4:"""""""""""""" Webegin this chapter by developing two concepts that are

similar to absolute risk aversion: relative risk aversion and partial riskaversion. Once these concepts have been presented, we will discuss somereasonable assumptions that can be made concerning them. In Section 4.4, weput these concepts and assumptions to work by applying them to so me specificutility functions. Using certain utility functions, we reconstruct the 'meanvariance' and 'safety first' criteria presented in Chapter 2 as special cases of theexpected utility model.

Other measures of risk aversion andsome associated assumptions

8. Lexicographic, one would say in more scientific terms.9. As we will see in Chapter 15, a stock represents for its owner the right

to a random flow of revenue (dividends plus capital gains). Theexpected value of this lottery exceeds its risk premium and thus thesale price demanded by the stockholder is strictly positive.

10. In Chapter 5 we will be much more precise concerning this subject.11. These topics are discussed in more detail in Section 4.l.12. Its properties are analysed in detail in Chapter 4.

Allais M. (1953), Le comportement de l'homme rationnel devant le risque,critique des postulats et axiomes de l'ecole americaine, Econometrica,vol. 21, 503-546.

Arrow K. (1965), Aspects 0/ the theory 0/ risk-bearing, Yrjo JahnssonSaatio, Helsinki.

Cramer G. (1728), Letter to Nicolas Bernoulli. Republished inEconometrica, (1954), vol. 22, 33-35.

Demers F. and M. Demers (1990), Price uncertainty, the competitive firmand the dual theory of choice under risk, European Economic Review,vol. 34, 1181-1200.

Lancaster K. (1968), Mathematical Economics, Macmillan, New York. Seeespecially section R.8.5, pp. 331-334.

Machina M. (1987a), Expected utility hypothesis, in The New Palgrave. ADictionary 0/ Economics, J. Eatwell, M. Milgate and P. Newman(eds), Macmillan Press, London, vol. 2, 232-239.

Machina M. (1987b), Choice under uncertainty: problems solved andunsolved. The Journal 0/ Economic Perspectives, vol. 1,121-154.

Munier B. (1989), Portee et signification de l'reuvre de M. Allais, Revued'economie politique, vol. 99, 1-27.

Pratt J. (1964), Risk aversion in the sm all and in the large, Econometrica,vol. 32, 122-136.

Quiggin, J. (1982), A theory of anticipated utility, Journal 0/ EconomicBehavior and Organization, vol. 3, 323-343.

Richter M. (1959-1960), Cardinal utility, portfolio selection, andtaxation, Review 0/ Economic Studies, vol. 27, 152-166.

Sinn H. (1983), Economic Decisions under Uncertainty, North-Holland,Amsterdam.

Yaari M. (1987), The dual theory of choice under risk, Econometrica, vol.55, 95-116.

The alert reader will have noticed that all of the discussion inChapter 3 was in a specific context: that of lotteries that are additive withrespect to wealth. Now, as noted in Chapter 1, one can also imagine multi-plicative lotteries. The concept of relative risk aversion and its generalisation(partial risk aversion) were invented for this kind of situation.

If a decision-maker places a fortune Wo in an asset for which the rate of returny is random and if the risk is to be resolved quickly, then the final wealth isdefined by:

As a result, if the decision-maker adopts the expected utility criterion, thesituation is evaluated before resolution of the risk by E[U(wo(1 + y))].

Imagine now that someone offers to relieve our decision-maker of this(multiplicative) risk in return for a fraction of wal. To see whether the offeris advantageous, it must be compared with the fraction -rr' defined by:

In other words, if the decision-maker must give up exactly -rr' Wo (inmonetary units) there would be indifference between this outcome and simply

holding lottery y. C?f course, if there is an organisation (for example, aninsurance company) 10 the market that offers to take on this risk for a fractionthat is less than 7[' ~ then th~ decision-maker will accept quickly in order toenjoy an (ex ante) IOcrease 10 welfare.

To define relat~ve risk aversion, apply Pratt's method of approximation toeach side of Equa~lOn (4.1). To keep the notation simple (and also to avoidsome minor techOlcal problems that are not of fundamental interest), assumethat the lottery Y is 'actuarially fair', that is, that E(y) = O.

The first-order approximation of the left hand-side around Wo is:

U(wo(1-7[')) == U(wo) -7['woU'(wo) (4.2)

The second-order approximation of U(wo(1 + y)) is:

U(wo(1 + y)) == U(wo) + ywoU'(wo) + (y2w6/2)U"(wo)

Taking the expected value of both sides finally yields:

E[U(wo(l + y))] == U(wo) + (w6a2/2)U"(wo)

Because E(y) = 0 by assumption, E(y2) is the variance of y (a2). Equatingthese approximations of the two sides of Equation (4.1) yields:

U(wo) - 7['WoU'(wo) == U(wo) + (w6a2/2)U"(wo)

which implies that:

7[' == (f)~[ -woU"(wo)/U'(wo)] (4.3)

The fraction of fortune that an individual is prepared to give up in orderto avoid the amount of risk is a function of two elements:

(a) the quantity of risk as measured by a2;

(b) a psychological element reflecting the nature of the utility function andmeasured by Ar(wo) = - Wo U"(wo)/ U' (wo), a coefftcient that is calledthe degree of relative risk aversion.

The terms 'absolute aversion' and 'relative aversion' are tied to the natureof the lottery. Absolute risk aversion applies to additive lotteries that aree~pre~sed in ~onetary units while relative risk aversion applies to multi-phcatlve lottenes expressed in rates or fractions.

I~ is interesting to note an obvious connection between relative riskaverSIOn (A ) and ab I . k . (A) If . .A . r so ute ns aversIOn a. we return to the defiOltlOn of

a de~lved from Equation (3.15) and assurne that the additive lottery isahetuanally fair (E( x) = p, = 0, then Aa = - U"(wo)/ U' (wo). It is then obvioust at we can write:

Wf = Wo(1 + y) = Wo + ywo

If we agree to write x = ywo, then we also have:

Wf= Wo +xand this permits us to shift from the multiplicative case to the additive case.We note in addition that:

E(x) = woE(y) = 0

due to the assumption on E (y). The two lotteries x and y are thus actuariallyfair. Furthermore,

Var(x) = w6Var(y)

Although they have the same expected values (namely, zero), lotteries xand y have very different variances (unless wo = 1).

