1 Asset Pricing: Bonds vs Stocks

Transcription

1 Asset Pricing: Bonds vs Stocks The historical data on financial asset returns show that one dollar invested in the Dow- Jones yields 6 times more than one dollar invested in U.S. Treasury bonds. The return on stocks is roughly 6% in real terms (i.e., after inflation) and the real return on bonds roughly % This difference in returns, around 5%, iscalledtheequity premium, i.e., the premium earned by a stock over a bond. What explains such large equity premium? Clearly, the explanation has to do with risk. Ifweabstractfrominflation (which affects both bonds and stocks in the same way), therateofreturnonstocksisvolatileandhencerisky,whileatreasurybondisasafe asset that delivers exactly what promised (except in the case of sovereign default, which is close to zero for the United States). The interesting question is: what are the features of this higher risk of stock returns that explain the premium on equity over bonds? In what follows, we try to answer this question.. A simple two period economy Consider the two-period problem of a household with CRRA preferences who faces some uncertainty about his period-two income 2 In particular, suppose that there are two states of the world in period 2, = ª, denoting the high state and the low state, with 2 2 For example, an expansion and a recession. The probability of the two states occurring are and with + = Hence, we can write preferences as + X () 2 () Compute now the coefficient of relative risk aversion for our utility function 00 () 0 () = = Hence measures how averse the individual is to consumption fluctuations. Recall that measures the elasticity of intertemporal substitution, i.e., the willingness of agents to substitute intertemporally. With these preferences one single parameter governs both risk aversion and elasticity of substitution, and both are independent of the level of consumption..2 Pricing the bond In period, the household can save for the next period into a risk-free bond, i.e. an asset whose price at time =is normalized to one which pays =+ for sure next

2 period, exactly like a Treasury bond. Budget constraints for periods = 2 are: + 2 = 2 = 2 + 2,withprob. 2 = 2 + 2,withprob. Which one of the two budget constraints for time 2 will arise depends on the realization of the state at time 2 It is useful to simplify this problem by substituting out savings 2 using the statecontingent period-two budget constraints into the period-one budget constraint: + 2 () + 2 () () The constrained maximization problem can be solved with the usual Lagrangian techniques: ( 2 ()) = + X " () 2 () µ + () + 2 () Thesolutionofthisproblemis # 2 () ( ) : ( 2 ()) : =0 = X () () = [ ()] 2 () =0 () 2 () = () () Adding up the two FOC s with respect to 2 () we obtain 2 () = [ ()] Note that the return on bonds is outside the expectation because it is certain. Combining the FOC s, we obtain the familiar Euler equation = 2 () Rearranging, we obtain an expression for the return on the safe bond: = ³ = 2 () h i (2) 2 () where is the marginal utility of consumption at Hence, given the expected growth in consumption (weighted by the risk-aversion coefficient ), we can determine and price the bond correctly. Finally, note that all we did so far is using the familiar Euler equation to derive an expression for the return on the Treasury bond. 2

3 .3 Pricing the Stock Now, suppose that the household invests in equity (or stocks), i.e., a risky asset, whose return () is contingent on the state of the world next period, exactly like income. Then, the household problem becomes max { 2 ()} + X () 2 () + 2 = 2 = 2 + 2,withprob. 2 = 2 + 2,withprob. The constrained maximization problem can be solved with the usual Lagrangian techniques: ( 2 ()) = + X " () 2 () µ + () + 2 () () # 2 () () Thesolutionofthisproblemis ( ) : ( 2 ()) : =0 = X () () = [ ()] 2 () =0 () 2 () = () () () Adding up the two FOC s with respect to 2 () and rearranging the two above: = ()( 2 ()) or " µ # 2 () () = (3) where now () cannotbemovedoutoftheexpectation, because it is no longer certain: it s a random variable..4 The Equity Premium Recall the definition of the covariance between the pair of random variables ( ) ( ) = () () () or () = () ()+ ( ) 3

