Solving optimal investment problems with structured products under CVaR constraints

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<ul><li><p>This article was downloaded by: [York University Libraries]On: 18 November 2014, At: 13:58Publisher: Taylor &amp; FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK</p><p>Optimization: A Journal ofMathematical Programming andOperations ResearchPublication details, including instructions for authors andsubscription information:</p><p>Solving optimal investment problemswith structured products under CVaRconstraintsRalf Korn a &amp; Serkan Zeytun b ca Department of Mathematics , University of Kaiserslautern andFraunhofer Institute for Industrial Mathematics Kaiserslautern ,67653 Kaiserslautern, Germanyb Institute of Applied Mathematics, Middle East TechnicalUniversity , 06531 Ankara, Turkeyc Fraunhofer Institute for Industrial Mathematics Kaiserslautern ,67653 Kaiserslautern, GermanyPublished online: 18 Mar 2009.</p><p>To cite this article: Ralf Korn &amp; Serkan Zeytun (2009) Solving optimal investment problems withstructured products under CVaR constraints, Optimization: A Journal of Mathematical Programmingand Operations Research, 58:3, 291-304, DOI: 10.1080/02331930902741739</p><p>To link to this article:</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (theContent) contained in the publications on our platform. However, Taylor &amp; Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor &amp; Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.</p><p></p></li><li><p>This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &amp;Conditions of access and use can be found at</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Yor</p><p>k U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rari</p><p>es] </p><p>at 1</p><p>3:58</p><p> 18 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p><p></p></li><li><p>Optimization</p><p>Vol. 58, No. 3, April 2009, 291304</p><p>Solving optimal investment problems with structuredproducts under CVaR constraints</p><p>Ralf Korna and Serkan Zeytunbc*</p><p>aDepartment of Mathematics, University of Kaiserslautern and Fraunhofer Institute forIndustrial Mathematics Kaiserslautern, 67653 Kaiserslautern, Germany; bInstitute of AppliedMathematics, Middle East Technical University, 06531 Ankara, Turkey; cFraunhofer Institute</p><p>for Industrial Mathematics Kaiserslautern, 67653 Kaiserslautern, Germany</p><p>(Received 24 January 2007; final version received 14 March 2008)</p><p>We consider a simple investment problem where besides stocks and bonds theinvestor can also include options (or structured products) into the investmentportfolio. The aim of the investor is to maximize the expected return undera conditional value-at-risk (CVaR) constraint. Due to possible intermediatepayments, we have to deal with a re-investment problem which turns the originalone-period problem into a multi-period one. For solving this problem, an iterativescheme based on linear optimization is developed.</p><p>Keywords: linear optimization; risk measures; linearization of conditional value-at-risk; optimal investment; stochastic processes</p><p>AMS Subject Classifications: 90C05; 60G70; 91B28</p><p>1. Introduction</p><p>As a consequence of the new Solvency II regulations, buying structured products offered</p><p>by various banks might be a reasonable strategy for insurance companies to hedge their</p><p>liabilities. Those structured products can have very sophisticated forms such as various</p><p>types of cliquet structures, interest rate derivatives linked to the equity market or even</p><p>instruments to hedge mortality and interest rate risk at the same time. However, they can</p><p>also be simpler options such as standard calls or hindsight options (see, e.g. [6]).There are various questions that institutional investors such as insurance companies</p><p>have to deal with if they are considering the use of structured products in their investment</p><p>portfolio. Among them are:</p><p>. How to decide which type of structured product to buy?</p><p>. How to judge the advantage of the use of a structured product?</p><p>. How to measure the risk of the investment into a portfolio containing structuredproducts?</p><p>. Is the structured product worth its price?</p><p>*Corresponding author. Email:</p><p>ISSN 02331934 print/ISSN 10294945 online</p><p> 2009 Taylor &amp; FrancisDOI: 10.1080/02331930902741739</p><p></p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Yor</p><p>k U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rari</p><p>es] </p><p>at 1</p><p>3:58</p><p> 18 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>While the last question can only be judged in connection with a pricing routine for</p><p>the structured product and while the first question is closely related to the structure of</p><p>the firms liabilities, the remaining two points shall be addressed in this article. We</p><p>will therefore take up the approach given in [6], refine the problem and analyse</p><p>the properties of the corresponding optimal solutions. In particular, we will consider the</p><p>(highly