The definitions of the basic concepts imply:

U(wo - 7[) = E [U(wo + x)]= E [U(wo + wOy)]= U(wo(l - 7['))

If the multiplicative risk is sufftciently small for Pratt's approximation to bevalid, then it can be verified that expression (4.5) implies:

(f)a2(x)Aa = 7[

= W07['

= wo(f )a2(y)Ar

Since a2(x) i w6a2(y), this implies that Ar = woAa•All of these manipulations may have made the reader a bit dizzy, but we

only wished to show that there is a dose relationship between the differentresults. Furthermore, we hope to leave a message that is not always dear frommany artides: to every multiplicative lottery there corresponds an additivelottery. This explains the strong connection between absolute risk aversionand relative risk aversion on the one hand, and between 7[ and 7[' on the other.Thus, as we will argue again in Section 4.2, assumptions about the behaviourof one of these two concepts automatically implies assumptions about thebehaviour of the other.

Ar=woAa (4.4)

1 The relationship between the two approaches developed thus far is madec earer by the following. Begin with a multiplicative lottery y and note that

4.1.2 Partial risk aversion

In finance theory, the concept of relative risk aversion is weil known and quite

natural given that one is often confronted with random rates of return.Surprisin.gly, an i?tere~ting generalisation of relative risk aversion, namely,partial nsk averSIOn, IS practically unknown. Apparently this notion wasproposed independently by two pairs of researchers at about the same time.2

The basic idea is very simple. Let there be a given total wealth Wo composedof two elements. The first, w6, is completely certain, while the second Wo issubject to a multiplicative risk that is assumed to be actuarially fair. 0/ cou~sewo = w6 + wo.. .The initial. situation is characterised by E [U (w6 + wo( 1 + 51))] and themdlVldual coDSlders what fraction of w{{ would be willingly paid in order toavoid the risk and maintain welfare. The problem then is to determine thefraction 'Ir" that satisfies:

and a2( 51) = ~' This lottery includes extremely spectacular outcomes since ify = -1 is realised then all of the risk capital is lost, while if y = 1 is realisedthen the amount of risk capital doubles. The final wealth of the individual willthus fall between a minimum of 2 and a maximum of 18 if this lottery, thatis about to be played, is retained. The expected utility from this lottery is:

E[U(Wf)] = ~:: [2+8(1+y)]112(!)dy

By applying a change of variable technique, which is naturally suggestedby economic intuition since it makes the substitution:

U(w6 + wo(l - 'Ir")) = E [U(w6 + wo(l + 51))] (4.6)

It follows immediately from Equation (4.6) that 'Ir" includes 'Ir' as a specialcase. Speci~cally, if w6 = 0 so that wo = Wo (i.e. all wealth is subject to risk),then Equatlon (4.6) reduces to Equation (4.1) and 'Ir' = 'Ir".

In order to make 'Ir" explicit, we take approximations of both sides ofEquation (4.6) around wo. Using the by now well-known technique, we findthat:

one can deduce that:

E [U(Wf)] = (1/16) L8

w[12 dWf

= (1/16) [(~)WiI2Ji8 = 3.0641

Given the definition of 'Ir", it is now a matter of solving:

'Ir"=: (!)a2[-woU"(wo)/U'(wo)] (4.7)

where the expression .in square brackets is called partial risk aversion; it isdenoted Ap or sometlmes Ap(wo, wo) to indicate that it depends on total~ealth and its division between its risky and certain components. It ismteresting (an~ comforting) to note again that if Wo = wo, then Ap equals Ar.~urthermore, simple algebraic manipulation establishes a simple and intuitive~mk between Ap and Ar. Multiply the definition of Ap by Wo and then divideIt by Wo to write:

which, after the obvious manipulations, yields:

Ap = - (wo/wo)wo [U"(wo)/U' (wo)]

Using the definition of relative risk aversion then yields:

Ap(wo, wo) = (WO/Wo)Ar(WO) (4.8)

. Par:ial risk aversion is thus proportional to relative risk aversion and it ismterestmg to note that the factor of proportionality equals the fraction of totalwealth that is subject to risk.

This means that the individual is indifferent between giving up 7.64 percent of the risk capital to avoid the lottery or, conversely, retaining the lottery

and awaiting its result.We now compare this 'exact' result with its approximation in Equation

(4.7). It was established above that, for Cramer's utility function,_ U"/U' =!w and thus, with Wo = 10,

- U"(wo)/U' (wo) = {ö

'Ir"=: (!)(~K28o)=0.0666

This result is a very good approximation to the true value.To pursue this example, consider Cramer's neighbour, whom we will call

Dr Barrois. Imagine that Barrois has the same utility function and anticipatesthe same distribution of 51 but that the total fortune Wo = 10 is investeddifferently, namely, with w6 = 6 and wo = 4. A few calculations furnish thefollowing results:

rr" = 0.0343 (the exact value)'Ir" = 0.0333 (the approximate value)

So the approximation formula gives as a result:

Returning to Cramer's utility function U = w[ 12, assurne that an initial wealthof 10 is divided as folIows: 2 is 'invested' in a safe security and the balance(wo = 8) is 'invested' in a risky security with a random rate of return yrepresented by a uniform distribution on [-1,1], so f( y) =! while E(y) = 0

Notice that the approximation is even better than before. Of course therisk inherent in y has not changed, but since it affects a smaller fraction of thewealth, there is a smaller dispersion of possible values of Wf which, as wehave seen, improves the quality of the approximation.

We also point out that an intermediate step in deriving these resultsestablishes that, for Dr Barrois, E[U(wd] = 3.145, which is larger than thesimilar value for Cramer. This is not surprising. Given that the utility functionthat both scientists have in common exhibits risk aversion, it is to be expectedthat Dr Barrois will be more 'comfortable' in facing less risk than Cramer whilehaving the same expected nnal wealth. Specincally, E(Wf) = 10 for bothindividuals because E ( y) = 0; but Barrois has a sm aller fraction of wealthinvested in the risky asset. Thus the position is necessarily less risky and, sinceE (Wf) is the same, the risk-averse decision-maker prefers this. This simpleassertion will be formalised in Chapter 5 when we study the notion of 'greaterrisk' and in Chapter 9 we will show that no risk-averse decision-maker shouldbuy that risky security with a zero mean.

