Risk Based Capital Allocation

The present contribution reviews the procedures (absolute, incremental and marginal capital allocation) as well as the general principles (proportional allocation, covariance-principle, conditional expectation-principle, conditional value-at-risk principle, Euler-principle) for risk based capital allocation. The approaches discussed are applicable for the insurance case, the investment case and as well for credit risks.

Risk Measures

The present review of (financial) risk measures, prepared for the Encyclopaedia of Actuarial Science, first distinguishes two conceptions of risk. Risk of the first kind conceives risk as the magnitude of (one- or two-sided) deviations from a target, whereas risk of the second kind conceives risk as necessary capital or necessary premium, respectively. Some important axiomatic characterizations of risk measures are reviewed, including a characterization of a correspondence between risk measures of the first kind and risk measures of the second kind. Finally, a detailed overview of different risk measures of the first and second kind is presented.

Self-Annuitization, Ruin Risk in Retirement and Asset Allocation: The Annuity Benchmark

The present paper considers a retiree of a certain age with an initial endowment of investable wealth facing the following alternative investment opportunities. One possibility is to buy a single premium immediate annuity-contract. This insurance contract pays a life-long constant pension payment of a certain amount, depending e.g. on the age of the retiree, the operating cost of the insurance company and the return the company is able to realize from its investments. The alternative possibility is to invest the single premium into a portfolio of mutual funds and to periodically withdraw a fixed amount, in the present paper chosen to be equivalent to the consumption stream generated by the annuity . The particular advantage of this self annuitization strategy compared to the life annuity is its greater liquidity. However, the risk of the second opportunity is to outlive the income stream generated by this investment. The risk in this sense is specified by considering the probability of running out of money before the uncertain date of death. The determination of this personal ruin probability with respect to German mortality and capital market conditions is the objective of the following paper.

Combined Accumulation- and Decumulation-Plans with Risk-Controlled Capital Protection

We base our analysis on an investor, usually a retiree, endowed with a certain amount of wealth W, who considers both his own consumption needs (fixed periodic withdrawals) and the requirement of his heirs (defined bequest). For this purpose he pursues the following in-vestment strategy. The part F is invested in a set of investment funds with the target to achieve an accumulated wealth at the end of a certain time horizon of at least the original amount of wealth W (or the fraction ), measured in real terms. As certain investment risks are implied, we allow for the probability of falling short of the target and implement it into our model as a risk control parameter. The remaining part MM of the original wealth is invested in money market funds in order to avoid additional investment risks and deliver fixed periodic withdrawals until the end of the respective time horizon. The optimal investment strategy is the investment fund allocation that satisfies the probability of shortfall and mini-mizes F, while maximizing the fixed periodic withdrawals. We outline this investment prob-lem in a mathematical model and illustrate the solution for a reasonable choice of empirical parameters.

Asset/Liability Management of German Life Insurance Companies: A Value-at-Risk Approach in the Presence of Interest Rate Guarantees

This contribution analyses the implications of two major determinants influencing the asset allocation decision of German life insurers, which are the capital market development on the one hand and the interest rate guarantees of the traditional life insurance policies on the other hand. The adverse development of the stock prices between 2000 and 2002 asks for a consideration of not only the �normal� volatility but also the worst-case developments in an asset/liability management. In order to meet the latter requirement, we technically apply the risk measures of Value-at-Risk and Conditional Value-at-Risk. German life insurance policies incorporate interest rate guarantees, which are granted on an annual basis. This specific �myopic� nature of guarantees creates � beyond the control of the shortfall risk in general � the necessity to manage the asset allocation on an annual basis to match the time horizon of assets and liabilities. A quantitative approach analyses the impacts on the asset allocation decision. In our research we do not only consider market valuation, but also institutional peculiarities (such as hidden reserves and accounting norms) of German life insurers. We reveal the possibility of a riskless one-year investment, either based on market values or on book values, to be crucial for guaranteeing interest rates on an annual basis.

On the Risks of Stocks in the Long Run:A Probabilistic Approach Based on Measures of Shortfall Risk

The present paper examines the long-term risks of a representative one-time investment in German stocks (DAX/0) in real terms relative to various risk free investments (returns of 0%, 2% and 4% in real terms) as well as relative to a representative investment in German bonds (REXP). As underlying risk measures the shortfall probability, the mean excess loss (conditional shortfall expectation) as well as the product of these two measures, the shortfall expectation have been used. One main structural result is that the mean excess loss is monotonously increasing over time. This reveals a long-term worst case-characteristic of a stock investment.

Theoretische Grundlagen des Minimum-Value at Risk-Hedges

Der vorliegende Beitrag befasst sich mit der Bestimmung der optimalen Hedge Ratio auf der Basis von Future-Kontrakten unter Zugrundelegung der Forderung, dass der Value at Risk der Hedge-Position minimiert werden soll. Unter Verwendung von Ergebnissen im Kontext von Quantilableitungen gelingt hier zunächst die Bestimmung einer allgemeinen strukturellen Lösung. Unter Ausnutzung der Eigenschaften von elliptischen Verteilungen gelingt darüber hinaus eine explizite Bestimmung der optimalen Hedge Ratio und damit eine systematische Verallgemeinerung der in der Literatur entwickelten Lösung für den Normalverteilungsfall

Safety first-Investoren : Separation, Performancemessung und Kapitalmarktgleichgewicht

Auf der Grundlage der Konzeption der Safety first-Investoren und einem damit verbundenen Separationstheorem wird eine risikoadjustierte Performancekennziffer entwickelt und diskutiert, die Safety first-Ratio. Diese Performancekennziffer wird zunächst im Rahmen eines Investmentkontexts verwendet, um eine Alternative (die Mean-Value at Risk-Ratio) zur Sharpe-Ratio theoretisch zu fundieren, die das Downside-Risiko berücksichtigt. In einer zweiten Anwendung wird im Kontext des Risikomanagements von Unternehmen eine alternative RORAC-Kennziffer entwickelt und theoretisch fundiert. Zugleich gelingt damit eine einheitliche Fundierung zweier Forschungsfelder, die in der Literatur bislang weitgehend separat nebeneinander stehen. Abschließend wird auf der Basis von Ergebnissen im Kontext von Quantilableitungen eine strukturelle Charakterisierung des Kapitalmarktgleichgewichts vorgenommen. Im Falle des Vorliegens von elliptischen Renditeverteilungen stimmt dieses mit dem CAPM überein

Quantile Maximizing Safety-First Investors: Separation, Performance Measurement and Capital Market Equilibrium

We combine the safety-first principle of Te lser (1955/56) and Arzac and Bawa (1977) with the principle of quantile maximization studied in Rostek (2010). While maintaining the short- fall constraint of the safety-first principle, we propose to maximize an upper quantile of the return distribution instead of maximizing its exp ected value. We study the implications of this new decision principle for portfolio selection and capital market equilibrium on one hand and for risk-adjusted performance measurement on the other hand