124 Performance Measurement in Finance

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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

Years

Source

: See Appendix.

Power

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.

5.3 WHAT ARE THE ALTERNATIVES?

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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