Hypothesis 4.1 An individual whose wealth increases fears a givenadditive lottery less (or at least not more), that is,

This assumption states that Aa, and thus the risk premium of any givenadditive risk, is a non-increasing function of wealth.3 In terms of risktolerance, Equation (4.9) means that, if the level of risk is unchanged, anincrease in Wo does not reduce the tolerance for this risk because Ta isinversely related to Aa• In some sense, the richer the decision-maker the betterable they are to handle a given additive risk.

4.2 Assumptions about the properties of the measures ofrisk aversion

4.2.2 The degree of relative risk aversion and changing wealth

We now ask the same question about multiplicative risk that we asked aboutadditive risk: given a multiplicative risk y, will an increase in wealth reducethe propensity of an individual to seil this risk? The ans wer requires anexamination of how Ar reacts to a change in Wo, as implied by Pratt'sapproximation Equation (4.3) which links 7r' and Ar. First, it is interesting torecall the unifying relationship between absolute and relative risk aversiondescribed by Equation (4.4), namely,

Ar(wo) = woAa(wo)

where the dependence of each of these coefficients on Wo is made dear. Bydifferentiating this expression with respect to Wo one obtains:

The three measures of risk aversion (Aa, Ar, Ap) that we havedenned overlap, as has been shown. It could not be otherwise because thedifference between the concepts is due not to the nature of the risk, but to theway that the risk is presented to the individual (additive or multiplicative).Under these conditions, it is not surprising that an assumption made about theproperties of one of the measures reduces the number of plausible assumptionsthat can be imposed on the behaviour of another of the measures. We will notignore these issues. The assumptions to be made will revolve around thefollowing question: what happens to the measures of risk aversion when,ceteris paribus, wealth increases? This kind of question is relevant forindividuals who are risk averse or, at most, are risk neutral.

4.2.1 The degree of absolute risk aversion and changing wealth

Long before the modern version of risk theory was weil developed, thereexisted in the business world a widely held idea that a nrm with a high er net:-V0rth could handle a given risk more successfully. Somewhat more precisely,It was said that an increase in wealth allows companies to retain their risk.This means that the risk premium of an additive risk must be a decreasingfunction of wealth, if it is to be the case that a wealthier individual has a higherpropensity to accept risk. Technically, this intuitive idea is captured byHypothesis 4.1.

Since Aa is non-negative, it is obvious that if Aa is increasing in Wo thenAr will be as weil. In other words, if there were increasing absolute risk aver-sion,4 it would not be consistent with decreasing relative risk aversion.Furthermore, if absolute risk aversion is constant (an assumption that isperfectly consistent with Hypothesis 4.1) then dAa/dwo is zero but dAr/dwois strictly positive. It follows that decreasing Ar is inconsistent with constantAa• The most difficult case (and certainly the most likely, according toHypothesis 4.1) is where absolute risk aversion is decreasing in wealth. In thiscase two opposite effects arise in Equation (4.10) and dAr/dwo can bepositive, negative or zero.

It is important to note the parallel between Equation (4.10) and itsequivalent in terms of the risk premium. Specincally, since:

7r' == (!)<?(y) [ - Wo U"(wo)/ u' (wo)] = (i wo)a2(x )Aa (wo)

where x = wOy is the corresponding additive risk, we have:

d7r' /dwo == (iwo)a2(x)(dAa/dwo) + (!)Aa[d(a2)(x)/wo) dwo](4.10a)

Similar reasoning about the ambiguity of the sign of d7f'''/dwo can be stated:the first term on the right-hand side of Equation (4.10a) represents the effecton the risk premium for a given additive risk. It is negative by Hypothesis 4.1.The second term, in contrast, is positive since a2(x )/wo = woa2(ji). Itrepresents the increase in the risk premium due to the implicit increase inadditive risk.

A deeply entrenched tradition in risk theory tilts the balance in favour ofa non-negative derivative as summarised in Hypothesis 4.2.

Hypothesis 4.2 If wealth increases, relative risk aversion does notdecrease, that is,

Since 71" and Ar are proportional for smaIl risks, Hypothesis 4.2 isequivalent to assuming that d7f"/dwo ;;:::0 for every lottery ji.

It is important to understand this assumption weIl. When Wo rises givena multiplicative lottery, there are two effects that appear clearly in thedecomposition of d7f"/dwo provided by Equation (4.10a):

(a) since Wo increases, the individual feels wealthier and if the riskremained constant their risk aversion would have to decrease due toHypothesis 4.1. This is the first term (which is negative) of the right-hand side of Equation (4.10a);

(b) however, since the lottery is multiplicative, the increase in Wo implies agreater risk for final wealth because a2(wf) = a2(ji )w6, and this tends toincrease aversion to the risk inherent in ji. This effect is captured bythe second term (which is positive) on the right-hand side of Equation(4.10a).

In imposing Hypothesis 4.2, we assurne that the second effect (the 'risk'effect) is never dominated by the first effect (the 'wealth' effect). One can moreeasily accept the assumption of increasing relative risk aversion in Wo byconsidering that when Wo increases in a multiplicative lottery, a2(wf) increasesmuch more quickly (to the second power, as we saw above), and thus the 'risk'effect quickly becomes very large. In light of the above discussion of the twoeffects, assuming that Ar is constant in Wo is equivalent to asserting that the'risk' effect exactly offsets the 'wealth' effect. This case is analysed in moredetail in Appendix 1.

Every increasing and concave utility function implies riskaversion (see Theorem 3.3). However, it would also be desirable for a utility

function to satisfy some apriori reasonable properties such as those aIludedto in Section 4.2. For this reason we consider various utility functions that arecommonly employed in risk theory and we judge them by various criteria.

