Optimal portfolios with downside risk

Monash CQFIS working paper

2017 – 11

Fima KlebanerSchool of Mathematics, Monash University

Zinoviy LandsmanDepartment of Statistics, University of Haifa,

Haifa, Israel

Udi MakovDepartment of Statistics, University of Haifa,

Haifa, Israel

Jing YaoFaculty of Economics, Vrije Universiteit

Brussels, Brussels, Belgium

Abstract

We show, first, that minimization of downside risk for portfolioswith pre-specified expected returns leads to the same solution asminimization of the variance. Hence such optimal portfolios aredefined by the Markovitz optimal solution. If the expected returnsare not pre-specified, we show that the problem of minimization ofdownside risk has an analytical solution and we present this solutiontogether with several illustrative numerical examples.

Centre for Quantitative Finance and Investment Strategies

http://www.monash.edu

Quantitative Finance, 2017Vol. 17, No. 3, 315–325, http://dx.doi.org/10.1080/14697688.2016.1197411

© 2016 iStockphoto LP

Optimal portfolios with downside risk

FIMA KLEBANER†, ZINOVIY LANDSMAN‡, UDI MAKOV‡ and JING YAO∗§

†School of Mathematical Sciences, Monash University, Melbourne, Australia‡Department of Statistics, University of Haifa, Haifa, Israel

§Faculty of Economics, Vrije Universiteit Brussels, Brussels, Belgium

(Received 13 November 2015; accepted 26 May 2016; published online 19 July 2016)

1. Introduction

Markowitz optimal portfolio theory (Markowitz 1987), alsoknown as the Mean-Variance theory, has had a tremendousimpact and hundreds of papers are devoted to this topic. Thistheory addresses the question of minimizing risk for a givenexpected return and the optimal solution is found under oneof the two assumptions: the distribution of the portfolio isnormal, or the utility function is quadratic. In this theory, inv-estor’s decision formulates a trade-off between the return andthe risk, in which the risk is measured by the variance ofthe returns. However, it has also been noted frequently in thepast, starting with Markowitz himself, that investors are moreconcerned with downside risk, i.e. the possibility of returnsfalling short of a specified target, rather than with the variance,which takes into account the favourable upside deviations aswell as the adverse downside parts. Moreover, such a classicMean–Variance framework does not consider the investor’sindividual preferences. Thus, alternatives are proposed in theliterature in the form of downside risk measures, such as targetshortfall and semivariance, or more generally, the so-calledlower partial moments; see, for example, Harlow and Rao(1989).

∗Corresponding author. Email: jingyao@vub.ac.be

The lower partial moments for stochastic returns, as down-side risk measures, are defined as the expectation of the nthpower of the return’s deviation below a pre-specified targetwhich depends on investor’s preference. The first- and thesecond-order lower partial moments are usually called targetshortfall and below target variance, respectively. Intuitively,these risk measures are asymmetric and focus on the left tailof the portfolio returns below a given target rate rather thanon their entire domain. The target shortfall is the expectationof portfolio returns dropping below the given target rate orbenchmark return. By contrast, the below target variance (itis often called the semivariance when the target is set as theexpectation of the return) considers the dispersion of returnbelow the target rate (Fishburn 1977). Both criteria aim tomeasure the extent that the portfolio fails to reach its manager’starget or benchmark return. In this regard, these downside riskmeasures are more appropriate for investment risk becauseinvestors are often more interested in losses relative to targetreturns. Moreover, unlike the variance, semivariance canremain the same together with higher ‘upside potential’. Formore details on downside risk measures, we refer to Harlow(1991), Nawrocki (1999) and Chapter 2 of McNeil et al. (2005).

A manager who does not wish the return of his portfolioto fall below the target rate would tend to compose portfoliosminimizing downside risk measures, which is the so-called

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316 Feature

downside optimization (Harlow 1991). Such portfolios withoptimal downside risks are attractive as they may lower therisks while maintaining or improving the expected returns inthe classic mean–variance framework. As a result, by cons-idering downside risk measures, portfolio managers areusually able to search for more profits in the trade-off betweenrisk and return. Empirical evidence also shows that downsidemeasures are more efficient than Mean–Variance measuresin this sense. Moreover, in asset pricing models, a downsiderisk framework can provide less downside exposure whilepreserving the same, or a greater level, of expected return, seeHarlow and Rao (1989). In fact, despite the complexity of thecomputation, it is clear from the literature that lower partial mo-ments, especially the target shortfall and below target variance,are not simply ad hoc measures, but are grounded in capitalmarket theory with both appealing theoretical and intuitivefeatures. For instance, Markowitz (2010) accords support tosemivariance as a comparable risk measure with variance andthere is mounting empirical evidence showing the superiorityof downside risk measures (Jarrow and Zhao 2006).

Due to the formulation of partial moments, downside opti-mization is naturally connected with utility theory and stochas-tic dominance. Bawa (1978) used lower partial moments withan α-degree and τ -threshold model to capture investor’s riskaversion. More recently, Cumova and Nawrocki (2014) pro-posed a ratio of upside partial moment and lower partial mo-ment to study investors’ behaviour. In contrast to the classicalMean–Variance model, which is consistent with a quadraticutility function, downside optimization is a more general frame-work as it allows investors to consider different orders n andto choose a favourable target. Hence, there is no doubt that thedownside risk framework should provide a useful set of toolsfor portfolio managers considering a broad set of problems.

