Search

Distorted expectations can be expressed as mixtures of quantiles. In this note, we show that this statement is essentially true, but that one has to be careful with the correct formulation of it. Furthermore, the proofs of the additivity property for distorted expectations of a comonotonic sum that appear in the literature often do not cover the case of a general distortion function. We present a straightforward proof for the general case, making use of the appropriate expressions for distorted expectations in terms of mixtures of quantiles. ispartof: FEB Research Report AFI_1273 pages:1-11 status: published

ispartof: FBE Research Report AFI_1265 pages:1-56 status: published

ispartof: FEB Research Report AFI_1274 pages:1-15 status: published

In this paper, we propose a new fear index based on (equity) option surfaces of an index and its components. The quanti¯cation of the fear level will be solely based on option price data. The index takes into account market risk via the VIX volatility barometer, liquidity risk via the concept of implied liquidity, and systemic risk and herd-behavior via the concept of comonotonicity. It thus allows us to measure an overall level of fear (excluding credit risk) in the market as well as to identify precisely the individual importance of the distinct risk components (market, liquidity or systemic risk). As a side result we also derive an upperbound for the VIX. ispartof: FBE Research Report AFI_1155 pages:1-17 status: published

ispartof: FBE Research Report AFI_1156 pages:1-13 status: published

We introduce a new and easy to calculate measure for systemic risk in financial markets. This measure is baptized the Herd Behavior Index (HIX). It is model-independent and forward looking, based on observed option data. In order to determine the degree of systemic risk or herd behavior in a financial market one should compare the observed market situation with the extreme (theoretical) situation under which the whole system is driven by a single factor. The Herd Behavior Index (HIX) is defined as the ratio of an option-based estimate of the risk-neutral variance of the market index and an option-based estimate of the corresponding variance of this extreme market situation. Using the theory of comonotonicity, the extreme situation can easily be backed out of the observed option quotes. The HIX can be determined for any market index provided an appropriate series of vanilla options is traded on this index as well as on its components. As an illustration, we determine historical values of the 30-days implied Herd Behavior Index for the Dow Jones Industrial Average, covering the period January 2003 to October 2009. ispartof: FBE Research Report AFI_1157 pages:1-31 status: published

For any random vector X = (X1; :::;Xn) on a given probability space ( ;F;Pr), one can always onstruct comonotonic modi cations of X, which are defined as random vectors with the same marginals as X but with the comonotonic copula describing their dependency structure. In this short note, we investigate whether it is always possible or not to define comonotonic modifications of X on its own probability space ( ;F; Pr). ispartof: FBE Research Report AFI_1151 pages:1-6 status: published

We investigate optimal buy-and-hold strategies for terminal wealth problems in a multi-period framework. As terminal wealth is a sum of dependent random variables, the distribution function of final wealth cannot be determined analytically for any realistic model. By calculating lower bounds in the convex order sense, we consider approximations that reduce the multivariate randomness to univariate randomness. These approximations are used to determine buy-and-hold strategies that optimize, for a given probability level, the Value at Risk and the Conditional Left Tail Expectation of the distribution function of final wealth. Finally, the accurateness of the different approximations is investigated numerically.

In this paper we discuss multiperiod portfolio selection problems related to a specific provisioning problem. Our results are an extension of Dhaene et al.(2005), where optimal constant mix investment strategies are obtained in a provisioning and savings context, using an analytical approach based on the concept of comonotonicity. We derive convex bounds that can be used to estimate the provision to be set up at a speci ed time in the future, to ensure that, after having paid all liabilities up to that moment, all liabilities from that moment on can be fulfilled, with a high probability. We give some interpretations of this additional reserve, and apply our results to optimal portfolio selection. ispartof: FBE Research Report AFI_0937 pages:1-19 status: published

