Chapter 6 Intertemporal Equilibrium Models CAPM assumes a myopic behavior of investors, who optimize the portfolio value at the next period only. An extension is to allow for dynamic portfolio management. This chapter relates asset prices to the consumption and investment and saving decisions of investors.

6.1Intertemporal equilibrium model Consumption: on a representative good with quantity C t and price q t at time t. Income: External income R t and α t shares on stocks with price S t. Budget constraint: q t C t = R t − (α t − α t−1 ) T S t. Intertemporal choice problem: Let δ be the subjective dis- count factor. An individual wants to maximize wrt α t the dis- counted expect utility E t ∞ δ j U (C t+j ), j=0 subject to the budget constraint.

Equivalent problem: With respect to portfolio allocation α t+j, max E t ∞ j=0 δjUδjU R t+j − (α t+j − α t+j−1 ) T S t+j q t+j. Note that α t appears in the first two terms. The first-order condition can be obtained by taking derivative wrt α t and setting it to zero, −E t U (C t ) · S t /q t + δE t U (C t+1 )S t+1 /q t+1 = 0. Euler condition: S t = E t [M t+1 S t+1 ], where M t+1 is Stochastic δU (C )q t q t+1 Remark: — The price of each asset S i,t = E t [M t+1 S i,t+1 ],1 = E t [M t+1 (1 + R i,t+1 )],

M t+1 /E t M t+1 = E t where R i,t+1 is the return at period t + 1. The price depends on *** inflation rate q t+1 /q t, which can sometimes be ignored. *** intertemporal rate of substitution δU (C t+1 )/U (C t ). — If there exists a risk-free asset with return r f, then E t M t+1 = (1 + r f,t+1 ) −1. An extension of this is the existence of “zero-beta” asset, which is defined as S 0,t satisfying: Cov t (S 0,t+1, M t+1 ) = 0. Hence, S 0,t = E t M t+1 S 0,t+1 = (E t M t+1 )(E t S 0,t+1 ). — The price of the i th stock is S i,t = E t S i,t r f,t+1 ∗ S i,t r f,t+1.

where E ∗ is the expectation with the conditional probability density of M t+1 /E t M t+1. Pricing probability or risk-neutral probability: The current asset price is the expected discounted future price with respect to the pricing probability dP t ∗ dP t = M t+1 E t M t+1.

6.2Comsumption-Based CAPM Individual demand: If assets and good prices are identical for all individuals, we have S t = E j,t [M j,t+1 S t+1 ]. — The stochastic discount factors vary across individuals. — The individuals have di ff erent expectations, assess to information various time preference and di ff erent income patters. — It is hard to aggregate individual demands for financial assets and consumptions goods to derive the condition of equilibrium of ag- gregate demand and supply. Representative investor and equilibrium condition:

Assume that there is a representative investor who has rational ex- pectation (The Euler condition): S t = E t [M t+1 S t+1 ], which can be applied at the aggregate level. Consumption-based CAPM(CCAPM): S t = E t q t q t+1 δ U (C t+1 ) U (C t ) S t+1. — q t is a retail price index, which can be ignored for simplicity. — C t is the aggregate consumption of physical goods and services. — E t, δ and U are those of the representative investor.

Note that S t = (E t M t+1 )(E t S t+1 ) + Cov t (M t+1, S t+1 ). If there exists a risk-free asset with return r f, then S t = (1 + r f ) −1 E t S t+1 + Cov(M t+1, S t+1 ). — Current price depends discounted present value and a risk pre- mium. — If an asset price evolves positively with the consumption growth (the discounted factor), investors need to pay a premium: The more assets are demanded for intertemporal transfers, the higher their prices.

Commonly-used utility functions: — Exponential utility function: U (C) = − exp(−AC). — Power utility: U (C) = C 1−γ −1 1−γ (when γ → 1, U (C) = log C), where γ is the coe ffi cient of relative risk aversion. For this utility function: U (C) = C −γ and S t = E t δ q t q t+1 C t+1 −γ C t S t+1 or δE t exp R t+1 − log q t+1 /q t − γ log C t+1 /C t = 1, where R t+1 is the vector of log-returns: R t+1 = log S t+1 /S t — componentwise division.

δ exp{E t Y i,t+1 + Var t (Y i,t+1 )} = 1. E t Y i,t+1 = − log δ − Var t (Y i,t+1 ). Remark: If Y = log X ∼ N (µ, σ 2 ), then X has a log-normal dis- tribution. Furthermore, EX = Ee Y = exp(µ + σ 2 /2) Let (1) Y t+1 = R t+1 − log(q t+1 /q t ) − γ log(C t+1 /C t ). If Y i,t+1 is normally distributed, with the power utility function, 1 2 or 1 2

