1 , . . . , xn ) is the vector of real numbers, one can write, TMVq (L)
p = TMVq (−R) = −µT x + λ1,q Var (xT X) + λλ2,q Var (xT X) √ = −µT x + λ1,q xT Σ x + λλ2,q xT Σ x, (1.7)
is equivalent to the problem
T
cn fX (x) = √
(1.4)
where λ1,q , λ2,q are some constant that depends on level q. For a multivariate normal model,
λ1,q =
ϕ(Zq ) , 1 − Φ (Zq )
λ2,q = 1 − λ1,q δq ,
δq = (λ1,q − Zq ),
(1.8)
where ϕ(x) and Φ (x) are density and distribution functions of the standard normal random variable Z , and Zq is its q-quantile. In the case of a more general multivariate elliptical setup, λ1,q , λ2,q look slightly more complicated and can be found in Furman and Landsman (2006). This setup will be considered in more detail in Section 4. So, the problem of the optimal portfolio selection in the Tail MV model is, in fact, equivalent to that of the problem of the minimization of the functional f (x) = −µT x + λ1,q
√
xT Σ x + λλ2,q xT Σ x,
(1.9)
Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
549
l
which is a combination of the linear functional, square root of a quadratic and a quadratic function with balance parameter λ > 0, subject to the linear constraint (1.2). This problem is considered in Section 2, where the analytical closed form solution is evaluated under a system of linear constraints. Section 3 is devoted to the application of the obtained results to stock data returns. A comparison between the TMV-optimal portfolios and efficient frontiers corresponding to different q-levels is provided. In Section 4 the results are extended to the multivariate elliptical model. Special attention was paid to the multivariate generalized Student-t model, which is often regarded to be an appropriate model for stock returns (MacDonald, 1996; Aparicio and Estrada, 2001). The analytic solution of the quartic equation, which is important for the closed form solution of the optimal problem being considered here, is presented in the Appendix.
f0 = cT (BΣ −1 BT )−1 c,
2. Minimization of the combination of linear, root of quadratic and quadratic functionals
1 Proof. Define vector d2 = B− 22 c. Then from system of constraints (2.12) and from (2.14) it follows that xT = (xT1 , dT2 − xT1 D12 )T and then straightforwardly
First we notice that the goal function
√
f (x) = µT x + β( xT Σ x + α xT Σ x),
(2.10)
α, β > 0, which is the main subject of this Section, is convex as a sum of the linear form and the convex functions. The convexity of the square root of the quadratic form follows from the fact that, for any u, v ∈ Rn and t ∈ R,
p p (u + tv)T Σ (u + tv) = vT Σ vt 2 + 2vT Σ ut + uT Σ u
Bx = c,
c 6= 0,
(2.12)
m,n bij i,j=1
where B = ( ) is m × n, m < n, is the rectangular full rank matrix, c is some m × 1 vector, and 0 is a vector-column of m zeros. We advance the approach given in Landsman (2008), where only the combination of linear and square root quadratic functionals were investigated, to the more complicated functional (2.10). Choosing the first n − m variables we have the natural partition of vector xT = (xT1 , xT2 ), x1 = (x1 , . . . , xn−m )T , x2 = (xn−m+1 , . . . , xn )T and the corresponding partition of vectors µT = (µT1 , µT2 ), 1T = (1T1 , 1T2 ), matrix Σ ,
Σ11 Σ= Σ21
Σ12 Σ22
(2.13)
and matrix B = B21
B22 ,
where matrices B21 and B22 are of dimensions m × (m − n) and m × m, respectively. As matrix B is of full rank, then suppose without loss of generality that matrix B22 is nonsingular. Define m × (n − m) and (n − m) × m matrices D12 = DT21
(2.14)
Q = Σ11 − Σ12 D21 − D12 Σ21 + D12 Σ22 D21 = ( )
m qij ni,− j =1
l=
1
and set
T
Q −1 1
x∗ = Σ −1 BT (BΣ −1 BT )−1 c +
w∗ l
(1T Q −1 , −1T Q −1 D12 )T , (2.18)
where w ∗ is the unique rational root of the quartic equation
w − 2kw + f0 + k − 4
3
1
2
4α 2
w 2 − 2kf0 w + k2 f0 = 0,
(2.19)
located in the interval (0, k). The analytic form for w ∗ exists and will be given in the Appendix.
xT Σ x = xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2 .
