Topic 7: Asset Pricing and the Macroeconomy

Transcription

1 Topic 7: Asset Pricing and the Macroeconomy Yulei Luo SEF of HKU November 15, 2013 Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

2 Consumption-based Asset Pricing Even if we cannot easily solve the full-fledged optimal consumption and portfolio choice model, we can still gain many interesting insights about the joint dynamics of the asset return and consumption dynamics by inspecting the Euler equation. We now assume that there are n risky assets such that R p t+1 = n j=1 α j R j t+1 + (1 n j=1 In this case, the Euler equations for all assets are where j = f, 1,, n. u (c t ) = 1 ] 1 + ρ E t R j t+1 u (c t+1 ), α j ) R f. (1) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

3 Model Implications Note that the Euler equations can be rewritten as 1 1 = 1 + ρ E t R j u ] (c t+1 ) t+1 u (c t ) ] E t R j t+1 M t+1, (2) where M t+1 is the stochastic discount factor applied at t to consumption in the following period. It is the intertemporal marginal rate of substitution, i.e., the discounted ratio of marginal utilities of consumption in any two subsequent periods. Using (2), we have the key result of consumption-based asset pricing: ] ] ( ) E t R j t+1 M t+1 = E t R j t+1 E t M t+1 ] + cov t R j t+1, M t+1 = ] E t R j 1 ( )] t+1 = 1 cov t R j t+1 E t M t+1 ], M t+1. (3) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

4 (conti.) In the case of the risk free asset, we have R f = 1 E t M t+1 ]. (4) Combining (3) with (4), we have ] ( ) E t R j t+1 R f = R f cov t R j t+1, M t+1, (5) which means that: In equilibrium, the risky asset j whose return has a negative correlation with the SDF yields an expected return higher than R f. This asset is risky for the investor because it yields lower returns when the marginal utility of consumption relatively high due to a relatively low level of consumption. In equilibrium investors are still willing to hold this asset only if such risk can be compensated by a premium determined by an expected return higher than the risk free rate R f. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

5 Implications of CRRA Utility 1 1 γ (conti.) Assume the CRRA utility c 1 γ t E t R j t+1, we have ( ] ( R f = R f cov t R j t+1, β ct+1 c t ) γ ). (6) Using the facts that (1) If c t+1 c t is small, c t log c t+1. (7) c t (2) If x is small, (1 + x) n 1 + nx, (8) (6) can be written as ] ( ) E t R j t+1 R f βr f cov t R j t+1, 1 γ log c t+1 ) cov t (R j t+1, γ log c t+1 (9) Note that βr f is close to 1 if c t+1 c t is small. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

6 First Look: The Equity Premium Puzzle Taking unconditional expectations on both sides of (9) gives ] ( ) E R j t+1 R f cov R j t+1, γ log c t+1 (10) ( ) ( ) = γ corr R j t+1, log c t+1 sd R j t+1 sd ( log c t+1 ) Mehra and Prescott (1985) show that it is diffi cult to reconcile observed returns on stocks and bonds with equation (9). As documented in Campbell (2003), in the U.S. data from : ( ) corr R j t+1, log c t+1 = 0.34, (11) ( ) sd R j t+1 = 15.6%, (12) sd ( log c t+1 ) = 1.1%, (13) ] E R j t+1 R f = 7%, (14) which means that γ = 120! It is highly unrealistic. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

7 To gain some idea about what plausible values of γ are, consider the following gamble: You must choose between a gamble in which you consume $50000 in the rest of your life with probability 0.5 and $ with probability 0.5, or consuming some amount X with certainty. The CRRA, γ, determines the value of X which would make you indifferent between consuming X or being exposed to the risky gamble. E.g., if γ = 0, then you have no risk aversion at all and will will be indifferent between $75000 with certainty and the 50/50 gamble with expected value of $ Here are the values of X associated with γ X different γ: Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

8 Consumption-based Capital Asset Pricing Model (C-CAPM) In reality, assets are traded every period, as new information becomes available, and decisions are made sequentially. Consequently, today s decisions affect tomorrow s opportunities. CCAPM can be used to capture these dynamic features and to price assets in such an environment. Another advantage: This theory provides a way to link the real economy (aggregate output and consumption) and financial markets (asset prices and returns). Lucas (1978) first developed this CCAPM theory. It is an endowment economy (i.e., no production decisions) and allows recursive security trading. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

