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From risk to opportunity Lecture 11

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  1. From risk to opportunityLecture 11 John Hey and Carmen Pasca

  2. Lecture 11 Implications of EUT • Finding your (EU) utility function… • … two different ways. • Defining risk aversion and risk loving. • Defining two indices of risk aversion. • Some special utility functions with nice properties. • Examples of its use in economics: • The theory of the competitive firm facing price uncertainty. • The life-cycle savings problem under income risk.

  3. Lecture 11 Implications of EUT: Finding your (EU) utility function 1 • Finding your utility function over [x, X]. Here we are using x to denote the lower bound and X the upper bound of the interval over which we are going to find your (EU) utility function. • There are lots of ways to find it. Here is just one. • Put u(x)=0 and u(X)=1. • To find the utility value for some intermediate amount xianswer the question: “what probability ui in the gamble [X,ui;x,(1-ui)] makes you indifferent between that gamble and xi?”. • It immediately follows that u(xi) = ui. Repeat for lots of different values of xi. • Example, put x= €0 and X= €100. Suppose you are risk-averse and you are indifferent between €50 and the gamble [€100,0.75; €0,0.25] then for you u(€50) = 0.75. (Note EX=€75>€50.)

  4. Lecture 11 Implications of EUT: Finding your (EU) utility function 2 • Here is another way: interpolation. • Suppose that you have already found xa and xb such that, for you, u(xa)=a and u(xb)=b. • To find your utility value half-way in-between answer the question: “what amount of money x(a+b)/2 makes you indifferent between that amount and the 50-50 gamble between xa and xb; that is the gamble [xa,½;xb,½]?”. • It immediately follows that u(x(a+b)/2) = (a+b)/2. • Example, suppose a=0.5, xa=25; b=0.7, xb=49 and you are indifferent between €36 and a 50-50 gamble between €25 and €49 then for you u(36) = 0.6. • Note this latter gamble has expected value €37 – you are risk-averse (and your function is concave between €25 and €49).

  5. Lecture 11 Implications of EUT: Certainty Equivalent • For a given individual we define his or her certainty equivalent, CE, of some lottery/gamble as the amount of money, which, if received with certainty, the individual regards as the same as the lottery. • So u(CE) = Eu(X) where X is the amount received in the lottery, CE denotes the Certainty Equivalent and where u(.) is the individual’s utility function. • Example: lottery is 50:50 chance of €16 or €4. (Note that EX = 10.) Suppose u(x) = x0.5. • Then Eu(X) = 0.5u(16) + 0.5u(4) = 0.5(4)+0.5(2) = 3. • And hence the CE is given by u(CE)=3. Hence CE = 9.

  6. Lecture 11 Implications of EUT: Risk Premium • For a given individual we define his or her risk premium, RP, for some lottery/gamble as the amount of money he or she would pay to convert the lottery into its expected value. • So RP = EX – CE, where CE is the individual’s certainty equivalent for the gamble. • Example: lottery is 50:50 chance of €16 or €4. (Note that EX = €10.) Suppose u(x) = x0.5. • Then Eu(X) = 0.5u(16) + 0.5u(4) = 0.5(4)+0.5(2) = 3. • And hence the CE is given by u(CE)=3. That is CE = €9. • And so the RP = 10 – 9 = 1; the individual would pay up to €1 to exchange the lottery for the certainty of €10.

  7. Lecture 11 Implications of EUT: Risk aversion • We define a risk-averse person as one who (always) prefers a certainty to a risk with the same expected value. • So his or her certainty equivalent for some lottery is (always) less than the Expected Value of the lottery; the risk premium is always positive. • This implies that his or her utility function is (everywhere) concave. • Let us continue with the example where u(x) = √x = x0.5(concave) and where the lottery is a 50:50 chance of 16 or 4. What is the expected value of this lottery? 0.5(16) + 0.5(4) = 10.And the CE? 9. See the next slide.

