Use (solution to) stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research

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1 TOCATIC CONUMPTION-AVING MODE: CANONICA APPICATION EPTEMBER 4, 0 s APPICATION Use (solution to stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research Certainty-equivalent consumption Assuming αc Quadratic period-utility uc ( = γ c Risk-free asset returns Risky period- income (with arbitrary distribution Risk aversion Precautionary savings Introduction to asset pricing eptember 4, 0

2 CERTAINTY EQUIVAENCE Assume quadratic utility αc αc vc (, c = uc ( + uc ( = γc + γc Assume interest rate is not state contingent r = r = r = r risk-free interest rate Insert in definition of solution to intertemporal problem M M c + a = y + ( + r a c + a = y + ( + r a c + a = y + ( + r a = 0 = 0 = 0 [ u c + r ] Euler eqn often the key u'( c = E '( ( [( ( ] γ αc = E γ αc + r γ αc = q( γ αc ( + r + p( γ αc ( + r + ( p q( γ αc ( + r M M γ αc = ( + r q( γ αc + p( γ αc + ( p q( γ αc M ( c + ( p q c = ( + r γ α qc + p c + a = y + ( + r a 0 0 eptember 4, 0 3 =? CERTAINTY EQUIVAENCE Assume quadratic utility αc αc vc (, c = uc ( + uc ( = γc + γc Assume interest rate is not state contingent r = r = r = r risk-free interest rate Insert in definition of solution to intertemporal problem M M c + a = y + ( + r a c + a = y + ( + r a c + a = y + ( + r a = 0 = 0 = 0 [ u c + r ] Euler eqn often the key u'( c = E '( ( [ ] γ α γ α [( ( ] γ αc = E γ αc + r γ αc = q( γ αc ( + r + p( γ αc ( + r + ( p q( γ αc ( + r M M γ αc = ( + r q( γ αc + p( γ αc + ( p q( γ αc M ( c + ( p q c = ( + r γ α qc + p c = ( + r Ec c = r ( r Ec α + + c + a = y + ( + r a 0 0 eptember 4, 0 4 γ

3 CERTAINTY EQUIVAENCE If not concerned with state-contingent solutions for c solution to consumer problem is an asset position and expected consumption profile c, Ec ; a that satisfies ( Period- budget constraint in expectation Ec = (+ r a + E y Euler equation c = γ r ( r Ec α + + Period- budget constraint ( c + a = y + ( + r a 0 0 taking as given r; y, a0, r0 and the stochastic distribution G(. of y Optimal period- consumption γ r ( + r + r = + ( y ( + r0 a0 + Ey α + ( + r + ( + r + + ( + r c A B C eptember 4, 0 5 CERTAINTY EQUIVAENCE Optimal period- (current consumption ( ( c = A + B y + + r a + C E y 0 0 Depends only on the mean of risky future income, E y Independent of second- and higher-moments of risky future income Distribution function G(. of period- income y = y probability q y probability p y probability -p-q E y = y ( ( Var y = q y y + ( p q y y Certainty Equivalence Mean-preserving spreads of G(. do not affect optimal choice of c E.g., (p =, q = 0 Period- income has no risk But c is identical s (period- savings is identical eptember 4, 0 6 3

4 CERTAINTY EQUIVAENCE A benchmark result in intertemporal consumption theory Result depends on Quadratic utility Riskless (aka non-state-contingent asset returns Only source of risk is income risk Only version of the intertemporal consumption model with analytical solution trong implication: risk about future (income does not affect current consumption and savings decisions Intuitively plausible? Empirically relevant? Probably not but why not? Model does feature both Income risk (Var y > 0 Risk averse utility with respect to consumption need to define formally eptember 4, 0 7 s APPICATION Use (solution to stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research Certainty-equivalent consumption Assuming αc Quadratic period-utility uc ( = γ c Risk-free asset returns Risky period- income (with arbitrary distribution Risk aversion Precautionary savings Introduction to asset pricing eptember 4, 0 8 4

5 Macro/Finance Fundamentals RIK AVERION Illustrate with simple static example Utility function u(c, with u (. > 0 and u (. < 0 Two possible consumption outcomes c with probability η c with probability -η Expected consumption is c = ηc + (-ηc eptember 4, 0 9 Macro/Finance Fundamentals RIK AVERION ow to measure? u(c u(cavg u(c c cavg = 0.5*(c+c c eptember 4, 0 0 5

