Utility Theory CHAPTER Single period utility theory

Transcription

1 CHAPTER 7 Utility Theory 7.. Single period utility theory We wish to use a concept of utility that is able to deal with uncertainty. So we introduce the von Neumann Morgenstern utility function. The investor makes choices consistent with maximizing the expected value of the utility function. Utility is defined over either consumption or wealth, but can also be defined over a bundle of goods. von Neumann-Morgenstern utility functions can be developed axiomatically. Suppose an individual cares about consumption c and define a lottery L as a set of consumption outcomes.c ; : : : ; c m / and an associated set of probabilities. ; : : : ; m /. Let L be the space of lotteries. Elements of the set of consumption outcomes can be elements of L. Axiom (Completeness): For every pair of lotteries, L % L 2 or L - L 2. Axiom 2 (Reflexivity): For every lottery, L % L. Axiom 3 (Transitivity): If L % L 2 and L 2 % L 3, then L % L 3. Axiom 4 (Continuity): For any x; y; 2 L, and f 2 Œ0; W..x; y/;.;. /// %. ; /g f 2 Œ0; W..x; y/;.;. /// -. ; /g are closed sets. Axiom 5 (Independence): If.x; /.y; /, then..x; /;.; //..y; /;.; //. Axiom 6 (Dominance): Let L D..x ; x 2 /;. ; // and L 2 D..x ; x 2 /;. 2 ; 2 //. If x x 2, then L L 2, > 2. Theorem 27. Under Axioms 6, there exists a utility function u./ such that the choice of lottery by a decisionmaker corresponds to the lottery with the highest EŒu./. PROOF. Let L h be the best lottery in L and L w be the worst lottery in L. Define u.l h / D and u.l w / D 0. To find the utility of an intermediate lottery L i 2 L, set u.l i / D p i, where p i is defined by..l h ; L w /;.p i ; p i //.L i ; /. We need to check two things: Existence of such a p i : The two sets fp 2 Œ0; W..L h ; L w /;.p; p// % L i g 56

2 7.2. RISK AVERSION 57 and fp 2 Œ0; W..L h ; L w /;.p; p// - L i g are closed (Axiom 4) and nonempty (since L h is best and L w is worst). Every point in Œ0; is in one or the other of the two sets (Axiom ). Since the unit interval is connected, there must be some p in both, but this will be just the desired p i. Uniqueness of such a p i : Suppose p i and pi 0 both satisfy the above definition. Then one must be larger than the other. By Axiom 6, the lottery that gives a bigger probability of L h must be preferred to the one that gives a smaller probability. Hence p i is unique and u./ is well defined. We next check that u./ has the expected utility property. This can be shown as follows. By Axiom 5,..L x ; L y /;.p; p//...l h ; L w /;.p x ; p x //;..L h ; L w /;.p y ; p y ///;.p; p//;..l h ; L w /;.p x p C p y. p/;. p x /p C. p y /. p/ pxp py. p/ //..L h ; L w /;.pu.l x / C. p/u.l y /; pu.l x /. p/u.l y ///; so u..l x ; L y /;.p; p// D pu.l x / C. p/u.l y /; as required. Finally we verify that u./ is a utility function. Suppose that L x L y. Then u.l x / D p x 3 L x..l h ; L w /;.p x ; p x //; u.l y / D p y 3 L y..l h ; L w /;.p y ; p y //; so by Axiom 6, u.l x / > u.l y /. A von Neumann Morgenstern (vnm) utility function is identified up to a positive linear transformation, i.e. u./ and a C bu./ with b > 0 are equivalent. If the agent prefers more to less, u./ is increasing Risk aversion A decision-maker is said to be risk averse at a particular consumption level if she is not prepared to accept any actuarially fair, immediately resolved consumption gamble. For a given vnm utility function u.c/, the utility function is risk averse at c if u.c/ > EŒu.c C "/ for all " satisfying EŒ" D 0 and varœ" > 0. Theorem 28. A decisionmaker is globally risk averse iff her vnm utility function is strictly concave at all consumption levels.

