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Abstract:
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.

Capital allocation, Research, Requirements, Investment, Investment portfolio, Portfolio, Agreements, Regulation, Stability, International, Framework, Methods, Consumption, Models, Model

Abstract:
This paper develops a unifying framework for allocating the aggregate capital of a financial firm to its business units. The approach relies on an optimization argument, requiring that the weighted sum of measures for the deviations of the business unit's losses from their respective allocated capitals be minimized. This enables the association of alternative allocation rules to specific decision criteria and thus provides the risk manager with flexibility to meet specific target objectives. The underlying general framework reproduces many capital allocation methods that have appeared in the literature and allows for several possible extensions. An application to an insurance market with policyholder protection is additionally provided as an illustration.

Capital allocation, risk measure, comonotonicity, Euler allocation, default option, Lloyd's of London

Abstract:
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 well.

Solvency II, Basel II, KMV, MKMV, asset correlation, credit risk, economic capital, VaR

Abstract:
Since the beginning of the development of the so-called embedded value methodology, actuaries have been using the present value of future profits as yardstick when valuing life insurance activities.

However, using profits as a fundamental input is subject to criticism because profits are no actual cash flows. In an attempt to create more transparency and robustness the CFO forum (2008) has set a definition for market consistent embedded value (MCEV).

Nevertheless, this definition refers again to the present value of future profits. In this note we show that such a definition is misleading and, instead of creating more transparency, it could end up in creating more confusion.

Embedded Value, MCEV, Fair Value, Cash Flow Projections, Business Valuation, Profits, Cash Flows

Abstract:
Copulas have become a buzzword in recent years in the academic community, and practitioners are paying more and more attention to the choice of a copula in risk management applications.

This paper gives a non-technical and pedagogical introduction to the topic of copulas and explains their role for economic capital calculations.

Risk professionals may be tempted to dress up models by using sophisticated tools like for instance copulas. This is because these toys give them the possibility to give a scientific flavour and "serieux" to their models, and as such may serve as "an umbrella" towards the different stakeholders involved.

We provide examples to show that models that involve complicated copulas are by no means better than simple but robust and transparent models and do not always add value. However, building a simple as possible, but not too simple, model requires significant actuarial training and expertise.

Copula, Economic Capital, Basel II, Solvency II, Correlations, Value-at-Risk

Abstract:
Cox & Leland (2000) used techniques from the field of stochastic control theory to show that in the particular case of a Brownian motion for the asset log-returns risk averse decision makers with a fixed investment horizon prefer path-independent pay-offs over path-dependent ones.

In this note we provide a novel and simple proof for the Cox & Leland result and we will extend it to general Levy markets in case pricing is based on the Esscher transform (exponential tilting). It is also shown that in these markets optimal path-independent pay-offs are increasing with the underlying final asset value. We provide examples that allow explicit verification of our theoretical findings and also show that the inefficiency cost of path-dependent pay-offs can be significant.

Our results indicate that path-dependent investment pay-offs, the use of which is widespread in financial markets, do not offer good value from the investor's point of view.

Financial Structured Product, CPPI, Asian Option, Optimal investment, Mean Variance, Markowitz, Lévy Process, Exponential tilting, CAPM, Esscher transform

Abstract:
The Credit Risk model is one of the industry standards for estimating the credit default risk for a portfolio of credit loans.

The natural parameterization of this model requires the default probability to be apportioned using a number of (non-negative) factor loadings. However, in practice only default correlations are often available but not the factor loadings. In this paper we investigate how to deduce the factor loadings from a given set of default correlations.

This is a novel approach and it requires the non-negative factorization of a positive semi-definite matrix which is by no means trivial. We also present a numerical optimization algorithm to achieve this.

Basel II, KMV, Asset Correlations, Default Correlations

Abstract:
systematic factors and the latter are responsible for default dependence between different firms. Another source of default dependence is structural links between firms. For example, a mother company may consist of different legal entities and a default of the former may be contagious and lead to the default of all others, i.e. strong dependence is present in this case. Conversely a possible default from one of the constituent companies may be prevented by the mother company.

In fact, such dependence or guarantee considerations are often made when assessing the individual default probabilities, and then typically result in assigning lower default probabilities to daughter companies.

While it is correct to consider these direct dependence relations when assessing the single default probabilities they also need to be considered when modelling the aggregate loss but it appears that this is ignored by the current state-of-the-art credit risk portfolio models.