9 Lucas (1978) s Asset Pricing Model Imagine the security as representing ownership of a fruit tree where the nonstorable output varies over time. Suppose that there is an economy with N identical agents with RE. That is, each agent s expectations are conditional on all available information (including the structure of the economy, the output process, etc.) Suppose that all output is obtained from an asset which produces a stochatic endowment of perishable consumption goods for each unit of the asset the agent owns at the beginning of t. If an agent owns z t units of the asset at the beginning of t, he receives an endowment of z t y t of the consumption good and y t is identical for each unit of the asset held by an agent and is an exogenous stochastic process. Assume that they have identical endowments. Since the agents have identical preferences, they will make the same decisions given the state of the economy. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

10 (conti.) The typical agent chooses optimal holdings of securities and consumes dividends: ] max E {c t,z t+1 } β t u(c t ) t=0, (15) s.t. c t + p t z t+1 z t y t + p t z t, t, (16) where p t is the period t real price of the security in terms of consumption and z t is the agent s beginning-of-period t holdings of security. The expectation operator applies across all possible states of y. In the this economy: Financial markets are in equilibrium iff at the prevailing price, supply equals demand and the equilibrium price is that price at which the agents wishes to hold exactly the amount of the securities present in the economy. Assume that the total supply of assets is N. With N agents wantin the same number of assets, we must have z t = 1 for all t and for every agent in equilibrium. The total net supply of I-owe-You (IOU) type of contract (inside bonds) must be zero; otherwise, there is no equilibrium. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

11 Output is an exogenously stochastic process. We may assume that the output process follows a n-state probability transition. E.g., y t could take three states: y 1 y 2 y 3 ] and it follows a probability transition matrix to switch from one state to another state: Table: Three-state probability transition matrix output in t + 1 π 11 π 12 π 13 output in t Π = π 21 π 22 π 23 π 31 π 32 π 33 where π ij = prob ( y t+1 = y j y t = y i ) for any t. Note this discrete-state distribution can be approximated from an AR(1) process. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

12 Approximating an AR(1) Process Given the following AR(1) process with continuous states, y t+1 = ρy t + (1 ρ) y + ε t+1, (17) we can numerically approximate it with finite discrete states. The Matlab codes posted on the course website can used to approximate the AR process: discretizationar1.m and Tauchen.m. E.g., given that ρ = 0.9, ω = 0.1, y = 1, and n = 3, running the discretization code yields: y 1 y 2 y 3 ] = ], (18) Π = (19) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

13 Using the Bellman equation to Solve the Model Define the value function as v(m s ) = max E s t=s ] β t s u(c t ) (20) s.t. c t + p t z t+1 z t y t + p t z t (21) where m s is the beginning-of-period s wealth: m s = (y s + p s ) z s, (22) and rewrite the budget constraint as c s + p s z s+1 = m s (23) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

14 The Bellman equation can be written as v(m s ) = max {c s } {u(c s ) + βe s v(m s+1 )]} (24) = max {u(c s ) + βe s v((y s+1 + p s+1 ) z s+1 )]} (25) {c s } = max {u(c s ) + βe s v((y s+1 + p s+1 ) m ]} s c s ) (26) {c s } p s The FOC is: u 1 (c s ) βe s V 1 (m s+1 ) y ] s+1 + p s+1 = 0 (27) p s The envelop theorem is: v 1 (m s ) = βe s v 1 (m s+1 ) y s+1 + p s+1 p s ], (28) which means that u 1 (c s ) = v 1 (m s ) and u 1 (c s+1 ) = v 1 (m s+1 ). Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

15 Substituting this Envelop condition into the FOC gives the Euler equation: u 1 (c s )p s = βe s u 1 (c s+1 ) (y s+1 + p s+1 )]. (29) Given the functional dependence on the output state variables, the Euler equation can be written as ( ( u 1 cs y i )) ( p s y i ) ( ( = β π ij u1 cs+1 y j )) ( y j + p s+1 (y j ) )], i j (30) Economic implications: The LHS: the utility loss in period t associated with the purchase of an additional unit of the security. The RHS: the expected discounted utility gain associated with selling the extra unit of the security. If this equality is not satisfied, the agent will try either to increase or to decrease his holdings of the security to increase his utility. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

16 In Equilibrium z t = z t+1 = = 1, in other words, every agents owns the same number of trees, 1. c t = y t, that is, ownership of the entire security entitles the agent to all the economy s output. u 1 (c s )p s = βe s u 1 (c s+1 ) (y s+1 + p s+1 )] are optimal given the prevailing prices. Substituting c t = y t into this Euler equation gives: u 1 (y s )p s = βe s u 1 (y s+1 ) (y s+1 + p s+1 )], (31) which is the fundamental equation of the consumption-based CAPM. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

