# Solving optimal investment problems with structured products under CVaR constraints

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This article was downloaded by: [York University Libraries]On: 18 November 2014, At: 13:58Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

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Solving optimal investment problemswith structured products under CVaRconstraintsRalf Korn a & 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.

To cite this article: Ralf Korn & 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

To link to this article: http://dx.doi.org/10.1080/02331930902741739

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Optimization

Vol. 58, No. 3, April 2009, 291304

Solving optimal investment problems with structuredproducts under CVaR constraints

Ralf Korna and Serkan Zeytunbc*

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

for Industrial Mathematics Kaiserslautern, 67653 Kaiserslautern, Germany

(Received 24 January 2007; final version received 14 March 2008)

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.

Keywords: linear optimization; risk measures; linearization of conditional value-at-risk; optimal investment; stochastic processes

AMS Subject Classifications: 90C05; 60G70; 91B28

1. Introduction

As a consequence of the new Solvency II regulations, buying structured products offered

by various banks might be a reasonable strategy for insurance companies to hedge their

liabilities. Those structured products can have very sophisticated forms such as various

types of cliquet structures, interest rate derivatives linked to the equity market or even

instruments to hedge mortality and interest rate risk at the same time. However, they can

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

have to deal with if they are considering the use of structured products in their investment

portfolio. Among them are:

. How to decide which type of structured product to buy?

. How to judge the advantage of the use of a structured product?

. How to measure the risk of the investment into a portfolio containing structuredproducts?

. Is the structured product worth its price?

*Corresponding author. Email: szeytun@metu.edu.tr

ISSN 02331934 print/ISSN 10294945 online

2009 Taylor & FrancisDOI: 10.1080/02331930902741739

http://www.informaworld.com

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While the last question can only be judged in connection with a pricing routine for

the structured product and while the first question is closely related to the structure of

the firms liabilities, the remaining two points shall be addressed in this article. We

will therefore take up the approach given in [6], refine the problem and analyse

the properties of the corresponding optimal solutions. In particular, we will consider the

(highly relevant) aspect of using a structured product that does not exactly match

the investors needs as it has a maturity that lies before the investors investment

horizon. The same situation would turn out if the structured product features

intermediate payments. Then, the one-period Martinelli et al. [6] problem getsa dynamic aspect, the problem of optimally reinvesting the payments resulting from

the structured product.In the process of measuring the risk, the question of which risk measure should be

taken into account has a critical importance. Several risk measures have been considered in

the literature. In the classical Markowitzs mean-variance model [5], the variance is used as

the measure of the risk. Variance is a measure of variability and therefore is a measure of

volatility. Although the risk of an investor is to face with a large negative return (loss)

realization, variance also takes into account the upward return realizations which are

desired by the investors. Contrary to the variance, Value-at-Risk (VaR) is a risk measure

which takes only the lower quantile of the return distribution into the account. VaR has

great popularity among banks and other financial institutions. It is the amount of money

that expresses the maximum expected loss from an investment over a specific investment

horizon for a given confidence interval. VaR does not give any information beyond this

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

several risk measures proposed so far in the literature, none of them have superiority in

all aspects.In recent years, particular stress is laid on the use of the so-called coherent risk

measures as a risk measure instead of the use of the variance as in the standard Markowitz

mean-variance model. A coherent risk measure is a risk measure that satisfies some desired

properties, namely monotonicity, sub-additivity, homogeneity, translational invariance

(for detailed information and explanation of the properties see [1]. We will examine the use

of the CVaR (a coherent risk measure) as is done in [6]. We will assume a financial market

with investment possibility in a bond, a stock and a structured product, and impose

a constraint for the CVaR as is done in [6]. However, the use of option-type securities in

the portfolio will result in a peculiar behaviour of the mean-CVaR optimal portfolio: using

the option with the higher strike leads to a higher expected return while keeping the riskconstant.

We will set up our investment problem in the next section which will also contain the

necessary theoretical background. The rest of this article is devoted to the numerical

solution of some concrete problems and the interpretation of the special forms of the

solutions.

2. Optimal investment with structured products

In [6], the authors considered an investment problem where the investor can choose

a buy-and-hold strategy in a riskless bond, a stock and an option on the stock. In

particular, the option is assumed to mature exactly at the investors investment horizon.

292 R. Korn and S. Zeytun

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