6 Macro/Finance Fundamentals RIK AVERION Illustrate with simple static example Utility function u(c, with u (. > 0 and u (. < 0 Two possible consumption outcomes c with probability η c with probability -η Expected consumption is c = ηc + (-ηc Definition: an individual is risk averse (with respect to consumption risk if uc ( > ηuc ( + ( η uc ( JENEN INEQUAITY Risk aversion A preference for certain (deterministic outcomes to risky (stochastic outcomes Embodied in strictly concave utility ow to measure risk aversion? Need to capture something about concavity of utility eptember 4, 0 Macro/Finance Fundamentals RIK AVERION ow to measure? u(c u(cavg u(c c cavg = 0.5*(c+c c eptember 4, 0 6

7 Macro/Finance Fundamentals RIK AVERION ow to measure? u(c u(cavg u(c c cavg = 0.5*(c+c c A candidate measure: -u (c But not invariant to positive linear transformations of u(. even though implied choices are invariant to any monotonically increasing transformation of u(. eptember 4, 0 3 Macro/Finance Fundamentals RIK AVERION Arrow-Pratt coefficient of absolute risk aversion (ARA u ( c Controls for linear ARA( c u transformations of u(. ( c ARA(c gets at idea of risk aversion in level gains or losses of c from E(c Increasing ARA: ARA (c > 0 Decreasing ARA: ARA (c < 0 Most empirically-relevant case Richer people can afford to take a chance eptember 4, 0 4 7

8 Macro/Finance Fundamentals RIK AVERION Arrow-Pratt coefficient of absolute risk aversion (ARA u ( c Controls for linear ARA( c u transformations of u(. ( c ARA(c gets at idea of risk aversion in level gains or losses of c from E(c Increasing ARA: ARA (c > 0 Decreasing ARA: ARA (c < 0 Most empirically-relevant case Richer people can afford to take a chance Perhaps also useful to have measure of risk aversion in percentage gains or losses of c from E(c Relative risk aversion (RRA cu ( c RRA( c ( = c ARA( c Adusts for level of u ( c consumption/wealth Controlling for income/consumption, richer people cannot afford to take a chance anymore than anyone else eptember 4, 0 5 Macro/Finance Fundamentals RIK AVERION CRRA σ σ c c vc (, c = + σ σ σ > 0 uc ( uc ( Continuing to assume utility is additively-separable over time Attitude of consumers toward smoothing consumption between time periods IE = /σ Attitude of consumers toward risky outcomes within a given time period cu ''( c RRA( c = =... u'( c u''( c ARA( c = =... u'( c eptember 4, 0 6 8

9 Macro/Finance Fundamentals RIK AVERION CRRA σ σ c c vc (, c = + σ σ σ > 0 uc ( uc ( Continuing to assume utility is additively-separable over time Attitude of consumers toward smoothing consumption between time periods IE = /σ Attitude of consumers toward risky outcomes within a given time period cu ''( c RRA( c = =... u'( c u''( c ARA( c = =... u'( c CRRA utility: σ governs both intertemporal attitudes and intratemporal (relative risk attitudes! Inverses of each other!! Must/should IE and RRA be so directly related in reality? Not at all Epstein-Zin (EZ utility function disentangles the two concepts eptember 4, 0 7 s APPICATION Use (solution to stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research Certainty-equivalent consumption Assuming αc Quadratic period-utility uc ( = γ c Risk-free asset returns Risky period- income (with arbitrary distribution Risk aversion Precautionary savings Introduction to asset pricing eptember 4, 0 8 9

10 PRECAUTIONARY AVING Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most important assumption: quadratic utility Other necessary assumptions Non-state-contingent asset returns Future income the only source of risk eptember 4, 0 9 PRECAUTIONARY AVING Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most important assumption: quadratic utility αc αc Risk aversion (within period with vc (, c = uc ( + uc ( = γc + γc? Obviously = 0! (whether RRA or ARA o why certainty equivalence? i.e., why does future income risk not matter for current choices? Euler eqn often the key = E [ u c + r ] γ αc = E [( γ αc ( + r ] u'( c '( ( γ αc = q( γ αc ( + r + p( γ αc ( + r + ( p q( γ αc ( + r M M γ αc = ( + r q( γ αc + p( γ αc + ( p q( γ αc M ( c + ( p q c = ( + r γ α qc + p [ ] γ α γ α c = ( + r Ec c = r ( r Ec α + + eptember 4, 0 0 γ 0

