Asset Pricing and Equity Premium Puzzle 1 E. Young Lecture Notes Chapter 13 1 A Lucas Tree Model Consider a pure exchange, representative household economy. Suppose there exists an asset called a tree. Trees do not depreciate and each household is endowed with 1 tree. Each period a stochastic quantity of fruits (dividends) are dropped from a tree. Fruits are perishable. We call a stock a claim to the stream of fruits from a given tree. With a representative household there can be no trade in trees, so we will search for prices that induce the household to hold 1 tree. Denote the price of these trees by p t, a household s current holding of trees by s t, and current dividends by d t.
2 A household solves the following problem subject to v (s t,d t )=max s t+1,c t {u(c t )+βe t [v (s t+1,d t+1 d t )]}, c t + p t s t+1 =(p t + d t ) s t. The goods and asset market clearing conditions are c t = d t, s t =1. The Euler condition, together with the market clearing conditions, is given by p t = E t β u0 (d t+1 ) u 0 (d t ) (p t+1 + d t+1 ). Note that M t+k t = β k u 0 (d t+k ) u 0 (d t ) is the intertemporal marginal rate of substitution between periods t and t + k, or the pricing kernel.
3 Iterating the pricing difference equation forward, p t = E t M t+1 t (p t+1 + d t+1 ) = E t M t+1 t Et+1 M t+2 t+1 (p t+2 + d t+2 ) + d t+1 = E t Et+1 M t+1 t M t+2 t+1 (p t+2 + d t+2 ) + M t+1 t d t+1. Applying the law of iterated expectations p t = E t M t+2 t (p t+2 + d t+2 )+M t+1 t d t+1 = E t M t+2 t d t+2 + Et M t+2 t p t+2 + M t+1 t d t+1 =... Xk = E t M t+k t d t+k + Et j=1 Mt+j t p t+j Then, we have hx i p t = E t j=1 Mt+j t p t+j, (1)
4 if lim E t M t+k t d t+k =0, k where the transversality condition implies that bubbles do not exist. (1) says that the price of a stock is the discounted expected future dividends where the discount factor is the intertemporal marginal utility of consumption between t and t + k. To examine the equity premium puzzle, we next introduce some risk-free bonds into this economy. Therefore, households solve the problem subject to v (s t,b t,d t )= max {u(c t )+βe t [v (s t+1,b t+1,d t+1 d t )]}, s t+1,b t+1,c t The bonds market clearing condition is c t + p t s t+1 + q t b t+1 =(p t + d t ) s t + b t. b t =0.
The Euler conditions, together with the market clearing conditions, are given by p t = E t β u0 (d t+1 ) u 0 (d t ) (p t+1 + d t+1 ), (2) u 0 (d t+1 ) q t = E t. u 0 (3) (d t ) Define the (net) returns for equity and bond Thus, the equity premium is given by R e t+1 = p t+1 + d t+1 p t 1, R b t+1 = 1 q t 1. R e t+1 R b t+1. 5
6 We then specify the utility function and the stochastic process of the dividends. Let u(c t )= c1 σ t 1 σ, and d t+1 = g t+1 d t, where g t+1 G = {λ 1,λ 2 },andd t+1 follows a 2-state Markov process Insert these into the Euler equation (2), p t (d t,g t )=E t "β Pr{g t+1 = λ j g t = λ i } = φ ij. µ dt+1 d t σ (p t+1 (d t+1,g t+1)+d t+1)#, (4) which is a linear stochastic difference equation in the unknown function p t (d t,g t ). We guess a solution of the form p t (d t,g t )= ½ ω1 d t g t = λ 1 ω 2 d t g t = λ 2.
