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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:
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:
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