4 Apply this definition to equation (3) with = () and = " µ # " 2 () () = [ ()] µ 2() ³ 2 () to obtain: # à µ + ()! 2() = and rearranging [ ()] = µ () ³ 2 () ³ 2 () which gives an expression for the expected return on the stock. Dividing both sides by the formula for the risk-free rate in equation (2) gives à µ [ ()] = ()! 2 () and rearranging: [ ()] à µ! = 2 () () (4) where the left hand side is the equity-premium, i.e., the excess expected returns of stocks over the return of a risk-free bond. Note first that to have a positive equity premium, () should vary across states. If the return on the stock is constant across states, that covariance is zero and the equity premium is zero because the stock is nothing else than a safe asset, exactly like the bond. Hence, a necessary condition for a positive equity premium is that must vary across states. But volatility of is not a sufficient condition for a positive equity premium. To generate a positive equity premium, the covariance should be negative, i.e., the return on the stock () should comove negatively with the marginal rate of substitution: suppose that in states where the marginal utility of consumption is high (and consumption is low), the stock pays very little and in states where the marginal utility of consuming is low (and consumption is high) the stock pays a lot. Households do not like this type of asset because instead of providing insurance for stochastic income fluctuations it exacerbates income fluctuations. So in order to hold such asset households require ³ a high ³ return. Put differently, to get a high positive equity premium the () 2 () should be positive. Why? In the high state the individual receives a positive income shock (and therefore has high consumption), and in the low state the individual receives a negative income shock (and therefore has low consumption). Ideally, a risk averse household would like an asset that pays a high (low) return in the low (high) state, so she can smooth consumption across states. Therefore, to hold an asset that instead has high 4

5 return precisely in the high state (like equity), the household requires a large premium over the bond. Hence, the equity premium is not only a result of the fact that the stock is volatile while the bond is not. The crucial feature that explains why the stock requires a higher return is that the return of stock is positively correlated with households income shocks. For example, if this correlation was negative, the stock could command a lower return than the bond because it helps insuring the agent ³ more ³ than a non-state contingent bond. Note also that for any given value of () 2 () a large coefficient of riskaversion will amplify consumption fluctuations and increase the covariance, raising the equity premium. An agent who is more risk averse will demand a higher premium to hold stocks. In the data, stocks yield high returns during expansions which happen to be the periods where income and consumption is high: this explains the fact that, empirically, the equity premium is positive. However, since empirically consumption is very smooth, i.e., ( 2 () ) does not vary a lot over time, one needs a huge risk aversion coefficient, around =50or higher, to generate an equity premium of 5%.5 Hansen-Jagannathan Bounds Consider the expression for the equity premium [ ()] = ( () 2 ()) (5) where we have used the definition of the MRS between today and state tomorrow (net of the discount factor which we moved outside the covariance) µ 2 () 2 () which, in the context of asset pricing, is called the stochastic discount factor and is the key object needed to price assets of any type. Using the definition of the correlation coefficient and rearranging ( () 2 ()) = ( () 2 ()) ( ()) ( 2 ()) ( () 2 ()) = ( () 2 ()) ( ()) ( 2 ()) into equation (5) we arrive at [ ()] = ( () 2 ()) ( ()) ( 2 ()) = ( () 2 ()) ( ()) ( 2 ()) 5

6 Now, because, the correlation is defined between - and, and because, wecan construct theoretical bounds for the equity premium ( ()) ( 2 ()) [ ()] ( ()) ( 2 ()) In particular, the equity premium is bounded above by ( ()) ( 2 ()) This result is useful to evaluate what is the maximum potential of a particular theory (summarized by the dynamics of the stochastic discount factor ) for generating a large equity premium. Now, let s do a back of the envelope calculation. ³ Historically, ( ()) = 020 and with =(i.e., log utility) ( 2 ()) = 2 () =004 therefore [ ()] = 0008 i.e. less that % whereas the equity premium in the data is at least 5%. To account for a higher equity premium, we need to crank up the risk aversion coefficient. One problem in doing this is that, with these CRRA preferences, a very high risk aversion (say =50) corresponds to an extremely low elasticity of intertemporal substitution, i.e. the sensitivity of consumption growth to the (risk-free) interest rate. But empirical estimates set this elasticity between /2 and. One solution to this dilemma is to free up the theory from the straightjacket of the CRRA preferences. There exist preferences ( recursive preferences ) which have two separate parameters, one measuring risk aversion and one measuring the elasticity of intertemporal substitution, which can be consistent with both a high response of consumption growth to the interest rate and a high equity premium. 6