ECON 581. Decision making under risk Instructor: Dmytro Hryshko 1 / 36
Outline Expected utility Risk aversion Certainty equivalence and risk premium The canonical portfolio allocation problem 2 / 36
Suggested readings for this lecture Danthine, J. P., Donaldson, J. (2005), Intermediate Financial Theory, Elsevier Academic Press. Chapters 3 and 4. [DD] Louis Eeckhoudt, Christian Gollier, and Harris Schlesinger. 2005. Economic and Financial Decisions under Risk. Chapter 1. http://press.princeton.edu/chapters/s7945.pdf 3 / 36
A bit of history Daniel Bernoulli, a Swiss mathematician, wrote in St. Petersburg a paper in 1738 titled Exposition of a new theory on the measurement of risk. It was translated and published in Econometrica in 1954. Uses 3 examples. Shows that two people facing the same lottery may value it differently. This contrasts with previous beliefs that the value of a lottery should equal its mathematical expectation and hence identical for all people, independent of their risk attitude. 4 / 36
St. Petersburg Paradox Consider the following game of chance: you pay a fixed fee to enter and then a fair coin is tossed repeatedly until a tail appears, ending the game. The pot starts at 1 dollar and is doubled every time a head appears. You win whatever is in the pot after the game ends. First T(ail) appears Event Payoff Prob. in round 1 1 T 1 2 1 2 HT 2 4 1 3 HHT 4 8. n H. }{{.. H} T 2 n 1 1 2 n n 1 times 5 / 36
St. Petersburg Paradox, contd. What would be a fair price to pay for entering the game? E = 1 1 1 + }{{} 2 }{{} 4 =prob(t) =prob(ht) 2 + 1 4 +... =! }{{} 8 =prob(hht) Bernoulli solved this problem by computing the logarithm of earnings (kind of utility) E = 1 2 log 1 + 1 4 log 2 + 1 8 log 4 +... = log 2 n=1 n 1 2 n = log 2 < 6 / 36
Expected utility (EU) We wish to study rational choice among alternative risky prospects Utility is not a primitive in economic theory; what is assumed is that each consumer can value goods in terms of subjective preferences Preferences satisfy certain axioms (unquestioned truths ) of choice and can be represented by a utility function (see, e.g., D&D, Ch. 3) Theory developed by John von Neumann and Oskar Morgenstern (axiomatization and reinterpretation of what Daniel Bernoulli did in the 18th century to resolve the St. Petersburg Paradox) 7 / 36
Attitude towards risk Fair gamble: a gamble with an expected payoff of zero. An individual is risk-averse if she dislikes a fair gamble (she will not accept it for free) Similarly, she is risk-seeking/loving if she likes it and risk-neutral if she is indifferent Jensen s inequality: Similarly, function f is convex iff E[f( x)] f[e( x)]. function g is concave iff E[g( x)] g[e( x)]. Furthermore, if the convexity/concavity is strict and x is not degenerate, i.e. Prob[ x E( x)] > 0, then the inequalities are strict. 8 / 36
Risk aversion Proposition A decision maker with utility function u is locally (globally) risk-averse if and only if u is strictly concave at a given wealth level W (all wealth levels). Proof. Let x be a fair gamble. We have strict concavity iff Eu[W + x] < u[w + E( x)] = u(w ) by Jensen s inequality where W is a given wealth. The global case is for all W > 0. 9 / 36
Notes A lottery with a non-zero expected payoff can be decomposed into its expected payoff E x plus a fair (zero-mean) lottery x E x Thus, a risk averse individual prefers the expected outcome of the lottery rather than the lottery itself. Thus, for any lottery x it must be true that Eu(W + x) < u(w + E x) 10 / 36