(2.20)
The goal-function f (x) = g (x1 ) = µT2 d2 + (µ1 − D12 µ2 )T x1
+β
q
xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2
+ α(
xT1 Q x1
+
2dT2
(Σ21 − Σ22 D21 )x1 +
dT2 Σ22 d2
) ,
is a function of n − m variables, x1 = (x1 , . . . , xn−m )T , and the problem reduces to the problem of finding the unconditional minimum min g (x1 ).
x1 ∈Rn−m
As a corollary of the well-known solution of the quadratic programming problem x0 = arg min xT Σ x =Σ −1 BT (BΣ −1 BT )−1 c,
(2.21)
Bx=c
and xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2 T
≥ x0 Σ x0 = f0 > 0,
x1 ∈ Rn−m ,
(2.22)
as matrix BΣ B > 0 and c 6= 0 (see, for example, Luenberger, 1984, Chapter 14.1). This means that the function q −1 T
xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2 is differentiable for
that function g (x1 ) is strictly convex on Rn−m . Denote by
.
dT dx1
= ( dxd1 · · · dxnd−m ) the vector-row of the first n − m derivatives and let x∗ and x0 be partitioned as follows x∗ = (x∗1T , dT2 − x∗1T D12 )T
(2.23)
and x = ( , − ) Then the vector x1 is the unique solution of the vector-equations (2.15)
In the cited paper it was proved that, together with Σ , matrix Q is positive definite. Denote by
q
(2.17)
Theorem 1. The problem of the minimization of function (2.10) subject to (2.12) has the following solution
0
and (n − m) × (n − m) matrix
1 = D12 µ2 − µ1 ,
,
any x1 ∈ Rn−m . For the same reasons as those given in (2.11) and taking into account the last inequality, one may conclude
1 D21 = B− 22 B21 ,
2αβ
we now state the following theorem:
(2.11)
is a strictly convex function of t (the square root of a quadratic univariate polynomial with a positive leading coefficient √ and negative discriminant is strictly convex) and consequently xT Σ x is strictly convex. We give the explicit closed form solution for the problem of minimization of a function f (x) subject to a system of linear restrictions presented in the matrix form
k=
(2.16)
d dx1
x0T 1
dT2
T x0T 1 D12
∗
g (x1 )
= (µ1 − D12 µ2 ) + β((xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2 )−1/2 + 2α)(Q x1 + (Σ12 − D12 Σ22 )d2 ) = 01 , where 01 is a vector-column of (n − m) zeros, taking into account (2.22). This equation can be rewritten in the form
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Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
(Q x1 + (Σ12 − D12 Σ22 )d2 ) = τ((xT1 Q x1 + 2dT2 (Σ21 − Σ22 D21 )x1 + dT2 Σ22 d2 )−1/2 + 2α)−1 , where 1
τ = (τ1 , . . . , τn−m )T =
1
(D12 µ2 − µ1 ) = − 1. β β
P4 (0) = k2 f0 > 0, (2.24)
Consider x∗1 in the form x∗1 = x01 + y∗ , where y∗T = (y1 , . . . , yn−m ) is the (n − m) dimension vector. Then, as x0 is a solution of problem (2.21), it follows that Q x01 + (Σ12 − D12 Σ22 )d2 = 01
A brief analysis of quartic equation (2.19) shows that it has a rational root on the interval (0, k). In fact, defining P4 (w) = w 4 − 2kw3 + (f0 + k2 − 4α1 2 )w 2 − 2kf0 w + k2 f0 , one has
(2.25)
∗
and y is the unique solution of the vector-equations
P4 (k) = −
k2
The root is unique in interval (0, k), because the considered optimization problem has a unique solution. The analytic explicit form for this root will be given in the Appendix, where Neumark’s (1965) approach to the solution of quartic equations is implemented. Substituting (2.31) into (2.28) we obtain, taking into account (2.29), (2.24) and (2.27),
y∗ = Q −1 τ((x∗1T Q x∗1 + 2dT2 (Σ21 − Σ22 D21 )x∗1
+ dT2 Σ22 d2 )−1/2 + 2α)−1 .