In Dhaene et al. (2005), multiperiod portfolio selection problems are discussed,using an analytical approach to find optimal constant mix investment strategies in a provisioning or savings context. In this paper we extend some of these results,investigating some specific, real-life situations. The problems that we consider in the first section of this paper are general in the sense that they allow for liabilities that can be both positive or negative, as opposed to Dhaene et al. (2005), where all liabilities have to be of the same sign. Secondly, we generalize portfolio selection problems to the case where a minimal return requirement is imposed. We derive an intuitive formula that can be used in provisioning and terminal wealth problems as a constraint on the admissable investment portfolios, in order to guarantee a minimal annualized return. We always apply our results to optimal portfolio selection. ispartof: FBE Research Report AFI_0933 pages:1-20 status: published

In the present paper, we give su¢ cient conditions for an ordering of De Pril approximations of the distribution of the number of claims in an insurance portfolio of independent policies. Possible extensions are discussed, both for the De Pril approximation and the Kornya approximation. A numerical example is given. ispartof: pages:1-25 status: published

Numeraire invariance is a well-known technique in option pricing and hedging theory. It takes a convenient asset as the numeraire, as if it were the medium of exchange, and expresses all other asset and option prices in units of this numeraire. Since the price of the numeraire relative to itself is identically 1 at all times, this reduces pricing and hedging to a market with zero-interest rates. A somewhat controversial implication is that the modelling focus should be more on the asset price ratios rather than on the asset price processes themselves. The idea of numeraire invariance is already implicit in Merton (1973), and since then many authors have contributed to its development. After a brief survey of its origins, we state and prove the numeraire invariance principle for general semimartingale price processes, following essentially Duffie [3]. We then present its application to unique pricing in arbitrage-free models and discuss nondegeneracy and unique hedging.

ispartof: Tijdschrift voor Economie en Management issue:2 pages:265-278 status: published

This paper considers different approximations for computing either the distribution function or various risk measures associated with a discrete sum of nonindependentlognormal random variables. In Dhaene et al. (2002a,b), convex upper and lower bounds for sums of non-independent lognormals were proposed.Of particular interest is the lower bound obtained by taking the conditional expectation of the original sum S with respect to a conditioning random variable. This has proven to provide excellent approximations for various risk measures associated with S. The choice of this conditioning variable plays a crucial rolein arriving at a good approximation and intuitively, it could be chosen so that, in some sense, it is 'close' to the original sum S. In the literature then, variouschoices for have been proposed. For example, Dhaene et al (2002b) propose to determine such that it can be viewed as a first-order approximation to theoriginal sum S. In another development, Vanduffel et al (2005) argue that the best is obtained when the variance of the resulting lower bound is as close aspossible to the variance of the original S. In this paper, we notice that these choices for are global in the sense that they lead to lower bounds for whichthe distribution function presents a global goodness-of-fit when compared with the original distribution function. In many financial and actuarial problems, however, we are usually only interested in the tails of the distribution function. Hence, in this paper, we derive lower bounds approximations considered to be'locally' optimal. ispartof: DTEW - AFI_603 pages:1-14 status: published

The Basel Accords represent landmark financial agreements for the regulation of commercial banks. The main purpose of the accords was to strengthen the soundness and stability of the international banking system by providing a minimum standard for capital requirements. In 2004, the Basel Committee proposed new guidelines, which have become known as Basel II. We give a short overview of the Basel II framework and present the different approaches which can be used to determine the amount of regulatory capital needed for equity exposures. These methods vary from simple, rather rule of thumb methods, to more sophisticated and economic-oriented approaches. We compare the regulatory capital consumption of two equity portfolios using the different Basel II-compliant methods. We provide evidence that, as far as regulatory capital consumption for equity exposures is concerned, there is no real incentive for banks to use the more sophisticated and economic-oriented models such as VaR or EVT models. ispartof: DTEW - AFI_0602 pages:1-10 status: published

info:eu-repo/semantics/published

ispartof: Tijdschrift voor Economie en Management issue:1 pages:103-114 status: published