This yields the Hansen-Singleton (1983) formula: assuming the inflation is negligible, where ∆C t+1 = log C t+1 /C t. Riskless return: For the riskfree asset, from (1), we have Var t (Y 0,t+1 ) ≈ Var(γ∆C t+1 ). Hence, by (2), r f,t+1 ≈ − log δ + γE t ∆C t+1 − γ22γ22 Var t (∆C t+1 ). It explains the factors that determine the riskless rate: the time preference rate (log-discount factor) “− log δ”, expected consumption

growth (incentive to borrow for future consumption) and volatility of the growth (the representative agent has precautionary motive for saving). Using the Hansen-Singleton formula (2), the expected excess return 1 E t (R i,t+1 − r f,t+1 ) ≈ − Var t (Y i,t+1 ) + Var t (Y 0,t+1 ) 2 1 = − Var t (R i,t+1 − γ∆C t+1 ) + Var t (−γ∆C t+1 ). 2 Excess return: E t [R i,t+1 − r f,t+1 ] = −σ i2 /2 + γσ ic, where σ i2 = Var t (R i,t+1 ) and σ ic = Cov(R i,t+1, ∆C t+1 ). If the asset is positively correlated with the consumption growth, its expected return tends to be higher.

Table 6.1: Moments of consumption growth and asset returns Variable Consumption growth Stock return CP return Stock − CP return Mean Standard deviation Correlation with consumption growth Covariance with consumption growth Consumption growth is the change in log real consumption of nondurables and services. The stock return is the log real return on the S&P 500 index since 1926, and the return on a comparable index from Grossman and Shiller (1981) before CP is the real return on 6-month commercial paper, bought in January and rolled over in July. All data are annual, Let us illustrate the above pricing formula by using the following empirical data, from The commercial papers with 6-month maturity are used the proxy of the riskless asset. The consumptions refer to those of nondurables and services. Expected excess return: log r E ∼ N (0.0418, )

Er E = exp( /2) = The real return has expected value about 6% with an SD 18%. Risk aversion coe ffi cient /2 = γ · = ⇒ ≈ 19. This explains somewhat the equity premium puzzle of Mehra and Prescott (1985): the average excess return on the US stock market is too high to be easily explained by standard asset pricing model. They considered value of γ up to 10, which is too small to fit the data.

Risk-free rate: = − log δ + γ · − γ22γ22 · = ⇒ δ ≈ This is called the riskfree rate puzzle by Weil (1989): — Given a positive average consumption growth, a low riskless interest rate and a positive rate of time preference, such investors would have a strong desire to borrow from the future. — A low riskless interest rate is possible in equilibrium only if investors have a negative rate of time preference that reduces their desire to borrow. In some other studies, γ needs to be much larger to fit the data. For

example, with γ = 29, δ < Mean-Variance frontier ∗ Value of a portfolio: W t = α Tt S t. Let W t+1 = α Tt S t+1. Then from CCAPM, S t = E t M t+1 S t+1, we have W t = E t M t+1 W t+1 = Cov t (M t+1, W t+1 ) + (E t M t+1 )(E t W t+1 ). If there is a risk free asset, E t M t+1 = (1 + r f,t+1 ) −1. Thus, (3) E t W t+1 − (1 + r f,t+1 )W t = Cov t (M t+1, W t+1 )(1 + r f,t+1 ). By the Cauchy-Schwartz inequality (E t W t+1 − (1 + r f,t+1 )W t ) 2 Var t (W t+1 ) ≤ Var t (M t+1 ) (E t M t+1 ) 2.

E ffi cient frontier: Excess gain (return) per unit risk (the Sharpe ratio) is bounded by Var t (M t+1 )/(E t M t+1 ) which is not always achievable. Question (Hansen and Jagannathan, 1991): Is there another stochastic discount factor M t ∗ +1 such that S t = E t [M t ∗ +1 S t+1 ], and the Sharpe ratio is attainable? Construction: Let M t ∗ +1 = α ∗ t T S t+1 where α ∗ t minimizes E t (M t+1 − α Tt S t+1 ) 2. Note that E t (M t+1 − α ∗ t T S t+1 )S t+1 = 0. Thus, E t M t ∗ +1 S t+1 = E t M t+1 S t+1 = S t. For any portfolio, its Sharpe ratio is bounded by (E t W t+1 − (1 + r f,t+1 )W t ) 2 Var t (W t+1 ) ≤ Var t (M t ∗ +1 ) (E t M t ∗ +1 ) 2.

Benchmark portfolio: M t ∗ +1 attains the maximum Sharpe ratio, called the benchmark portfolio. Excess return: E t W t+1 − (1 + r f,t+1 )W t = Cov t (M t ∗ +1, W t+1 )(1 + r f,t+1 ). Excess gain of benchmark portfolio: E t M t ∗ +1 − (1 + r f,t+1 )M t ∗ = Var t (M t ∗ +1 )(1 + r f,t+1 ). Hence E t W t+1 − (1 + r f,t+1 )W t = t∗t∗ Cov t (M t ∗ +1, W t+1 ) Var t (M +1 ) [E t M t ∗ +1 − (1 + r f,t+1 )M t ∗ ]. Beta pricing: The excess gain of any portfolio is β times the excess gain of the benchmark port- folios. In terms of return, E t R t+1 − r f,t+1 = t∗t∗ Cov t (M t ∗ +1, R t+1 ) Var t (M +1 ) [E t M t ∗ +1 − (1 + r f,t+1 )M t ∗ ]. Sharpe-Lintner CAPM can be regarded as a stochastic discount model with the discount factor M t ∗ +1 being the return of the market portfolio.