(Modified) example from Bernoulii Sempronius owns goods at home worth a total of 4,000 ducats and in addition possesses 8,000 ducats worth of commodities in foreign countries from where they can only be transported by sea. However, our daily experience teaches us that of two ships one perishes. Lottery x = (4000, 1/2; 12000, 1/2). E z = 1 2 4000 + 1 2 12000 = 8000 ducats. If Sempronius is risk averse (utility is strictly concave) 1 2 u(12, 000) + 1 u(4, 000) }{{ 2 } =Eu( x) < u(8, 000) }{{} =u[e( x)] 11 / 36
Risk-averse preferences 1.2. Definition and Characterization of Risk Aversion 7 f e utility c d a 4000 8000 12 000 wealth Figure 1.1. Measuring the expecting utility of final wealth (4000, 2 1 ; 12000, 2 1 ). 1.2 Definition and Characterization of Risk Aversion Source: Louis Eeckhoudt, Christian Gollier, and Harris Schlesinger. 2005. WeEconomic assume that the and decision Financial maker lives Decisions for only one under period, Risk. which implies that he immediately uses all his final wealth to purchase and to consume goods and services. c = elater ed, in this c = book, a + webc will = disentangle a + ed wealth c = and 1/2(a consumption + e). by allowing the agent 12 / 36
Risk premium and certainty equivalent A risk-averse agent dislikes zero-mean risks However, a risk-averse agent may like risky lotteries if the expected payoffs that they yield are large enough (e.g., risk-averse investors may want to purchase risky assets if their expected returns exceed the risk-free rate) One way to measure the degree of risk aversion of an agent is to ask how much she is ready to pay to get rid of a zero-mean risk x 13 / 36
Definition Risk premium Π is defined by E[u(W + x)] = u(w Π). It is the amount of money which the individual is willing to pay to avoid gamble x. 14 / 36
Risk premium for mean-zero risk 1.3. Risk Premium and Certainty Equivalent 11 utility Π 4000 8000 Π 8000 wealth 12 000 Figure 1.2. Measuring the risk premium P of risk ( 4000, 1 2 ; 4000, 1 2 ) when initial wealth is w = 8000. that E z = 0. Using a second-order and a first-order Taylor approximation for the Source: Louis Eeckhoudt, Christian Gollier, and Harris Schlesinger. left-hand side and the right-hand side of equation (1.4), respectively, we obtain that 2005. Economic and Financial Decisions under Risk. 15 / 36
When risk x has an expectation that differs from zero, we usually use the concept of the certainty equivalent. 16 / 36
Definition Certainty equivalent (CE) of gamble x is defined by E[u(W + x)] = u(w + CE). It is the sure increase in wealth which gives the individual the same welfare as having to bear risk x. Clearly, CE = Π for mean-zero gambles. If risk-averse, CE < E( x). 17 / 36
Bernoulli s example, contd. Assume the utility function is quadratic. CE is defined from 1 1 4000 + 12000 = 4000 + CE 2 2 CE 3, 464 < E[ x] = 4000, where x = 0 with prob. 1/2 and 8000 with prob. 1/2 x = E[ x] + z where z = 4000 with prob. 1/2 and 4000 with prob. 1/2. The risk premium of playing gamble z with wealth W + E[ x] = 8000 will equal Π = 536. 1 1 4000 + 12000 = 8000 Π 2 2 18 / 36
Risk aversion in the small Pratt (1964, Econometrica) [and independently also Arrow] develops a local measure of risk aversion (hence in the small ). Let x in be a small fair gamble now. E[u(W + x)] = u(w Π). Take the first-order Taylor approximation of the RHS around 0 to obtain u(w Π) u(w ) u (W ) Π We take the second-order Taylor approximation of u on the LHS of the previous expression in terms of x around E( x) = 0 to obtain E[u(W + x)] u(w ) + u (W ) E( x) + 1 2 u (W ) σ 2 Equating, this results in the following approximation of the risk premium. 19 / 36
Absolute risk aversion and risk premium where Π σ2 2 A(W ) A(W ) = u (W ) u (W ) is the Arrow-Pratt function of absolute risk aversion. Two related measures of risk attitudes are the functions of risk tolerance and relative risk aversion, respectively: T (W ) = 1/A(W ), R(W ) = W A(W ) 20 / 36
