A.1.7.3 Generalising Sharpe Ratios

The violation of stochastic dominance shown in some circumstances by the Sharpe ratio rule (or, equivalently, by the mean–variance criterion) is more than a mere curiosity. Often the risks to be compared are not similarly distributed. In finance, the dynamics of asset prices are often modelled with a constant-volatility geometric Brownian process implying that log-returns are normally distributed (see Section I.A.8.7). But there is ample empirical evidence to suggest that volatility is not constant and is difficult to forecast, that log-returns have heavier tails than normal distributions (positive excess kurtosis) and are often asymmetrical (non-zero skew) – see Sections II.B.5.5 and II.B.5.6. Even if basic assets were similarly distributed, many portfolios contain

21 The certain equivalent of the investment is maximised when (µ – r)/ƱƳ is maximised, or, substituting ơ + Ƣµmrfor(µ – r), when ơƱƳ + SRm is maximised, which amounts to maximisingơƢ. Note the special case when Ƣ

= 0: the Treynor ratio becomes infinite, indicating that any investment not correlated to the market but yielding an expected return above the risk-free rate is infinitely desirable. Obviously, if one began to invest heavily in that opportunity, its specific risk (risk not correlated with the market) would become significant and impossible to diversify;

the diversification assumption supporting the Treynor ratio would break down.

22 See Huebner (2003).

instruments such as options that are non-linear in the basic assets or are actively managed so as to create skewed and heavy-tailed returns. Most securities are also vulnerable to default of their issuer, creating a small probability of a large loss, that is, a negative skew. Finally, many risks are not obviously linked to market returns; there are many business and operational risks with heterogeneous distributions.

Academics and practitioners have therefore tried to design choice criteria that would be simpler to implement than maximising expected utility (this requires a utility function and full probability distributions) and yet would have a wider range of applicability than the mean–variance criterion or its equivalent for ranking investments, the Sharpe ratio.

One avenue of research is to base preferences on simple risk–reward value functions that, on the one hand, are simpler to evaluate than expected utility and, on the other hand, are richer than mean–variance functions in that they would capture other risk characteristics of the choice portfolios. A simple family of risk–reward functions consists of linear combinations of a reward metric, typically the expected value, and a more ‘appropriate’ risk metric than variance. A host of candidate risk metrics have been put forward, some ad hoc such as value-at-risk (see Chapters III.A.2, III.B.6 and III.C.3) and some more soundly grounded, such as semi-variance (see Section II.B.5.4). We give a few examples in the next section.

Two arguments, often but not necessarily compatible, have been used to devise an appropriate risk metric. One is that the value function in which it will be used should be compatible with the existence of a utility function (i.e. the value function should lead to the same preference order).

The other is that a risk metric should have a few intuitive properties to account for aggregation of risks, diversification, hedging, etc. This second approach has led to the specification of coherent risk measures with desirable properties (see Artzner et al., 1999).²³

I.A.1.7.3.1 The Generalised Sharpe Ratio

Compatibility with the existence of a utility function has led to two types of generalisations of the Sharpe ratio. One, pioneered by Hodges (1997), seeks to extend the Sharpe ratio to apply to the

23 Artzner et al. (1999) propose four axioms for the coherence of a risk metric ƱGiven gambles X and Y with cumulative distribution functions FX and FY, we require:

i. Sub-additivity Ʊ(X + Y) ” Ʊ(X) + Ʊ(Y);

ii. Homogeneity Ʊ(aX) = aƱ(X), for any scalar a;

iii. Monotonicity: Ʊ(X) • Ʊ(Y) if, for any scalar u , FX(u)• FY(u) (stochastic dominance);

iv. Risk-free condition: Ʊ(X +b) = Ʊ(X) – b, for any scalar b.

The first axiom ensures that the risk of an aggregate portfolio is no greater than the sum of the risks of its constituents, a property that enables risk budgeting. The first two axioms ensure that the risk of a diversified portfolio is no greater than the corresponding weighted average of the risks of the constituents. The third axiom ensures that stochastic dominance is preserved. The last axiom, which we have rewritten in a time-independent context, indicates that the risk measure is defined on a monetary scale and can be offset by cash amounts (e.g. capital).

comparison of any return distributions. Hodges assumes an exponential utility function (see Appendix I.A.1.B) and seeks the optimal quantity q to invest in a risky portfolio financed at the risk-free rate r. If the return of the risky investment is normally distributed with mean µ and variance Ƴ² (see Section II.E.4.4), then an investment q produces the same certain equivalent CE(q) as in (I.A.1.5) and therefore the same optimum value of q. The corresponding maximum EU is

EU^* = (1/ƫ)[1 – exp(–½((µ – r)/Ƴ)²)]. (I.A.1.7)

Therefore, as we have already found, investors maximising EU should prefer to invest in the portfolio that maximises the Sharpe ratio whatever their risk toleranceƫ. Since (I.A.1.7) can be rewritten as

SR = [ –2ln(1 – ƫEU^*]^1/2

and since utility functions are defined only within a positive linear transformation, Hodges defines a generalised Sharpe ratio (GSR) as:

GSR = [–2ln(–EU^*)]^1/2. (I.A.1.8)

The GSR is equivalent to the traditional Sharpe ratio for ranking portfolios with normally distributed returns and when the utility function is exponential. But its range of applicability extends to any type of return distribution. The drawback, of course, is that it is restricted to exponential utility functions²⁴ and it requires an expected utility maximisation. However, its advantage is that it produces a dimensionless index for ranking risky portfolios of any type in a manner consistent with expected utility and without having to specify a coefficient of risk tolerance. The GSR is also a coherent risk measure according to the axioms of Artzner et al. (see footnote 24).

I.A.1.7.3.2 The Adjusted Sharpe Ratio

The second approach to compatibility with utility theory uses a Taylor series expansion of a utility function to account for higher moments of the return distribution than just mean and variance. The derivation of such an adjusted Sharpe ratio (ASR) is beyond the scope of the PRM exam and we simply state it here:

ASR = SR[1 + (µ3/6)SR – ((µ4 –3)/24)SR²], (I.A.1.9)

where µ3and µ4 the third and fourth standardised central moments of the return distribution, that is, the skewness and the kurtosis (see Section II.B.5.5 and II.B.5.6). The ASR exhibits not only

24 Similar generalisations of the Sharpe ratio can be obtained for other classes of utility functions such as the members of the power family described in Appendix I.A.1B.

variance aversion but also aversion to negative skewness (µ3 < 0) and to positive excess kurtosis (µ4 – 3 > 0).

Example I.A.1.4:

We calculate the ASR and GSR for the two investments A and B in Figure I.A.1.7. The results are compared in Table I.A.1.1.

Table I.A.1.1: RAPMs for choosing between two mutually exclusive investments Comparison of the Sharpe ratio (SR), adjusted Sharpe ratio (ASR) and generalised Sharpe ratio (GSR) for the

two risky investments in Figure I.A.1.7

SR ASR GSR

InvestmentA InvestmentB

0.707 0.587

0.685 0.709

0.691 0.718

Both modified Sharpe ratios indicate that investment B should be preferred to investment A, as logic dictates. Note, however, that ASR is a simple adjustment to the Sharpe ratio and is not guaranteed to be always consistent with maximum expected utility or even with stochastic dominance.