  8. Lecture 11 Implications of EUT: Concave utility Gamble pays €4 with probability ½ and €16 with probability ½. Expected Value is €10 Certainty equivalent is €9 because u(9) = 3 = ½ u(4) + ½ u(16) = EU(X) Risk Premium = €1 = €10 - €9 = EX- CE

  9. Lecture 11 Implications of EUT: risk attitudes • An individual is everywhere risk-averse (-neutral, -loving)… • …if his or utility function is always concave (linear, convex) • … if his or her certainty equivalent for some risk is always less than (equal to, more than) the expected value of the risk. • … if he or she is always willing to pay a positive (a zero, a negative) amount to turn a risk into a certainty with the same expected value. • The degree of concavity (convexity) indicates the degree of risk-aversion (loving).

  10. Lecture 11 Implications of EUT: measuring risk attitude • The degree of concavity indicates the degree of risk aversion. • Concavity of a function is to do with its second derivative. • But as the function is unique only up to a linear transformation, it has to be divided by the first derivative. • Absolute risk aversion index = -u”(x)/u’(x) • Relative risk aversion index = -xu”(x)/u’(x)

  11. Lecture 11 Implications of EUT: CARA and CRRA • For one who has Constant Absolute Risk Aversion: • If we add some constant to all the outcomes of a gamble, the CE of that gamble rises by the same constant and hence the Risk Premium stays the same. • From -u”(x)/u’(x) = r we get u(x) is proportional to –e-rx • [unless r=0 in which case is proportional to x] • If X is N(μ,σ2) then Eu(X) proportional to –exp(-rμ+r2σ2/2). • For one who has Constant Relative Risk Aversion: • If we multiply by some constant to all the outcomes of a gamble, the CE of that gamble is multiplied by the same constant and hence the Risk Premium is multiplied by the same constant. • From -xu”(x)/u’(x) = r we get u(x) is proportional to x1-r • [unless r=1 in which case is proportional to ln(x)] • Note that the proportionality results from the fact that the utility function is unique only up to a linear transformation.

  12. Lecture 11 Implications of EUT: the perfectly competitive firm • Consider the perfectly competitive firm under output price uncertainty. • p, the price, is risky with known density function. • The cost function c(.) is known. • The firm wants to maximise the Expected Utility of profits = π = px – c(x) by its choice of x, the output. • Choose x to maximise Eu(π)=Eu[px – c(x)]. • FOC is that E{u’(π)[p-c’(x)]} = 0. • From this we can show c’(x) < Ep • Firm produces less under risk. • See Hey JD Uncertainty in Economics, Martin Robertson 1979 (now way out of print).

  13. Lecture 11 Implications of EUT: life-cycle savings • Life-cycle consumption/savings problem under income risk. • Objective to maximise u(C1)+ ρu(C2) + ρ2u(C3) + … subject to Wt+1 = R(Yt – Ct + Wt) for all t. • C, Y and W are consumption, income and wealth; ρand R are the discount rate and the rate of return (1 plus the rate of interest). • In general it can be shown that the optimal strategy is C* = a + b W, and that • b=(R-1)/R • So the marginal propensity to consume (out of wealth) depends only on the rate of interest/return. • When the utility function is CARA, with r the index of absolute risk aversion and the distribution of income is N(μ,σ2) it can also be shown (assuming r > 0) that • a = μ – ½r(R-1)σ2 –ln(Rρ)/[r(R-1] • So the intercept of the consumption function depends positively on the mean of the income distribution and negatively on the variance; also if Rρ < 1 then increases in r and in R both lead to decreases in the intercept. • Hey J D, “Optimal Consumption under Income Uncertainty”, Economic Letters, 5, 1980, 129-133.

  14. Lecture 11 Implications of EUT: Conclusions • The great joy of EUT is its elegance and tractability. • It is easy to find your (EU) utility function. • It is concave (linear, convex) where you are risk-averse (-neutral, -loving). • The degree of risk-aversion can be measured by the degree of concavity of the utility function (using either an absolute or a relative measure). • CARA and CRRA are to useful special cases… • … which lead to insightful results.

  15. Lecture 11 • Goodbye!

  16. Lecture 11 Implications of EUT