17 Asset Prices A recursive substitution of (31) into itself yields p s = E s β j u 1(y s+j ) u 1 (y s ) y s+j j=1 ], (32) where M s,s+j = β j u 1 (y s+j ) u 1 (y s is the stochatic discount factor (i.e., the ) intertemporal marginal rate of substitution) and assume that the price is bounded. (32) means that the stock price is the sum of all expected discounted future dividends. Example: If the utility function displays risk neutrality (i.e., u 1 ( ) is constant), p s = E s ] β j y s+j = E s j=1 j=1 y s+j (1 + r f ) j ], (33) which means that the stock price is the sum of expected future dividends discounted at the constant risk free rate. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

18 The difference between (32) and (33) is the necessity of discounting the flow of expected dividends at a rate higher than the risk free rate, so as to include a risk premium. Which factors affect risk premium is a central issue in finance. Another examples: If the utility function is log, the price is p s = ( ) E s β j y ) s y s+j = E s j=1 y s+j (β j y s j=1 (34) = β 1 β y s, (35) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

19 Calculating the equilibrium price function. The Euler equation, u 1 (y s )p s = βe s u 1 (y s+1 ) (y s+1 + p s+1 )], implicitly defines the equilibrium price. And we can calculate the actual equilibrium prices once specifying parameter values and function forms: select β, the utility function u(c), and the transition matrix Π. Specifically, we can solve for { p(y j ), j = 1,, N } as the solution to a system of linear equations: u 1 (y 1 )p ( y 1) = β ( u 1 (y N )p y N ) = β N j=1 π 1j u1 (y j ) ( y j + p ( y j ))] (36) (37) N j=1 with unknowns { p(y j ), j = 1,, N }. π Nj u1 (y j ) ( y j + p ( y j ))] (38) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

20 Numerica Example Suppose that β = 0.96, u(c) = ln(c), (y 1, y 2, y 3 ) = (1.5, 1, 0.5), and the transition matrix Π :. Three-state probability transition matrix output in s + 1 (y s+1 ) output in s (y s ) Π = where π ij = prob ( y s+1 = y j y s = y i ) for any s. Hence, we have three equations and three unknowns { p(y j ), j = 1, 2, 3 } and can solve for the equilibrium prices: p(1) = 24; p(1.5) = 36; p(0.5) = 12. (39) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

21 Link Asset Prices to Asset Returns Define the return of security j from period t to t + 1 as 1 + r j,t+1 = p j,t+1 + y j,t+1 p j,t. (40) Using this definition, the above Euler equation can be rewritten as ] u1 (c t+1 ) 1 = βe t u 1 (c t ) (1 + r j,t+1) (41) Let q t denote the price in t of a one-period risk free bond in zero net supply, which pays one unit of consumption in every state in t + 1. Hence, q t u 1 (c t ) = βe t u 1 (c t+1 ) 1], (42) where q t is the equilibrium price at which the agent desires to hold zero units of the security, and thus supply equals demand. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

22 Since (1 + r f,t+1 ) q t = 1, we have r f,t+1 = q t = βe t ] u1 (c t+1 ), (43) u 1 (c t ) which means that in the risk neutrality case, the risk free rate must be a constant. Combining the above two pricing equations, we have 1 = βe t u1 (c t+1 ) u 1 (c t ) ] E t 1 + r j,t+1 ] + β covar t u1 (c t+1 ) u 1 (c t ), 1 + r j,t+1 where we use the fact that for two random variables: covar x, y] = E xy] E x] E y]. Denote E t 1 + r j,t+1 ] = 1 + r j,t+1, we have 1 = 1 + r j,t r f,t+1 + β covar t ] u1 (c t+1 ) u 1 (c t ), r j,t+1 ], (44) (45) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

23 Rearranging gives ] u1 (c t+1 ) r j,t+1 r f,t+1 = 1 β (1 + r f,t+1 ) covar t u 1 (c t ), r j,t+1 (46) This equation is the central relationship of the CCAPM and is very similar to the pricing equation obtained in the last lecture. The LHS is the risk premium on security j. It means that the risk premium ] will be large when u1 (c covar t+1 ) t u 1 (c t ), r j,t+1 is large and negative, that is, for those securities paying high returns when consumption is high (i.e., u 1 ( ) is low), and low returns when consumption is high (i.e., u 1 ( ) is high). These securities are not very desirable for reducing consumption risk because they pay high returns when investors don t need them and low returns when they are most needed. Since they are not desirable, they have a low price and high expected returns for compensation. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