In this paper we present in a general setting lower and upper bounds for the stop-loss premium of a (stochastic) sum of dependent random variables. Therefore, use is made of the methodology of comonotonic variables and the convex ordering of risks, introduced by Kaas et al. (2000) and Dhaene et al. (2002a, 2002b), combined with actuarial conditioning. The lower bound approximates very accurate the real value of the stop-loss premium. However, the comonotonic upper bounds perform rather badly for some retentions. Therefore, we construct sharper upper bounds based upon the traditional comonotonic bounds. Making use of the ideas of Rogers and Shi (1995), the first upper bound is obtained as the comonotonic lower bound plus an error term. Next this bound is refined by making the error term dependent on the retention in the stop-loss premium. Further, we study the case that the stop-loss premium can be decomposed into two parts. One part which can be evaluated exactly and another part to which comonotonic bounds are applied. As an application we study the bounds for the stop-loss premium of a random variable representing the stochastically discounted value of a series of cash flows with a fixed and stochastic time horizon. The paper ends with numerical examples illustrating the presented approximations. We apply the proposed bounds for life annuities, using Makeham's law, when also the stochastic nature of interest rates is taken into account by means of a Brownian motion. ispartof: DTEW Research Report 0523 pages:1-26 status: published

ispartof: Tijdschrift voor Economie en Management issue:1 pages:23-49 status: published

A simple static superhedging strategy for the payoff of an arithmetic Asian option in terms of a portfolio of European options based on stop-loss transforms is applicable under general stock price models. The hedge obtained is optimal in some sense. Numerical illustrations of the hedging performance are given for various Lévy models calibrated to market data of the S&P 500. ispartof: The Journal of derivatives vol:12 issue:3 pages:63-72 status: published

We investigate a liability driven methodology for determining optimal asset mixes. We study the effect on the optimal investment stategy when changing the duration of the liability cash flow stream, changing the certainty level and changing the correlation matrix. It is shown that the methodology leads to results which are in accordance with intuition. Keywords: Liability driven investing, strategic asset allocation

For a series of cash flows, its stochastically discounted or compounded value is often a key quantity of interest in finance and actuarial science. Unfortunately, even for most realistic rate of return models, it may be too difficult to obtain analytic expressions for the risk measures involving this discounted sum. Some recent research has demonstrated that in the case where the return process follows a Brownian motion, the so-called comonotonic approximations usually provide excellent and robust estimates of risk measures associated with discounted sums of cash flows involving log-normal returns. In this paper, we derive analytic approximations for risk measures in case one considers the continuous counterpart of a discounted sum of log-normal returns. Although one may consider the discrete sums as providing a more realistic situation than its continuous counterpart, considering in this case, the continuous setting leads to more tractable explicit formulas and may therefore provide further insight necessary to expand the theory and to exploit new ideas for later developments. Moreover, the closed-form approximations we derive in this continuous set-up can then be compared more effectively with some exact results, thereby facilitating a discussion about the accuracy of the approximations. Indeed, in the discrete setting, one always must compare approximations with results from simulation procedures which always give rise to room of debate. Our numerical comparisons reveal that the comonotonic 'maximal variance' lower bound approximation provides an excellent fit for several risk measures associated with integrals involving log-normal returns. Similar results as we derive here for continuous annuities can also be obtained in case of continuously compounding which therefore opens a roadmap for deriving closed-form approximations for the prices of Asian options. Future research will also focus on optimal portfolio slection problems. ispartof: DTEW Research Report 0518 pages:1-18 status: published

Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable at- tention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables defined on some measurable space to the (extended) real line. Economically, a risk measure should capture the preferences of the decision-maker. In incomplete financial markets, prices are no more unique but depend on the agents' attitudes towards risk. This paper complements the study initiated in Denuit, Dhaene & Van Wouwe (1999) and considers several theories for decision under uncertainty: the classical expected utility paradigm, Yaari's dual approach, maximin expected utility theory, Choquet expected utility theory and Quiggin rank-dependent utility theory. Building on the actuarial equivalent utility pricing principle, broad classes of risk measures are generated, of which most classical risk measures appear to be particular cases. This approach shows that most risk measures studied recently in the financial literature disregard the utility concept (i.e. correspond to linear utilities), causing some deficiencies. Some alternatives proposed in the literature are discussed, based on exponential utilities. ispartof: pages:1-26 status: published