11 PRECAUTIONARY AVING Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most important assumption: quadratic utility αc αc Risk aversion (within period with vc (, c = uc ( + uc ( = γc + γc? Obviously = 0! (whether RRA or ARA o why certainty equivalence? i.e., why does future income risk not matter for current choices? Euler eqn often the key = E [ u c + r ] γ αc = E [( γ αc ( + r ] u'( c '( ( γ αc = q( γ αc ( + r + p( γ αc ( + r + ( p q( γ αc ( + r M M γ αc = ( + r q( γ αc + p( γ αc + ( p q( γ αc M ( c + ( p q c = ( + r γ α qc + p Because of linear marginal utility!!! [ ] γ α γ α c = ( + r Ec c = r ( r Ec α + + eptember 4, 0 γ PRECAUTIONARY AVING Certainty-equivalent consumption Current consumption depends only on the mean of future risky income Most important assumption: quadratic utility αc αc Risk aversion (within period with vc (, c = uc ( + uc ( = γc + γc? Obviously = 0! (whether RRA or ARA o why certainty equivalence? Marginal utility function of order one (or lower implies risk on future income doesn t matter for current consumption Contrapositve Risk on future income matters for current consumption implies marginal utility function must be strictly convex u (c > 0 necessary for breaking certainty-equivalence result (Given u (. > 0 and u (. < 0 u (. > 0 u (. increasing in c u (. decreasing less quickly as c Not satisfied by quadratic utility eptember 4, 0

12 PRECAUTIONARY AVING Assume utility with u (c > 0 vc (, c = uc ( + uc ( Assume interest rate is not state contingent r = r = r = r risk-free interest rate eptember 4, 0 3 PRECAUTIONARY AVING Assume utility with u (c > 0 vc (, c = uc ( + uc ( Assume interest rate is not state contingent r = r = r = r risk-free interest rate Insert in definition of solution to intertemporal problem M M c + a = y + ( + r a c + a = y + ( + r a c + a = y + ( + r a = 0 = 0 = 0 [ u c + r ] Euler eqn often the key u'( c = E '( ( [ '( ] u' ( c = (+ r E u c M u'( c = ( + r qu'( c + pu'( c + ( p q u'( c c + a = y + ( + r a 0 0 = E c, so none of the subsequent steps with quadratic u(. follow u (c > 0 current consumption depends on distribution G(. of future risk i.e., on first- and (in principle all higher-order moments of G(. eptember 4, 0 4

13 PRECAUTIONARY AVING u (c > 0 current consumption depends on distribution G(. of future risk Is optimal c larger or smaller than certainty-equivalent c? For a given G(. eptember 4, 0 5 PRECAUTIONARY AVING u (c > 0 current consumption depends on distribution G(. of future risk Optimal c is smaller than certainty-equivalent c Proof: Implication: optimal s is larger than certainty-equivalent s Precautionary avings Risk about the future induces prudent (cautious choices in the present Desire to build up a buffer stock of assets to ensure c does not fall too low in future Risk aversion a necessary, but not sufficient, feature of preferences trictly convex marginal utility the key feature of preferences Classic papers: Kimball (990 Econometrica, andmo (970 Review of Economic tudies ow to measure precautionary savings motive? Need to capture something about convexity of marginal utility Kimball (990 provides clever insight eptember 4, 0 6 3

14 PRECAUTIONARY AVING ow to measure? c c avg = 0.5(c + c c c c avg = 0.5(c + c c A candidate measure: u (c Analogy with measures of risk aversion eptember 4, 0 7 PRECAUTIONARY AVING ow to measure? u'(c u'(c 0.5*( u (c + u (c u'(cavg u'(c u'(cavg u'(c Kimball (990: Define v(c = -u (c. Then can apply standard theory of risk aversion to v(c! -u'(c -u'(cavg -u'(c -u'(c -u'(cavg -0.5*( u (c + u (c -u'(c eptember 4, 0 8 4

