UTILITY AND RISK BEHAVIOUR

3.2 UTILITY AND RISK BEHAVIOUR

An expected utility provides a quantitative expression of a decision makers’ de- sires for higher rewards as well as his attitude towards the ‘risks’ of such rewards. Say that {R, P(.)} is a set of rewards R assumed to occur with probability P(.) and let u(.) define a utility function. The basic utility theorem states that the expected utility provides an objective index to evaluate the desirability of rewards, or:

E (u(R)) = u (R)P(R) dR; R ∈ ℜ

Given uncertain prospects, a rational decision-maker will then select that prospect whose expected utility is largest. For example, the EU of an alternative prospect

i with probability outcomes (π ij , p ij ) is:

and the optimal alternative k is found by:

k ∈ Max i ∈[1,n] {EU i }

In this decision approach, the function u(.), stands for the investor’s psychology. For example, we might construe that u ′ (.) > 0 implies greed, u ′′ (.) < 0 implies fear , while risk tolerance and prudence are implied by the signs of the third derivative u ′′′ (.) > 0 and u ′′′ (.) < 0 respectively. Given a probability distribution for rewards, P(R), the basic assumptions regarding continuous utility functions are that alternative rewards:

(1) can be compared (comparability). (2) can be ranked such that preferred alternatives have greater utility. (3) have strong independence. (4) have transitive preferences (transitivity). (5) are indifferent if their utilities are equal.

UTILITY AND RISK BEHAVIOUR

3.2.1 Risk aversion

Expected utility provides an investor preference for uncertain payoffs, expressing thereby his attitude toward the risk associated with such payoffs. Three attitudes are defined: (1) risk aversion (2) risk loving and (3) risk neutrality. Risk aversion expresses a risk-avoidance preference and thus a preference for more conservative gambles. For example, a risk-averse investor may be willing to pay a premium to reduce risk. A risk lover would rather enjoy the gamble that an investment risk provides. Finally, risk neutrality implies that rewards are valued at their objective value by the expectation criterion (EMV). In other words, the investor would be oblivious to risk. For risk-averse investors, the desire for greater rewards with smaller probabilities will decrease (due to the increased risk associated with such rewards); such an attitude will correspond to a negative second derivative of the utility function or equivalently to an assumption of concavity, as we shall see below. And, vice versa, for a risk loving decision-maker the second derivative of the utility function will be positive. To characterize quantitatively a risk attitude, two approaches are used:

r Risk aversion directly relates to the risk premium, expressed by the difference between the expected value of a decision and its certainty (riskless) equivalent

reward. r Risk aversion is expressed by a decreasing preference for an increased risk,

while maintaining a mean preserving spread. These two definitions are equivalent for concave utility functions, as we shall see

below.

Certainty equivalence and risk premium

Assume an uncertain reward ˜ R whose expected utility is E(u( ˜ R )). Its equivalent sure amount of money, given by the expected utility of that amount, is called the certainty equivalent which we shall denote here by ¯ R and is given by

R=u ¯ −1 {E[u( ˜R)]} Note that the certainty equivalent is not equal to the expected value ˆ R = E( ˜R) for

u (¯ R ) = E(u( ˜ R ))

and

it embodies as well the cost of risk associated with the uncertain prospect valued by its expected utility. The difference ρ = ˆR − ¯R, expresses the risk premium a

decision maker would be willing to pay for an outcome that provides for sure the expected return compared to the certainty equivalent. It can be null, positive or negative. In other words, the risk premium is:

Risk premium (ρ) = Expected return ( ˆR) − The certainty equivalent ( ¯R) An alternative representation of the risk premium can be reached by valuing the

expected utility of the random payoff: ˜ R = ˆR + ˜ε where E(˜ε) = 0, var(˜ε) = σ 2 and σ 2 denotes the payoff spread. In this case, note that a Taylor series expansion

44 EXPECTED UTILITY

around the mean return yields: u ′′

2 Similarly, a first-order Taylor series expansion of the certainty equivalent utility

around the mean return (since there are no uncertain elements associated with it) yields:

u (¯ R ) = u( ˆR − ρ) = u( ˆR) − ρu ′ (ˆ R ) Equating these two equations, we obtain the risk premium calculated earlier but

expressed in terms of the derivatives of the utility function and the return variance, or:

This risk premium can be used as well to define the index of risk behaviour suggested by Arrow and Pratt. In particular, Pratt defines an index of absolute risk aversion expressing the quantity by which a fair bet must be altered by a risk- averse decision maker in order to be indifferent between accepting and rejecting the bet. It is given by:

u ′′ (ˆ R )

ρ a (˜ R )= 2 =− σ / 2 u ′ (ˆ R )