In addition to the literature mentioned above, there areongoing efforts devoted to various applications of downsiderisks and related risk measures in finance. We list a few ofthem. Zhu et al. (2009) discuss robust portfolio selection withrespect to parameter uncertainty under a downside riskframework. Moreover, the target shortfall is directly related tothe Conditional Value-at-Risk (CVaR), which has beenfrequently applied as the optimizing objective in many port-folio selection studies. For more details on the risk measureCVaR, we refer to Artzner et al. (1999) and Rockefeller andUryasev (2002). Mansini et al. (2007) consider optimal port-folios with respect to the CVaR and obtain the solution bylinear programming. Sawik (2008) formulates the portfolio op-timization problem as multi-objective mixed integer program.Ogryczak and Sliwinski (2011a, 2011b) use duality to improvethe efficiency of linear programming in portfolio selection.Sawik (2012a) compares three different bi-criteria portfolio op-timization models based on Value-at-Risk, CVaR and variance.Sawik (2012b) studies multi-objective portfolio optimizationwith downside risk approaches and Sawik (forthcoming) pro-poses further work in multi-objective models. Cumova andNawrocki (2014) use both upside and downside partial mo-ments to account for an investor’s utility. A literature reviewon downside risk measures can be found in Nawrocki (1999)and more account of robust portfolio optimization with respectto various risk measures is in Gabrel et al. (2014).

However, most of the aforementioned works have to relyon numerical method to find the solutions to the optimizationproblem. In fact, as Jarrow and Zhao (2006) commented, an-alytical solutions to optimal portfolio’s weights are generallyout of reach for such downside measures. One could eitheremploy numerical techniques to search for the optimal solutionor implement optimization based on empirical estimation andsimulation, see, for instance, Nawrocki (1991) and Cumovaand Nawrocki (2014). Nevertheless, in the absence of massivecomputational efforts such as in simulation and numerical op-timization, analytical solutions are always preferable in boththeoretical studies and practical applications, and further, theydo not suffer from computational errors.

Landsman (2008) finds an analytical solution to optimalportfolio weights in the classical Mean–Variance frameworkwith additional constraints on the returns. Based on his re-sults, we adopt a similar approach and derive the analyti-cal solutions to the downside optimization in the context ofnormally distributed returns. This work offers several novelcontributions. Firstly, we show that optimal portfolios withrespect to downside risk are the same as those generated by theclassical Markowitz Mean–Variance analysis in a frameworkin which the expected portfolio return is pre-specified. Thus,in this case the downside risk framework can not improve theclassical results and investors cannot expect additional profitsfrom the downside approach assuming normal distributions ofthe returns. Particularly, Jarrow and Zhao (2006) empiricallyobserved that optimal portfolios with respect to mean–varianceand downside risk frameworks are much alike. Our findingsprovide a theoretical proof for their results.

Secondly, we consider a more general portfolio selectionwith downside risk measure when the expected return of theportfolio is not pre-specified. We show that such downsideoptimization is a convex optimization and has a unique so-lution. We further derive analytical formulae for the solutionand obtain optimal weights. Consequently, we can easily selectoptimal portfolios with respect to downside risk measures forany inputs of means and covariances. As mentioned above,such results are useful both theoretically and practically.

Thirdly, we present numerical examples to illustrate ourresults. We first observe that the general principal that an assetwith higher return should have higher risk holds true in thedownside risk measure framework. Moreover, we notice that aportfolio with minimal target shortfall can have very differentweights from the one with minimal below target variance and itis plausible to argue that one could always outperform the other.Hence, instead of optimizing expected shortfall or below targetvariance separately, we propose a new downside optimizationthat targets a combination of the two downside risk measures.In this regard, an investor can choose his/her preference be-tween the downward deviation from the target and the risk todrop below the target, and the corresponding analytical weightsare still available. Similar to the classical results, we also showthat one can obtain an analogue to the Mean–Variance efficientfrontier based on the Mean-Downside-risk framework.

The rest of the paper is organized as follows. In section 2, wepresent the formulae for the risk associated with downside riskmeasures and state some useful properties. Section 3 providesthe downside optimization and finds an analytical solution.

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We offer numerical examples in section 4. We discuss furtherpotential developments and conclude our paper in section 5.

2. Downside risk

To facilitate the expressions of the equations and formulaein the sequel, we first present a short list of the notions andnotations that are useful as below.

• X ∼ N (μ, σ 2). X denotes stochastic return of an assetthat is normally distributed with mean μ and variance σ .

• X ∼ N (μ, �). X denotes stochastic returns of multipleassets that are multi-normally distributed with means μand covariance matrix �.

• �(x) and ϕ(x) are the cumulative distribution functionand density function of the standard normal random vari-able, respectively.

• For a fixed target K , we consider the downside risksbelow K

Ri (μ, σ ) := E[((X − K )−)i ]

= σ i∫ K−μ

σ

−∞

(K − μ

σ− x

)iϕ(x)dx

for i = 1 and i = 2, i.e. R1(μ, σ ) is the target shortfalland R2(μ, σ ) is the below target variance.

