The most popular framework for quantitative performance evaluation is that of mean- variance, which leads to the CAPM (Chapter 5). The most important assumptions underlying mean-variance models and hence mean-variance measures of performance evaluation are that:
- investors can make their portfolio decisions solely on the basis of means,variances and co-variances which are stationary over time;
- investors have homogenous information i.e. investors share common knowledge about the population means, variances and co-variances of all assets in the economy;
- there are no taxes or transaction costs; and
- all assets are marketable and traded.
These assumptions lead to an economy with three important characteristics:
- all investors hold some combination of the market portfolio and can borrow or lend at a single risk-free rate;
- the risk of any asset solely depends on the co-variance between the asset's returns and the returns on the market portfolio - its systemic risk; and
- the expected return of any asset solely depends on its systemic risk, unsystemic or idiosyncratic risk is not rewarded.
Active management has no practical role or use in this framework, as the assumptions of the CAPM require that all investors have homogenous information.6 Nevertheless, many academics and practitioners still rely on the mean-variance CAPM framework as a benchmark for normal performance, implicitly assuming that - the amount of actively managed money is small (literally infinitesimal) relative to the aggregate amount of wealth in the economy.
Figure 4.1 below plots the expected return/standard deviation trade-off facing investors in a standard CAPM world. As previously noted in section 5.2.1, the standard CAPM assumptions result in all investors choosing some combination of the risk-free asset and the market portfolio.
The dashed line in figure 4.1 is the efficient frontier of all risky assets and the risk-free asset. Investors can choose any point along the dashed line, known as the Capital Market Line ("CML"), by combining the market portfolio (M) with either risk-free lending (points to the left of M) or risk-free borrowing (points to the right of M). Under the CAPM assumptions, investors could form portfolios that would plot below the dashed line, such as Portfolio B, but no investor would choose such an inefficient portfolio. No feasible portfolios exist above the line, and all portfolios that people hold should plot on the dashed line.
The CML is given by the following equation:
where Rf denotes the risk-free rate-of-return, Rm,the return on the market portfolio. E[.] the expected value, σm the standard deviation of the market portfolio's returns, and σp the standard deviation of the portfolio's returns. The latter is also a measure of the portfolio's total risk. The equation predicts that for a total risk of σp, the portfolio (if efficient) should have an excess expected return (above the risk-free rate) of:
It follows that a portfolio with superior mean-variance performance, such as portfolio A in figure 4.1 on the top before which plots above the CML, will have expected returns which satisfy:
The SML is given by the following equation:
Where βp, denotes the β of the portfolio i.e.
The equation predicts that for systemic risk, βp, the portfolio (whether efficient or not) should have an excess expected return - above the risk-free rate - of:
It follows that a portfolio with positive risk-adjusted returns, such as portfolio A in figure 4.2 above before which lies above the SML, will have expected returns which satisfy:
Two popular measures of performance evaluation, which adjust only for systemic risk,
where and Rf are as in the Sharpe measure, and βp is the β coefficient of the portfolio. This measure of portfolio performance was first suggested by Treynor , and is often referred to as the Treynor measure. It is identical to the Sharpe measure, although in expected return β space. It is also usually compared to the average excess return (above the risk-free rate):
for a passive index, such as the Financial Times All-Share Index.
It measures whether the sample return of the investment (above the risk-free rate) was sufficient for the systemic risk being borne.7
A second popular measure of performance evaluation, which adjusts only for systemic risk, is the Jensen measure (Jensen ):
A Jensen measure greater than zero indicates above-average returns; less than zero indicates below-average returns.
One problem with the Treynor and Jensen measures is that they do not meet the requirement that the quantitative performance measures of investments A, B and C be identical. A slight modification of these measures, however, enables them to meet this criterion:
where σcp is the standard deviation of the investment's residual returns. This measure is referred to as the Treynor-Black Appraisal ratio.
It can be shown that a sufficient condition for a portfolio to have a positive Treynor measure, Jensen measure or Treynor-Black Appraisal Ratio, is that it has a positive Sharpe measure. However, a positive Treynor measure, Jensen measure or Treynor-Black Appraisal Ratio does not ensure a positive Sharpe measure. The intuition behind this is that if a portfolio has positive excess returns after adjusting for total risk. it will certainly have positive excess returns after adjusting for systemic risk only. However, if a portfolio has positive excess returns after adjusting for systemic risk only, it mayor may not have a positive excess return after adjusting for total risk. Note, however, that a negative excess return after adjusting for systemic risk is sufficient to ensure a negative excess return after adjusting for total risk.
6Roll  has thus argued that it is logically inconsistent to make use of the CAPM framework for performance evaluation - see section 5.3.8.
7Srivastava & Essayyad  obtained a composite measure of β, by combining the traditional CAPM and MSV models; see section 5.3.6). The β estimated was then used in Trevnor's measure to rank a sample of US international unit trusts. Their results showed that the composite β was a statistically significant parameter.