Prudence and robustness

When a decision-maker’s expected utility is not (or is mildly) sensitive to other sources of risk, we may state that the expected utility is ‘robust’ or expresses

a prudent attitude by the decision-maker. A prudent investor, for example, who adopts a given utility function to reach an investment decision, expresses both his desire for returns and the prudence he hopes to assume in obtaining these returns, based on the functional form of the utility function he chooses. Thus, an investor with a precautionary (prudence) motive will tend to save more to hedge against the uncertainty that arises from additional sources of risk not accounted for by the expected utility of uncertain returns. This notion of prudence was first defined by Kimball (1990) and Eeckoudt and Kimball (1991) and is associated with the optimal utility level (measured by the relative marginal utilities invariance), which is, or could be, perturbed by other sources of risk. Explicitly, say that (w, ˜ R ) is the wealth of a person and the random payoff which results from some investment. If we use the expected marginal utility, then at the optimum investment decision:

Eu ′ ′ (w) if u (w + ˜R) > u ′ is convex Eu ′

(w + ˜R) < u ′ (w) if u ′ is concave The risk premium ψ that the investor pays for ‘prudence’ is thus the amount of

money required to maintain the marginal utility for sure at its optimal wealth

45 level. Or:

UTILITY AND RISK BEHAVIOUR

(w − ψ) = Eu ′−1 (w + ˜R) and ψ = w − u [Eu ′ (w + ˜R)] Proceeding as before (by using a first term Taylor series approximation on the

marginal utility), we find that:

1 u ′′′ (w) ψ = var( ˜ R ) −

2 u ′′ (w)

The square bracket term is called the degree of absolute prudence. For a risk- averse decision maker, the utility second-order derivative is negative (u ′ ≤ 0) and therefore prudence will be positive (negative) if the third derivative u ′′′ is positive (negative). Further, Kimball also shows that if the risk premium is positive and decreases with wealth w, then ψ > π . As a result, ψ − π is a premium an investor would pay to render the expected utility of an investment invariant under other sources of risks.

The terms expected utility, certainty equivalent, risk premium, Arrow–Pratt index of risk aversion and prudence are used profusely in insurance, economics and financial applications, as we shall see later on.

3.2.2 Expected utility bounds

In many instances, calculating the expected utility can be difficult and therefore bounds on the expected utility can be useful, providing a first approximation to the expected utility. For risk-averse investors with utility function u(.) and u ′′ (.) ≤ 0, the expected utility has a bound from above, known as Jensen’s inequality. It is given by:

Eu (˜ R ) ≤ u( ˆR) when u ′′ (.) ≤ 0 Eu (˜ R ) ≥ u( ˆR) when u ′′ (.) ≥ 0

and vice versa when it is the utility function of a risk-loving investor (i.e. u ′′ (.) ≥ 0). When rewards have known mean and known variance however, Willasen (1981, 1990) has shown that for risk-averse decision-makers, the expected utility can be bounded from below as well. In this case, we can bound the expected utility above and below by:

2 /ˆ R )/α 2 ;α 2 = E( ˜R ) The first bound is, of course, Jensen’s inequality, while the second inequality

u (ˆ R

2 ) ≥ Eu( ˜R) ≥ ˆR 2 u (α

provides a best lower bound. It is possible to improve on this estimate by using the best upper and lower Tchebycheff bounds on expected utility (Willasen, 1990). This inequality is particularly useful when we interpret and compare the effects of uncertainty on the choice of financial decisions, as we shall see in the example below. Further, it is also possible to replace these bounds by polynomials such that:

Eu (˜ R ) ≤ E A( ˜R); Eu( ˜R) ≥ E B( ˜R)

46 EXPECTED UTILITY

where A(.) and B(.) are polynomials of the third degree. To do so, second- and third-order Taylor series approximations are taken for the utility functions (using thereby the decision-makers’ prudence). For example, consider the following

portfolio prospect with a mean return of ˆ R and a variance σ 2 . Say that mean returns are also a function of the variance, expressing the return-risk substitution, with:

R = ˆR(σ ), ∂ ˆR(σ )/∂σ >, ˆR(0) = R ˆ f

where R f denotes the riskless rate of return. It means that the larger the returns uncertainty, the larger the required expected payoff. Using the Jensen and Willasen inequalities, we have for any portfolio, the following bounds on the expected utility:

u (ˆ R (σ )(1 + ν)) 2 σ

(˜ ) = ˆR +σ Thus, lower and upper bounds of the portfolio expected utility can be constructed

2 2 ≤ Eu( ˜R) ≤ u( ˆR(σ )); ν= 2 E R 1+ν

by maximizing (minimizing) the lower (upper) bounds over feasible ( ˆ R, σ ) port- folios. Further, if we set ˆ R=R f + λσ where λ is used as a measure for the price of risk (measured by the return standard deviation and as we shall see subsequently), we have equivalently the following bounds:

u ((R f + λσ )(1 + ν)) ≤ Eu( ˜R) ≤ u(R f + λσ )

The definition of an appropriate utility function is in general difficult. For this reason, other means are often used to express the desirability of certain outcomes. For example, some use targets, expressing the desire to maintain a given level of cash, deviations from which induce a dis-utility. Similarly, constraints (as they are defined by specific regulation) as well as probability constraints can also be used to express a behavioural attitude towards outcomes and risks. Such an approach has recently been found popular in financial circles that use ‘value at risk’ (VaR) as an efficiency criterion (see Chapter 10 in particular). Such assumptions re- garding decision-makers’ preferences are often used when we deal with practical problems.