Relative risk aversion Risk aversion driven by the fact that marginal utility decreases with wealth (extra dollar worth less to you if you re a millionaire than if you re an academic) A(W ) = [u (W )] /u (W ) is the rate of decay of marginal utility it says by what percentage ( marginal utility ) decreases if wealth is increased by one dollar = d dw [u (W )] u (W ) R(W ) (aka proportional risk aversion) is the elasticity of marginal utility w.r.t. wealth: R(W ) = W u (W ) u (W ) It says by what percentage ( marginal utility ) decreases if wealth is increased by one percent = du (W )/u (W ) dw/w. Note that A(W) and hence T(W) and R(W) are invariant to positive affine transformations of the VNM utility function 21 / 36
Risk aversion in the large Now we focus on any risk, i.e. not just in the small but also in the large. Proposition The following three conditions are equivalent: (A) Individual 1 is more risk-averse than individual 2, i.e. at any common wealth level, the former would be willing to pay a larger premium to avoid any risk than the latter. (B) For all W, A 1 (W ) A 2 (W ). (C) The utility function u 1 is a concave transformation of u 2 : φ : φ > 0 and φ 0 such that u 1 (W ) = φ[u 2 (W )] for all W. 22 / 36
Risk aversion in the large, cont d Proof. Let us first show that (B) (C). We have and u 1(W ) = φ [u 2 (W )] u 2(W ) u 1(W ) = φ [u 2 (W )] [u 2(W )] 2 + φ [u 2 (W )] u 2(W ), which implies A 1 (W ) = A 2 (W ) φ [u 2 (W ) φ [u 2 (W )] u 2(W ). Thus, A 1 uniformly greater than A 2 iff φ increasing concave 23 / 36
Risk aversion in the large, cont d Proof, cont d. By Π σ2 2 A(W ), (A) (B). Now it is enough to show that (C) (A). Any gamble x gives rise to risk premia Π 1 and Π 2 for individuals 1 and 2, respectively. u 1 (W Π 1 ) = Eu 1 (W + x) = Eφ[u 2 (W + x)] and denoting ỹ = u 2 (W + x), Jensen s inequality implies u 1 (W Π 1 ) = Eφ(ỹ) φ[e(ỹ)] = φ[u 2 (W Π 2 )] = u 1 (W Π 2 ) Because u 1 > 0, we must have Π 1 Π 2. 24 / 36
Bernoulli s example, contd. If u(w) = w, we showed that the risk premium Π of the lottery x = (0, 1 2 ; 4000, 1 2 ) at W = 4000 is 536. For this utility function, A u (w) = 1/4w 3/2 = 1 1/2w 1/2 2w. If, however, v(w) = log(w), A v (w) = 1 w 1 2w = A u(w) for all w risk premium for v (log-preferences) is higher than for u. The risk premium of x for log-preferences is about 1,071: 1 2 log(4, 000) + 1 2 log(12, 000) log(4, 000 + 4000 1, 071). 25 / 36
The portfolio allocation problem There are two assets in the economy a risky one with (net) return r and a riskfree one with return r f An investor with wealth W decides what amount a to invest in a risky asset Her final wealth is W =(1 + r f ) (W a) + (1 + r) a = W (1 + r f ) + a ( r r f ) She must solve the following portfolio problem: max V (a) where V (a) = Eu( W ) a Note that if a / [0, W ], then borrowing/short-selling takes place By selecting a, given her preferences, she structures the most desirable distribution of W subject to the investment opportunities 26 / 36
Portfolio allocation problem, cont d The FOC of the problem is { V (a ) = E u ( W } ) ( r r f ) = 0 We assume such a solution exists. Note that since u > 0, this implies Prob( r > r f ) (0, 1). The SOC is satisfied as V concave (effectively a weighted average of concave functions). Proposition Given u > 0 and u < 0, let a be the solution satisfying the expression above. Then sign a = sign[e( r) r f ]. 27 / 36
Portfolio allocation problem, cont d Proof. By u < 0, V (a) = E {u ( W } ) ( r r f ) 2 < 0 and so V is strictly decreasing. At a = 0, V (0) = Eu [W (1 + r f )]( r r f ) = u [W (1 + r f )]E( r r f ). For a to be positive it must be the case that V (0) > 0 or E[ r r f ] > 0. Therefore, to satisfy the FOC of the problem, it must be that sign a = sign { u [W (1 + r f )][E( r) r f ] }. Since u > 0, the proof is complete. The Proposition tells us that a > 0 iff E( r) > r f, i.e. if the risk premium (expected risky return over and above the riskfree rate) is positive, a risk-averse investor will buy at least a little bit of the risky asset. 28 / 36