relevant) aspect of using a structured product that does not exactly match</p><p>the investors needs as it has a maturity that lies before the investors investment</p><p>horizon. The same situation would turn out if the structured product features</p><p>intermediate payments. Then, the one-period Martinelli et al. [6] problem getsa dynamic aspect, the problem of optimally reinvesting the payments resulting from</p><p>the structured product.In the process of measuring the risk, the question of which risk measure should be</p><p>taken into account has a critical importance. Several risk measures have been considered in</p><p>the literature. In the classical Markowitzs mean-variance model [5], the variance is used as</p><p>the measure of the risk. Variance is a measure of variability and therefore is a measure of</p><p>volatility. Although the risk of an investor is to face with a large negative return (loss)</p><p>realization, variance also takes into account the upward return realizations which are</p><p>desired by the investors. Contrary to the variance, Value-at-Risk (VaR) is a risk measure</p><p>which takes only the lower quantile of the return distribution into the account. VaR has</p><p>great popularity among banks and other financial institutions. It is the amount of money</p><p>that expresses the maximum expected loss from an investment over a specific investment</p><p>horizon for a given confidence interval. VaR does not give any information beyond this</p><p>amount of money. Conditional Value-at-Risk (CVaR) is an extension of VaR andexpresses the expected loss of an investment beyond its VaR value. Although there are</p><p>several risk measures proposed so far in the literature, none of them have superiority in</p><p>all aspects.In recent years, particular stress is laid on the use of the so-called coherent risk</p><p>measures as a risk measure instead of the use of the variance as in the standard Markowitz</p><p>mean-variance model. A coherent risk measure is a risk measure that satisfies some desired</p><p>properties, namely monotonicity, sub-additivity, homogeneity, translational invariance</p><p>(for detailed information and explanation of the properties see [1]. We will examine the use</p><p>of the CVaR (a coherent risk measure) as is done in [6]. We will assume a financial market</p><p>with investment possibility in a bond, a stock and a structured product, and impose</p><p>a constraint for the CVaR as is done in [6]. However, the use of option-type securities in</p><p>the portfolio will result in a peculiar behaviour of the mean-CVaR optimal portfolio: using</p><p>the option with the higher strike leads to a higher expected return while keeping the riskconstant.</p><p>We will set up our investment problem in the next section which will also contain the</p><p>necessary theoretical background. The rest of this article is devoted to the numerical</p><p>solution of some concrete problems and the interpretation of the special forms of the</p><p>solutions.</p><p>2. Optimal investment with structured products</p><p>In [6], the authors considered an investment problem where the investor can choose</p><p>a buy-and-hold strategy in a riskless bond, a stock and an option on the stock. In</p><p>particular, the option is assumed to mature exactly at the investors investment horizon.</p><p>292 R. Korn and S. Zeytun</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Yor</p><p>k U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rari</p><p>es] </p><p>at 1</p><p>3:58</p><p> 18 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>The utility criterion considered is the minimization of the CVaR of the</p><p>portfolio for a given level of expected return. The authors can then make use of a very</p><p>convenient property of CVaR that allows to solve this problem via</p><p>a combination of linear optimization and Monte Carlo simulation, a fact that has been</p><p>pointed out in [7].To keep our article self-contained, we will recall both this way of solving the problem</p><p>as well as the definition of the CVaR.</p><p>Definition 1</p><p>(a) Let Lw be the loss of an investor using the portfolio vector w, and denote the</p><p>probability of Lw not exceeding a threshold by</p><p> w, P Lw :</p><p>Then, we define the value-at-risk VaR (Lw,) of the loss with a confidence level of2 [0, 1] (for short: VaR) via</p><p>VaRLw, minf 2 R : w, g:</p><p>(b) In addition, let Lw have a finite expected value. Then its conditional value-at-risk</p><p>CVaR (Lw,) with a confidence level of (for short: CVaR) is given as</p><p>CVaRLw, ELwjLw VaRLw,:</p><p>Note that LwRw, where Rw is the return associated with the portfolio vector w, andwe define the return as</p><p>Rw final wealthinitial wealth</p><p> 1:</p><p>Note also that in contrast to variance CVaR is a so-called coherent risk measure.