In this paper, we consider different approximations for computing the distribution function or risk measures related to a sum of non-independent lognormal random variables. Approximations for such sums, based on the concept of comonotonicity, have been proposed in Dhaene et al. (2002a,b). These approximations will be compared with two well-known moment matching approximations: the lognormal and the reciprocal Gamma approximation. We find that for a wide range of parameter values the comonotonic lower bound approximation outperforms the two classical approximations. ispartof: DTEW Research Report 0418 pages:1-13 status: published

We investigate multiperiod portfolio selection problems in a Black & Scholes type market where a basket of 1 riskless and m risky securities are traded continuously. We look for the optimal allocation of wealth within the class of 'constant mix' portfolios. First, we consider the portfolio selection problem of a decision maker who invests money at predetermined points in time in order to obtain a target capital at the end of the time period under consideration. A second problem concerns a decision maker who invests some amount of money (the initial wealth or provision) in order to be able to fullfil a series of future consumptions or payment obligations. Several optimality criteria and their interpretation within Yaari's dual theory of choice under risk are presented. For both selection problems, we propose accurate approximations based on the concept of comonotonicity, as studied in Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002 a,b). Our analytical approach avoids simulation, and hence reduces the computing effort drastically. ispartof: DTEW Research Report 0408 pages:1-44 status: published

In this paper we examine and summarize properties of several well-known risk measures that can be used in the framework of setting solvency capital requirements for a risky business. Special attention is given to the class of (concave) distortion risk measures. We investigate the relationship between these risk measures and theories of choice under risk. Furthermore we consider the problem of how to evaluate risk measures for sums of non-independent random variables. Approximations for such sums, based on the concept of comonotonicity, are proposed. Several examples are provided to illustrate properties or to prove that certain properties do not hold. Although the paper contains several new results, it is written as an overview and pedagogical introduction to the subject of risk measurement. The paper is an extended version of Dhaene et al. (2003). ispartof: DTEW Research Report 0416 pages:1-33 status: published

Knowledge of the distribution function of the stochastically compounded value of a series of future (positive and/or negative) payments is needed for solving several problems in an insurance or finance environment, see e.g. Dhaene et al. (2002 a,b). In Kaas et al. (2000), convex lower bound approximations for such a sum have been proposed. In case of changing signs of the payments however, the distribution function or the quantiles of the lower bound are not easy to determine, as the approximation for the random compounded value of the payments will in general not be a comonotonic sum. In this paper, we present a method for determining accurate and easy computable approximations for risk measures of such a sum, in case one first has positive payments (savings), followed by negative ones (consumptions). This particular cashflow pattern is observed in 'saving - consumption' plans. In such a plan, a person saves money on a regular basis for a certain number of years. The amount available at the end of this period is then used to generate a yearly pension for a fixed number of years. Using the results of this paper one can find accurate and easy to compute answers to questions such as: 'What is the minimal required yearly savings effort a during a fixed number of years, such that one will be able to meet, with a probability of at least (1 - e), a given consumption pattern during the withdrawal period ?' ispartof: DTEW Research Report 0436 pages:1-17 status: published

In the last few years the properties of risk measures that can be considered as suiting 'best practice' rules in insurance have been studied extensively in the actuarial literature. In Artzner (1999) so-called coherency axioms were proposed to be satisfied for risk measures that are used for providing capital requirements. On the other hand Goovaerts et al. (2003a), (2003b),(2003c) argue that the choice of appropriate set of axioms should depend on the axiomatic 'situation at hand'. In this contribution, we show that so-called concave distortion risk measures are not always consistent with some well-known dependency measures such as Pearson's r, Spearman's p and Kendall's T, i.e. higher dependency between random variables does not necessary lead to higher risk measure of corresponding sums. We also test numerically to what extend risk measures are consistent with certain dependency measures and how stable the consistency level is for different one-parametric families of distortion risk measures. ispartof: DTEW Research Report 0435 pages:1-37 status: published