This is a very popular utility function because it is easily manipulated and itprovides good intuition for several specific results from finance theory.Unfoftunately, it has a very undesirable property, as we shaIl see.

The quadratic utility function is given by:

,-:'"

from which it foIlows that U' = 1 - 2Wf and U" = - 2ß· e:--For U to be concave, it is sufficient for coefficient ß to be strictly posi-

tive.5 Note that for function U to be increasing everywhere in Wf, ß shouldnot be too large. Let WM be the largest value of Wf, then 1 - 2ßWM must bepositive if U' is positive everywhere in its domain of definition. This impliesthat ß must be less than !WM.

When ß is chosen to satisfy this legitimate constraint, one easily obtains:

and it is apparent that absolute risk aversion is increasing in wealth. Thisresult contradicts the seemingly reasonable assumption, Hypothesis 4.1.Hence, to accept the quadratic utility function is to assurne that the richer anindividual is the more they fear a given additive lottery. Since Aa is increasingin wealth given a quadratic utility, so is Ar, which is consistent withHypothesis 4.2. . .

The balance sheet for this utility function is thus rather ummpreSSlve,which is aIl the more regrettable because quadratic utility provides a justi-fication for the intuitive 'mean variance' criterion discussed in Chapter 2.SpecificaIly, we will show that if U is quadratic then individuals evaluate t~eirsituation exclusively by the mean and variance of final wealth. From Equatlon(4.12) we can derive:

V(wE) = E [U(wE)J = E [Wf - ßwrJ

= E(wE) - ß[(E(wE))2 + a2(wE)J

48 THE EVALUATION OF RISK SITUATIONS

. . I~ fOIl?WS.fro~ Equation (4.13) that if V is quadratic then the value of themdividual s sItuatIOn is a function of E(wE) and of q2(Wf) alone, as ispostulated by the 'mean variance' model. Furthermore, notice that:

Unfortunately, the utility function proposed in Equation (4.14) does notsatisfy Hypotheses 4.1 and 4.2. For example, consider the behaviour ofabsolute risk aversion. For Wf < tone obtains:

V' = 1 + 2k(t - wE) > 0V" = -2k

- V"/V' = 2k/(1 + 2k(t - wE))

from which it follows that the degree of risk aversion is strictly positive andgrowing for w < t. Above t it is zero because risk neutrality prevails. A similarsituation arises for Ar, which is increasing below t and zero above.

This corresponds to the situation where /I > 0 and fz < 0 and thus results inupward sloping indifference curves in mean variance space as long as ß> O.

The mediocre performance of the quadratic utility relative to Hypothesis4.1 on the one hand and its natural association with the me an variancecriterion on the other have often provided an argument against the latter. Forcompleteness we point out that this evaluation criterion can be justifiedwithout quadratic utility. For example, the mean variance criterion is adequateif Wf is normally distributed.6 More recently, in a very interesting article,J. Meyer (1987) showed that the criterion is appropriate for comparinglotteries that differ only by a location parameter and a spread parameter(which includes normal distributions as a special case).

In summary, we have reservations concerning the ability of a quadraticutility to validly represent risk-averse individuals. However, these are not suffi-cient to reject the me an variance model despite its association with quadraticutility because it can be justified in other ways.

We point out that the 'safety-first' criterion (see Chapter 2) also hasfoundations in expected utility that are not far from those underlying the meanvariance model. SpecificaIly, if one writes7:

4.3.2 The logarithmic utility function

We now consider a utility function that is famous because it was proposed byBernoulli hirnself to resolve the 'Saint Petersburg Paradox' by demonstratingthe inadequacy of the expected value criterion. This utility function is givenby:

V= Wf

V = Wf - k (t - wdif Wf ~ t

if Wf :::;;t and k > 0

V = In Wf

which implies Wf > 0 if the function is weIl defined. For additive lotteries thismeans that if the worst possible outcome, denoted Xm, is negative, then itmust be smaller in absolute value than w~.We thus implicitly assurne that thelottery can never result in a loss greater than the individual's initial wealth.This assumption is very reasonable in many cases, but it can nevertheless failto be satisfied (more and more frequently, by the way) for certain risks facedby firms when the notion of legalliability for damages is introduced.

For this utility function we have:

V' = 1/Wf > 0 (because Wf> 0

V" = -1/wfand hence Aa = 1/Wf is a decreasing function of wealth. Thus the logarithmicutility function satisfies Hypothesis 4.1. Furthermore, Ar = 1 is a non-decreasing function of wealth for wh ich the derivative is everywhere zero. Sothe logarithmic utility function also satisfies Hypothesis 4.2.

then one easily derives the safety-first criterion. To see this, it is sufficient tocalculate E [V(WE)] using this utility function, that is,

= C: wf/(wE) dWf- k Loo (t- wE)2f(wE) dWf

= E(wE) - k~-(t)

4.3.3 The power function

We have already seen a special case of this function in V = WI/2

. Here wegeneralise the special case to write:

V= sgn(ß)wß

from which it follows that V' = sgn( ß )ßwß - I. It is clear that this utilityfunction is increasing in wealth because sgn( ß)ß > O.

where the reader will have recognised t as the threshold above which thedecision-maker is risk neutral and below which the decision-maker is riskaverse.

Furthermore, U"=sgn(ß)ß(ß-1)wß-2, so risk aversion implies thatß< 1. Of course ß = 0.5 yields the utility function attributed to Cramer, whileß = 1 implies risk neutrality.

The above results imply that Aa(w) = (1 - ß)/w and Ar = 1 - ß, whichshows that if ß = 0 then the power function exhibits the same coefficients ofrisk aversion as does the logarithmic.

It is easily demonstrated that the power function satishes Hypotheses 4.1and 4.2.

and the resulting negative values of U sometimes baffle newcomers to risktheory. In reality, the sign of U has no particular signihcance. Indeed, as statedin Theorem 3.1, one can apply any increasing linear transformation to a utilityfunction without altering its properties. Thus, if one wishes to make Upositive, it suffices to add a sufficiently large constant - 1 is large enough ifWf is positive - to Equation (4.15) and the result will be a positive utilityfunction with the same interpretation as before. Although one should notworry about the sign of U, the sign of U' must be absolutely positive. Herethis is guaranteed when ß is positive because U' = ße - ßWf. Furthermore,U" = - ß2e-ßWf and thus Aa = ß.