(2.26)
1 ∗ As τ = 01 (means µ1 = B12 B− 22 µ2 ) it results trivially that y = 01 . n−m −1 Suppose vector τ 6= 0, then as matrix Q = (δij )i,j=1 is nonsin-
gular (positive definite) there exists row δTi = (δi1 , . . . ., δin−m ) of Q −1 such that δTi τ 6= 0. Suppose for convenience, and without loss of generality, that i = 1. Then, using the following partition of matrix Q −1 into the 2 matrices Q1−1 and Q2−1 , Q
−1
Q1−1 Q2−1
=
y∗ = y∗1 (1, rT )T ,
(2.28)
where r = Q2−1 τ/Q1−1 τ.
(2.29)
Substituting (2.28) into the first equation of (2.26), we get straightforwardly y∗1 = Q1−1 τ((x∗1T Q x∗1 + 2dT2 (Σ21 − Σ22 D21 )x∗1
+ dT2 Σ22 d2 )−1/2 + 2α)−1
−1
1 f0 + y1 (1, rT )Q (1, rT )T ∗2
+ 2α
−1
1 f0 +
y ∗2 l2
1 (Q1−1 τ)2 β 2
+ 2α
.
(2.30)
The latter was done by taking into account (2.25), (2.20), the right hand side of (2.22) and equality
(1, rT )Q (1, rT )T =
1
(Q1−1 τ)2
τ T Q −1 τ,
obtained in Landsman (2008, Eq.(29)). After substituting
w = ∗
l
y1 , ∗
β Q1−1 τ
(2.31)
the Eq. (2.30) reduces to
√
1
f0 + w ∗
=
l
βw∗
−1
βw∗ l
1,
(Q2−1 τ)T
!T
Q1 τ −1
=
w∗ l
Q −1 1.
This finishes the proof of the Theorem, taking into account that from (2.23) and (2.21) x∗ = ((x01 + y∗ )T , (d2 − D21 x01 − D21 y∗ )T )T
= x0 + (y∗T , −y∗T D12 )T .
(2.27)
where Q1−1 is simply the first row of Q −1 (i.e., Q1−1 = δT1 ) and Q2−1 consists of other (n − m − 1) rows of Q −1 , we have from (2.26)
= Q1−1 τ r
y = Q1 τ ∗
3. Application to stock data returns
,
= Q1−1 τ q
< 0.
4α 2
− 2α,
w ∗ > 0.
The latter is equivalent to the quartic equation (2.19) together with conditions βwl ∗ − 2α > 0, w ∗ > 0. This means that w ∗ is a root of Eq. (2.19) in interval (0, k), recalling (2.17).