• AK = (1,μ − K 1)T where 1 is a n-dimensional vectorwith identical unit components. δ = ∣∣AK �−1 ATK ∣∣ is thedeterminant of AK �−1 ATK .• q1 = δ−11T �−11, q2 = δ−1(μ − K 1)T �−11, q3 =δ−1(μ − K 1)T �−1(μ − K 1).

In the context of normality, we compute the downside riskfunctions as the follows.

Proposition 1

R1(μ, σ ) = E[(X − K )−]= σϕ

(K − μ

σ

)+ (K − μ)�

(K − μ

σ

), (1)

R2(μ, σ ) = E[((X − K )−)2]= (σ 2 + (K − μ)2)�

(K − μ

σ

)+ σ(K − μ)ϕ

(K − μ

σ

)(2)

Proof Let X = σ Z + μ, where Z ∼ N (0, 1).R1(μ, σ ) = E[(σ Z + μ − K )−]

=∫ K−μ

σ

−∞σ

e− z22√

2πdz + (μ − K )

∫ K−μσ

−∞e− z

22√

2πdz

= σϕ(

K − μσ

)+ (K − μ)�

(K − μ

σ

).

R2(μ, σ ) = E[((X − K )−)2]

= σ 2 E⎡⎣((Z + μ − K

σ

)−)2⎤⎦= σ 2 E

[((Z + μ − K

σ

)2IZ< K−μ

σ

)]

= σ 2(

E[

Z2 IZ< K−μσ

]− 2 K − μ

σE[

Z IZ< K−μσ

]+ (K − μ)

2

σ 2P

(Z <

K − μσ

)).

Using dϕ(z) = −zϕ(z)dz and integration by parts, we haveR2(σ ) = (σ 2 + (K − μ)2)�

(K − μ

σ

)+ σ(K − μ)ϕ

(K − μ

σ

).

�Theorem 2 Downside risk functions (1) and (2) are increas-ing functions of σ .

Proof Assume now that μ is given. We show that R′1(σ ) > 0and R′2(σ ) > 0. Direct calculations using the property φ′(x) =−xφ(x) gives

R′1(σ ) = ϕ(

K − μσ

)> 0,

and similarly,

R′2(σ ) = 2σ�(

K − μσ

)> 0.

�Obviously, the conclusion of this theorem is that for a pre-

specified mean of return, minimizing each of these risk mea-sures is equivalent to minimizing the variance. Hence there isa clear equivalence to the mean–variance principle.

3. Downside risk for portfolios

In this section we establish optimal portfolios with minimaldownside risk measures. We assume that short selling is per-mitted. Let X denote the vector of returns in the portfolio andα be the vector of corresponding weights that adds up to unity,then αTX is the return of portfolio. Note that in our settingthe components of α are not necessarily positive. Assumingmultivariate normal distribution for X, N (μ, �) , we have thatαTX ∼ N (αT μ,αT �α).

Theorem 2 indicates that finding a portfolio that minimizesE[(αTX−K )−] or E[((αTX−K )−)2] is equivalent to findingα that minimize αT �α. Formally, given any expected returnc on a portfolio, the solutions to

minα

E[(αTX − K )−] subject to Bα = c, (3)and

minα

E

[((αTX − K

)−)2]subject to Bα = c, (4)

are the same as

minα

αT �α subject to Bα = c, (5)where

B =(

1, 1, . . . , 1μ1, μ2, . . . , μn

)2×n

, c =(

1c

).

The constant B allows us, in addition to requiring all weightsto sum to 1, to put a constraint (c) on the expected return of

318 Feature

the portfolio. We now present the following proposition andcorollary.

Proposition 3 (Landsman (2008)) A portfolio with returnαTX, X ∼ N (μ, �), that minimizes αT �α with given ex-pected return αTμ = c is defined by weights

α = �−1 BT (B�−1 BT )−1c.Corol lary 4 A portfolio with return αTX, X ∼ N (μ, �),that minimizes either E(αTX− K )− or E((αTX− K )−)2 withgiven expected return αTμ = c is defined by weights

α = �−1 BT (B�−1 BT )−1c.Note that in proposition 3 and corollary 4, we tacitly assume

that the means of X are inhomogeneous. In case that all Xi ,i = 1, 2, . . . , n have a common mean μ, i.e. μ = μ1, where1 is a n-dimensional vector with identical unit components,B�−1 BT becomes a singular matrix which is not invertible.Also note that αT μ ≡ μ due to the constraint αT 1 = 1 forhomogeneous means. Thus c can only be μ and (5) reduces tothe classic quadratic optimization:

minα

αT �α subject to αT 1 = 1,whose solution is well known as

α = �−11

1T �−11,

see Landsman (2008) and Luenberger and Ye (1984). Withoutloss of generality, we shall further assume that the means of Xare inhomogeneous.

Let us further consider a more general optimization problemfor downside risk portfolios where the expected return on theportfolio is not pre-specified. That is, we drop off the secondrow in B and only reserve the unity constraint,

minα

E[(αTX − K )−], subject to αT 1 = 1. (6)We obtain the following results.