24 A Discussion The standard CAPM in finance tells us a security is relatively undesirable and thus commands a high return when it covaries positively with the market portfolio. The Consumption CAPM adds some some further degree of precision: from the viewpoint of consumption smoothness and risk diversification, an asset is desirable if it has a high return when consumption is low and vice versa. C-CAPM is more convincing in the multiperiod context because the value of an asset is to provide the investor intermediate consumption over time; consequently, the key to an asset s value is its covariation with the marginal utility of consumption. An unappealing feature of the above CCAPM is that the marginal utility of consumption is not observable. We can eliminate this feature by adopting a specific utility function. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

25 An Example Let u(c) = ac t b 2 c2 t, (47) which a, b > 0. It follows that u 1 (c) = a bc t ; substituting it into the CCAPM gives r j,t+1 r f,t+1 = ] a bct+1 1 β (1 + r f,t+1 ) covar, r j,t+1 (48) a bc t = b 1 β (1 + r f,t+1 ) covar c t+1, r j,t+1 (49) ] a bc t = 1 + βb (1 + r f,t+1) covar c t+1, r j,t+1 ] (50) a bc }{{ t } >0 Implications: Since the term in front of the covariance is positive, if the next-period consumption covaries in a large positive way with r j,t+1, then the risk premium on asset j will be high. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

26 Solving the CCAPM with growth So far, we have assumed that output (=dividend=consumption in equilibrium) is stationary. In reality, consumption and output are growing over time. Here we assume that output growth rather than output itself follows a distribution with possible finite states: (x 1,, x N ) whose realizations are governed by a stochastic process with transition matrix Π. Then for whatever x i is realized in period t + 1: d t+1 = x t+1 y t = x t+1 c t = x i c t. (51) Using the CRRA utility, in equilibrium (y t = c t ): y γ t p(y t, x i ) = β p(y t, x i ) = β N j=1 N j=1 π ij (x j y t ) γ x j y t + p(x j y t, x j )] or (52) π ij (x j ) γ x j y t + p(x j y t, x j )], (53) which means that the SDF is determined exclusively by the consumption growth rate x j. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

27 Just like Mehra and Prescott (1985), we guess p(y t, x i ) = v i y t, (54) for a set of constants {v 1,, v N }, each should be identified with the corresponding growth rate. With this functional form, the asset pricing equation reduces to v i y t = β v i = β N j=1 N j=1 π ij (x j ) γ x j y t + v j x j y t ] or (55) π ij (x j ) γ x j + v j x j ] = β N j=1 π ij (x j ) 1 γ (1 + v j )(56), which is again a system of linear equations in the N unknowns {v 1,, v N }. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

28 Thus, for any state (y, x j ) = (c, x j ), the equilibrium asset price is p(y t, x j ) = v j y t, (57) if we assume that the current state is (y, x i ) while next period it is (x j y, x j ), then the one-period return is r ij = p(x jy t, x j ) + x j y p(y, x i ) p(y, x i ) (58) = v jx j y t + x j y v i y v i y t = x j (1 + v j ) v i 1, (59) Hence, the mean or expected return, conditional on state i, is r i = N j=1 π ij r ij,and the unconditional equity return is given by E r] = N j=1 π j r j, where π j are long-run stationary probability of each state. The price of the risk free asset is p f (c, x i ) = β N j=1 π ij (x j ) γ. (60) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

29 The Empirical Validity of the CCAPM A few key empirical observations regarding financial returns in US markets: ] E R j t+1 R f = 7%, based on Campbell (2003) s U.S. data from The equity premium puzzle found by Mehra and Prescott (1985): the standard CCAPM is completely unable to replicate the high observed equity premium once reasonable parameter values are inserted in the model. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

30 The Reasoning of the Puzzle According to the CCAPM theory, the only factors determining the characteristics of asset returns are the RA s utility function, the subjective discount factor, and the consumption process. The utility function: CRRA c 1 γ 1 γ and empirical studies have placed γ in the range of 1, 5). The consumption process. In reality, consumption is growing over time. If there were no uncertainty in the model, and if the constant growth rate of consumption were to equal to its long-run historical average (around ), the asset pricing equation would reduce to ( ) ] γ ct+1 1 = βe t R t+1 = βg γ R, (61) c t where R t+1 is the gross rate on capital, g and R are historical averages of consumption growth and the gross rate. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