Consider a portfolio of 10 stocks from NASDAQ/Computers (ADOBE Sys. Inc., Top Image System (TISA), Hauppauge Digit (HAUP), Immerson Corp (IMMR), Logithch Int. Sa. (LOGI), NVIDIA Corp., O2micro Intl Lt (OIIM), Universal Displ (PANL), SCM Microsystem (SCMM), Stratasys Inc (SSYS)) for the year 2007, and denote by X = (X1 , . . . , Xn )T with n = 10 stock weekly returns. The vector of means and the covariance matrix of weekly returns are given in Tables 1 and 2. Substituting B, c and µ in (2.18) with λ
1T , 1 and −µ, respectively, and setting β = λ1,q , α = λ λ2,q , 1,q we obtain the explicit solution of the problem of minimization of functional (1.9) under constraint (1.2). For the investigated data we found l = 0.430761 and the TMV-optimal solutions are reported in Table 3 for q = 0.6 and q = 0.95, respectively. In this table we also present the classical MV-solution, that corresponds to the case q = 0. In Fig. 1, we give the graphic illustration of these solutions, where cases q = 0.6 and q = 0.95 are presented by dashed and solid lines, respectively. For comparison, we plotted the limit cases, MV solution (q = 0) and minimum variance solution (q = 1), denoted by big-dotted and small-dotted lines, respectively. From the explicit solution of minimization problem (2.18) it follows that the ‘‘amplitude’’ of changing the corrected term ∗ ∗ depends on value wl . In Fig. 2 one can see that wl decreases when q increases from 0 to 1. The maximal value of wl corresponds to the case q = 0 (classical MV-optimal portfolio) and for the case ∗ q = 1 (minimum variance-optimal portfolio), value wl vanishes. Now, using the formula for optimal solution (2.18) and taking into account (1.7) we can write the equation for the efficient frontier in the (σq2,x , µq,x ) system of coordinates, where µq,x = ∗
TCEq (Rx ), σq2,x = TVq (Rx ), and Rx = xT X is a return on portfolio
xT X. Recall x0 = Σ −1 BT (BΣ −1 BT )−1 c and denote by u = (1T Q −1 , −1T Q −1 D12 )T , r0 = µT x0 , r1 = µT u, a = xT0 Σ x0 , b = uT Σ u. Then the efficient frontier can be given by
q λ1,q 2 σq2,x − aλ2,q + σ . λ2,q q,x bλ2,q
µq ,x = r 0 + p
r1
In Fig. 3 we present the graphs of efficient frontiers for the TMV model with q = 0.8 and q = 0.9, respectively. For comparison we present also the frontier of the MV model (q = 0).
Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
σ11 = V (X1 ), and fZ ∗ (z ) is the density of some spherical random
Table 1 Portfolio mean returns. Stock Mean
−0.0016
ADBE
Stock Mean
551
TISA 0.0092
HAUP 0.0053
IMMR −0.0148
−0.0059
LOGI
NVDA
OIIM
−0.0093
−0.0080
PANL −0.0085
SCMM −0.0023
−0.0121
SSYS
variable Z ∗ , the distribution of which is called the distribution associated with the elliptical family. Finally, r (z ) =
F Z ∗ (z ) F Z (z )
(see details in Landsman, 2006; Furman and Landsman, 2006). D
Notice that for a multivariate normal model, Z ∗ = Z , and then ¯ (z ), r (z ) = 1, and λ1,q , λ2,q coincide h(Z , Z ∗ ) = h(z ) = ϕ(z )/Φ with those given in (1.8). Suppose that the vector of returns X = (X1 , . . . , Xn )T has the multivariate generalized Student-t distribution (GST), which is the natural generalization of the multivariate normal family. GST has a density of the form (1.5) with density generator gn (u) = 1 + u/kn,p
−p
,
(4.32)
where the power parameter p > n/2 and the normalized constant
kn,p =
p − (n + 2)/2, p > (n + 2)/2 1/2, n/2 < p ≤ (n + 2)/2.
We write X ∼ tn µ, Σ ;p, kn,p (see details in Landsman and Valdez, 2003; Landsman, 2004, 2006). Taking, for example, p = (n + m) /2, where m is a number of the degrees of freedom, and kn,p = m/2, we get the traditional form of the multivariate Student t distribution with the density
Fig. 3. Efficient frontiers for TMV and MV models.
4. Elliptical underlying model Let the vector of asset returns X = (X1 , X2 , . . . , Xn ) now have the multivariate elliptical distribution, X ∼ En (µ, Σ , gn ), where gn is a density generator. As mentioned in the Introduction, property (1.6) guarantees that TMVq (L) = TMVq (−R) is represented by the form of (1.7), i.e., in fact, it is still a combination of linear, root of quadratic and quadratic functionals. So, one may use Theorem 1 to evaluate explicitly the TMV-optimal portfolio in the elliptical context. Here constants λ1,q , λ2,q have a more complicated form T
λ1,q = h(Z , Z ∗ ) λ2,q = r zq + hZ ,Z ∗ zq zq − hZ ,Z ∗ zq , where h(Z , Z ∗ ) =
fZ ∗ (z ) F Z (z )
is the elliptical distorted hazard function, FZ (z ) is the distribution √ function of standardized random variable Z = (X1 − µ1 )/ σ11 ,
fX (x) =
Γ ((n + m)/2) √ (π m)n/2 Γ (m/2) |6| −(n+m)/2 (x − µ)T 6−1 (x − µ) × 1+ .