We thus have, for the hrst time, a utility function with a constant degree of~ absolute risk aversion. It follows that Hypothesis 4.1 is satished. Hypothesis

4.2 is also satished because, with exponential utility, Ar(w) = ßw is anincreasing function of w. Nevertheless, since relative risk aversion increaseswithout bound in w, it can be the case for large values of Wf that 7f' exceeds1, which me ans that the individual is prepared to abandon with certainty morethan 100 per cent of the initial wealth rather than face an actuarially fair multi-plicative lottery (see Appendix 2 for an example). Since such a situation isdifficult to imagine in practice, the exponential function has an inherentweakness' in that in certain cases it generates 'abnormal' phenomena.

This weakness aside, the utility function has many interesting properties.For example, when the utility function is a negative exponential, it turns outthat individuals who maximise expected utility, given normally distributedhnal wealth, behave as if they were using the mean variance criterion. We willprove this result in Chapter 9 (see also Exercise 3.4). In addition, in the caseof exponential utility and a normal distribution, the Arrow-Pratt approxi-mation formula for the risk premium is exact, even for large risks. This is oneof the consequences of the derivations found in Appendix 2.

4.3.5 A generalisation and summary

It is interesting to note that practically all of the utility functions presented thusfar can be derived as special cases of a dass of functions known as 'hyperbolicabsolute risk aversion utility functions'. This generalisation is as follows:

U(wfl = 1 - "I [ aWf + b]'Y"I 1~'Y

where the domain of dehnition of the function is those values of Wf thatmake the expression in parentheses positive, that is, Wf satisfyingaWf + b (1 - "I) > O.

Applying the rules of differentiation gives us:

U' (wfl = a [aWf + b] 'Y - 1

1-"1

Aa = - U"/U' = al[t::~+ b]

The name given to this dass of utility functions is easily understood bynoticing that, as is the case with all hyperbolic functions, Wf appears linearlyin the denominator of Aa• Notice also that if one were interested in risktolerance then it could be written as:

which is linear in Wf. For this reason utility functions in this dass are alsocalled 'linear risk tolerance utility functions'.

The importance of the functional form in Equation (4.16) is that itfurnishes an interesting generalisation of all the utility functions so fardiscussed. It also provides a method of remembering them. To illustrate howEquation (4.16) is a generalisation, it is sufficient to compare the absolute riskaversion of each utility function with that found in Equation (4.17). As shownin T able 4.1, a specihc utility function corresponds to particular values of theparameters a, band "I.

It is unnecessary to comment in detail on Table 4.1. The middle column(Aa) lists the results obtained for the specihc utility functions. The lastcolumn lists the values that a, band "I must take on to generate the middlecolumn from Equation (4.17). Thus there is a perfect equivalence betweenthe general utility function with appropriate parameter values and thecorresponding specihc utility functions, since they yie1d the same value for the

Value of parameters

Quadratic

Logarithmlc

Power

Negative exponential

2ßI( I - 2ßWf) b = II/wf 0= I

(I - ß)/wf 0 = I - ß-ß 'Y=-oo

0=2ß

'Y=Ob=Oolb = ß

degree of absolute risk aversion and thus reflect the same psychologicalattitude of a decision-maker facing risk.

To the careful reader who would like to derive the utility functions (andnot their coefficients of absolute risk aversion) by substituting the parametersfrom the last column into Equation (4.16), we point out that this exercise isnot very interesting. It poses some interesting mathematical questions but itreveals no new economic intuition.

Finally, to condude this section, we point out that even the linear utilityfunction is a special ca se of Equation (4.17). It is derived by setting 'Y = 1. Inthis case, the denominator of Equation (4.16) is infinite and thus the value ofAa is zero. Since the formulation in Equation (4.17) can genera te linear orquadratic utility, it can, of course, also be used to construct the utility functionthat implies 'safety first' behaviour (see Chapter 2) which is a mixture ofquadratic utility up to a point and linear utility thereafter.

Our study of measures of risk aversion now permits us tointroduce two somewhat advanced subjects. The first concerns the comparisonof degrees of risk aversion and it allows an elegant transition into Chapter 5,which is dedicated to the measurement of the 'degree of risk'. The second dealswith the relationship between the asking price and the bid price of a lottery,a theme that was furtively approached previously.

4.4.1 The comparison of degrees of absolute risk aversion

The question can - as is often the case in microeconomics - arise in twodifferent contexts and yet have the same kind of answer .

The first context can be called 'cross-sectional', using the language ofeconometrics. In this context, one compares two decision-makers at the samepoint in time who are in every way similar but one: their attitude toward risk.

In the second context, one adopts the philosophy of time series. Onefocuses on an individual and asks how their risk aversion changes over timeif, for example, the initial wealth changes.

In wh at folIows, we will adopt the first point of view but, as was seen inthe discussion surrounding Hypothesis 4.1, this can be easily translated interms of the second approach.

How can one compare two decision-makers' attitudes toward risk at apoint in time? Can one make a precise comparison so as to be able to makestatements like 'A is 3.42 times as risk averse as B'?

It turns out that if two individuals have utility (or evaluation) functionsthat belong to the same family, then in so me cases one can develop precisecomparisons of the kind discussed below. We already saw one such ca se whenwe presented the linear version of the me an variance model of evaluation (seeChapter 2). Here is another that belongs to the dass of expected utility models.If the first decision-maker, A, is characterised by:

UA(wE) = _e-6.82wf

and if the second decision-maker, B, also has a negative exponential utilityfunction:

UB(wE) = _e-3.41wf

then one can say that A is twice as risk averse as B. Specifically, the degree ofabsolute risk aversion for A is 6.82 and for B it is 3.41. In this case it is thuspossible to assert without ambiguity that one decision-maker is more riskaverse than the other, independently of their wealth, and one can make theintensity of the difference precise.

