Some results on Denault's capital allocation rule.

Denault (2001) introduces a capital allocation principle where the capital allocated to any risk unit is expressed in terms of the contribution of that risk to the aggregate conditional tail expectation. Panjer (2002) derives a closed-form expression for this allocation rule in the multivariate normal case. Landsman & Valdez (2003) generalize Panjer's result to the class of multivariate elliptical distributions. In this paper we provide an alternative and much simpler proof for the allocation formula in the elliptical case. Further, we show how to derive accurate closed-form approximations for Denault's allocation formula in case of lognormal risks.Capital allocation;

Comonotonicity.

Comonotonicity;

On the evaluation of 'saving-consumption' plans.

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 ?'Research; Evaluation; Knowledge; Distribution; Value; Problems; Insurance; Approximation; Sign; Quantile; Risk; Risk measure; Consumption; Probability;

Comparing approximations for risk measures of sums of non-independent lognormal random variables.

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.Approximation; Choice; Comonotonicity; Criteria; Decision; Distribution; Dual theory; Lognormal; Market; Moment matching; Optimal; Order; Portfolio; Problems; Random variables; Reciprocal gamma; Research; Risk; Risk measure; Selection; Simulation; Theory; Time; Value; Variables;

On the suboptimality of path-dependent pay-offs in Lévy markets.

Cox & Leland (2000) use techniques from the field of stochastic control theory to show that in the particular case of a Brownian motion for the asset returns all risk averse decisionmakers with a fixed investment horizon prefer path-independent payoffs over path-dependent ones. We will provide a novel and simple proof for the Cox&Leland result and we will extend it to general, not necessarily complete, Lévymarkets. It is also shown that in these markets optimal path-independent pay-offs have final values increasing with the underlying asset value. Our results imply that path-dependent investment payoffs, the use of which is widespread in financial markets, do not appear to offer good value for risk averse decisionmakers with a fixed investment horizonResearch; Approximation; Distribution; Risk; Risk measure; Lognormal; Random variables; Variables; Lower bounds; Choice; Variance; Goodness of fit; Actuarial; Problems; Framework; Requirements; Credit; Portfolio; Impact; Software; Value; Data; Markets; Market; Field; Control; Control theory; Theory; Brownian motion; Investment; IT; Optimal;

Closed-form approximations for constant continuous annuities.

Abstract: 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.Approximation; Choice; Comonotonicity; Criteria; Decision; Distribution;

Optimal Payoffs under State-dependent Preferences

Most decision theories, including expected utility theory, rank dependent utility theory and cumulative prospect theory, assume that investors are only interested in the distribution of returns and not in the states of the economy in which income is received. Optimal payoffs have their lowest outcomes when the economy is in a downturn, and this feature is often at odds with the needs of many investors. We introduce a framework for portfolio selection within which state-dependent preferences can be accommodated. Specifically, we assume that investors care about the distribution of final wealth and its interaction with some benchmark. In this context, we are able to characterize optimal payoffs in explicit form. Furthermore, we extend the classical expected utility optimization problem of Merton to the state-dependent situation. Some applications in security design are discussed in detail and we also solve some stochastic extensions of the target probability optimization problem

Asset correlations: shifting tides.

The Basel II accord outlines a general framework for determining regulatory capital requirements for credit risk portfolios. Different obligors usually operate independent socio-economic environments and these structural correlations are the main reason why regulatory capital is needed. Therefore, it is not surprising that an important component of the regulatory regime for capital is the asset correlation between obligors. Basel II has set a range for corporate asset correlations from 8 to 24 %, the exact value depending on several individual firm characteristics.We use monthly asset value data to calculate asset correlations and compare these with Basel II as well as results from other papers. Our results are in line with literature but a clear difference is found between the majority of these results and the results from Basel II and some major software providers. We discuss these differences and offer some explanations as an attempt to reconcile the differences. The impact of horizon is considered as wellResearch; Approximation; Distribution; Risk; Risk measure; Lognormal; Random variables; Variables; Lower bounds; Choice; Variance; Goodness of fit; Actuarial; Problems; Framework; Requirements; Credit; Portfolio; Impact; Software; Value; Data;

Optimal portfolio selection for cash-flows with bounded capital at risk.

Optimal; Optimal portfolio selection; Portfolio; Selection; Cash flow; Capital at risk; Risk;

The hurdle-race problem.

We consider the problem of how to determine the required level of the current provision in order to be able to meet a series of future deterministic payment obligations, in case the provision is invested according to a given random return process. Approximate solutions are derived, taking into account imposed minimum levels of the future random values of the reserve. The paper ends with numerical examples illustrating the presented approximations.Processes; Value;