Theorem 5 The downside risk function (6) of a portfolio withreturn αTX, X ∼ N (μ,�), subject to αT 1 = 1, is minimizedat

α∗ = �−1 ATK (AK �−1 ATK )−1c∗,where c∗ = (1, c∗)T and c∗ is the solution to the equation

ϕ

(−c√

q1c2 − 2q2c + q3

)(q1c − q2)

=√

q1c2 − 2q2c + q3�(

−c√q1c2 − 2q2c + q3

).

c∗ exists and is unique.

Proof Note that (6) can be recasted as

minα

E[(αTY)−], subject to αT 1 = 1 andY = X − K 1, Y ∼ N (μY, �), μY = μ − K 1. (7)

We know that for any given αTμY = c, E(αTY)− is anincreasing function at αT �α and is minimized at

αc = arg minα

αT �α = �−1 ATK (AK �−1 ATK )−1c, (8)

where ATK = (1,μ − K 1)T and c = (1, c)T . It is straightfor-ward to show that

AK �−1 ATK =

(1T �−11 1T �−1μYμTY�

−11 μTY�−1μY

),(

AK �−1 ATK

)−1 = 1δ

(μTY�

−1μY −1T �−1μY−μTY�−11 1T �−11

),

αTc �αc =(

1T �−11c2 − 2μTY�−11c+ μTY�−1μY

)δ−1, (9)

where δ = 1T �−11 × μTY�−1μY − 1T �−1μY × μTY�−11 isthe determinant of AK �−1 ATK . Thus, the minimum of (16),say α∗TY, should always have mean c and standard deviationthat satisfy (9). Consequently, taking (1) into account, (16) isequivalent to the univariate minimization,

minc

E(αTc Y)− := min

cf (c), where

f (c) =√

q1c2 − 2q2c + q3ϕ(

−c√q1c2 − 2q2c + q3

)

− c�(

−c√q1c2 − 2q2c + q3

)(10)

with q1 = δ−11T �−11, q2 = δ−1μTY�−11 , q3 =δ−1μTY�−1μY. We compute the first- and second-order deriva-tives of f (c):

f ′(c) =ϕ

(−c√

q1c2−2q2c+q3

)(q1c − q2)√

q1c2 − 2q2c + q3− �

(−c√

q1c2 − 2q2c + q3

),

f ′′(c) = ϕ(

−c√q1c2 − 2q2c + q3

)×(

q1c2 − 2q2c + q3

)−5/2g(c),

g(c) = c2(

q22 − q1q22 + q21 q3)

+ 2c(

q32 − q3q2 − q1q2q3)

+ q23 − q22 q3 + q1q23 . (11)Due to the positive definity of the covariance matrices, wehave q1 > 0, q3 > 0 and q1c2 − 2q2c + q3 > 0. Also note thatq1c2 − 2q2c + q3 > 0 implies that q1q3 > q22 . Then, we have

g(c) = c2(

q22 − q1q22)

+ 2c(

q32 − q3q2)

+ q23 − q22 q3+ q1q3

(q1c

2 − 2q2c + q3)

> c2(

q22 − q1q22)

+ 2c(

q32 − q3q2)

+ q23 − q22 q3+ q22

(q1c

2 − 2q2c + q3)

= c2q22 − 2cq3q2 + q23 = (cq2 − q3)2 ≥ 0.Thus, we see that f (c) is convex and has unique minimumpoint c∗, which is the solution to the equation f ′(c∗) = 0. �

Following theorem 5, we are able to establish that the opti-mal portfolio which solves the following minimization,

minα

E[((αTX − K )−)2], subject to αT 1 = 1. (12)

Feature 319

However, to make the proof easier, we would first present thefollowing lemma.

Lemma 6 The function

h(t) := (1 + t)�(−1√

t

)− √tϕ

(−1√t

)(13)

is increasing and positive for all t > 0.

Proof Taking the derivative of h(t), we have

h′(t) = �(−1√

t

)> 0

Thus, for t > 0, h(t) are non-decreasing and attain theminimum at 0 :

limt→0 h(t) = 0,

which concludes the proof. �The following theorem presents the optimal portfolio that

minimizes (2).

Theorem 7 The downside risk function (12) of a portfolioαTX with X ∼ N (μ,�), subject to αT 1 = 1, is minimized at

α∗ = �−1 ATK (AK �−1 ATK )−1c∗,

where c∗ = (1, c∗)T , c∗ is the solution to the equation

ϕ

(−c√

q1c2 − 2q2c + q3

)√q1c2 − 2q2c + q3

= ((q1 + 1)c − q2)�(

−c√q1c2 − 2q2c + q3

).

c∗ exists and is unique.

Proof Similar to theorem 5, we first recast (12) as

minα

E[((αTY)−)2], subject to αT 1 = 1 andY = X − K 1, Y ∼ N (μY, �), μY = μ − K 1, (14)

and transform it to an equivalent univariate minimizationaccording to (2), (9) and corollary 4, i.e.

minc

E[((αTc Y)−)2] := minc f (c), wheref (c) = ((q1 + 1)c2 − 2q2c + q3)

× �(

−c√q1c2 − 2q2c + q3

)− c

√q1c2 − 2q2c + q3

× ϕ(

−c√q1c2 − 2q2c + q3

).