31 For γ = 1, g = , and R = 1.04, we can solve for the implied β (Note that the economywide debt-to-equity ratios are not very different from 1.) Since we use an annual estimate for g, the resulting β must be viewed as an annual or yearly subjective discount factor. Similarly, on quarterly basis, β = If assume that γ = 2, the implied β = 0.99 annually and a quarterly β even closer to 1. Specifically, assuming higher rates of risk aversion would be incompatible with maintaining the hypothesis of a time discount factor β < 1. At the root of this diffi culty is the low return on the risk free asset (1%). Highly risk averse individuals want to smooth consumption over time (1/γ low if γ high), meaning they want to transfer consumption from good times to bad times. Hence, when consumption is growing predictably, the good times lies in the future. Agents want to borrow now against their future income. In the RA model, it is hard to reconcile with a low rate on borrowing: everyone is on the same side of the market and then inevitably forces a higher rate. As a result, we need either an independent explanation for the low average risk free rate or accept β > 1. Here, we just limit γ 2. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

32 Further Inspection With these assumptions, we can obtain the equations used to indirectly test the model. The key step is to guess and verify that the asset price takes the form: p t = vy t (62) where v is a constant coeffi cient. That is, the stock price is proportional to the dividend. Given this guess, we have vy t = E t β u ] 1(c t+1 ) u 1 (c t ) (vy t+1 + y t+1 ). (63) If the growth rate x t+1 = c t+1 /c t is iid through time, drop the conditional expectation and have v = E βx γ y ] t+1 (v + 1) y t = E βx 1 γ (v + 1) ] = βe x 1 γ] 1 βe x 1 γ ] = A constant. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

33 We can rewrite the equity return as which means that R t+1 = 1 + r t+1 = y t+1 + p t+1 = v + 1 x t+1, (64) p t v E R t+1 ] = v + 1 E x t+1 ] = v E x t+1] ]. (65) βe x 1 γ t+1 The risk free rate is R f,t+1 = 1 q b t = (E t β u ]) 1(c t+1 ) 1 = 1 u 1 (c t ) β 1 E x γ ], (66) which is also a constant, i.e., R f,t+1 = R f. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

34 where ρ is the correlation coeffi cient between ln x and ln y. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56 Some Properties of the Lognormal Distribution Definition A random variable x is said to follow a lognormal distribution if ln x is normally distributed. Let ln x N(µ x, σ 2 x ). If this is the case, ( E x] = exp µ x + 1 ) 2 σ2 x ( E x α ] = exp aµ x + 1 ) 2 a2 σ 2 x (67) (68) var x] = exp ( 2µ x + σ 2 x ) exp ( σ 2 x 1 ), (69) suppose further that x and y are two variables that are independently and identical lognormally distributed; then we also have ( E x a y b) ( = exp aµ x + bµ y + 1 ) ( a 2 σ 2 x + b 2 σ 2 ) y + 2ρabσx σ y, (70) 2

35 Applying the above results to consumption growth: x is lognormally distributed, that is, ln x N(µ x, σ 2 x ). We know that E x] = and var x] = (0.0357) 2. To identify (µ x, σ 2 x ), we need to solve to get ( = exp µ x + 1 ) 2 σ2 x (71) (0.0357) 2 = exp ( 2µ x + σ 2 x ) exp ( σ 2 x 1 ) (72) σ 2 x = and µ x = (73) Directly use these values and the value of γ to solve: ] ( E x γ t = exp γµ x + 1 ) 2 γ2 σ 2 x = (74) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

36 Another application: Computing the standard deviation of the SDF m t = βx γ t where x follows log normal distribution. Given a derived E m t ], one can estimate { k } var m t ] 1 ] 2 β (x i ) γ E m t ] ; (75) k i=1 for ln x N(µ x, σ 2 x ), where σ 2 x = and µ x = , and k is suffi ciently large (say 10, 000). For γ = 2, we have var m t ] = ( ) 2, or (76) std m] = (77) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