(4.33)
m
Notice that for X ∼ tn µ, Σ ;p, kn,p , the univariate marginal Xi ∼
t1 µi , σii ;p1 , k1,p , where p1 = p − (n − 1)/2 and for p1 > 1.5, the covariance exists. The normal distribution is the right limit case, p1 → ∞, of the GST distribution. In Fig. 4 one can see that when p1 increases from 1.5 to ∞, w∗ , the ‘‘amplitude’’ of the corrected term of the explicit solution l (2.18), decreases from ∞ and approaches the normal case. The ∗ limit value of wl corresponds to the case p1 = ∞, the normal underlying distribution. In Fig. 5 we present a graphic illustration of TMV-optimal solutions corresponding to q = 0.6 and GST underlying distributions with different power parameters. Here cases p1 = 1.501, 1.7 are presented by dashed and solid lines, respectively.
Table 2 Portfolio covariance return. ADBE
TISA
HAUP
IMMR
LOGI
ADBE TISA HAUP IMMR LOGI
0.0012088 0.0001915 0.0001390 0.0005170 0.0005389
0.0001915 0.0030343 −0.0003295 0.0008143 0.0001490
0.0001390 −0.0003295 0.0050096 0.0005269 0.0005666
0.0005170 0.0008143 0.0005269 0.0053998 0.0003593
0.0005389 0.0001490 0.0005666 0.0003593 0.0018776
NVDA
OIIM
PANL
SCMM
SYSS
ADBE TISA HAUP IMMR LOGI
0.0006330 0.0000345 0.0007979 0.0007018 0.0010585
0.0004924 −0.0001516 0.0001053 0.0012359 0.0006486
0.0008320 0.0004867 0.0004440 −0.0000263 0.0011565
−0.0002446
0.0008472
0.0000974 0.0006469 −0.0000338 0.0004212
−0.0002585
NVDA
OIIM
PANL
SCMM
SSYS
NVDA OIIM PANL SCMM SSYS
0.0024724 0.0007340 0.0007735 0.0001867 0.0003532
0.0007340 0.0033428 0.0007140 −0.0006152 0.0007267
0.0007735 0.0007140 0.0034164 −0.0004335 0.0016879
0.0001867 −0.0006152 −0.0004335 0.0034052 −0.0007061
0.0003532 0.0007267 0.0016879 −0.0007061 0.0045641
0.0011251 0.0007976 0.0006734
552
Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
Table 3 TMV-optimal portfolios. Stock TMVq=0.95 TMVq=0.6 MV
ADBE 0.2741 0.2099 −0.1621
TISA 0.1159 0.0544 −0.3014
HAUP 0.0232 −0.0197 −0.268
IMMR 0.0296 0.0724 0.3207
LOGI 0.0334 0.0317 0.0214
Stock TMVq=0.95 TMVq=0.6 MV
NVDA 0.079 0.1247 0.3899
OIIM 0.1289 0.1272 0.1172
PANL 0.026 0.0624 0.2737
SCMM 0.2091 0.2271 0.3313
SSYS 0.081 0.1099 0.2773
At the same time, the Mean–Variance risk measure uses all the information about the distribution of risks and does not pay any special attention to their tail behavior. There is a growing interest in risk measures based on the tails of distributions among insurance and investment experts. This results in the multipurpose use of the Tail Condition Expectation, TCEq (X ) = E (X |X > VaRq (X )), and the recently introduced Tail Variance, TVq (X ) = E (X |X > VaRq (X )). In the present paper we proposed the Tail Mean–Variance model, based on TMVq (R) = TCEq (−R) + λTVq (R), as a subject for optimal portfolio selection. We have shown that under multivariate normal underlying distribution the TMVq (R) reduces to a functional that is more complicated than quadratic, TMVq (R) = −µT x+λ1,q
√
xT 6x + λλ2,q xT 6x.
