Consumption-Savings Decisions and State Pricing

Transcription

1 Consumption-Savings Decisions and State Pricing Consumption-Savings, State Pricing 1/ 40

2 Introduction We now consider a consumption-savings decision along with the previous portfolio choice decision. These decisions imply a stochastic discount factor (SDF) based on marginal utilities of consumption at di erent dates. This SDF can value all traded assets and can bound assets expected returns and volatilities. The SDF can also be derived by assuming market completeness and no arbitrage. We can modify the SDF to value assets using risk-neutral probabilities. Consumption-Savings, State Pricing 2/ 40

3 Consumption and Porftolio Choices Let W 0 and C 0 be an individual s initial date 0 wealth and consumption, respectively. At date 1, the individual consumes all of his wealth C 1. The individual s utility function is: h i U (C 0 ) + E U ec1 where = 1 1+ is a subjective discount factor. A rate of time preference > 0 re ects impatience for consuming early. There are n assets where P i is the date 0 price per share and X i is the date 1 random payo of asset i, i = 1; :::; n. Hence R i X i =P i is asset i s random return. (1) Consumption-Savings, State Pricing 3/ 40

4 Consumption and Porftolio Choices cont d The individual receives labor income of y 0 at date 0 and random labor income of y 1 at date 1. Let! i be the proportion of date 0 savings invested in asset i. Then the individual s intertemporal budget constraint is C 1 = y 1 + (W 0 + y 0 C 0 ) P n i=1! i R i (2) where (W 0 + y 0 C 0 ) is date 0 savings. The individual s maximization problem is max U (C 0) + E [U (C 1 )] (3) C 0 ;f! i g subject to P n i=1! i = 1. Substituting in (2), the rst-order conditions wrt C 0 and! i, i = 1; :::; n are Consumption-Savings, State Pricing 4/ 40

5 Consumption and Porftolio Choices cont d U 0 (C 0 ) E U 0 (C 1 ) P n i=1! i R i = 0 (4) E U 0 (C 1 ) R i = 0, i = 1; :::; n (5) where 0 = (W 0 + y 0 C 0 ) and 0 is the Lagrange multiplier for the constraint P n i=1! i = 1. From (5), for any two assets i and j: E U 0 (C 1 ) R i = E U 0 (C 1 ) R j Equation (6) implies that the investor s optimal portfolio choices are such that the expected marginal utility-weighted returns of any two assets are equal. (6) Consumption-Savings, State Pricing 5/ 40

6 Consumption and Porftolio Choices cont d To examine the optimal intertemporal allocation of resources, substitute (5) into (4) U 0 (C 0 ) = E U 0 (C 1 ) P n i=1! i R i = P n i=1! i E U 0 (C 1 ) R i = P n i=1! i = (7) Then substituting = U 0 (C 0 ) into (5) gives or, since R i = X i =P i, E U 0 (C 1 ) R i = U 0 (C 0 ), i = 1; :::; n (8) P i U 0 (C 0 ) = E U 0 (C 1 ) X i, i = 1; :::; n (9) Consumption-Savings, State Pricing 6/ 40

7 Consumption and Porftolio Choices cont d Equation (9) implies the individual invests in asset i until the marginal utility of giving up P i dollars at date 0 just equals the expected marginal utility of receiving X i at date 1: Equation (9) for a risk-free asset that pays R f (gross return) is U 0 (C 0 ) = R f E U 0 (C 1 ) (10) With CRRA utility U (C) = C =, for < 1, equation (10) is " C0 # 1 1 = E (11) R f implying that when the interest rate is high, so is the expected growth in consumption. C 1 Consumption-Savings, State Pricing 7/ 40

8 Consumption and Porftolio Choices cont d If there is only a risk-free asset and nonrandom labor income, so that C 1 is nonstochastic, equation (11) is C1 1 (12) R f = 1 C 0 Note C = 1 1 C 0 C1 C 0 = (1 ) R f C 1 C 0 (13) Consumption-Savings, State Pricing 8/ 40

9 Intertemporal Elasticity So that the intertemporal elasticity of substitution is R f C 1 C C 1 C 0 ln (C 1=C 0 ) = ln (R f ) 1 (14) Thus for CRRA utility, is the reciprocal of the coe cient of relative risk aversion. When 0 < < 1, exceeds unity and a higher interest rate raises second-period consumption more than one-for-one. Conversely, when < 0, then < 1 and a higher interest rate raises second-period consumption less than one-for-one, implying a decrease in initial savings. Consumption-Savings, State Pricing 9/ 40

