THE SHORTFALL RISK APPROACH
Last week we looked at alternative measures for risk measurement. In particular, we reviewed the advantages of lower partial moments compared to the classic risk measures of variance or standard deviation. In this week's article, we will go into more detail on the construction and selection of risk/return-efficient portfolios. For the sake of simplicity, we will restrict discussions to the zero-order lower partial moment. Zero-order lower partial moment defines the probability of a shortfall against the given target return. For this risk measure we will consider a range of analytical approaches for identifying efficient portfolio combinations. The following considerations can of course be applied to the higher-order lower partial moments discussed last week.
MEAN-SHORTFALL RISK EFFICIENCY
Having set the target return and selected the most appropriate risk measure for the particular involvement, an investor can construct an efficiency graph by positioning all potentially available portfolios in a two-dimensional diagram. The vertical axis of this graph represents each portfolio's expected return, and the horizontal axis the value of the risk measure, in relation to the set target return. Figure 1 gives an example of this for the shortfall probability towards a fixed target return.
For the actual portfolio selection process, the investor should now consider only the investment combinations that lie on the efficient part of the curve thus generated. All other portfolios have a higher risk for the same expected return than a portfolio on the efficienct upper part of the curve. The portfolio located at the peak of the efficiency curve has the lowest possibility of all portfolios of falling short of the target return.
TRADE-OFF BETWEEN TARGET RETURN AND SHORTFALL RISK
Presentations of the type shown in Figure 1 are particularly suited for asset managers who are in a position to set the appropriate target return directly. In practice, however, it is often the case that at the beginning of the investment process many investors do not know the best target return for them, and thus to analyze the effect of various target returns on the optimum design of their portfolios. In these cases, a presentation of the type shown in Figure 3 is helpful. Here, all available investment combinations are presented so that the asset manager can directly read off the effect of a variation of the target return on the shortfall risk. The emphasis with this presentation is on focusing the investor on the trade-off between target return and shortfall risk. Figure 2 shows that the lower the aspirations regarding the target return selected, the lower the probability is they will not be achieved. Again, only portfolios on the new efficient frontier generated through this process are relevant for further considerations.
With its focus on the trade-off between target return and shortfall risk, the form of presentation shown in Figure 2 does, however, have a significant disadvantage. It is now no longer possible to read off the expected return of the investment alternatives.
THREE-DIMENSIONAL SHORTFALL RISK ANALYSIS
In order to combine the advantages of the types of presentation shown in Figures 1 and 2, we can adopt a three-dimensional presentation. Figure 3 shows the expected return, the set target return and the shortfall probability for all the available investment combinations. This offers the investor a tool ideally matched to his particular requirements, enabling him to see at a glance what effect variations of any one variable will have on the others.
After introducing shortfall risk measures last week, we have now looked at their use in selecting risk/return-efficient portfolios. It may be said that lower partial moments represent a significant alternative to classic risk measures. By providing an individual choice for target return and statistical order of the lower partial moment, they enable asset managers to perform investor-specific risk analyses and consequently the selection of their optimum portfolio. Applications of the concepts described above can be already observed in practice, not only in institutional asset management, but also in serving the needs of private banking customers.
This week's Learning Curve was written by Jochen V. Kaduff, consultant with McKinsey & Company, Switzerland.