If g t = λ 1 (the economy is at state 1 today), then by (4) ω 1 d t = E t β (gt+1 ) σ (p t+1 (d t+1,g t+1 )+g t+1 d t ) = φ 11 β (λ1 ) σ (ω 1 d t+1 + λ 1 d t ) +(1 φ 11 ) β (λ 2 ) σ (ω 2 d t+1 + λ 2 d t ). Then, ω 1 = φ 11 β (λ1 ) σ (ω 1 λ 1 + λ 1 ) +(1 φ 11 ) β (λ 2 ) σ (ω 2 λ 2 + λ 2 ) i h i = φ 11 hβ (λ 1 ) 1 σ (ω 1 +1) +(1 φ 11 ) β (λ 2 ) 1 σ (ω 2 +1). (5) Similarly, if g t = λ 2 (the economy is at state 2 today), then i h i ω 2 = φ 21 hβ (λ 1 ) 1 σ (ω 1 +1) +(1 φ 21 ) β (λ 2 ) 1 σ (ω 2 +1). (6) We can solve for ω 1 and ω 2 from (5) and (6). To quantify the equity premium, we need to know the values of these model parameters φ 11,φ 21,β,λ 1,λ 2,σ, which is a simplified exercise of calibration. 7
8 (1) Set λ 1 =1+μ + δ and λ 2 =1+μ δ, whereμ is to match the long-term growth rate of aggregate consumption, i.e., μ =0.018 from data, and δ is to match the standard deviation of growth rate of aggregate consumption, i.e., δ = 0.036 from data. (2) Set φ 11 = φ 21 = φ (the durations of good states and bad states of dividend growth are equally the same). Let φ match the 1st-order autocorrelation in the growth rate of aggregate consumption,i.e., φ =0.957. (3) Finally, β and σ are parameters in the utility function. We can directly assign values that are found in the literature if we simply want to simulate the impulse responses of the model. But the purpose here is to see how the theoretical model can match the observed equity premium. Therefore, we will allow β and σ to vary, and see if the values of β and σ that match the observed equity premium are within reasonable ranges.
We now calculate asset returns. For the net realized returns on equity,given today s state is i and tomorrow s turns out to be j, r e ij = p t+1 + d t+1 p t 1= g t+1d t + ω j d t+1 ω i d t 1 = λ i + ω j λ j ω i 1. The expected net return given that today s state being i (g t = λ i )is R e i = φ i1 r e i1 +(1 φ i1 ) r e i2. For a symmetric Markov process the stationary distribution implies that consumption growth (or dividend growth) spends half of its time in each state t 1 φ φ 0.5 0.5 lim =. t φ 1 φ 0.5 0.5 Thus, the unconditional expectation of returns on equity is E [R e ]=0.5R e 1 +0.5R e 2. 9
10 Similarly, the realized bond price, given today s state is i and tomorrow s turns out to be j, is r b ij = 1 q i 1. The expected net return given that today s state being i is R b i = φ i1 r b i1 +(1 φ i1 ) r b i2 = 1 q i 1= = 1 φ i1 β (λ 1 ) σ σ 1. + φ i2 β (λ 2 ) The unconditional expectation of returns on bonds is E R b =0.5R b 1 +0.5R b 2. 1 ³ 1 σ E t β dt+1 d t gt = λ i
We allow the combinations of values for (σ, β) to vary within the rectangle [0, 10] [0, 1], which are considered reasonable in the literature. It turns out that largest equity premium we can find is about 0.3% (which occur when σ =10and β =0.5); however, the historical equity premium is 6%. Clearly the model has difficulties in explaining the size of the equity premium with reasonable ranges of parameter values. On the other hand, if we raise the equity premium implied by the model, we unfortunately also increase the risk-free return. For example, for the premium to be the largest value 0.3%, the bond return will turn out to be 4%. But the data shows that the risk-free bond return is only 1%. The former problem is called Risk Premium Puzzle and the latter Risk-Free Rate Puzzle. 11
Why does the model fail to match the high risk premium and the low risk-free rate at thesametime? Given our utility function u(c t )= c1 σ t 1 σ, the Arrow-Pratt coefficient of relative risk aversion is given by u00 (c)c u 0 (c) = σc σ 1 c c σ = σ. This means that how much a consumer will demand as compensation for holding stocks is related directly to σ. Thus, that is why we could raise the equity premium by increasing σ. 12
Unfortunately, the intertemporal elasticity of substitution (IES), which measures the willingness to save, turns out to be 1/σ for this utility function. To see this, recall the Euler equation " µ σ p t = E t β u0 (d t+1 ) u 0 (d t ) (p ct+1 t+1 + d t+1 ) = E t β (p t+1 + d t+1)#. Ignoring the expectation operator and taking logs µ ct+1 ln = 1 σ ln β + 1 σ ln er t+1, c t where er t+1 =(p t+1 + d t+1 ) /p t is the gross returns on investing in equity. Then ³ d ln ct+1 c t IES = 1 d ln er t+1 σ. This says that, to raise the equity premium by increasing σ, we lower the willingness to save, which then requires the risk-free rate to rise in order to induce households to hold bonds in equilibrium. c t 13
14 On the other hand, if we allow combinations of values for (σ, β) to go beyond the rectangle [0, 10] [0, 1], say,β =1.12 and σ =20, we obtain an equity premium of 7% and a risk-free return on 0.6%. However, using empirically implausible parameter values is not a good resolution to the puzzle. Efforts have been made to resolve this puzzle. Some introduce incomplete markets for risk, heterogeneous agents, transactions costs, small possibilities for disaster states, etc..