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69

In an uncertain environment risk preferences can still be expressed rationally if decision-makers accept a few elementary rules of behaviour which, to most, will appear natural

appropriate RAPM to use is the Sharpe ratio (SR) given by the slope of the line (7.9):

SR = (Rq - Rf)/aq.

If risk-free returns are assumed to be zero the information ratio (IR) is the appropriate RAPM, given by the slope of the line from the origin to the point (j.v, rjY) in risk-return space that represents the optimal portfolio X, that is,

IR = u,/<v

In either case the efficient utilization of capital should allocate funds to activities or investments with the highest RAPM.

7.2.4 Modelling Attitudes to Risk

Which is regarded as best of all efficient portfolios will depend on the risk attitude of the investor. If an investor is very averse to taking risks, he is more likely to choose a low-return, low-risk portfolio over a high-risk, high-return portfolio. To formulate the problem in a mathematical framework, and hopefully find an analytic solution, it is convenient if the preferences of an investor are represented by a utility function.16 In an uncertain environment risk preferences can still be expressed rationally if decision-makers accept a few elementary rules of behaviour which, to most, will appear natural. The three elementary rules are as follows:

> Transitive preferences: If an investor says he prefers outcome A to outcome and he also prefers to C, then he should prefer A to C.

*- Independence: If the investor is indifferent between outcomes A and B, then he is also indifferent between the two gambles {A with probability p and with probability 1 - p} and {B with probability /; and with probability 1 - p) for any outcome C.

*- Certainty equivalence: For any gamble there is a certainty-equivalent value such that the investor is indifferent between the gamble and the certainty equivalent.

These three rules are sufficient to prove the existence of a utility function U( W) that assigns a real number to any monetary amount Wand such that if an investor has a choice between two risk investments he should always prefer the investment that has maximum expected utility. The expected utility is defined by

EU(P)= 2Zp,U(W,),

If an investor has a choice between two risk investments he should always prefer the investment that has maximum expected utility

16 A utility function is a function i : -< : from the space of all possible outcomes Q to the real numbers :1t. Thus a utility function assigns a real number to every possible outcome, and outcomes may be ranked in order of preference by the size of their utility.



where W\, . . ., Wn are the possible outcomes in terms of end of period wealth resulting from an investment P, and these occur with probabilities pu . . ., p„.

Note that the expectation operator is a linear operator, so a utility function is only unique up to an increasing affine transformation. That is, if we define V(x) = a + bU(x), b > 0, then the choices resulting from the use of U and V will be the same.

Usually we assume that more is better - that is, U > 0. The risk preference of an investor is expressed by the curvature of the utility function - in particular, the convexity U". To see this, return first to the concept of certainty equivalence, that is, the monetary value QP that has the same utility as the expected utility of an investment P:

U(QP) = EU(P).

Call ytp and ap the mean and the variance of the investment P; then, using a second-order Taylor series expansion of U around \iP and taking expectations, we obtain

The risk preference of an investor is expressed by the curvature of the utility function

EU(P) « .

«-2 j j" <5pU

Thus when:

U" = 0 locally, QP = \iP and the investor is risk-neutral (that is, he is willing to play the average because his certainty equivalent is the mean return); U" < 0 locally, QP < nP and the investor is risk-averse (the certainty equivalent of an investment is less than its expected value); U" > 0 locally, QP > nP and the investor is risk-loving (because of the pleasure in gambling the investor puts a greater certainty equivalent on an investment than its expected value).

Diminishing marginal utility of wealth implies the investor is risk-averse. The degree of risk aversion - that is, the concavity of the utility function - is commonly measured by the coefficient of absolute risk aversion A(W) = -U"/U, or the coefficient of relative risk aversion R(W) = - WU"/U. These coefficients are not independent of the level of wealth. The investor may have increasing absolute risk aversion, A > 0, so that as his wealth increases he holds less in risky assets in absolute terms, or increasing relative risk aversion, R > 0, so that as his wealth increases he holds proportionately less in risky assets. Or he may have constant or decreasing absolute or relative risk aversion, depending on the functional form assumed for the utility function.

The utility function can be encoded by recording the choices that the investor or the decisionmaker for the firm would make when presented with simple investment alternatives

The quantification of risk attitude through the construction of a utility function is a useful discipline that allows decision-makers to choose more consistently between risky opportunities. The utility function of an investor, or of a firm, can be encoded by recording the choices that the investor or the decision-maker for the firm would make when presented with simple



investment alternatives. For example, the first encoding of a utility function may show domains where the investor is risk-averse and some domains where he is risk-loving. It is not uncommon to find investors showing risk aversion for a range of positive returns but a risk-loving attitude for extremely positive or negative returns. This may be the result of cognitive biases or the consequence of conflict between the incentive of an individual and those of the firm for which he is acting. For example, it may not matter to an individual what losses he makes beyond a certain threshold because his career would already have been ruined; or an individual may take a very small chance of winning a return that will change his life.

If a decision-maker is risk-averse for some levels of returns and risk-loving for others, then a more consistent decisionmaker can take advantage of this pattern

When examining a utility function and realizing the different degrees of risk attitude expressed for different levels of returns, a decision-maker has the opportunity to revise his preferences and perhaps choose to follow a more consistent pattern of behaviour. To illustrate, if a decision-maker is risk-averse for some levels of returns and risk-loving for others, then a more consistent decision-maker can take advantage of this pattern; he could make systematic gains at the expense of the first by offering a series of transactions that the first will find attractive, even though they will result in a net loss for him. Most organizations would want to avoid falling into this trap and should therefore adopt a utility function that is always convex (U" < 0); it is normally expected that organizations are risk-averse.

Other, stronger conditions on a utility function may lead to specific functional forms. For example, the Swiss mathematician Daniel Bernoulli (who was the first to promote the concept of utility functions) argued strongly for the logarithmic utility function U( W) = In W defined on positive wealth W. He argued that a man, no matter how poor, still has some positive net worth as long as he is alive. The main property of a logarithmic utility is that of constant relative risk aversion (CRRA): if Q is the certainty equivalent of an uncertain state of wealth W then XQ is the certainty equivalent of the uncertain wealth XW for any X > 0. The other utility function commonly used to reflect CRRA is the power utility function U(W) - 1¥]~</(1 - ), W > 0.

CRRA may well reflect the risk attitude of decision-makers faced with extreme circumstances that may have a considerable effect on their state of wealth. However, it is more common that investment decisions will affect the wealth of the decision-maker only marginally. If that is the case many decision-makers like to adopt a constant absolute risk aversion (CARA) that is independent of their state of wealth. CARA corresponds to the following delta property: if Q is the certainty equivalent of an uncertain state of wealth W then Q + 5 is the certainty equivalent of an uncertain state of wealth W +5, where 5 is any known change to the state of wealth. The only utility functions showing the delta property of CARA are the linear utility function and the exponential utility function

U{W) = - ~ .

(7.11)



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