We compute the first three order derivatives of f (c) :

f ′(c)2

= ((q1 + 1)c − q2)�(

−c√q1c2 − 2q2c + q3

)

− ϕ(

−c√q1c2 − 2q2c + q3

)√q1c2 − 2q2c + q3,

f ′′(c)2

= (q1 + 1)�(

−c√q1c2 − 2q2c + q3

)

− ϕ(

−c√q1c2 − 2q2c + q3

)(cq1 − q2)

×(

c2q1 − 3cq2 + 2q3) (

q1c2 − 2q2c + q3

)−3/2,

f ′′′(c)2

= ϕ(

−c√q1c2 − 2q2c + q3

)(q1c

2 − 2q2c + q3)−7/2

× (cq2 − q3)g(c), whereg(c) = c2(q22 − 3q1q22 + 3q21 q3)

+ 2c(3q32 − q3q2 − 3q1q2q3)+ q23 − 3q22 q3 + 3q1q23 .

Due to the fact that q1 > 0, q3 > 0, q1c2 − 2q2c + q3 > 0 andq1q3 > q22 , we have

g(c) = 3q1q3(q1c2 − 2cq2 + q3) − 3q2(q1c2 − 2cq2 + q3)+ c2q22 − 2cq2q3 + q23

> c2q22 − 2cq2q3 + q23 = (cq2 − q3)2 ≥ 0.

Then, we establish the positivity of f ′′(c) via the followingstatement.

(1) q2 = 0; f ′′(c) is decreasing w.r.t. c and has minimum at+∞;

limc→+∞

f ′′(c)2

= (1 + q1)�( −1√

q1

)− √q1ϕ

( −1√q1

), q1 > 0. (15)

It is easy to see that (15) is of the form of (13). Thus, bylemma 6, we have f ′′(c) > 0.

(2) q2 < 0; f ′′(c) is increasing for c < q3/q2 and decreasingfor c > q3/q2. Thus, f ′′(c) has maximum at c = q3/q2and minimum at +∞ or −∞. But

limc→−∞

f ′′(c)2

= (1 + q1)�(

1√q1

)+ √q1ϕ

(1√q1

)> lim

c→+∞f ′′(c)

2.

Thus, similar to the case q2 = 0, we see that f ′′(c) ispositive according to (15) and lemma 6.

(3) q2 > 0; f ′′(c) is decreasing for c < q3/q2 and increasingfor c > q3/q2. Thus, f ′′(c) has minimum at c = q3/q2,

320 Feature

Table 1. Expected returns (means) of the 10 stocks.

Adobe Compuware NVDIA Staples VeriSign−0.0061022899 0.0081117935 0.0095500842 −0.0057958520 −0.0064370369Sandisk Microsoft Citrix Intuit Symantec0.0198453317 −0.0001754431 0.0037884516 0.0041225491 −0.0061032938

i.e.

f ′′(q3/q2)2

= (1 + q1)�⎛⎝ −√q3√

q1q3 − q22

⎞⎠− ϕ

⎛⎝ −√q3√q1q3 − q22

⎞⎠√

q1q3 − q22√q3

.

Moreover, note that q1 > (q1q3 − q22 )/q3, we have

f ′′(q3/q2)2

> (1 + q1q3 − q22

q3)�

⎛⎝ −√q3√q1q3 − q22

⎞⎠− ϕ

⎛⎝ −√q3√q1q3 − q22

⎞⎠√

q1q3 − q22√q3

,

which is again of the form of (13) with t = (q1q3 −q22 )/q3 > 0. Thus, f

′′(c) ≥ f ′′(q3/q2) > 0.We therefore conclude that f (c) is convex and has unique

minimum c∗, which is the solution to the equation f ′(c) = 0.�

With theorems 5 and 7, we are able to find the optimalportfolios with respect to the downside risk approach. Suchmethods are obviously more appropriate to the objectives ofthe downside risk philosophy as compared with the classicmean–variance framework. However, one may consider thetwo downside risk functions simultaneously by using a linearcombination of the risk measures as a new goal function. Morespecifically, for any given 0 < λ < 1, we aim to solve

minα

E[λ((αTX − K )−)2 + (1 − λ)(αTX − K )−],subject to αT 1 = 1. (16)

According to the proofs of theorem 5 and 7, we can see thatthe analytical solution to the optimal portfolio of (16) is againavailable. We formulate this in the following corollary.

Corol lary 8 The mixed downside risk function (16) of aportfolio with return αTX, X ∼ N (μ,�), subject to αT 1 = 1,is minimized at

α∗ = �−1 ATK (AK �−1 ATK )−1c∗,

where c∗ = (1, c∗)T , c∗ is the solution to the equation

ϕ

(−c√

q1c2 − 2q2c + q3

)((λ − 1)(q1c − q2)√

q1c2 − 2q2c + q3+ 2λ

√q1c2 − 2q2c + q3

)= �

(−c√

q1c2 − 2q2c + q3

)× (λ(2q1c + 2c − 2q2 + 1) − 1).

c∗ exists and is unique.