3 as 7.2. RISK AVERSION 58 Absolute risk aversion. For a utility function u./, the absolute risk aversion measure is defined ARA.c/ D u00.c/ u 0.c/ : It measures the dollar amount that the decision-maker would pay to avoid an actually fair gamble (per unit of dollar gamble variance), i.e. EŒu.c C "/ u.c 2 ARA.c/ varœ" q where EŒ" D 0. To show this, use Taylor series expansions of both sides: EŒu.c/ C "u 0.c/ C 2 "2 u 00.c/ u.c/ /; ) 2 varœ" u00.c/ qu 0.c/; qu 0.c/ q D 2ARA.c/ varœ" : Relative risk aversion. For a utility function u./, the relative risk aversion measure is defined as RRA.c/ D u00.c/ u 0.c/ c: It measures the amount (as a fraction of consumption) that the decision-maker would pay to avoid an actuarially fair gamble (per unit of variance of the gamble expressed as a fraction of consumption), i.e. EŒu.c C e/ u.c where EŒ" D 0. To show this, notice that c 2RRA.c/ varœ"=c //; 2 crra.c/ varœ"=c D 2ARA.c/ varœ" : Comparison of ARA.c/ and RRA.c/. Consider: CARA: constant ARA.c/, so RRA.c/ is increasing in c. CRRA: constant RRA.c/, so ARA.c/ is decreasing in c. q Fix ", vary c, and consider the dollar payment the decision-maker would pay: q CARA.c/ D 2CARA varœ" a constant; q CRRA.c/ D CRRA 2 varœ" c which is decreasing in c: Fix "=c, vary c, and consider the payment as a fraction of hat the decision-maker would pay: q CARA.c/=c D 2cCARA varœ"=c which is increasing in c; q CRRA.c/=c D 2CRRA varœ"=c a constant:

4 7.3. HARA UTILITY FUNCTION HARA utility function The functional form of the hyperbolic absolute risk aversion (HARA) utility function is ' w ' w u.w/ D a Ow C b; ' 0; Ow > 0; a > 0; ' ' ' with w u 0.w/ D a ' u 00.w/ D ARA.w/ D RRA.w/ D a w ' u00.w/ u 0.w/ D u00.w/w u 0.w/ Ow ' > 0; Ow ' 2 < 0; w ' D w w ' Ow ; Ow : HARA specializes to a number of important utility specifications. Power utility. If Ow D 0, we obtain power utility. Defining D ' and setting a D. '/ C', ' > 0, then u.w/ D w ; > 0; w > 0; RRA D : Log utility. If Ow D 0, a D. '/ C' and b D =', letting '! 0, we obtain log utility as the limit since u.w/ D w' ; ' so using L Hôpital s rule, lim u.w/ D lim '!0 '!0 d d'.e' ln w / d d' ' D lim '!0 w ' ln w D ln w: Note that since u 0.w/ D, a power or log utility agent does not want to hold a wealth portfolio with a positive probability of a return of. Consequently, when a power or log utility agent has access to risky assets whose returns are multivariate normal and a riskless asset, the agent chooses to hold a portfolio 00% invested in the riskless asset. Quadratic utility. If ' D 2 and Ow < 0, we obtain quadratic utility. Setting a D and b D 0 and defining w D Ow, we get u.w/ D 2.w w/ 2 ; w < w :

5 Exponential utility. If ' Ow D ' ; 7.4. UTILITY THEORY WITH MULTIPLE PERIODS 60 ' ' a D ' ; b D 0; ' then letting '!, we obtain exponential utility since ' ' '. '/ w u.w/ D ' ' ' ' ' ' D ' w ' ' ' D and so lim u.w/ D lim '! '! w ' ; ' ' ' ' w ' D e w ; ARA.w/ D : ' 7.4. Utility theory with multiple periods Use a time separable (or additive) utility function " X max E ı u.c / ˇ F t ; D0 where u./ is a vnm utility function and ı is a patience parameter. Intertemporal elasticity of substitution. The intertemporal elasticity of substitution is the percentage change in consumption growth in response to a percentage change in one plus the interest rate. With power utility, the intertemporal elasticity of substitution for consumption is equal to the inverse of RRA. Here are two illustrations. Ignore uncertainty and consider a 2-period problem: c t max ;C C ı c tc The first-order condition is c t ctc ) ı. C R/c tc D 0 D ı. C R/ s.t. C D.W t ) C D ı =. C R/ = c t ctc ) ln D ln ı C ln. C R/: Now the elasticity is given by d ln C d ln. C R/ D ; /. C R/:

6 7.5. USING DYNAMIC PROGRAMMING TO SOLVE MULTIPERIOD PROBLEMS 6 as required. Why? d ln. C / d ln. C R/ D d ln C d C d ln. C R/ d. C R/ d C C ı ctc d. C R/ D d d. C R/=. C R/ : Consider a 2-period problem and assume consumption growth is log normal conditional on F t : c t max ;C C ı E tœ c tc s.t. C D.W t /. C R/: The first-order condition is c t ı. C R/ E t Œc tc D 0 ctc ) ı E t Œ. C R/ D ) E t Œe ln.ı/ ln.c= /Cln.CR/ D ) e ln.ı E t Œln.C = / Cln.CR/C0:5 2 t Œln.C= / D ) ln.ı/ E t Œln.C = / C ln. C R/ C 0:5t 2 Œln.C= / D 0 ) E t Œln.C = / D ln.ı/ C 0:5 ln. C R/ C t 2 Œln.C= / D 0: So the elasticity is given by de t Œln. C / D d ln. C R/ ; as required Using dynamic programming to solve multiperiod problems Finite-lived investor. Define " X T (27) V t.w t ; s t / D max E ı t u.c / j s t ; W t fc g T Dt ;f g T Dt subject to (28) W C D.W c /. 0.R C i N R f C / C Rf C /; c ; 2 F.s ; W / Dt for D t; : : : ; T, where s is the vector of state variables that affect the investor s happiness at time for D t; : : : ; T. We would like to convert the multiperiod problem into a single period problem. We can solve recursively working back from T. At time.t /, subject to V T.W T ; s T / D max c T ; T EŒu.c T / C ıu.w T / j s T ; W T W T D.W T c T /. 0T.R T i N R f T / C Rf T /; c T ; T 2 F.s T /:

7 7.5. USING DYNAMIC PROGRAMMING TO SOLVE MULTIPERIOD PROBLEMS 62 At T 2, ( " V T 2.W T 2 ; s T 2 / D max E u.c T 2 / C ıu.c T / C ı 2 u.w T / fc g T 2 ; ˇ f g T 2 subject to W C D.W c /. 0.R C i N R f C / C Rf C /; c ; 2 F.s / s T 2 W T 2 ) for D T 2; T. This time-.t 2/ value function can be written as: V T 2.W T 2 ; s T 2 / D 2 " s T E u.c T / C ıu.w T / ˇ W T u.c T 2 / C ı max s.t. W c T D.W T c T / max E T ; T c T 2 ;. 0T T 2.R T i N R f T / C Rf T /; 6 c 4 T ; T 2 F.s T / ˇ s T 2 W T 2 ƒ Ṽ T.W T ;s T / subject to W T D.W T 2 c T 2 /. 0T 2.R T i N R f T / C Rf t /; c T 2; T 2 2 F.s T 2 /: Thus, the time-.t 2/ value function becomes ( " V T 2.W T 2 ; s T 2 / D max E u.c T 2 C ıv T.W T ; s T / c T 2 ; T 2 ˇ subject to s T 2 W T 2 W T D.W T 2 c T 2 /. 0T 2.R T i N R f T / C Rf T /; c T 2; T 2 2 F.s T 2 /: ) This holds for any D t; : : : ; T, so (29) V.W ; s / D max feœu.c / C ıv C.W C ; s C / j s g c ; L.c ; IW ;s / subject to.28/w C D.W c /. 0.R C i N R f C / C Rf C / ; c ; 2 F.s /; V T./ D u./: PROOF. Fix fc gt Dt. Let fc gt ; f gt ; f gt Dt (30) V T.W T ; s T / D max fc g T R W C be the solution to " X T E as any arbitrary choice of c and for D t; : : : ; T ı.t / u.c / ˇ s T W T