3.2.3 Some utility functions

A utility function is selected because it represents the objective of an investor faced with uncertain payoffs and his attitude towards risk. It can also be selected for its analytical convenience. In general, such a selection is difficult and has therefore been one of the essential reasons in practice for seeking alternative approaches to decision making under uncertainty. Below we consider a number of analytical utility functions often used in theoretical and practical applications.

(1) The exponential utility function : u(w) = 1 − e −aw , a> 0 is a concave function. For this function, u ′ (w) = a e −aw >

0, u ′′

(w) = −a 2 e −aw < 0 while the index of absolute risk aversion R A is constant and given by: R A (w)

47 = −u ′′ / u ′

UTILITY AND RISK BEHAVIOUR

= a > 0. Further u 3 ′′′ (w) = a e −aw > 0 and therefore the degree of prudence is a while the prudence premium is, ψ = 1

2 a var( ˜ R ).

(2) The logarithmic utility function : u(w) = log(β + γ w), with β > 0, γ > 0 is strictly increasing and strictly concave and has a strictly decreasing absolute risk aversion. Note that, u ′

(w) = γ /(β + γ w) > 0, u 2 ′′ (w) = −γ / (β + γ w) 2 <

0 while, R A (w) = γ /(β + γ w) = u ′ (w) which is decreasing in wealth. (3) The quadratic utility function

: u(w) = w − ρw 2 is a concave function for all ρ ≥ 0 since u ′ = 1 − 2ρw, u ′ ≥ 0 → w ≤ 1/2ρ and u ′′ = −2ρ ≤ 0. As a

result, the Arrow–Pratt index of absolute risk aversion is

u ′′ [E(w)]

R A (w) = −

1 − 2ρw and the prudence is null (since the third derivative is null).

[E(w)]

3 2 2 : u(w) = w 2 − 2kw + (k +g )w, k 2 > 3g 2 is strictly increasing and strictly concave and has a decreasing absolute risk aversion

(4) The cubic utility function

2 if 0 ≤ w ≤ 1

3 k− 2 k 2 − 3g 2 . (5) The power utility function

: u(w) = (w − δ) β , 0 < β < 1 is strictly increas- ing and has a strictly absolute risk aversion on [δ, ∞) since: u ′

= β (w − δ) β −1 , and u β ′′ = −(1 − β)β (w − δ) −2 . The risk aversion index is thus, R

A (w) = −(1 − β) (w − δ) −1 .

(6) The HARA (hyperbolic absolute risk aversion) has a utility function given by:

while its first and second derivatives as well as its index of absolute risk aversion are given by:

b + aw/(1 − γ )

This utility function includes a number of special cases. In particular, when γ tends to one, we obtain the logarithmic utility.

3.2.4 Risk sharing

Two firms sign an agreement for a joint venture. A group of small firms organize

a cooperative for marketing their products. The major aerospace companies in the US west coast set up a major research facility for deep space travel. A group of 70 leading firms form a captive insurance firm in the Bahamas to insure their managers against kidnappings, and so on. These are all instances of risk sharing. Technically, when we combine together a number of (independent) participants and split among them a potential loss or gain, the resulting variance of the loss or gain for each of the participants will be smaller. Assuming that this variance

48 EXPECTED UTILITY

is an indicator of the ‘risk’, and if decision makers are assumed to be risk averse, then the more partners in the venture the smaller the individual risk sustained by each partner. Such arguments underly the foundations of insurance firms (that create the means for risk sharing), of major corporations based on numerous shareholders etc. Assuming that our preference is well defined by a utility function U (.), how would we know if it is worthwhile to share risk? Say that the net

benefits (profits less costs) of a venture is $ ˜ X whose probability distribution is p (˜ X ). If we do nothing, nothing is gained and nothing is lost and therefore the ‘value’ of doing nothing is U(0). The venture with its n participants, however, will have an expected utility EU ( ˜ X /n ). Thus, if sharing is worthwhile the expected utility of the venture ought to be greater than the utility of doing nothing! Or,

EU (˜ X /n ) > U (0).

Problems

(1) Formulate the problem of selecting the optimal size of a risk-sharing pool. (2) How much does a member of the pool benefit from participating in sharing.