10 Intertemporal Elasticity cont d The individual s response re ects two e ects from an increase in interest rates. 1 A substitution e ect raises the return from transforming current consumption into future consumption, providing an incentive to save more. 2 An income e ect from the higher return on a given amount of savings makes the individual better o and, ceteris paribus, would raise consumption in both periods. For > 1, the substitution e ect outweighs the income e ect, while the reverse occurs when < 1. When = 1, the income and substitution e ects exactly o set each other. Consumption-Savings, State Pricing 10/ 40

11 Equilibrium Asset Pricing Implications The individual s consumption - portfolio choice has asset pricing implications. Rewrite equation (9): P i = U 0 (C 1 ) E U 0 (C 0 ) X i = E [m 01 X i ] (15) where m 01 U 0 (C 1 ) =U 0 (C 0 ) is the stochastic discount factor or state price de ator for valuing asset returns. In states of nature where C 1 is high (due to high portfolio returns or high labor income), marginal utility, U 0 (C 1 ), is low and an asset s payo s are not highly valued. Conversely, in states where C 1 is low, marginal utility is high and an asset s payo s are much desired. Consumption-Savings, State Pricing 11/ 40

12 Stochastic Discount Factor The SDF or pricing kernel may di er across investors due to di erences in random labor income that causes the distribution of C 1, and hence U 0 (C 1 ) =U 0 (C 0 ), to di er. Nonetheless, E [m 01 X i ] = E [U 0 (C 1 ) X i =U 0 (C 0 )] is the same for all investors who can trade in asset i since individuals adjust their portfolios to hedge individual-speci c risks, and di erences in U 0 (C 1 ) =U 0 (C 0 ) re ect only risks uncorrelated with asset returns. Utility depends on real consumption, C 1. If Pi N and Xi N are the initial price and end-of-period payo measured in currency units (nominal terms), they need to be de ated by a price index to convert them to real quantities. Consumption-Savings, State Pricing 12/ 40

13 Real Pricing Kernel Let CPI t be the consumer price index at date t. Equation (15) becomes Pi N U 0 (C 1 ) X N i = E CPI 0 U 0 (16) (C 0 ) CPI 1 If we de ne I ts = CPI s =CPI t as 1 plus the in ation rate between dates t and s, equation (16) is 1 Pi N U 0 (C 1 ) = E I 01 U 0 (C 0 ) X i N h i = E M 01 Xi N where M 01 (=I 01 ) U 0 (C 1 ) =U 0 (C 0 ) is the SDF for nominal returns, equal to the real pricing kernel, m 01, discounted at the (random) rate of in ation between dates 0 and 1. (17) Consumption-Savings, State Pricing 13/ 40

14 Risk Premia and Marginal Utility of Consumption Equation (15) can be rewritten to shed light on an asset s risk premium. Divide each side of (15) by P i : 1 = E [m 01 R i ] (18) = E [m 01 ] E [R i ] + Cov [m 01 ; R i ] = E [m 01 ] E [R i ] + Cov [m 01; R i ] E [m 01 ] Recall from (10) that for the case of a risk-free asset, E [U 0 (C 1 ) =U 0 (C 0 )] = E [m 01 ] = 1=R f. Then (18) can be rewritten R f = E [R i ] + Cov [m 01; R i ] (19) E [m 01 ] or Consumption-Savings, State Pricing 14/ 40

15 Risk Premia and Marginal Utility of Consumption cont d E [R i ] = R f Cov [m 01 ; R i ] E [m 01 ] = R f Cov [U 0 (C 1 ) ; R i ] E [U 0 (C 1 )] (20) An asset that tends to pay high returns when consumption is high (low) has Cov [U 0 (C 1 ) ; R i ] < 0 (Cov [U 0 (C 1 ) ; R i ] > 0) and will have an expected return greater (less) than the risk-free rate. Investors are satis ed with negative risk premia when assets hedge against low consumption states of the world. Consumption-Savings, State Pricing 15/ 40