Now consider two individuals characterised by the power functions

UA = wf/2 and UB = wf/3

Their degrees of absolute risk aversion are respectively !Wf and } Wf. If thetwo individuals have the same level of wealth, then B is more risk averse thanA and one can say, for a given level of wealth, that the ratio of their degreesof risk aversion is (} )!(!) = 1. B is thus '1.33 times' as risk averse as A whenthey have the same wealth. While for exponential utility comparison ofintensity was possible independently of the wealth of each individual, with thepower function we must be careful to make comparisons of individuals withthe same wealth. It is intuitively dear that the precise comparison in relativeterms of the degrees of risk aversion is very difficult if not impossible when thedecision-makers have utility functions that do not belong to the same family.The problem comes from the fact that absolute ri~ aversion is a local concept.At this higher level of generality comparison of the degrees of risk aversion canonly be made 'ordinally' by saying that one individual is more risk averse thananother at every level of wealth but without being able to express the intensityof the difference. This 'ordinal' definition was proposed by Pratt - once again!- in his remarkable artide of 1964. One says that the utility function L(Wf)exhibits more risk aversion than another utility function U(Wf) at every level

of wealth9 if L can be written as an increasing concave transformation of U,that is, if there exists a function k with k' > 0 and k" < 0 such that:

The transformation of one power function into anotherInitially, Investor U is a riskophobe and has utility function of U = wr /2. Aneighbour, Investor L, has a utility function that is a concave transformationof U - indeed apower function of U - namely,

L = k(U) = UI/3

from which it follows that k' = W U-2/3 > 0 and k" = (- ~)U-S/3 ~ O.Since L = UI/3 and U = wr/2, one can write L = wr/6 and, using the

results from the previous section (see T able 4.1),

A~(wr) = t Wf > A~(wr) = iWf

für all Wf> 0, which corresponds to the domain of the functions.We have thus reconstructed with the help of transformation k an example

that had been given previously to motivate the notion of an increasing andconcave transformation of U.

L' (wr) = k' [U(wr)] U' (wE)

L"(wr) = k" [U(wr)] [U' (wr)] 2 + k' [U(Wf)] U"(wE)

From the assumptions that have been imposed on k" and k' , and since U'is necessarily positive, the first term on the right-hand side of Equation (4.18)is strictly positive and thus:

The transformation of logarithmic ULet U = In Wf and consider L = k ( U) = UI/ 2

. It is easily shown that this trans-formation is increasing and concave, and thus A~(wr) must exceed A~(wr).Note that L can be written as L = [ln(wr)] 1/2 and thus the function L is onlydefined if In Wf> 0, that is, if Wf > 1.

To derive AL we calculate in turn:

which can be interpreted as saying that for all Wf the degree of absolute riskaversion is higher for L than for U. This is denoted by A~(wE) > A~(wr)where A~ is the degree of absolute risk aversion given the utility function Land A~ is the corresponding expression given U.

For brevity it is sometimes said that if the utility function L of anindividual is 'more concave' than the utility function U of another individual,then the first decision-maker exhibits more risk aversion than the second.

Since the above proof of this result is somewhat abstract, we illustrate itwith three examples.

L'(Wf) = W(1n wr)-1/2(lfwr)

L"(wr) = (-iwt)(1n wr) - (~wf)(1n wr)-3/2from which it follows that:

-L"(wr)fL'(wr) = (1fwr) + (iwr)(1n wr)-l

Recalling that U = In Wf implies Aa = 1fwf (see Table 4.1) and noting that1f(2wf In wr) is strictly positive for Wf> 1, one obtains as predicted:

A~ > A~

The comparison of linear U to concave LIf an individual is risk neutral, their utility function is linear (see Theorem 3.2)and can thus be written as10

:

If their neighbour has a utility function, L, that can be expressed byL = k(U) = U1I2, then k' = (0.5) U-1/2 > 0 and k" = (- 0.25) U-3/2 < O. ButL = UI/2 and U = Wf obviously imply that:

L = wr/2Our previous results show that A~ = 0 and Al. = i Wf > 0, (for Wf> 0,

which corresponds to the domain of definition of L) and thus we haveA~ > A~.

In some sense the change from risk neutrality to risk aversion implies aconcave transformation of an initially linear utility and thus a higher degreeof absolute risk aversion than initially observed, and this holds for all Wf.

4.4.2 Bid and asking prices o( a lottery

We briefly alluded to the difference between these two notions in Chapter 3and we will now consider this theme in more detail. Surprisingly, the differencebetween the two concepts is little analysed and yet it is not uninteresting. Thediscussion here owes much to the excellent article by I. La Vallee (1968) whichis, however, written for a mathematically sophisticated group. As usual webegin with two examples that we then generalise.

To derive the asking price of a lottery (pa) for an individual, one assurnesthat initially the decision-maker holds a lottery that is intended for disposal.Thus one passes from uncertainty toward certainty when calculating the

asking price. If, for example, U = wI12 with Wo = 5 and a lottery characterisedby:

x = -4

x = +4

P _1-4

P = ~

to define pa and

_e-Y(S)= _(~) e-y(l-Pb)_(~) e-y(9-Pb)

to determine Pb.Using the well-known property that ea + b = eaeb, and carrying out so me

obvious simplifications, one can rewrite these two equations as folIows:

e-sy e-YP" = 0) e-Y + W e-9ye-sy = eYPb[(~) e-Y - W e-9y]

and it folIows, from multiplying both sides of the first equation by eY- P., andcomparing the result with the second equation, that eYPh = eYP", whichimplies Pb = pa. In this case, with exponential utility characterised byconstant absolute risk aversion,

then the asking price is the solution to:

(5 + Pa)1/2 = 0)(1)112 + W(9)1I2 = 2.50

Ir follows that pa = 1.25, implying a risk premium equal to 0.75 smceE(x)=2.