Proof The proof is similar to that of theorems 5 and 7. �Clearly, λ reflects the manager’s preference in the choice

of downside risk measures, i.e. the target shortfall and belowtarget variance. As mentioned previously, by choosing λ = 1,the manager’s only concern is the variance of the portfoliobelow the threshold. The portfolio is, therefore, expected to bemore ‘stable’ in the target zone. On the other hand, for λ = 0,the manager’s portfolio has the minimal expected shortfall inmind. We, therefore, anticipate its return to be closer to thetarget. Intuitively, portfolios with minimal expected shortfallshould outperform the ones with minimal below target vari-ances in the sense of the expected returns for the same target. Incontrast, portfolios with minimal below target variances shouldgenerally be less risky in the sense of lower variability. In nextsection, we demonstrate these two alternatives using numericalexamples.

4. Numerical illustration

In this section, we present a numerical example to illustrateour the results. We consider a portfolio of 10 stocks from theNASDAQ stock exchanges (ADOBE Sys. Inc., CompuwareCorp., NVIDIA Corp., Starles Inc.,VeriSign Inc., SandiskCorp, Microsoft Corp., Symantec Corp., Citrix Sys Inc., IntuitInc.) for the year 2005, and denote by X = (X1, . . . , Xn)T ,n = 10, stocks’ weekly returns. We implement theKolmogorov–Smirnov test to test the normality assumptionand since the statistics is 0.15486 with p-value = 0.1484, wedo not reject the hypothesis of the normality.

Morover, our approach could also provide an analytical ap-proximation for non-normal data. For example, in the case thatthe stochastic return r is modelled by log-normal distribution,we could take the well-known approximations that log(1+r) ≈r , r � 1 into account. Then, we have approximately a normaldistribution again. In fact, such an approximation is prettyprecise when r is small, which is particularly true for short-term returns such as in daily and weekly trades.

Feature 321

Tabl

e2.

Cov

aria

nce

Mat

rix

ofth

e10

stoc

ks.

Ado

beC

ompu

war

eN

VD

IASt

aple

sV

eriS

ign

Sand

isk

Mic

roso

ftC

itrix

Intu

itSy

man

tec

Ado

be0.

0061

0155

70.

0011

7317

10.

0001

1811

10.

0005

1285

70.

0001

2069

70.

0010

0433

4−0

.000

1199

540.

0003

9590

0.00

0134

857

0.00

0536

54C

ompu

war

e0.

0011

7317

20.

0033

1020

20.

0010

4726

10.

0004

9846

90.

0008

4744

80.

0004

2813

50.

0003

0932

40.

0006

7374

0.00

0550

476

0.00

0807

46N

VD

IA0.

0001

1811

10.

0010

4726

10.

0021

4510

40.

0001

2203

50.

0007

7198

40.

0004

6873

30.

0002

5590

80.

0005

5918

0.00

0479

347

0.00

0269

85St

aple

s0.

0005

1285

80.

0004

9846

90.

0001

2203

50.

0029

4047

6−0

.000

5465

40.

0010

5321

50.

0000

2870

20.

0003

9449

0.00

0254

851

0.00

0435

42V

eriS

ign

0.00

0120

698

0.00

0847

448

0.00

0771

984

−0.0

0054

654

0.00

3486

011

0.00

0131

760.

0001

1536

20.

0007

9373

0.00

0665

291

0.00

0905

85Sa

ndis

k0.

0010

0433

50.

0004

2813

50.

0004

6873

30.

0010

5321

50.

0001

3176

0.00

4012

978

−3.2

6811

E−0

50.

0008

4372

0.00

0131

403

8.32

98E−0

5M

icro

soft

−0.0

0011

995

0.00

0309

324

0.00

0255

908

2.87

029E

−05

0.00

0115

362

−3.2

6811

E−0

50.

0004

8526

60.

0002

1972

0.00

0167

364

6.18

376E

−05

Citr

ix0.

0003

9590

40.

0006

7374

40.

0005

5918

50.

0003

9449

90.

0007

9373

50.

0008

4372

80.

0002

1972

0.00

1365

090.

0003

9711

80.

0004

4450

Intu

it0.

0001

3485

70.

0005

5047

60.

0004

7934

70.

0002

5485

10.

0006

6529

10.

0001

3140

30.

0001

6736

40.

0003

9711

0.00

0876

343

2.71

609E

−05

Sym

ante

c0.

0005

3654

30.

0008

0746

90.

0002

6985

10.

0004

3542

60.

0009

0585

18.

3298

E−0

56.

1837

6E−0

50.

0004

4450

2.71

609E

−05

0.00

2542

27

322 Feature

Tabl

e3.

Opt

imal

port

folio

sw

ithre

spec

tto

Opt

imiz

atio

n(6

)an

dO

ptim

izat

ion

(12)

for

diff

eren

ttar

gets

.Sum

ofth

ew

eigh

tseq

uals

to10

.Exp

ecte

dre

turn

san

dva

rian

ces

ofth

epo

rtfo

lioar

ere

port

edco

rres

pond

ingl

yin

the

last

two

row

s.

K=

−0.0

5K

=−0

.01

K=

0K

=0.

01K

=0.

