This article will provide an introduction to target-oriented risk measures in portfolio construction and optimization. In this first part, the various measures and the input required to calculate them are explained. The second part will describe their use to determine efficient portfolios in risk-return terms.
MODERN PORTFOLIO THEORY VERSUS ALTERNATIVE RISK MEASURES.
Modern portfolio theory describes complex investment opportunities, and portfolios created through them, along only two dimensions--risk and return--and attempts to identify portfolios that offer the optimum relationship between these two dimensions. The first step is to define and quantity the two dimensions. For the return dimension, the expected return, expressed by the mean of the return from the investment opportunities, is usually employed. For the risk dimension, the variance or the standard deviation of the random variable describing the returns is normally applied.
However, this form of risk quantification is open to criticism. Firstly, variances or standard deviations of returns do not correspond to the way investors perceive risk. Whereas these measures value all deviations from the mean negatively, investors do not generally take a symmetrical view of the variability of returns. Downward deviations are perceived as economic risks but upward deviations are regarded positively. A second criticism is that with the exclusive use of variance or standard deviation to define the overall risk of a portfolio, valuable information on the return distribution is lost. This is particularly the case when the normal distribution assumption is replaced by considerations of asymmetrical return distributions. It is precisely these distributions that are gaining in importance because derivatives in combination with traditional investments, such as stocks or bonds, systematically result in asymmetrically distributed overall portfolio returns.
Alternatives have been developed, such as probability of loss, loss expectation, maximum loss, and semivariance. These measures are aimed at recording negative returns and are often grouped as downside risk measures. This includes the so-called lower partial moments or target-oriented measures, which are aimed at analyzing the return distribution tail lying below a minimum return requirement set by the investor.
To make alternative risk measures easier to understand, take a sample portfolio of stocks with an asymmetrical return distribution caused by the inclusion of derivatives (see Figure 1).
The shortfall probability or lower partial moment of order zero now shows the probability of a shortfall against a pre-specified target return. It can be calculated mathematically using an integral, and corresponds in graphic terms to the shaded area below the probability curve up to the target return. The target return may be selected individually depending on the investor's specific situation. Nor is it restricted to deterministic returns, such as a fixed rate of 4%. Stochastic returns may be appropriate, as in many cases asset managers may have to meet obligations not known at the beginning, such as achieving a return at least equal to the rate of inflation in the coming year.
For a wide range of investment purposes, the information provided by the lower partial moment of order zero may not be sufficient. For many asset managers it is not enough just to know the probability of failing to reach their target return. They also want to be able to say how great the loss will be if the target return is not met, or what variance of loss may be expected. In such cases, it becomes necessary to turn to higher-order lower partial moments. These measures offer the advantage of a more precise description of the probability distribution below the target return than the shortfall probability alone.
Figure 2 shows the return distribution of two portfolios which have an identical shortfall probability in terms of a given target return, but entirely different return expectations in the event of a shortfall against the target return. With Portfolio 1 the probability mass of returns below the target return is concentrated below the target return. This indicates that in a shortfall, no great deviation from the target is expected. With Portfolio 2, the probability distribution of the returns in the event of a shortfall against the target return implies that returns substantially below the target return must be expected. The two forms of return distribution shown here are for illustrative purposes only, but can easily be replicated in unlimited variety through the use of derivatives. The mathematical calculation of the expected shortfall against the target return or the loss variance can be done by integrating the appropriately modified density function of the return distribution up to the target return.
Classic risk measures such as variance or standard deviation are unsuitable for describing the risk of a portfolio in cases where, through the use of derivatives, asymmetrical overall return distributions result. In such cases, measures that focus on the distribution of returns below the specified target return offer asset managers a better picture of the risks, and one tailored to their specific situations.
This week's Learning Curve was written by Jochen V. Kaduff, consultant with McKinsey & Company, Switzerland.