16 Relationship to the CAPM Suppose there is a portfolio with a random return of e R m that is perfectly negatively correlated with the marginal utility of date 1 consumption, U 0 ec1, so that it is also perfectly negatively correlated with m 01 : Then this implies U 0 ( ~C 1 ) = e R m ; > 0 (21) Cov[U 0 (C 1 ); R m ] = Cov[R m ; R m ] = Var[R m ] (22) and Cov[U 0 (C 1 ); R i ] = Cov[R m ; R i ] (23) Consumption-Savings, State Pricing 16/ 40

17 Relationship to the CAPM cont d From (20), the risk premium on this portfolio is E[R m ] = R f Cov[U 0 (C 1 ); R m ] E[U 0 (C 1 )] = R f + Var[R m] E[U 0 (C 1 )] (24) Using (20) and (24) to substitute for E[U 0 (C 1 )], and using (23), we obtain E[R m ] R f = Var[R m] E[R i ] R f Cov[R m ; R i ] (25) and rearranging: E[R i ] R f = Cov[R m; R i ] Var[R m ] (E[R m ] R f ) (26) Consumption-Savings, State Pricing 17/ 40

18 Relationship to the CAPM cont d Equation (26) is the CAPM relation E[R i ] = R f + i (E[R m ] R f ) (27) Note that under CAPM assumptions the market portfolio is perfectly negatively correlated with consumption: 1 There is no wage income, so end of period consumption derives only from asset portfolio returns. 2 With a risk-free asset and normally distributed asset returns, everyone holds the same risky asset (market) portfolio. Hence, the only risk to C 1 is the return on the market portfolio. Consumption-Savings, State Pricing 18/ 40

19 Bounds on Risk Premia m 01 U 0 (C 1 ) =U 0 (C 0 ) places a bound on the Sharpe ratio of all assets. Rewrite equation (20) as E [R i ] = R f m01 ;R i m01 Ri E [m 01 ] (28) where m01, Ri, and m01 ;R i are the standard deviation of the discount factor, the standard deviation of the return on asset i, and the correlation between the discount factor and the return on asset i, respectively. Rearranging (28) leads to E [R i ] R f m01 = m01 ;R i Ri E [m 01 ] (29) Consumption-Savings, State Pricing 19/ 40

20 Hansen-Jagannathan Bounds Since 1 m01 ;R i 1, we know that E [R i ] Ri R f m 01 E [m 01 ] = m 01 R f (30) Equation (30) was derived by Robert Shiller (1982) and generalized by Hansen and Jagannathan (1991). If there exists a portfolio of assets whose return is perfectly negatively correlated with m 01, then (30) holds with equality. The CAPM implies such a situation, so that the slope of the capital market line, S e E [Rm] R f Rm, equals m01 R f. Consumption-Savings, State Pricing 20/ 40

21 Ex: Bounds with Power Utility If U (C) = C = so m 01 (C 1 =C 0 ) 1 = e ( 1) ln(c 1=C 0 ) and C 1 =C 0 is lognormal with parameters c and c, then qvar e ( 1) ln(c 1=C 0 ) m01 E [m 01 ] = = = = E e ( q E e 2( 1) ln(c 1=C 0 ) E e ( 1) ln(c 1=C 0 ) 2 1) ln(c 1=C 0 ) E e ( q E e 2( 1) ln(c 1=C 0 ) =E e ( 1) ln(c 1=C 0 ) 2 1 q q e 2( 1) c +2( 1)2 2 c =e 2( 1) c +( 1)2 2 c 1 = e ( 1)2 2 c 1 1) ln(c 1=C 0 ) ( 1) c = (1 ) c (31) The fourth line evaluates expectations assuming C 1 log-normality, E (X ) = e The fth line takes a two-term approximation of the series e x = 1 + x + x 2 2! + x 3 + :::, which is reasonable for small positive x. The (+) 3! solution is negative for < 1. Consumption-Savings, State Pricing 21/ 40

22 Ex: Bounds with Power Utility Hence, with power utility and lognormal consumption: E [R i ] R f (1 ) c (32) Ri For the S&P500 over the last 75 years, E [R i ] R f = 8:3% and Ri = :17, implying a Sharpe ratio of E [R i ] R f Ri = 0:49. U.S. per capita consumption data implies estimates of c between 0.01 and Assuming (32) holds with equality for the S&P500, = 1 E [Ri ] R f Ri = c is between and -48. Other empirical estimates of are -1 to -5. The inconsistency of theory and empirical evidence is what Mehra and Prescott (1985) termed the equity premium puzzle. Consumption-Savings, State Pricing 22/ 40