We have further shown that under general linear constraints the solution of the optimization problem still exists and in the case where short selling is possible we provide the analytical closed form solution, which coincides with the classical MV when q → 0. The results were extended for more general multivariate elliptical distributions of risks. Appendix. Analytic solution of the quartic equation (2.19)
Fig. 4. Changing of ‘‘amplitude’’ wl with increasing power parameter of univariate GST p1 from 1.5 to ∞; q = 0.6, λ = 5. ∗
The solution of a quartic equation is actually based on the solution of some cubic equation, called the resolvent. Neumark (1965) suggests a special form of the resolvent, which in the case of (2.19) looks like
1 w 3 − 2 f0 + k2 − 2 w2 4α 2 k2 f0 1 + f0 + k2 − 2 w + 2 = 0. 4α α
(6.35)
A brief analysis of Eq. (6.35) shows that it has only a negative rational root. In fact, when w ≥ 0, polynomial P3 (w) = w 3 − k2 f
k2 f
2bw 2 + b2 w + α 20 = w(w − b)2 + α 20 > 0, b = (f0 + k2 − 4α1 2 ). For the analytic solution of this cubic equation we use the k4 f 2
approach suggested by Nickalls (1993) and, denoting D = α 40 + 4k2 f0 (f + k2 − 4α1 2 )3 , the discriminant of Eq. (6.35), we obtain the 27α 2 0 rational root,
Fig. 5. TMV-optimal solutions corresponding q distributions with different power parameters.
= 0.6 and GST underlying
For comparison, we plotted the solution under normal distribution of returns (p1 = ∞) and MV-solution (small-dotted line and bigdotted line, respectively). 5. Conclusions In classical optimal portfolio selection theory, the Mean–Variance (MV) risk measure of the portfolio return R MV (R) = E (−R) + λV (R) = −µ x+λx 6x, T
T
(5.34)
was minimized subject to certain linear constraints. Here R = xT X, X = (X1 , . . . , Xn )T is a vector of random variables with expectations EX =µ, covariance matrix cov(X) = 6; λ > 0 is the balancing coefficient. The corresponding model reduces to the quadratic programming problem (see (5.34)), which is quite convenient for the technical realization, and, for the case where short selling is possible, even provides the analytic solution.
w R = x0 q√ q√ 1 3 3 D − y − D + y D ≥ 0, 0 0 , 2 + 2 1 1 y0 f0 + k2 − cos arccos − , 3 4α 2 3 hR
D < 0, k2 f
2 where x0 = 32 (f0 + k2 − 4α1 2 ), hR = 27 (f0 +k2 − 4α1 2 )3 , y0 = hR + α20 . The solution of the quartic equation (2.19), located in interval (0, k), can be derived using wR . First, recall that in our case wR is always negative. Then k2 − wR ≥ 0 and
G = −k +
p
g = −k −
p
k2 − wR ≥ 0,
(6.36)
k2 − wR ≤ −2k.
Denote h1 =
1 2
((f0 + k2 −
(6.37) 1 4α 2
) − wR ), h2 = k √
f0 −h1 k2 −wR
,H =
h1 + h2 , h = h1 − h2 . Then the quartic equation (2.19) has two real solutions.
w1,2
p 1 1 (−G ∓ G2 − 4H ), if G2 − H ≥ 0 4 = 2 p 1 (−g ∓ g 2 − 4h), if 1 G2 − H < 0. 2
4
Z. Landsman / Insurance: Mathematics and Economics 46 (2010) 547–553
For case 14 G2 − H ≥ 0, we have, taking into account (6.36), that w1 < 0 and the resulting solution of our quartic equation 1
r
w∗ = w2 = − G + 2
For the case w2 > k and
w∗ = w1 =
1 2 G 4
1 2
1 4
G2 − H .
− H < 0 we have, taking into account (6.37),
(−g −
p
g 2 − 4h).
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