8 7.5. USING DYNAMIC PROGRAMMING TO SOLVE MULTIPERIOD PROBLEMS 63 subject to (28) for D T ; : : : ; T. Let fc gt By definition " T X E ı.t / u.c ˇˇˇˇˇ / s T E Dt W T " T X ; f gt ı.t / u.c which means using the law of iterated expectations, " T X TX E ı t u.c ı.t! / u.c ˇˇˇˇˇ / E " T X Dt ı t u.c / C ıt t TX be any other arbitrary choice. / s t W t ˇ s T W T ı.t / u.c Since this holds for any fc gt Dt, it holds for fc gt Dt, the optimal solution to (27). Hence the solution to (30) is also the solution to (27) for D T ; : : : ; T, so dynamic programming works for u t D u. ; ; : : : ; s /: which is habit persistence. But does it work for u t D u. ; C ; : : : ; Cs /? It can be shown that if u./ is increasing and concave, then V.W ; s / is increasing and concave in W. Envelope condition. We can show that u 0.c.W ; s // D V ;W.W ; s /; where c.; / and.; / solve (29). (3) Now PROOF. The first-order condition for (29) wrt to c is given ; s /;.W ; s /I W ; s D 0 u 0.c.W ; s // ı EŒV C;W.W C ; s C R W C j s D 0: The first-order condition for (29) wrt to is given ; s /;.W ; s /I W ; s / D EŒV C;W.W C ; s C /.R C R f C i N / j s D 0: V ;W.W ; s / D dl.c.w ; s /;.W ; s /I W ; s / / C D ı EŒV C;W.W C ; s C /R W C j s : D0 /! ˇˇˇˇˇ s t W t :

9 So using (3) 7.6. CAMPBELL VICEIRA APPROXIMATE SOLUTION FOR THE MULTI-PERIOD AGENT S PROBLEM 64 u 0.c.W ; s // D V ;W.W ; s /: Infinite-lived investor. The problem at time t is " X V t.w t ; s t / D max E ı u.c / j s t D0 subject to (28) for D t; t C ; : : :. The problem at time t C is " X V tc.w tc ; s tc / D max E ı u.cc / j s tc D0 subject to (28) for D t C ; t C 2; : : :. The problem at t looks the same as the problem at t C, so if the solution to the right-hand side is finite, the Bellman equation becomes V.W ; s / D max feœu.c / C ıv.w C ; s C / j s g c ; subject to (28) for D t; t C ; : : : Campbell Viceira approximate solution for the multi-period agent s problem s.t. Consider a multi-period investor who solves the following problem: ( " ) X T V t.w t ; x t / D max ; E t ı t C ; C ; 2 F 8: fc g T Dt f g T Dt Dt (32) W C D.W C /R W;C and (33) R W;C D t.r ;C R f / C R f : Let r ;C D ln.r ;C /, r W;C D ln.r W;C /, r f ln.w C /. The return generating process satisfies: D ln.r f /, c C D ln.c C / and w C D E Œr ;C r f D x ; x C D C '.x / C C ; where 0 < ' < and are constants and C is normal i.i.d. white noise with zero mean and standard deviation of. The conditional mean of.e Œr ;C r f / is given by the state variable x which evolves through time as an AR() process. Moreover, r ;C is conditionally homoskedastic. Letting u C D r ;C E Œr ;C, assume that var Œu C D 2 u ; cov Œu C ; C D u :

10 7.6. CAMPBELL VICEIRA APPROXIMATE SOLUTION FOR THE MULTI-PERIOD AGENT S PROBLEM 65 The Euler equation for R i;tc that comes out of the first order condition for this investor s problem for i D, W and f says (34) D E t ı CtC C t R i;tc : Let V x;y;t cov t Œx tc ; y tc. Assume r ;C and c C ln C C are multivariate normal conditional on time- information. Then (34) implies that for i D W, E t Œr ;tc D 0 D ln ı E t Œ C C E t Œr ;tc C t Œ C C 2 t Œr ;tc ln ı C E t Œ C t Œ C r f D ln ı C ı E t C t Œ C ; 2 t Œ C ; r ;tc /; 2 2 t Œr ;tc C t Œ C ; r ;tc ; and so (35) E t Œr ;tc r f C 2 V 2 ;;t D V ;c;t: But c C and r ;C are not multivariate normal conditional on time- information even if r ;C is conditionally normal. Even if r ;tc and C are not multivariate normal conditional on the time-t information, it is possible to obtain (35) using log-linearizations. Let w C D w C w. We can use a log-linearization of the budget constraint (32) to obtain an approximate expression for w C that is linear in r W;C and.c w /. We log-linearize the budget constraint by a first order Taylor expansion of w t around Nc Nw. Letting D exp. Nc Nw/, (36) W tc D R W;tC.W t C t /; C t W tc =W t D R W;tC W t w tc D r W;tC C ln. exp. w t // exp. Nc Nw/ D r W;tC C ln. exp. Nc Nw// C exp. Nc Nw/. w t. Nc Nw// D r W;tC C. w t / C ln./ ln. / :