15 PRECAUTIONARY AVING Coefficient of absolute prudence: Coefficient of relative prudence: u'''( c u''( c cu '''( c u''( c Measures of the sensitivity of optimal choice to risk Governed by marginal utility function ARA and RRA measure the sensitivity of welfare to risk Governed by the utility function eptember 4, 0 9 PRECAUTIONARY AVING Coefficient of absolute prudence: Coefficient of relative prudence: u'''( c u''( c cu '''( c u''( c Measures of the sensitivity of optimal choice to risk Governed by marginal utility function ARA and RRA measure the sensitivity of welfare to risk Governed by the utility function CRRA utility u'''( c σ + = σ u''( c c c uc ( = σ cu '''( c = σ + u''( c Displays constant relative prudence Displays constant relative risk aversion Absolute prudence Relative prudence eptember 4,

16 s APPICATION Use (solution to stochastic two-period model to illustrate some basic results and ideas in Consumption research Asset pricing research Certainty-equivalent consumption Assuming αc Quadratic period-utility uc ( = γ c Risk-free asset returns Risky period- income (with arbitrary distribution Risk aversion Precautionary savings Introduction to asset pricing eptember 4, 0 3 Macro/Finance Fundamentals AET MARKET Risk about the future (period requires adopting a view about the nature of asset markets Continue with example of risky period- income y = y y probability q probability p y probability -p-q eptember 4, 0 3 6

17 Macro/Finance Fundamentals AET MARKET Risk about the future (period requires adopting a view about the nature of asset markets Continue with example of risky period- income y = y y probability q probability p y probability -p-q But now three distinct assets available for purchase in period Asset a : purchase price R in period, pays off one unit in period if y, zero else M Asset a : purchase price R in period, pays off one unit in period if y, zero else Asset a : purchase price R in period, pays off one unit in period if y, zero else Arrow-Debreu securities, aka contingent claims NOT EQUIVAENT to state-contingent asset returns on a single asset eptember 4, 0 33 TATE-CONTINGENT COICE Consumer problem max uc ( + qu( c + pu( c M + ( p q u( c + λ y+ a M 0 c R a Ra Ra M M M + qλ y + a c pλ y a c ( p q λ y a c FOCs eptember 4,

18 BAIC OF AET PRICING Consumer problem max uc ( + qu( c + pu( c M + ( p q u( c + λ y M + a0 c R a Ra Ra M M M + qλ y + a c pλ y a c ( p q λ y a c FOCs Asset prices qλ qu ( c R = = λ u ( c R pλ pu ( c ( p q λ ( p q u ( c M M = = R = = λ u ( c λ u ( c u'( c / u ( c is willingness to intertemporally substitute consumption between period and state in period intertemporal MR (IMR Contingent claims prices (aka Arrow-Debreu prices, aka state prices reflect IMR (if markets functioning well In principle, allow for inferences about Risk aversion Prudence Market participants assessment of probabilities of event occurring eptember 4, 0 35 BAIC OF AET PRICING Generalize the period- risk structure : number of possible realizations of y (in richer models, risk in other primitives R : period- price of AD security that pays off one unit in state, zero otherwise p : probability of state occurring in period, with ifetime expected utility uc ( + Eu ( c = u( c + puc ( Period- budget constraint tate- period- budget constraint AD price for state (compute FOCs = + = + 0 = c R a y a = p = { } c + a,,,3,..., = y + a = 0 eptember 4,

19 BAIC OF AET PRICING Generalize the period- risk structure : number of possible realizations of y (in richer models, risk in other primitives R : period- price of AD security that pays off one unit in state, zero otherwise p : probability of state occurring in period, with ifetime expected utility uc ( + Eu ( c = u( c + puc ( Period- budget constraint tate- period- budget constraint AD price for state (compute FOCs p λ p u ( c R = = λ u'( c f u ( c u ( c Define R R = p = E = = u ( c u ( c = + = + 0 = c R a y a Is the price of a one-period riskless bond = p = { } c + a,,,3,..., = y + a = 0 eptember 4, 0 37 BAIC OF AET PRICING One-period riskless bond Purchase price R f in period Pays off one unit ( face value in all states of the world in period (Can scale to any arbitrary face value: $00 bonds, $000 bonds, etc. Introduce in model Period- budget constraint tate- period- budget constraint c + b + a = y + b + a,,,3,..., FOC on b f + Rb+ = + 0 = c R a y a = 0 = 0 b : bond holdings carried from period to { } period eptember 4,