37 Taking advantage of the above lognormality hypothesis, we can combine (65) with (66): E R t+1 ] = E x] βe x γ ] R f,t+1 βe x 1 γ = exp ( γσ 2 ) x, (78) ] 1 where σ 2 x is the variance of ln x. Taking logs both sides gives ln (E R t+1 ]) ln (R f,t+1 ) = γσ 2 x. (79) Now we can confront the model with the data: feeding in the return characteristics of the US economy and solve for γ : ln (E R t+1 ]) ln (R f,t+1 ) σ 2 x = = 53 = γ, (80) (0.0357) 2 which is too high to be consistent with independent empirical evidence! An agent with γ = 50 would be too fearful to take a bath or to cross the street. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

38 Alternatively, if we assume that γ = 2, we can obtain an equity premium as follows 2(0.0357) 2 = = ln (E R t+1 ]) ln (R f,t+1 ) E R t+1 ] R f,t+1, (81) which means that given reasonable risk aversion, the resulting equity premium is much lower than the observed premium around 7%. 1 Given R f = 1 β E x γ ], E x γ ] = 0.97; given γ = 2, and R f = 1.008, we have β = 1.02 > 1! (82) This problem is due to the fact that we are using a even lower risk free rate (0.8%) rather than the long-run rate of return on capital of 4% used in the prior reasoning. In the present context, the problem is proposed by Weil (1989) and is called the risk free rate puzzle. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

39 Another Way to Test the CCAPM: Hansen-Jagannathan (HJ) Bounds The bound proposed by HJ (1991) leads to a falsification of the standard CCAPM and can also be applied in other asset pricing formulations. For all homogeneous agent economies, the equilibrium asset pricing can be rewritten as: p(s t ) = E t m t+1 (s t+1 )X (s t+1 )], (83) where s t is the state today, X (s t+1 ) is the total return in next period, and m t+1 (s t+1 ) is the SDF: m t+1 (s t+1 ) = β u 1(c t+1 (s t+1 )). (84) u 1 (c t ) Suppress the state dependence, the pricing equation: p t = E t m t+1 X t+1 ] 1 = E t m t+1 R t+1 ], (85) where R t+1 is the gross return. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

40 Since the equation holds for each state s t, it also holds unconditionally: 1 = E mr], (86) where E denotes the unconditional expectation. For any two assets, i and j, E m(r i R j )] = 0, or E mr i j ] = 0, (87) which implies that E m]e R i j ] + covar m, R i j ] = 0 = (88) E m]e R i j ] + ρ m, R i j ] sd m] sd R i j ] = 0 = E R i j ] sd R i j ] + ρ m, R sd m] i j ] E m] = 0. (89) Since ρ m, R i j ] 1, sd m] E m] E R i j ] sd R i j ] (90) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

41 The inequality is referred as the Hansen-Jagannathan lower bound on the SDF. If i is the market portfolio and j is the risk free asset, we have: std m] E m] E R M R f ] std R M f ] = E R M R f ] std R M ] = = (91) We now can check whether this bound is satisfied for the standard CCAPM in which m = β(x t ) γ. Given ( E m] = β exp γµ x + 1 ) 2 γ2 σ 2 x = 0.96 if γ = 2, (92) for the HJ bound to be satisfied, the standard deviation of the SDF cannot be much lower than = (93) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

42 Given the information about x (lognormal distribution), it is straightforward to compute that std m] = 0.002! (94) which is much lower than what is required for reaching the HJ bound (0.355). Intuition: aggregate consumption is just too smooth and the MU of consumption doesn t vary suffi ciently to satisfy the HJ bound implied by asset data. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

43 Summary Reviewing the source of the failure of CCAPM in matching the data. Recall the original pricing equation ] u1 (c t+1 ) r M,t+1 r f,t+1 = β (1 + r f,t+1 ) covar t u 1 (c t ), r M,t+1 = (1 + r f,t+1 ) ρ β u ] 1(c t+1 ) u 1 (c t ), r M,t+1 std m] std R M Implications: the equity premium depends on The standard deviation of the SDF The standard deviation of the market portfolio The correlation between the two variables. Hence, for the US and other industrial countries, the problem with the CCAPMs is that aggregate consumption does not vary much at all. To make this model better fit the data, we must modify it in a way that will increase the standard deviation of the relevant SDF. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

44 Habit Formation Main objective: admit utility functions that exhibit higher rates of risk aversion and thus can translate small variations in consumption into a large variability of the SDF. One way to achieve this objective without causing the risk free rate puzzle (which is exacerbated if we simply assume a higher RRA γ): introducing some form of habit formation. Habit formation: the agent s utility today is determined not by her absolute consumption level, but rather by the relative position of her current consumption. The stock of habit can summarizes either her past consumption history (with more or less weight placed on distant or old consumption levels) or the history of aggregate/average consumption (summarizing in a sense the consumption habits of her neighbors: a keeping up with the Joneses effect). Utility of consumption is primarily dependent on departures from prior consumption history, either one s own or that of a social reference group. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