To derive the bid price, one must consider the situation where initiallythe decision-maker does not own the lottery (certainty) but is consideringacquiring the lottery by paying a price, Pb, that will be disbursed in anycase independently of the realisation of X. The bid price implies a shift fromcertainty toward uncertainty and, in the ca se considered here, it is the solutionto:

Pb = paThe two examples just considered suggest that the relationship between

the bid price and the asking price of the same lottery is determined by thebehaviour of absolute risk aversion. This conclusion from the examples canbe generalised in the following theorem.

as is seen by applying the general definition proposed in Equation (3.7). Theleft-hand side represents the decision-maker's appraisal of the initial certainsituation. The right-hand side is an expected utility of final wealth composedof the 'receipts' Wo and x, and the 'expenditure' Pb that must be paidindependently of the outcome of X.

To begin, we prove that in general the asking price does not equal the bidprice. If it did one could substitute Pb = pa = 1.25 into (4.19) and verify theequality. Yet this is absolutely impossible since it would require taking thesquare root of a negative number (- 0.25). So it is immediately obvious fromEquation (4.19) that Pb cannot exceed unity.ll With Pb = 1,

(5)112 = 2.23607 > ~(0)1I2 + (m8)112 = 2.12132

So since the right-hand side is a decreasing function of Pb, there will exist avalue of Pb less than 1 that will achieve the equality of the two terms. Asearch procedure establishes that Pb = 0.8541. Observe also that with thepower utility function, for which absolute risk aversion is decreasing thelottery considered here exhibits the following relationship: '

0< Pb < Pa

d .~onsider now, in the context of the same lottery, the attitude of theheClslOn-maker characterised by the negative exponential utility function. We

ave, respectively,

-e-Y(s+p,,)= -(~) e-y(l)_(~) e-y(9)

Theorem 4./ If absolute risk aversion is decreasing then:

either 0 < Pb < paor 0> Pb > pa

If absolute risk aversion is increasing then:

either 0 < pa < Pbor 0> pa > Pb

Finally, in the case of constant absolute risk aversion Pb = pa. Afterproving the first part of the theorem, we will give an intuitive justification andpropose a few applications.

We consider the ca se where Aa is decreasing and we prove the result forpa > 0, that is, we assurne the lottery is favourably perceived by the risk-averse decision-maker. After the formal proof, we will provide some intuitionfor the result. The proof is somewhat peculiar. Thus we develop it in two stepsusing a somewhat cumbersome but precise notation.

Step I (If pa > 0 then Pb > 0). From the usual definitions

U[WO+Pa(WO,x)] =E[U(wo+x)] (la)

U(wo) = E [U(wo + x - Pb(WO,X))] (lb)

where the notation Pa(WO, x) and Pb(WO, x) re fleet that pa and Pb referto the prices calculated for an individual with initial wealth Wo who

owns (ar considers buying for Pb) lottery X. If pa is positive, it followsfrom U' > 0 that U(wo + Pa) > U(wo) and hence that E [U(wo + x)] >E [U(wo + X - pb(WO,X)]. This implies that Pb(WO,X) is also positive.Notice that Step 1 does not rely on the behaviour of Aa, which willplaya decisive role in Step 2.Step 2 By inspection of Equation (1b) - especially its right-hand side- we observe that it evaluates the expected utility of a lottery Xcombined with a certain (non-random) amount of wealth equal toWo - pb(WO,X). We can thus ask what would be the asking price for Xassociated with this wealth Wo - Pb(WO, x). Applying to this case thedefinition of the asking price we can write:

U[Wo - Pb(WO,X) + pa(WO - Pb(WO,X),x)]

= E [U(wo - Pb(WO,X) + x)]

Combining Equations (2a) and (lb) we can write:

Pb(WO, x) = Pa(WO - Pb(WO, x), x)

Since pa > 0 implies Pb > 0 from Step 1, if absolute risk aversion isdecreasing then we have:

because the richer an individual is the less the fear of lottery x and thehigher will be the price demanded for it. It fallows, of course, fromEquations (2b) and (2c) that:

Pb(WO,X) < pa(WO,x)

Ta obtain this result we assumed that pa is positive and that Aa isdecreasing in w.

The other parts of Theorem 4.1 are proven in exactly the same way.Readers can approach them theoretically or convince themselves by example.In the ca se where U = wl /2 and Wo = 5, consider a lottery y:

It can be seen that pa < 0 implies Pb < 0 and that, consistent withTheorem 4.1,0> Pb > pa.

We now turn to the intuition underlying Theorem 4.1. In same sense, itallows us to better grasp the nation of decreasing risk aversion. Specifically,the theorem essentially says this: an individual exhibiting decreasing Aa who

is endowed with Wo and a favourable lottery x, will demand a high er price toseil x than they would be willing to invest in order to buy it if there was onlywo. With decreasing Aa, the risk aversion, given only Wo, is relatively largerthan it is given Wo augmented by a favourable lottery, and the person willthus invest less to buy favourable lotteries ( Pb < Pa) when endowed with onlyWO. The assumption of decreasing absolute risk aversion implies that anindividual will demand a higher price to seil a favourable lottery that they ownthan they are prepared to invest in its bid if this is not the case.

Surprisingly, the concept of the bid price of a lottery and its relationshipwith the asking price have received little attention in the literat ure dedicatedto the economic theory of risk. We hope that these few pages will whet theappetite of researchers because we think that these concepts are importantsince they distinguish the strategies of risk reduction (sale of a lottery, purchaseof insurance) from strategies of risk acquisition (purchase of a lottery, financialinvestment, physical investment). Furthermore, we think that these conceptswould be useful in the study of markets for risk. Indeed, for a trade to bebeneficial to two parties (and thus realised) it must be that the (minimum)asking price demanded by the owner of the risk be less than the (maximum)bid price that the potential purchaser or purchasers are prepared to pay. Taour knowledge, the literature so far has little used the nations of asking priceand bid price.

These few reflections complete our lang study of the measures of riskaversion and of their properties. We now turn to a complementary subject: themeasurement, or at least the camparison, of levels of risk.

4.1 In a highly stimulating paper D. Bell (1988) has convincingly argued infavour of a utility function that combines linear and negative exponentialutility functions, that is:

Questions(a) Show that absolute risk aversion is decreasing.(b) Does Bell's utility function belang to the dass of hyperbolic utility

functions?(c) In the same fashion as in (a), show that the combination of two

exponentials,

The outcome + 6 is the increase in wealth if no accident (or illness) occurs andthe outcome - 3 is the reduction in wealth when an accident occurs. Hence+ 9 is the total opportunity cast of the accident.