05

min

R1

min

R2

min

R1

min

R2

min

R1

min

R2

min

R1

min

R2

min

R1

min

R2

Ado

be0.

0646

365

0.18

1915

4−0

.001

813

0.15

2419

1−0

.020

668

0.14

4445

5−0

.040

4826

0.13

6222

3−0

.129

726

0.10

0751

8C

ompu

war

e−0

.078

993

−0.2

3703

20.

0105

504

−0.1

9728

40.

0359

588

−0.1

8653

90.

0626

5868

−0.1

7545

870.

1829

182

−0.1

2766

0N

VD

IA0.

7040

149

0.55

9836

60.

7857

061

0.59

6098

20.

8088

861

0.60

5900

60.

8332

4444

0.61

6009

90.

9429

572

0.65

9616

1St

aple

s−0

.521

878

−0.2

1659

7−0

.694

849

−0.2

9337

7−0

.743

930

−0.3

1413

2−0

.795

5065

−0.3

3553

8−1

.027

810

−0.4

2786

9V

eriS

ign

−0.7

6679

9−0

.485

576

−0.9

2614

0−0

.556

305

−0.9

7135

3−0

.575

425

−1.0

1886

48−0

.595

144

−1.2

3286

2−0

.680

198

Sand

isk

1.56

9215

51.

2622

548

1.74

3138

91.

3394

571

1.79

2490

11.

3603

267

1.84

4349

691.

3818

498

2.07

7931

91.

4746

889

Mic

roso

ft4.

7980

679

5.07

2258

94.

6427

118

5.00

3298

44.

5986

292

4.98

4656

64.

5523

059

4.96

5431

34.

3436

598

4.88

2503

3C

itrix

−0.0

4299

9−0

.082

311

−0.0

2072

4−0

.072

424

−0.0

1440

4−0

.069

751

−0.0

0776

27−0

.066

995

0.02

2152

0−0

.055

105

Intu

it3.

4131

130

3.05

8321

3.61

4137

13.

1475

537

3.67

1178

13.

1716

753

3.73

1118

53.

1965

521

4.00

1097

43.

3038

573

Sym

ante

c0.

8616

230

0.88

6930

0.84

7283

90.

8805

655

0.84

3215

10.

8788

449

0.83

8939

50.

8770

704

0.81

9681

80.

8694

163

Port

folio

mea

n0.

0525

970.

0377

370.

0610

170.

0414

750.

0634

060.

0424

850.

0659

170 .

0435

273

0.07

7224

0.04

8021

Port

folio

vari

ance

0.03

8591

0.03

4286

0.04

1687

0.03

523

0.04

2651

0.03

5501

0.04

3706

0.03

5788

00.

0489

800.

0371

08

Feature 323

Tabl

e4.

Opt

imal

port

folio

sw

ithre

spec

tto

Opt

imiz

atio

n(1

6)fo

rdi

ffer

entλ

.Sum

ofth

ew

eigh

tseq

uals

to10

.Exp

ecte

dre

turn

san

dva

rian

ces

ofth

epo

rtfo

lioar

ere

port

edco

rres

pond

ingl

yin

the

last

two

row

s.

K=

−0.0

1K

=0.

01

λ=

0.05

λ=

0.5

λ=

0.95

λ=

0.05

λ=

0.5

λ=

0.95

Ado

be0.

0014

7659

0.04

2804

60.

1342

750

−0.0

3640

650.

0134

727

0.11

6759

9C

ompu

war

e0.

0061

1682

−0.0

4957

4−0

.172

834

0.05

7165

94−0

.010

048

−0.1

4923

2N

VD

IA0.

7816

6133

0.73

0854

20.

6184

039

0.82

8233

420.

7669

136

0.63

9936

3St

aple

s−0

.686

2854

−0.5

7870

7−0

.340

607

−0.7

8489

63−0

.655

059

−0.3

8619

9V

eriS

ign

−0.9

1825

08−0

.819

150

−0.5

9981

3−1

.009

0907

−0.8

8948

52−0

.641

813

Sand

isk

1.73

4527

431.

6263

573

1.38

6946

61.

8336

8104

1.70

3129

21.

4327

89M

icro

soft

4.65

0404

054.

7470

263

4.96

0878

64.

5618

3564

4.67

8450

34.

9199

293

Citr

ix−0

.021

8276

−0.0

3568

1−0

.066

342

−0.0

0912

91−0

.025

848

−0.0

6047

1In

tuit

3.60

4183

793.

4791

586

3.20

2443

13.

7187

8747

3.56

7893

03.

2554

297

Sym

ante

c0.

8479

9390

0.85

6912

00.

8766

502

0.83

9819

150.

8505

825

0.87

2870

7

Port

folio

mea

n0.

0606

00.

0553

60.

0437

70.

0654

00.

0590

80.

0459

9Po

rtfo

liova

rian

ce0.

0415

20.

0395

50.

0358

50.

0434

80.

0409

30.

0364

9

324 Feature

0.1 0.2 0.3 0.4 0.5

0.00

0.05

0.10

0.15

0.20

0.25

Means

Expe

cted

Sho

rtfal

l

0.05 0.10 0.15 0.20 0.250.

000.

020.

040.

060.

080.

100.