45 Habit Formation and The Equity Premium Puzzle The RA s preference takes the following form: u(c t, c t 1 ) = u(c t χc t 1 ) = (c t χc t 1 ) 1 γ, (95) 1 γ where χ 1 is a parameter. When χ = 1, the utility depends only on the deviation of current consumption c t from the previous period s consumption c t 1. Note that a general specification of habit formation can be written as where u(c t, x t ) = (c t χx t ) 1 γ, (96) 1 γ x t = (1 θ) x t 1 + c t 1. (97) Actual data indicate that aggregate consumption in the US and other developed countries is very smooth. This implies that (c t c t 1 ) is likely to be very small most of the time. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

46 For this specification, the agent s effective relative risk aversion reduces to R R (c t ) = c tu ( ) u = γ, (98) ( ) s t where s t = 1 χ (c t 1 /c t ). (99) With c t 1 c t, the degree of effective risk aversion R R (c t ) could thus be very high, even with a low γ, and the RA will appear as though he is very risk averse. This opens the possibility for a very high return on the risky asset. Note that when s t 0, R R (c t ). With habit, the SDF can be written as ( ) γ ( ) γ ct+1 st+1 SDF = β, (100) c t ( ) γ where st+1 s t could be very volatile and correlated. If so, habit formation might help explain the equity premium puzzle. See Campbell and Cochrane (JPE1999) for details. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56 s t

47 Epstein-Zin Recursive Utility Epstein and Zin (1989, 1991) propose a class of utility functions that allows each dimension to be parameterized separately. Specifically, preferences are defined recursively over current (known) consumption and a certainty equivalent of the next period s total utility: U t = U(c t, c t+1, c t+2, ) = W (c t, R t (U t+1 )), (101) where R t (U t+1 ) CE t+1 denotes the certainty equivalence in terms of period-t consumption of the uncertain total utility in the future periods. Consider the CES aggregator function: U(c t, CE t+1 ) = (1 β)c 1 1/ρ t + β (CE t+1 ) 1 1/ρ] 1 1 1/ρ. (102) R t (U t+1 ) = G 1 (E t G (U t+1 )]), (103) where W and G are increasing and concave. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

48 Asset Pricing Implications Consider a CRRA specification for the certainty equivalent (CE). For a random variable U t+1, the period-t + 1 onward utility, the certainty equivalent CE t (U t+1 ) (= CE t+1 ) is defined as follows: ] CE t (U t+1 )] 1 γ = E t U 1 γ when γ > 0 and γ = 1, (104) t+1 = CE t (U t+1 ) (= CE t+1 ) = (E t U 1 γ t+1 ]) 1/(1 γ) (105) (102) can be rewritten as: U(c t, CE t+1 ) = (1 β)c 1 γ θ t ] θ + β (CE t+1 ) 1 γ θ 1 γ, (106) where γ, ρ > 0, γ = 1, θ = 1 γ 1 1/ρ. If γ = 1/ρ (i.e., θ = 1) or if consumption is deterministic, we have the usual standard time-separable expected utility setting with the discount factor β and IES ρ (RRA γ = 1/ρ). Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

49 Asset Pricing Implications If γ = 1/ρ, recursive substitution to eliminate U t+j gives ] 1/(1 γ) U t = (1 β)e t β j c 1 γ t+j, (107) j=0 which is equivalent to the standard EU: E t j=0 β j c 1 γ t+j 1 γ. Epstein and Zin (1989,1991): derived the following asset pricing equation: ( ) ] 1/ρ θ ] ct θ E t β R j t+1 = 1, (108) c t R p t+1 where R p t+1 is the period-(t + 1) return on the market portfolio, and R j t+1 is the return on some asset in it. Note that when θ = 1 (γ = 1/ρ), the above pricing equation reduces to the standard time-separable CCAPM case. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

50 Note that in the EZ model, the SDF is a geometric average (with weights θ and 1 θ) of the SDF of the standard CCAPM ( ) (β ct+1 γ) c t and the SDF of the log case ( 1 R p ). t+1 Hence, two covariances matter for an asset s return pattern: 1 the covariance of asset returns with consumption growth; 2 the covariance of asset return with the return on the market portfolio. The covariance with consumption growth captures its risk across successive time periods (intertemporally, as in the standard CCAPM, while the covariance with the market portfolio captures its atemporal systematic risk (as in the standard static CAPM). Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

