Efficient risk/return frontiers for credit risk

FALL 2000 T he risk/return trade-off has been a central tenet of portfolio management since the seminal work of Markowitz [1952]. The basic premise, that higher (expected) returns can only be achieved at the expense of greater risk, leads naturally to the concept of an efficient frontier. The efficient frontier defines the maximum return that can be achieved for a given level of risk or, alternatively, the minimum risk that must be incurred to earn a given return. Traditionally, market risk has been measured by the variance (or standard deviation) of portfolio returns, and this measure is now widely used for credit risk management as well. For example, in the popular CreditMetrics methodology (J.P. Morgan [1997]), the standard deviation of credit losses is used to compute the marginal risk and risk contribution of an obligor. Kealhofer [1998] also uses standard deviation to measure the marginal risk and, further, discusses the application of meanvariance optimization to compute efficient portfolios. While this is reasonable when the distribution of gains and losses is normal, variance is an inappropriate measure of risk for the highly skewed, fat-tailed distributions characteristic of portfolios that incur credit risk. In this case, quantile-based measures that focus on the tail of the loss distribution more accurately capture the risk of the portfolio. In this article, we construct credit risk efficient frontiers for a portfolio of bonds issued in emerging markets, using not only the variance but also quantile-based risk measures such as expected shortfall, maximum (percentile) losses, and unexpected (percentile) losses. In an earlier article (Mausser and Rosen [1999]), we considered several scenario optimization models for credit risk with a particular emphasis on the expected regret measure. A primary motivation for using this measure is the fact that expected regret is both relevant and tractable. Relevant measures capture key properties of the loss distribution while tractable measures can be optimized using computationally efficient methods such as linear programming. In contrast to expected regret, variance is generally not a relevant credit risk measure, while maximum losses and unexpected losses are relevant but not tractable. Arvanitis et al. [1998] present a useful example of these concepts: they show that the minimum-variance portfolio is far from efficient with respect to unexpected losses. However, given that the latter measure is not tractable, they can only approximate the efficient frontier by evaluating randomly generated portfolios. Efficient frontiers for expected regret can be readily constructed, as demonstrated in Dembo [1998] and Dembo and Rosen [1999], for example. Furthermore, Mausser and Rosen [1999] showed that minimizing expected regret effectively reshapes the loss distribution, typically producing significant reductions in other risk measures as well. This suggests an attractive alternative to random search for constructing (approximate) efficient frontiers for intractable risk measures: use a tractable meaEfficient Risk/Return Frontiers for Credit Risk