Changes of risk aversion with wealth Increasing/constant/decreasing absolute risk aversion is abbreviated as IARA, CARA, DARA Increasing/constant/decreasing relative risk aversion is abbreviated as IRRA, CRRA, DRRA Regarding the absolute measure, Arrow postulates DARA, which sounds intuitively reasonable: In such a case, the risky asset is a normal good (as the individual s wealth increases, her demand for it rises; in contrast, it would be an inferior good under IARA) Regarding the relative measure, it seems less clear (IRRA/CRRA/DRRA imply that as wealth increases, the fraction of wealth allocated to the risky asset decreases/remains constant/increases) 29 / 36
Arrow Theorem Let a (W ) solve the portfolio problem we just considered and let η(w ) = W a da (W ). dw Then and < 0 (i.e. DARA) da W : A dw (W ) > 0 (W ) = 0 (i.e. CARA) da dw (W ) = 0 > 0 (i.e. IARA) da dw (W ) < 0 < 0 (i.e. DRRA) η(w ) > 1 W : R (W ) = 0 (i.e. CRRA) η(w ) = 1 > 0 (i.e. IRRA) η(w ) < 1 30 / 36
Arrow Theorem, cont d Proof. We prove the DARA case (CARA and IARA are similar). Define F (W, a) E { u [W (1 + r f ) + a( r r f )]( r r f ) } and note that the FOC implies that F (W, a ) = 0. We assume E( r) > r f, a > 0 (check proposition we saw today). By the Implicit Function Theorem, [ ] da F dw = W = F a (1 + r f )E u ( W )( r r f ) E [u ( W ] )( r r f ) 2 Since u < 0, sign da dw [u = sign E ( W ] )( r r f ). 31 / 36
Arrow Theorem, cont d Proof, cont d. When r r f, W W (1 + rf ) and so DARA implies A( W ) A[W (1 + r f )]. Similarly, when r < r f, W < W (1 + rf ) and hence A( W ) > A[W (1 + r f )]. In each case, multiplying both sides by u ( W )( r r f ) gives: when r r f and when r < r f. u ( W )( r r f ) A[W (1 + r f )]u ( W )( r r f ) u ( W )( r r f ) > A[W (1 + r f )]u ( W )( r r f ) 32 / 36
Arrow Theorem, cont d Proof, cont d. Remember that for the FOC to hold, we had Prob( r < r f ) (0, 1). So the two previous expressions imply: [ E u ( W ] [ )( r r f ) > A[W (1 + r f )]E u ( W ] )( r r f ). By the FOC, the expectation on the RHS is zero. Using the result to sign the expression proves the DARA case. Now we prove the IRRA case (CRRA and DRRA similar). Let us rewrite η = da W da dw a = 1 + dw W a a. 33 / 36
Arrow Theorem, cont d Substituting for da dw obtain η 1 = [ (1 + r f )W E from the IFT result and rearranging, we u ( W ] )( r r f ) a E [u ( W ] )( r r f ) 2 + a E [u ( W ] )( r r f ) 2 [ where the numerator can be rewritten as E u ( W ) W ] ( r r f ). Since u < 0, sign (η 1) = sign E [ u ( W ) W ] ( r r f ). Under IRRA, R( W ) R[W (1 + r f )] when r r f and R( W ) < R[W (1 + r f )] when r < r f. 34 / 36
Arrow Theorem, cont d Proof, cont d. Multiplying both sides by u ( W )( r r f ) gives u ( W ) W ( r r f ) R[W (1 + r f )]u ( W )( r r f ) when r r f and u ( W ) W ( r r f ) < R[W (1 + r f )]u ( W )( r r f ) when r < r f. Again, this and the FOC imply: [ E u ( W ) W ] ( r r f ) < 0, which gives η < 1. 35 / 36
Commonly used utility functions Quadratic utility: u(w ) = W b 2 W 2 for W < 1 b (implies increasing absolute risk aversion) CARA utility: u(w ) = e αw α where α is the coefficient of absolute risk aversion (implies increasing relative risk aversion) CRRA utility: { W 1 γ 1 γ for γ 1 u(w ) = log W for γ = 1 where γ is the coefficient of relative risk aversion (implies decreasing absolute risk aversion) All of the above examples are special cases of HARA (hyperbolic absolute risk aversion) utility, aka LRT (linear risk tolerance) utility 36 / 36