First compute the traditional notions mentioned above (this should be noproblem). Then consider the following question: how much money would yoube willing to pay for a safety design that would reduce to zero the probabilityof an accident. If you denote this amount of money by r, you have to solve:

Show that V3 exhibits decreasing absolute risk aversion if VI andV2 do.Hint: See Pratt (1964), Equation (29).

4.2 Consider Bell's utility function defmed at the beginning of Exercise 4.1together with utility:

L(W ) = - e - e W,

ln(20 - r) = (0.25)ln 11 + (0.75)ln 20

since with the new safety system and the price paid for it you obtain (20 - r)with certainty.

Compare the values and the defmitions of pa, 7r and r. You shouldobserve that pa measures how much you would pay (or how much you wouldrequire) to get rid of the lottery while r indicates how much you are ready topay in order to obtain with certainty the best outcome of the lottery. Thenotion of WTP is also very dose to that of the risk premium which measureshow much one is willing to pay to obtain with certainty the expected value ofthe lottery. Hence WTP (as defined so far) and 7r are very dose notions thatdiffer only by the gap between the expected value of a binary lottery and itsbest outcome. As a result, WTP so far looks like a rather uninteresting notion.Its interest lies in the following (rather realistic) consideration. In practice, itis almost impossible to reduce to zero the prob ability of an accident. Hence,wonder now how much you would invest (r') in a system that would reducefrom 0.25 to 0.10 the probability of an accident.Hint: solve by successive approximations

Questions(a) Show that L is more concave than V.(b) More generally, define V3 == VI + V2, where V2 is more concave than

VI. Show that V3 is more concave than VI, and less concave than V2.

4.3 Return to the data of Exercises 3.1 and 3.2, and compute the various bidprices.

4.4 Now apply Exercise 4.3 to the data of Exercises 3.3 and 3.4. Are yousurprised that the bid price is always equal to the asking price?

4.5 Derive the analytical expression of the fraction r' of wealth that one isprepared to give up in order to avoid rate risk Si when V = sgn( ß) W~, ß ~ 1.Show that this expression is larger than - E (ji).

4.6 So far, we have strongly emphasised the certainty equivalent, the askingprice and the risk premium of a lottery. However, in many applied fields(health economics, safety at the working place, radiological risk management,etc.) people often refer to the willingness to pay (WTP) for the reduction inthe probability of an accident. The purpose of this exercise is to illustrate thisnotion and relate it to those you already know. Consider an individual withutility:

How do rand r' campare? Explain intuitively why one of them is necessarilylarger than the other.

p(x)

0.250.75

endowed with certain wealth equal to 14 and a binary lottery x representingthe change in wealth in the states of 'no accident' and of 'accident':

By selecting a few other values of the probability of an accident sm allerthan 0.25 and different from 0 or 0.10, complete Figure 4.1 that relates WTPto various levels of P when the initial probability of an accident amounts to0.25.

While the WTP notion adds something to what you already knew, youshould realise that it uses the same basic ingredients as the previous concepts:a utility function and the description of a lottery (as weil as of itstransformation) .Remark: A pioneering article in the field is that of Jones-Lee (1974). Noticethat this author discusses WTP in the more general framework of 'statedependent' utilities which include as a special case our discussion in this book.

1. Since the risk is expressed as arate, the premium must be expressed inthe same way, that is, as a fraction. While absolute risk aversionexpresses results in monetary units and is thus sensitive to changes inthe units of measurement, relative risk aversion is not affected by thechoice of monetary unit because it is expressed exclusively as apercentage.

2. The two pairs are C. Menezes and D. Hanson (1970) andR. Zeckhauser and E. Keeler (1970).

3. Another way to express this assumption is to assert that if a lottery isaccepted by an individual with wealth Wl then it will be accepted byan individual with wealth W> Wl (see Yaari (1969) and D. Bell (1988)for a nice extension).

4. We use the conditional clause deliberately, since this case is ruled outby Hypothesis 4.1.

5. Note that if ß = 0 then there is risk neutrality and the expected utilitycriterion reduces once again to the expected value criterion.

6. This will be examined in more detail when we study the capital assetpricing model (CAPM) in Chapter 15.

7. k = 0 would lead once again to risk neutrality.8. The analogous condition for multiplicative lotteries is Ym > -1.9. The converse result can be proved: if, for all Wf, absolute risk aversion

is high er for L than for U then L = k (U) with the properties of kstated below (see Appendix 3).

10. It is not necessary to use the more general form U = a + bWf becausethese utility functions are equivalent (see Theorem 3.1 or Remark 5 atthe end of Chapter 3).

11. If the positive outcome of the lottery is increased without changing thenegative outcome, it turns out nevertheless that the purchase price cannot be larger than 1. This upper bound on Pb may seem surprising at

first sight. However, it is a result of the specific character of the utilityfunction considered here. Specifically, U' approaches - 00 as W

approaches zero, and thus the decision-maker is relatively much moreconcerned by negative results than by positive results no matter howlarge they are.

Bell D. (1988), One switch utility functions and a measure of risk,Management Science, vol. 34, 1416-1424.

Jones-Lee M. (1974), The value of changes in the prob ability of death orinjury, journal 0/ Political Economy, vol. 99, 235-849.

La Vallee I. (1968), On cash equivalence and information evaluation indecisions under uncertainty, part I, basic theory, journal 0/ theAmerican Statistical Association, vol. 63, 252-275.

Menezes C. and D. Hanson (1970), On the theory of risk aversion,International Economic Review, vol. 11, 481-487.

Meyer J. (1987), Two-moment decision models and expected utility,American Economic Review, vol. 77, 421-430.

Pratt J. (1964), Risk aversion in the small and in the large, Econometrica,vol. 32, 122-136.

Yaari M. (1969), Some remarks on measures of risk aversion and on theiruse, journal 0/ Economic Theory, vol. 1,315-329.

Zeckhauser R. and E. Keeler (1970), Another type of risk aversion,Econometrica, vol. 38, 661-665.