In this paper we will use the CreditRisk model to stress-test the direct (intra-)group dependences or contagion effects by making these as strong as possible while leaving the other characteristics of the portfolio unchanged. Then, we show how this model can still be readily applied without major modifications. We also show that the CreditRisk model will allow us to derive the loss distribution function explicitly.

Dependence, correlations, credit risk, contagion, group risk, Panjer's recursion

Abstract:
This is a PhD thesis about the concept of comontonicity and its applications in Finance and Insurance.

Risk measures, Coherency, Comonotonicity, Lognormal, Pension, Annuities

Abstract:
Tasche (1999) introduces a capital allocation principle where the capital allocated to each risk unit can be expressed in terms of its contribution to the conditional tail expectation (CTE) of the aggregate risk.

Panjer (2002) derives a closed-form expression for this allocation rule in the multivariate normal case. Landsman & Valdez (2003) generalise Panjer's result to the class of multivariate elliptical distributions.

In this paper we provide an alternative and simpler proof for the CTE based allocation formula in the elliptical case. Furthermore, we derive accurate and easy computable closed-form approximations for this allocation formula for sums that involve normal and lognormal risks.

Capital allocation, CTE, risk measure, coherent allocation, elliptical

Abstract:
In this paper we investigate approximations for the distribution function of a sum S of lognormal random variables.

These approximations are obtained by considering the conditional expectation E[S | Lambda ] of S with respect to a conditioning random variable Lambda.

The choice for Lambda is crucial in order to obtain accurate approximations. The different alternatives for Lambda that have been proposed in literature to date are 'global' in the sense that Lambda is chosen such that the entire distribution of the approximation E[S | Lambda ] is 'close' to the corresponding distribution of the original sum S.

In an actuarial or a financial context one is often only interested in a particular tail of the distribution of S. Therefore in this paper we propose approximations E[S | Lambda ] which are only locally optimal, in the sense that the relevant tail of the distribution of E[S | Lambda ] is an accurate approximation for the corresponding tail of the distribution of S. Numerical illustrations reveal that local optimal choices for Lambda can improve the quality of the approximations in the relevant tail significantly.

We also explore asymptotic properties of the approximations E[S | Lambda] and investigate links with results from Asmussen & Royas-Nandayapa (2005). Finally, we briefly adress the sub-optimality of Asian options from the point of view of risk averse decision makers with a fixed investment horizon.

Lognormal, random sum, Asian options, conditional expectation, Lower bound, Annuities,

Abstract:
In this paper we consider a portfolio of risks with known marginal distributions. It is well-known that when the risks are non-compensating, i.e. co-monotonic then its sum will be the largest one with respect to convex order, meaning that all risk averse decision makers agree that comonotonicity reflects the most dangerous dependence structure of the underlying risks.

The comonotonic upper bound also exhibits the largest possible variance and in this paper we extend results from Cheung (2008b) to show that under some mild conditions a maximal variance for the random sum implies comonotonicity.

Next, we study a portfolio where besides the marginal distributions of the risks also the variance for the sum is provided. Intuitively one expects that adding this information may give rise to a bound that will be sharper than the comonotonic upper bound and thus a possible recognition of diversifcation when using it to measure the risk of the portfolio.

In this paper we show that such upper bound does not even exist, meaning that risk averse decision makers will never be all agreeable on what the most risky situation of such portfolio is. However, we also show that within such portfolio the so-called upper comonotonic dependence structures exists with the property that the sum behaves like a convex largest sum in the upper tail, and we relate this to a notion of tail convex order.

It also follows that using an upper comonotonic sum to determine capital requirements still does not guarantee that diversification effects between the risks will be accounted for.

comonotonicity, copula, dependence, solvency, Basel II, Solvency II, Value-at-Risk, Tail Value-at-Risk

Abstract:
In this paper we consider different approximations for computing the distribution function or risk measures related to a discrete sum of nonindependent lognormal random variables.

Comonotonic upper bound and lower bound approximations for such sums have been proposed in Dhaene et al. (2002a,b). We introduce the comonotonic "maximal variance" lower bound approximation. We also compare the comonotonic approximations 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 "maximal variance" lower bound approximation outperforms the other approximations.

Lognormal, Sum of random variables, Reciprocal Gamma, Annuities, Value-at-Risk

Abstract:
Even in case of the Brownian motion as most natural rate of return model it appears too difficult to obtain analytic expressions for most risk measures of constant continuous annuities. In literature so-called comonotonic approximations have been proposed but these still require the evaluation of integrals.