11 7.6. CAMPBELL VICEIRA APPROXIMATE SOLUTION FOR THE MULTI-PERIOD AGENT S PROBLEM 66 We can log-linearize the portfolio return equation (33) by a first order Taylor expansion of r W;tC with respect to r ;tc around zero and r f around zero: (37) (38) R W;tC D t.r ;tc R f / C R f ; r W;tC D ln. t exp.r ;tc / t exp.r f / C exp.r f // D ln. t exp.0/ t exp.0/ C exp.0// t exp.0/ C t exp.0/ t exp.0/ C exp.0/.r ;tc 0/. t/ exp.0/ C t exp.0/ t exp.0/ C exp.0/.r f 0/ D t.r ;tc r f / C r f : We can use the approximate expressions derived in (36), (38) and (35) to show that the following approximate relation holds: (39) t D E tœr ;tc r f C 2 2 u 2 u Combine (36) and (38) to get (40) w tc t.r ;tc r f / C r f C Using (40) and cov t Œr ;tc ; C (4) C D.C w tc /. w t / C w tc ; 2 u w tc :. w t / C k: we can show that V c;t can be written as a function of V ;c w;t, t and V ;;t : V ;c;t D cov t Œr ;tc ; C D cov t Œr ;tc ; C w tc cov t Œr ;tc ; w t C cov t Œr ;tc ; tr ;tc C. t/r f C. D V ;c w;t C tv ;;t : /. w t / C k Using (35), E t Œr ;tc r f C 2 V ;;t D.V ;c w;t C tv ;;t /; t D E tœr ;tc r f C 2 V ;;t V ;c w;t V ;;t V ;;t as required. Suppose that the investor s life ended at time-.t C/: i.e., t D T. How would the formula for t in (39) change? If the investor s life ended at time-.t C/, then C tc D W tc and so C w tc D 0 and V ;c w;t D 0. Consequently, the the single-period agent s risky-asset allocation, which is known as the myopic demand, is given by: t D E tœr ;tc r f C 2 V ;;t V ;;t

12 7.6. CAMPBELL VICEIRA APPROXIMATE SOLUTION FOR THE MULTI-PERIOD AGENT S PROBLEM 67 The difference between the the risky-asset allocations by a multi-period agent and a single-period V agent, which is given by ;c w;t V ;;t, is known as the hedging demand and converges to 0 as t converges to T. Suppose u D 0. Since x t is the only state variable for the agent s problem and recalling that the power utility problem is homogeneous in wealth, we know that C w tc only depends on x tc. Thus, if u D 0, then V ;c w;t D 0 and the hedging demand is zero. So the risky-asset return from t to t C needs to be F t -conditionally correlated with the state variable at t C, x tc, for the hedging demand to be non-zero. It is possible to obtain an explicit expression for t as a function of x t. For simplicity, suppose that the agent is infinitely lived: i.e., T D. It is possible to show that the solution for satisfies (42) w t D b 0 C b x t C b 2 x 2 t ; where b 0, b and b 2 are constants. We can use (39) and the result x 2 C E Œx 2 C D.2 C 2 / C.2. '/ C 2'x / C to show that the solution for t is linear in x t : t D a 0 C a x t : We can obtain expressions for a 0 and a in terms of, 2 u, u,, ', b and b 2. Taking (42) as given and noting that it follows that x 2 tc E t Œx 2 tc D.2 tc 2 / C.2. '/ C 2'x t/ tc ; V ;c w;t D cov t Œr ;tc ; C w tc D covœu tc ; b.x tc E t x tc / C b 2. 2 tc 2 C.2. '/ C 2'x t/ tc D covœu tc ; b tc C b 2.2. D.b C b 2.2. '/ C 2'x t // u ; '// C 2'x t / tc and so t D x C 2 u 2 u 2 D 2 2 u 2 u 2 u.b C b 2.2. '/ C 2'x t / u b 2 2. '/ u a0 C b 2 2' u u 2 Ÿu 2 x t a as required. It is worth noting that the analysis can be extended to: multiple risky assets.