20 BAIC OF AET PRICING One-period riskless bond Purchase price R f in period Pays off one unit ( face value in all states of the world in period (Can scale to any arbitrary face value: $00 bonds, $000 bonds, etc. Introduce in model Period- budget constraint tate- period- budget constraint c + b + a = y + b + a,,,3,..., FOC on b f λ u ( c R = E = E λ u '( c u ( c p R = u ( c = = = f + Rb+ = + 0 = c R a y a = 0 = 0 { } Price of riskless bond reflects expected IMR and by no-arbitrage equals sum of state prices. Result: risk-free bond price can be decomposed into state prices A complete set of AD securities spans the risk space which makes b a redundant asset; consumer can synthesize b himself Any asset can be decomposed into state prices (Cochrane, Chapter 3. eptember 4, 0 39 b : bond holdings carried from period to period BAIC OF AET PRICING One-period riskless bond Purchase price R f in period Pays off one unit ( face value in all states of the world in period (Can scale to any arbitrary face value: $00 bonds, $000 bonds, etc. Introduce in model Period- budget constraint tate- period- budget constraint c + b + a = y + b + a,,,3,..., FOC on b f λ u ( c R = E = E λ u '( c u ( c p R = u ( c = = = f + Rb+ = + 0 = c R a y a = 0 = 0 { } Price of riskless bond reflects expected IMR and by no-arbitrage equals sum of state prices. Result: risk-free bond price can be decomposed into state prices A complete set of AD securities spans the risk space which makes b a redundant asset; consumer can synthesize b himself ow do these asset structures affect consumer s intertemporal life? eptember 4, 0 40 b : bond holdings carried from period to period 0

21 CONUMPTION, AVING, AND AET PRICE Consumption smoothing a primitive feature of preferences (u (.>0, u (.<0 Nature of asset markets affects ability to achieve consumption smoothing Two dimensions of consumption smoothing Intertemporal consumption smoothing: concavity of u(. implies preference for low time-series-variance of consumption R f u ( c = E u '( c Expected IMR = price of riskfree bond f R u'( c = Eu'( c eptember 4, 0 4 CONUMPTION, AVING, AND AET PRICE Consumption smoothing a primitive feature of preferences (u (.>0, u (.<0 Nature of asset markets affects ability to achieve consumption smoothing Two dimensions of consumption smoothing Intratemporal consumption smoothing: concavity of u(. implies preference for low cross-state variance of consumption within any period that has risk p λ pu ( c R = = λ u'( c eptember 4, 0 4

22 CONUMPTION, AVING, AND AET PRICE Consumption smoothing a primitive feature of preferences (u (.>0, u (.<0 Nature of asset markets affects ability to achieve consumption smoothing Two dimensions of consumption smoothing Intratemporal consumption smoothing: concavity of u(. implies preference for low cross-state variance of consumption within any period that has risk p λ pu ( c R = = λ u'( c A high state price R reflects igh probability of state igh u (. in state i.e., low consumption in state Or both View as intratemporal optimality condition across future state-contingent c R / p u ( c MR across states, k = (riskadusted relative state price =, k, {,,3,..., k k k } R / p u ( c eptember 4, 0 43 CONUMPTION, AVING, AND AET PRICE Define m = R /p as discount factor for state Intratemporal optimality condition m m k u = u c ( c,, k k,,3,..., ( { } Intertemporal optimality between period and state in period u ( c m =,,,3,... u'( c { } eptember 4, 0 44

23 CONUMPTION, AVING, AND AET PRICE Define m = R /p as discount factor for state Intratemporal optimality condition m m k u = u c ( c,, k k,,3,..., ( { } Intertemporal optimality between period and state in period u ( c m =,,,3,... u'( c Expected IMR between period and period Terminology: Em p m tochastic discount factor (DF = R = u ( c f = E u '( c { } eptember 4,