23 Ex. Bounds on R f Even if high risk aversion is accepted, it implies an unreasonable value for the risk-free return, R f. Note that and therefore 1 = E [m 01 ] (33) R f = E he i ( 1) ln(c 1=C 0 ) = e ( 1) c ( 1)2 2 c ln (R f ) = ln () + (1 ) c 1 2 (1 )2 2 c (34) If we set = 0:99, and c = 0:018, the historical average real growth of U.S. per capita consumption, then with = 11 and c = 0:036 we obtain: Consumption-Savings, State Pricing 23/ 40

24 Ex. Bounds on R f cont d ln (R f ) = ln () + (1 ) c 1 2 (1 )2 2 c = 0:01 + 0:216 0:093 = 0:133 (35) which is a real risk-free interest rate of 13.3 percent. Since short-term real interest rates have averaged about 1 percent in the U.S., we end up with a risk-free rate puzzle: the high results in an unreasonable R f. So a SDF derived from the marginal utility of consumption doesn t t the data. However, we can derive a SDF of the form P i = E 0 [m 01 X i ] using another approach. Consumption-Savings, State Pricing 24/ 40

25 Complete Markets Assumptions An alternative SDF derivation is based on the assumptions of a complete market and the absence of arbitrage. Suppose that an individual can freely trade in n assets and assume that there is a nite number, k, of end-of-period states of nature, with state s having probability s. Let X si be the cash ow returned by one share (unit) of asset i in state s. Asset i s cash ows can be written as: 2 3 X i = 6 4 X 1i. X ki 7 5 (36) Consumption-Savings, State Pricing 25/ 40

26 Complete Markets Assumptions cont d Thus, the per-share cash ows of the universe of all assets can be represented by the k n matrix 2 3 X 11 X 1n 6 X = (37) X k1 X kn We will assume that n = k and that X is of full rank, implying that the n assets span the k states of nature and the market is complete. An implication is that an individual can purchase amounts of the k assets that return target levels of end-of-period wealth in each of the states. Consumption-Savings, State Pricing 26/ 40

27 Complete Markets Assumptions cont d To show this complete markets result, let W be an arbitrary k 1 vector of end-of-period levels of wealth: 2 3 W = 6 4 W 1. W k where W s is the level of wealth in state s: 7 5 (38) To obtain W, at the initial date the individual purchases shares in the k assets. Let the vector N = [N 1 : : : N k ] 0 be the number of shares purchased of each of the k assets. Hence, N must satisfy XN = W (39) Consumption-Savings, State Pricing 27/ 40

28 Complete Markets Assumptions cont d Since X is a nonsingular, its inverse exists so that is the unique solution. N = X 1 W (40) Denoting P = [P 1 : : : P k ] 0 as the k 1 vector of beginning-of-period, per-share prices of the k assets, then the initial wealth required to produce the target level of wealth given in (38) is P 0 N. The absence of arbitrage implies that the price of a new, redundant security or contingent claim that pays W is determined from the prices of the original k securities, and this claim s price must be P 0 N. Consumption-Savings, State Pricing 28/ 40

29 Arbitrage and State Prices Consider the case of a primitive, elementary, or Arrow-Debreu security which has a payo of 1 in state s and 0 in all other states: e s = 6 4 W 1. W s 1 W s W s+1. W k = (41) Consumption-Savings, State Pricing 29/ 40

30 Arbitrage and State Prices Then p s, the price of elementary security s, is p s = P 0 X 1 e s, s = 1; :::; k (42) so a unique set of state prices exists in a complete market. These elementary state prices should each be positive, since wealth received in any state will have positive value when individuals are nonsatiated. Hence (42) and p s > 0 8s restrict the payo s, X, and the prices, P, of the original k securities. Note that the portfolio composed of the sum of all elementary securities gives a cash ow of 1 unit with certainty and determines the risk-free return, R f : Consumption-Savings, State Pricing 30/ 40

31 Arbitrage and State Prices cont d kx p s = 1 (43) R f s=1 For a general multicash ow asset, a, whose cash ow in state s is X sa, absence of arbitrage ensures its price, P a, is P a = kx p s X sa (44) s=1 Consider the connection to state probabilities, s, by de ning m s p s = s. Since p s > 0 8s, then m s > 0 8s when s > 0. Consumption-Savings, State Pricing 31/ 40