</p><p>Coherency is considered to be a desirable property since a risk measure having this</p><p>property, in particular, favours diversified investments (as a consequence of sub-</p><p>additivity). The use of CVaR in portfolio optimization problems as a measure of allowed</p><p>risk is particularly attractive because, as formulated thus, the optimization problem</p><p>reduces to a linear optimization problem with linear constraints. This problem can then be</p><p>solved by the standard simplex method.We will illustrate this by looking at a one-period problem similar to the one in [6].</p><p>We assume that we can invest into a stock with return RS, a bond with return RB, and</p><p>an option (or a structured product) with maturity T and return RO. Choosing the</p><p>investment portfolio w (wS,wB,wO) where wS, wB and wO represent the weights of thestock, the bond and the option in the portfolio, respectively, then leads to a portfolio</p><p>return of</p><p>RwT wSRST wBRBT wOROT :</p><p>To take the risk under the control, we want the CVaR of the loss to be bounded by</p><p>a constant C. We are trying to maximize the expected return over all portfolios w with</p><p>a return that has a CVaR that does not exceed an upper bound C. We therefore consider</p><p>the following problem.</p><p>Optimization 293</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Yor</p><p>k U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rari</p><p>es] </p><p>at 1</p><p>3:58</p><p> 18 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>Problem (RCVaR):</p><p>maxw2R3</p><p>ERwT,</p><p>such that:</p><p>RwT wSRST wBRBT wOROT ,wS wB wO 1, wS,wB,wO 0, CVaRRwT, C:</p><p>Besides the CVaR-constraint this is a linear optimization problem in w. However, the</p><p>CVaR-constraint seems to be highly non-linear in w. Fortunately, in [7], a linearization</p><p>procedure for CVaR is given. In their paper, they showed that using the function F(w,)defined by</p><p>Fw, 1 1Zy2Rm</p><p>Lw, y p ydy</p><p>one can calculate the -CVaR associated with the portfolio vector w by the formula</p><p>CVaRLw, min2R</p><p>Fw,:</p><p>where L(w, y) is the loss function associated with the portfolio vector w, and y2Rm is theset of uncertainties which determine the loss function. Moreover, the CVaR can be</p><p>minimized over all possible portfolio vectors w2X by</p><p>minw2X</p><p>CVaRLw, minw,2X,R</p><p>Fw,:</p><p>Here, (w*, *) is a solution of the right-hand-side minimization problem if and only if w* isa solution of the left-hand-side minimization problem and *2 arg min2R F(w,). Whenthe set of the arg min2R F(w,) reduces to a single point, then * gives the correspondingVaR with a confidence level of .</p><p>Furthermore, Rockafellar and Uryasev proposed to approximate the integral</p><p>appearing in F(w,) by using a sample from the distribution of the uncertaintyvector y. Then, the integral can be replaced by a summation, and in this case minimization</p><p>of F(w,) is equivalent to the minimization of the linear expression</p><p> 1N1 </p><p>XNi1</p><p>zk</p><p>subject to linear constraints</p><p>zi 0 and Lw, yi zi for i 1, 2, . . . ,N</p><p>where zi, i 1, 2, . . . ,N are dummy variables and N is the size of the sample.Later, Krokhmal, Palmquist and Uryasev [4] showed that, for two functions R(x) and</p><p>(x) depending on decision vector x, the optimization problems</p><p>minx2X</p><p>x s:t: Rx </p><p>294 R. Korn and S. Zeytun</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Yor</p><p>k U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rari</p><p>es] </p><p>at 1</p><p>3:58</p><p> 18 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>and</p><p>minx2X</p><p>Rx s:t: x "</p><p>generate the same efficient frontier if (x) is convex, R(x) is concave and set X is convex,provided that the constraints have internal points.</p><p>In our problem, CVaR(Lw,) is convex since the loss function Lw is linear w.r.t. w (forthe proof, see [7]), the reward function (expected return) is linear, therefore concave, and</p><p>the constraints are assumed to be linear. Therefore, the minimization of CVaR under an</p><p>expected return constraint can be replaced by the maximization of the expected return</p><p>under a CVaR constraint.Krokhmal et al. [4] also proved that solution of the optimization problems</p><p>minx2X</p><p>Rx s:t: CVaRLw, " 1</p><p>and</p><p>minx2X</p><p>Rx s:t: Fw, " 2</p><p>give the same minimum value. If the CVaR constraint in (1) is active, then (w*, *)minimizes (2) if and only if w* minimizes (1) and *2 arg min2RF(w,). When *2 argmin2R F(w,) reduces to a single point, then * gives the corresponding VaR witha confidence level of .</p><p>The problem Problem (RCVaR) can be converted to a linear optimization problem by</p><p>using the information proposed in [7], and [4] which is mentioned above. It mainly consists</p><p>of two steps:</p><p>. Step 1: Simulate N paths of the market prices of the stock, bond and the option.</p><p>. Step 2: Set up a suitable linear problem on those simulated paths that can besolved by the well-known simplex method.</p><p>More precisely, we can consider the following problem.Problem (LRCVaR):</p><p>maxw2R3</p><p>1</p><p>N</p><p>XNi1</p><p>RwT,i,</p><p>such that:</p><p>RwT,i wSRST,i wBRBT,i wOROT,i, i 1, . . . ,N,wS wB wO 1, wS,wB,wO 0,RwT,i zi 0, i 1, . . . ,N,</p><p> 1N1 </p><p>XNi1</p><p>zi C,</p><p>zi 0, i 1, . . . ,N:</p><p>Here, is the g...</p></li></ul>