# Performance

# Performance Evaluation

In order for a comparison to be made between investments with different levels of risk, a risk/return trade-off needs to be defined. For reasons of simplicity, the issues pertaining to the performance evaluation process - discussed in section 4.5 and of particular relevance here - have been ignored. Four main performance measures have been proposed in the literature to achieve this. They are:

- the coefficient-of-variation;
- the Sharpe measure (Sharpe [1966] );
- the Treynor measure (Treynor [1965] ); and
- Jensen's measure (Jensen [1968]).

The first and second measures above are of total risk. The others measure nondiversifiable risk.

As outlined in Chapter 4, these measures differ in the definition of risk and their assumptions about the ability of investors to adjust the level of risk by which an investment may affect their portfolios.

This section proposes the use of the above DRT model to evaluate performance, and seeks to compare it with the Sharpe measure. This approach is taken because of the difficulties - discussed in Chapter 5 - in evaluation of two-parameter portfolio performance. For example, Roll [1977; 1978; 1980] demonstrated that any ranking of fund performance is a function of the choice of the market index, as well as the risk-return properties of the fund. Thus, performance measures based on the SML cannot be used.

The performance measures of Treynor and Jensen are subject to serious errors in ranking portfolio performance.^{19} The inability to use these performance measures also disables a set of measures that permit statistical tests of performance. The latter would have allowed a distinction to be made between chance performance and truly abnormal performance. This leaves only total risk measures - the coefficient-of-variation and Sharpe's measure - that are not subject to Roll's criticism. Neither can the points summarised in section 5.4 be applied to them.

In this section, a sample of real estate companies quoted on the ISE over the periods 1984-1994, are ranked on the basis of the Sharpe^{20} and proposed DRT measures. The statistical significance of differences in ranks are subsequently judged by the Wilcoxon signed ranks test, and the Kendall *τ*, Spearman's R and Gamma correlation statistics. Before presenting the data and results in this section, a brief illustration of the two measures is provided.

Figure 7.11 plots the hypothetical returns distribution from two investments. **A**, with a mean return of 12% and standard deviation of 1%, and **B**, with a mean return of 16% and standard deviation of 4%. Assuming the risk-free rate to be 10%, the Sharpe measure ranks investment** A**

**B**. However, under both the DRT(RFR, 1) and DRT(RFR, 8) measures, **B**

**A**.

The ranking of **A**

**B** by the Sharpe measure is due to three underlying assumptions. It assumes that an investor could combine unlimited risk-free borrowing with investment in **A** in order to achieve a distribution superior to that of investment **B**. This is subject to criticism on three counts. Firstly it depends on the unrealistic assumption of unlimited risk-free borrowing, which in this case would amount to several times the value of investment in **A**. This is discussed further in section 4.5.3. Secondly, when evaluation *ex post* performance it assumes prior knowledge of the future returns distribution, in order to allow a prospective investor to borrow or lend at the risk-free rate as required. Thirdly, it assumes there is a linear relationship between risk and return, with deviations below the mean equal to those above it.

The DRT measure is less restrictive as it assumes only, as in the Sharpe measure, that returns are normally distributed. As discussed in section 7.2, an assumption that may be relaxed. As outlined, however, it relies on a different measure of risk. This assumes that risk may be defined in terms of downside-variance below, and tolerance above a target rate: in this case that is the risk-free rate.

Figure 7.12 on the following page includes the plots of two more hypothetical returns distributions. **C**, with a mean return of 9% and standard deviation of 1%, and **D**, with a mean return of 13% and standard deviation of 4%. Again the risk-free rate is assumed to be 10%. Sharpe's measure ranks **A**

**B** **D**; it cannot be used to rank investment **C**, as it has a negative excess return. ^{21} This problem - discussed further in section 4.4.1 - is a weakness of the Sharpe measure.

A DRT(RFR, 1) investor, on the other hand, would rank **B**

**D** **A** **C**, while a DRT(RFR, 8) investor would rank **B** **A** **D C**. Note, both the difference in order between the two DRT measures, brought about by an increase in downside risk-aversion, and the relative attractiveness of each investment.

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^{19}The construction of Sharpe [1966], Treynor [1965] and Jensen's [1968] performance measures were discussed in Chapter 4.

^{20}For a detailed discussion of the Sharpe measure, see Sharpe [1994].

^{21}Although by definition investment C would be ranked last, the problem arises when attempt is made to rank more than one inve