12Means

Belo

w T

arge

t Var

ianc

es

Figure 1. Examples of mean-downside risk efficient frontier.

We present the corresponding means and covariance in ta-bles 1 and 2. Optimal weights that minimize R1 and R2 withrespect to different targets K are presented in table 3. Table 4presents the results regarding the portfolios that minimize thecombined objectives.†

According to tables 3 and 4, we can see that for the samegiven target K , portfolios that have minimal expected short-fall have higher expected returns as well as higher variances,compared with the ones that minimize the below target vari-ances. Moreover, we observe that both the expected return andvariance of the optimal portfolio rise with the target K . Onthe one hand, minimal expected shortfall keep the expectedreturn of the portfolio close to the target; on the other hand,higher target results in an increase in volatility. After all, sucha scenario agrees with the common sense that higher returnsgo hand in hand with higher risks.

Furthermore, the consistency between the expected returnand the variance also implies a Mean–Variance efficient fron-tier for the optimal portfolio selections. The Mean–Varianceefficient frontier is as important as it is useful for establishingthe Capital Asset Pricing Model and for measuring the per-formance of the portfolio, see for example Sharpe (1971) and

†For the computational convenience, we consider the initial constrainton the weight as αT 1 = 10. In fact, for any initial wealth W0 �= 0,we are to solve

minα

E[(αTX − K )−], subject to αT 1 = W0.

We can rewrite it as

minβ

E[(βTX̂ − K )−], subject to βT 1 = 1, X̂ = W0X,

then it is easy to see that the results obtained in section 3 are applicableas X̂ is again multivariate normal distributed.

Joro and Na (2006). In contrast to the classic Mean-Varianceefficient frontier, we can work out the Mean-Downside riskefficient frontiers, see figure 1. Thus, we may also use suchMean-Downside risks efficient frontiers to measure the per-formance of the portfolios.

5. Conclusions and further discussion

In this paper, we discuss optimal portfolios with respect todownside risk, i.e. E((X − K )−)β, where β = 1, 2, in thecontext of the multivariate normal distribution. We show thatthe two downside risks are monotone increasing in σ , whichimmediately suggests that the solutions to the optimal mini-mization problems

minα

E(αTX − K )− subject to 1T α = 1, μT α = cand

minα

E((αTX − K )−)2 subject to 1T α = 1, μT α = c,coincides with those of the classical mean-variance optimiza-tion. Hence, considering downside risk, though it appears to bemore attractive for many investors compared to the classicalvariance, cannot improve the optimal gain when the expectedreturn of the portfolio is fixed. This proffers theoretical supportto the empirical findings in literature such as Jarrow and Zhao(2006).

Moreover, if we drop off the assumption of a pre-specifiedreturn, i.e. we consider the following optimization problems

minα

E(αTX − K )− subject to 1T α = 1,and

minα

E((αTX − K )−)2 subject to 1T α = 1,

Feature 325

then the optimal solutions differ. In the context of normality, weobtain the analytical solutions to the two general problems andshow that the solution to each optimization exists and is unique.We also provide a numerical illustration of the results. Accord-ing to the numerical results, we find that the optimal portfolioswith respect to the two minimizations can be very different.Hence, we further propose a new downside optimization thattargets the combination of the two downside risk measures asfollows,

minα

E[λ((αTX − K )−)2 + (1 − λ)(αTX − K )−],subject to αT 1 = 1,

where λ reflects the investors’ preference between the twodownside risk measures. Again, we obtain an analytical so-lution to this combined optimization and show that it existsand is unique.

As a matter of fact, there are a vast number of referencesconcerning optimal portfolios. Due to the complexity of theobjective function, most of them rely on numerical techniquesto find the optimal solutions. This is particularly the case fordownside risks, which has been often discussed in the litera-ture. In contrast to a numerical solution, the analytical solutionguarantees the existence and uniqueness of the solution. Onthe one hand, it is easy to calculate and to implement; andon the other hand, it does not suffer from unavoidable com-putational errors nor the great computational effort involved.Clearly, analytical solution is simply superior. In this regard,our results provide useful methodology and new insights intothe literature on optimal portfolios with respect to downsiderisks, and it gives both theoretical and practical contributionsto such problems.

We rely on the assumptions of normality to obtain our results.Although the normal distribution is probably one of the mostfrequently applied probability laws in modeling the stochasticreturns of assets, empirical evidence sometimes suggest thenon-normality of such data, especially for short-term returns.While our methodology offers a sound approximation to non-normal returns, it is our aim to further extend ourresults to the case where the stochastic returns are modelledby non-normal distributions in general, and the elliptical dis-tribution (Fang et al. 1990) in particular, which is much moreflexible and richer than the normal distribution.

Acknowledgements

The authors wish to thank the Israel Zimmerman Foundationfor the Study of Banking and Finance for financial support.Jing Yao also acknowledges the support from and FWO.

Disclosure statement

No potential conflict of interest was reported by the authors.

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WP_CQFIS_downside_risk.pdfOptimal portfolios with downside risk.pdf�1. Introduction2. Downside risk3. Downside risk for portfolios4. Numerical illustration5. Conclusions and further discussionAcknowledgementsDisclosure statementReferences