51 Proof. We first rewrite the definition of the recursive utility together with the certainty equivalent (104) as follows: U 1 γ t = (1 β)c 1 1/ρ t + β (E t which can be rewritten as U 1 γ t+1 ]) (1 1/ρ)/(1 γ) ] 1 γ 1 1/ρ, (109) V t (c t, E t V t+1 ]) = (1 β)c 1 1/ρ t + β (E t V t+1 ]) (1 1/ρ)/(1 γ)] 1 γ 1 1/ρ (110) Further, the budget constraint is A t+1 = R p t+1 (A t c t ), for any t. (111) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

52 Proof. (conti.) In an optimum, the optimal consumption decision can be characterized by the following consumption Euler equation V 1,t = E t R p t+1 V ] 2,tV 1,t+1 (112) where V i,t denotes the derivative of the aggregator function w.r.t its i th argument, evaluated at (c t, E t V t+1 ]). For the standard EU function, V 2,t is constant and equal to β: V 1,t = E t βr p t+1 V 1,t+1]. (113) As for optimal portfolio allocation, it must be the case that for any two distinct asset i and j which are held by our consumer, ] ] E t R i t+1 V 1,t+1 = Et R j t+1 V 1,t+1. (114) Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

53 Proof. (conti.) For any asset i which is held (which includes the market portfolio): ] V 1,t+1 E t V 2,t Rt+1 i = 1, V 1,t SDF = V 2,t V 1,t+1 = = = β = V 1,t V (θ 1)/θ t β (E t V t+1 ]) 1/θ 1] V (θ 1)/θ t+1 (1 β) (1 γ) c (1 γ)/θ 1 t+1 V (θ 1)/θ t (1 β) (1 γ) c (1 γ)/θ 1 t ] β (E t V t+1 ]) 1/θ 1] V (θ 1)/θ t+1 c (1 γ)/θ 1 t+1 ( ct+1 β c t ( ct+1 c t c (1 γ)/θ 1 t ) 1/ρ ( ) (1/ρ γ)/(1 γ) Vt+1 (11 E t V t+1 ] ) ] 1/ρ (1 γ)/(1 1/ρ) (R p (1 γ)/(1 1/ρ) 1 t+1). Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

54 Proof. (conti.) SDF = β ( ct+1 c t ) 1/ρ ] θ (R p t+1) θ 1. where V 1,t = θ (1 β)c (1 γ)/θ t +β (E t V t+1 ]) 1/θ = V (θ 1)/θ t (1 β) (1 γ) c (1 γ)/θ 1 t ; ] θ 1 (1 β) ((1 γ) /θ) c (1 γ)/θ 1 t and V 2,t = θ (1 β)c (1 γ)/θ t + β (E t V t+1 ]) 1/θ] θ 1 β (1/θ) (Et V t+1 ]) 1/θ 1 = (1 β)c (1 γ)/θ t + β (E t V t+1 ]) 1/θ] θ 1 β (Et V t+1 ]) 1/θ 1 = V (θ 1)/θ t β (E t V t+1 ]) 1/θ 1. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

55 Proof. (conti.) (115) is based on solving the following Bellman equation: { J (A t ) = max (1 β)c 1 1/ρ t + β (E t J(A t+1 )]) (1 1/ρ)/(1 γ)] 1 γ } 1 1/ρ, {c t,α t } (116) where J (A t ) is the value function. Since the procedure is complicated, I ignore it here. For those who are interested in the details, I encourage you to refer to Appendix A of Giovannini and Weil (1989, Risk aversion and intertemporal substitution in the capital asset pricing model, NBER working paper 2824) available at Luo, Y. (SEF of HKU) Macro Theory November 15, / 56

56 Implications If we assume that consumption growth and the return to the risky asset are jointly normally distributed, (108) can be written in the following log-linear form: ] E t r j t+1 r f + ω2 j 2 = θ ( ) ( ) ρ cov t r j t+1, c t+1 + (1 θ) cov t r j t+1, r p t+1 (117) which means that the expected excess return on the risky asset is a weighted average of the risky asset s covariance with consumption growth (divided by the IES ρ) and the asset s covariance with the market portfolio return. Luo, Y. (SEF of HKU) Macro Theory November 15, / 56