32 Arbitrage and State Prices cont d Then equation (44) can be written as P a = P k s=1 p s s = P k s=1 sm s X sa = E [m X a ] s X sa (45) where m denotes a stochastic discount factor whose expected value is E [m] = P k s=1 sm s = P k s=1 p s = 1=R f, and X a is the random cash ow of the multicash ow asset a. In terms of the consumption-based model, m s = U 0 (C 1s ) =U 0 (C 0 ) where C 1s is consumption at date 1 in state s, p s is greater when C 1s is low. Consumption-Savings, State Pricing 32/ 40

33 Risk-Neutral Probabilities De ne b s p s R f. Then P a = P k s=1 p s X sa (46) = 1 R f P k s=1 p sr f X sa = 1 P k s=1 R b s X sa f Now b s ; s = 1; :::; k, have the characteristics of probabilities because they are positive, b s = p s = P k s=1 p s > 0, and they sum to 1, P k s=1 b P s = R k f s=1 p s = R f =R f = 1. Consumption-Savings, State Pricing 33/ 40

34 Risk-Neutral Probabilities cont d Using this insight, equation (46) can be written P a = 1 R f P k s=1 b s X sa = 1 R f b E [Xa ] (47) where b E [] denotes the expectation operator using the "pseudo" probabilities b s rather than the true probabilities s. Since the expectation in (47) is discounted by the risk-free return, we can recognize b E [X a ] as the certainty equivalent expectation of the cash ow X a. Consumption-Savings, State Pricing 34/ 40

35 Risk-Neutral Probabilities cont d Since m s p s = s and R f = 1=E [m], b s can be written as b s = R f p s = R f m s s m s = E [m] s (48) In states where the SDF m s is greater than its average, E [m], the pseudo probability exceeds the true probability. Note if m s = 1 R f = E [m] then P a = E [mx a ] = E [X a ] =R f so the price equals the expected payo discounted at the risk-free rate, as if investors were risk-neutral. Consumption-Savings, State Pricing 35/ 40

36 Risk-Neutral Probabilities cont d Hence, b s is referred to as the risk-neutral probability. be [], also often denoted as E Q [], is referred to as the risk-neutral expectations operator. In comparison, the true probabilities, s, are frequently called the physical, or statistical, probabilities. Consumption-Savings, State Pricing 36/ 40

37 State Pricing Extensions This complete markets pricing, also known as State Preference Theory, can be generalized to an in nite number of states and elementary securities. Suppose states are indexed by all possible points on the real line between 0 and 1; that is, the state s 2 (0; 1). Also let p(s) be the price (density) of a primitive security that pays 1 unit in state s, 0 otherwise. Consumption-Savings, State Pricing 37/ 40

38 State Pricing Extensions cont d Further, de ne X a (s) as the cash ow paid by security a in state s. Then, analogous to (43), Z 1 0 p(s) ds = 1 R f (49) and the price of security a is P a = Z 1 0 p(s) X a (s) ds (50) Consumption-Savings, State Pricing 38/ 40

39 State Pricing Extensions cont d In Time State Preference Theory, assets pay cash ows at di erent dates in the future and markets are complete. For example, an asset may pay cash ows at both date 1 and date 2 in the future: let s 1 be a state at date 1 and let s 2 be a state at date 2. States at date 2 can depend on which states were reached at date 1. Suppose there are two events at each date, economic recession (r) or economic expansion (boom) (b). Then de ne s 1 2 fr 1 ; b 1 g and s 2 2 fr 1 r 2 ; r 1 b 2 ; b 1 r 2 ; b 1 b 2 g. By assigning suitable probabilities and primitive security state prices for assets that pay cash ows of 1 unit in each of these six states, we can sum (or integrate) over both time and states at a given date to obtain prices of complex securities. Consumption-Savings, State Pricing 39/ 40

40 Summary An optimal portfolio is one where assets expected marginal utility-weighted returns are equalized, and the individual s optimal savings trades o expected marginal utility of current and future consumption. Assets can be priced using a SDF that is the marginal rate of substitution between current and future consumption. A SDF can also be derived based on assumptions of market completeness and no arbitrage. A risk-neutral pricing formula transforms physical probabilities to account for risk. Consumption-Savings, State Pricing 40/ 40