124 Performance Measurement in Finance
0.1 3.5 6.9 10.3 13.8 17.2 20.6 24.0 27.4 30.8 34.3 37.7 41.1
: See Appendix.
Figure 5.3 Power function probability of correctly detecting abnormal performance
These figures are derived from Figure 5.3. The power of the test measures
the probability of correctly rejecting the null hypothesis that the fund manager
generates no abnormal performance. It is clear from the figures that it takes
a long time to detect with reasonable confidence that the performance of the
fund manager is abnormal. And this result is dependent on an unchanging
investment opportunity set which is in itself an unlikely eventuality over a
22-year time horizon.
Recently, the academic literature has begun to investigate alternative bench-
marks, based on extensions to the Capital Asset Pricing Model (CAPM).
They help to identify the sources of any under- or outperformance by fund
managers. There are also fixed benchmarks.
5.3.1 Single-index benchmarks with time-varying coefficients
The external benchmarks considered above are single-index benchmarks that
can be justified by the CAPM, invented by Nobel prize winner Bill Sharpe
and now one of the cornerstones of modern finance theory.
What is the CAPM?
The CAPM decomposes the expected return on a fund into two parts. The
first is the return on a riskless asset such as Treasury bills: all professional
investors should be expected to generate a return exceeding that on Treasury
Performance benchmarks for institutional investors 125
bills! The second is the additional return from taking on ‘market risk’. This,
in turn, has two components: the ‘market risk premium’ (otherwise called the
‘excess return on the market’ or the ‘market price of risk’), and the ‘quantity’
of market risk assumed by a particular fund as measured by that fund’s ‘beta’.
The market risk premium is measured by the difference between the
expected return on the market index and the risk-free rate. The principal
market index in the UK is the FTA All Share Index and many equity fund
managers have this index as their single-index benchmark. The historical
long-run market risk premium for the UK is about 6% p.a.
The beta of a fund measures the degree of co-movement between the return
on the fund and the return on the market index. Technically the beta is cal-
culated as the ratio of the covariance between the returns on the fund and
the market to the variance of the return on the market. It is also equal to the
product of the standard deviation of the return on the fund and the correlation
between the returns on the fund and the market. These are exactly the same
formulae as the slope or beta coefficient in a time-series regression of the
excess return on the fund on an intercept and the market risk premium, which
explains how a beta coefficient is so named. If the standard deviation of the
return on the fund or the correlation between the returns on the fund and the
market are high, then the fund’s beta will be high. The beta of the market
index itself is unity. If the fund beta exceeds unity, the fund is more volatile
than the market: a beta of 1.1 implies that the fund is 10% more volatile than
the market so that if the market rises or falls by 20%, the fund will rise or
fall by 22%.
The CAPM can be expressed as follows:
Excess return on fund = Alpha + Beta of fund × Market risk premium
= Alpha + Market risk of fund (5.4)
where the excess return on the fund is the difference between the realized
return on the fund and the risk-free rate. The CAPM is illustrated in Figure 5.4.
If the excess return on the fund exceeds the market risk of the fund, then
the fund has generated an above-average performance. The difference between
the excess return on the fund and the market risk of the fund is called the
fund ‘alpha’ (sometimes it is called the ‘Jensen alpha’ after its inventor). A
successful fund manager therefore generates a positive alpha. However, it is
important to recognize that a fund return exceeding the market index return
does not necessarily imply a positive alpha. It is possible for a fund to take
on a lot of market (i.e. beta) risk and generate a return higher than the market
index return, but nevertheless generate a negative alpha: this would indicate
that the market risk assumed by the fund manager was not fully rewarded.

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