In this paper we show that these integrals can sometimes be computed, and we obtain explicit approximations for some popular risk measures for annuities.

Next, we show how these results can be used to obtain fully analytic expressions for lower and upper bounds for the price of a continuously sampled European-style Asian option with fixed exercise price.

These analytic lower bound prices are as sharp as those from Rogers & Shi (1995), if not sharper, but in contrast do not require any longer the evalution of a two-dimensional or a onedimensional integral.

Asian option, closed- form, analytical, annuity, fast approximation, lognormal, maximal variance, conditional expectation

Abstract:
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).

risk measures, coherency, CTE

Abstract:
This is a condensed overview paper on the concept of comonotonicity and some of its applications in finance and insurance.

comontonicity, dependence, copula, correlation, sums of random variables

Abstract:
In this paper we analyze and evaluate a standard approach financial institutions use to calculate their so-called total economic capital.

If we consider a business that faces a total random loss S over a given one-year horizon then economic capital is traditionally defined as the difference between the 99.97% percentile of S and its expectation. The standard approach essentially assumes that the different components (risks) of S are multivariate normally distributed and this highly facilitates the computation of the total aggregated economic capital.

In this paper we show that this approach also holds for a more general framework which encompasses as a special case the multivariate normal (and elliptical) setting. We question also the assumption of multivariate normality since for many risks one often assumes other than normal distributions (e.g. a lognormal distribution for insurance risk). Assuming that risks are either normal or lognormal distributed we propose, using the concept of comonotonicity, an alternative aggregation approach.

economic capital, aggregation, Solvency II, Basel II, VaR

Abstract:
We consider a single period portfolio of n dependent credit risks that are subject to default during the period.

We show that using stochastic loss given default random variables in conjunction with default correlations can give rise to an inconsistent set of assumptions for estimating the variance of the portfolio loss.

Two sets of consistent assumptions are provided, which it turns out, also provide bounds on the variance of the portfolio’s loss. An example of an inconsistent set of assumptions is given.

default correlation, loss correlation, comonotonicity, credit risk, LGD, Solvency II, Basel II

Abstract:
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.

Abstract:
Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002a,b) have studied convex bounds for a sum of dependent random variables and applied these to sums of log-normal random variables. In particular, they have shown how these convex bounds can be used to derive closed-form approximations for several of the risk measures of such a sum.

In this paper we investigate to which extent their general results on convex bounds can also be applied to sums of log-elliptical random variables which incorporate sums of log-normals as a special case.

Firstly, we show that unlike the log-normal case, for general sums of log-ellipticals the convex lower bound does no longer result in closed form approximations for the dierent risk measures.

Secondly, we demonstrate how instead the weaker stop-loss order can be used to derive such closed form approximations. We also present numerical examples to show the accuracy of the proposed approximations.

Abstract:
We consider the problem of determining appropriate solvency capital requirements for an insurance company or a financial institution. We demonstrate that the subadditivity condition that is often imposed on solvency capital principles can lead to the undesirable situation where the shortfall risk increases by a merger. We propose to complement the subadditivity condition by a regulator's condition. We find that for an explicitly specified confidence level, the Value-at-Risk satisfies the regulator's condition and is the "most efficient" capital requirement in the sense that it minimizes some reasonable cost function. Within the class of concave distortion risk measures, of which the elements, in contrast to the Value-at-Risk, exhibit the subadditivity property, we find that, again for an explicitly specified confidence level, the Tail-Value-at-Risk is the optimal capital requirement satisfying the regulator's condition.

risk measure, coherency, solvency, value-at-risk, subadditive

Abstract:
We investigate the influence of the dependence between random losses on the shortfall and on the diversification benefit that arises from merging these losses.

We prove that increasing the dependence between losses, expressed in terms of correlation order, has an increasing effect on the shortfall, expressed in terms of an appropriate integral stochastic order. Furthermore, increasing the dependence between losses decreases the diversification benefit.

We also consider merging comonotonic losses and show that even in this extreme case a strictly positive diversification benefit will often arise.

Abstract:
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 alpha during a fixed number of years, such that one will be able to meet, with a probability of at least (1 - epsilon), a given consumption pattern during the withdrawal period?"

Abstract:
We investigate multiperiod portfolio selection problems in a Black & Scholes type market where a basket of 1 riskfree 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.

Optimal Portfolio selection, Comonotonicity, asset allocation, Merton, constant mix