Asset Pricing Week 6 PDF

Week 6: Asset Pricing Guanglian Hu University of Sydney Overview ▶ State Space Model ▶ Consumption-Based Asset Pricing: Theory State Space Model Suppose next period there are S states of the world (state of nature), which are labeled by s, s = 1, 2, 3,...S, representing...

Week 6: Asset Pricing Guanglian Hu University of Sydney Overview ▶ State Space Model ▶ Consumption-Based Asset Pricing: Theory State Space Model Suppose next period there are S states of the world (state of nature), which are labeled by s, s = 1, 2, 3,...S, representing different economic conditions. One of the most important frameworks for modeling investments under uncertainty, also known as the time-state model of Arrow (1964) and Debreu (1959) State Space Model ▶ State s occurs with probability πs , where 0 < πs < 1 and PS 1 πs = 1. [π1 , π2 , π3 ,...,πS ] are physical probabilities. ▶ Consider Arrow-Debreu securities that pay off $1 if state of nature s occurs next period and zero otherwise. Suppose we have one AD security for each state of the world. ▶ For example, the AD security for state 1 has the following payoff vector 1    0  0    .    .    .     0 ▶ Let ws be today’s price of such claim. Basic arbitrage arguments dictate that 0 < ws < 1. The prices of AD securities [w1 , w2 , w3 ,...,wS ] are also referred to as state prices. State Space Model ▶ Once we know the prices of Arrow Debreu securities, we will be able to price any risky asset with a random payoff Y :   y1  y2  .    Y =  .  yS ▶ Note that the random payoff Y can be replicated by buying y1 units of AD claim for state 1, y2 units of AD claim for state 2, y3 units of AD claim for state 3....... 1 0 0        0   1   0   0   0   0        .  + y2 .  +..... + yS .        y1       .  .  .  .  .  .        0 0 1 State Space Model ▶ The current price of Y , by arbitrage, must be given by adding up the values of its component pieces: S X p= ws ys (1) 1 ▶ We can also use AD securities to infer the price of a risk-less bond (B) that pay off $1 regardless of which state occurs. S X B= ws 1 State Prices and SDF ▶ Rewrite equation (1), S X ws p= πs ys (2) πs 1 ▶ Denote ws πs as Ms , we have S X p= πs Ms ys = E (Ms ys ) (3) 1 ▶ Ms is called stochastic discount factor (SDF) or pricing kernel and equal to state prices divided by physical probabilities. The value of an asset is equal to the expected value of discounted future cash flow. State Prices and Risk Neutral Probabilities ▶ Let ws ws π̃s = = PS (4) B 1 ws ▶ Note that π̃s have all of the properties of probabilities: 0 < π̃s < 1 and S1 π̃s = 1. P ▶ Using equation (4) and equation (1) on the previous slide, we can rewrite the price of Y as S S X ws X p=B ys = B π̃s ys B 1 1 State Prices and Risk Neutral Probabilities ▶ Lastly, using B = 1 1 + rf , we have S 1 X 1 p= π̃s ys = Ẽ (ys ) 1 + rf 1 + rf 1 ▶ The value of a random cash flow Y is calculated as the expected value of cash flow under the risk neutral probabilities, discounted by the risk-free rate. ▶ Note that S 1 X p ̸= πs ys 1 + rf 1 Summary ▶ We consider the state space model of Arrow (1964) and Debreu (1959) in which there exists a set of elementary contingent claims (AD securities), each paying one dollar in one specific state of nature and nothing in any other states. ▶ In this set-up, we can break the payoffs or cash flows of any risky asset down ’state-by-state’, and then pricing it as a bundle of AD securities. ▶ Valuation can be done under both physical and risk neutral probability measure. ▶ Stochastic discount factor equals state prices divided by physical probabilities ▶ Risk neutral probabilities are state prices (the prices of AD securities) divided by the price of the risk-free asset. Inferring State Prices From Option Prices ▶ State prices can be inferred from prices of options with different strikes. ▶ Suppose the state of nature is summarized by the value of the market portfolio which has a discrete probability distribution with possible values of: M = $1.00, $2.00,... , $N. ▶ Denote the vector of payoffs of a European call option on the market with a strike price of X as C(X); its price today will be denoted as c(X) Market Portfolio C(0) C(1) C(2) M=1 1 0 0 M=2 2 1 0 M=3 3 2 1............ M=N N N-1 N-2 Inferring State Prices from Option Prices ▶ The payoffs of AD securities can be replicated by combining call options on the market portfolio with various strike prices. ▶ For example, the claim having a payoff of $1.00 only if M(T) = $1 may be constructed as [C(0) - C(1)] - [C(1) - C(2)] 1 0 1        1   1   0   1   1   0        . −.  = .        .  .  .        .  .  .        1 1 0 ▶ In general, replicating the claim giving $1 only if the market is M consists of one long call with X =M-1,one long call with X=M+1, and two short calls with X=M (if you recall, this is butterfly spread). Inferring State Prices from Option Prices: Example ▶ For example, if N = 3 (only three states) C(0) C(1) C(2) M=1 1 0 0 M=2 2 1 0 M=3 3 2 1 ▶ Suppose the prices of calls are c(0) = $1.7, c(1) = $0.8, and c(2) = $0.1 ▶ Then the respective state prices are: ▶ P(M = 1) = $0.2, c(0)+c(2)-2c(1) ▶ P(M = 2)=$0.6, c(1)-2c(2) ▶ P(M = 3) = $0.1, c(2) ▶ Also, the price of a riskless discount bond paying $1.00 would be $0.2 + $0.6 + $0.1 = $0.9 Consumption Based Asset Pricing Consumption-Based Asset Pricing ▶ Equity Risk Premium Puzzle ▶ Risk-Free Rate Puzzle ▶ Hansen-Jagannathan Bound Intertemporal Choice Problem ▶ Consumption-based asset pricing models start with the intertemporal choice problem of an investor, who wants to maximize the expected value of the life time utility: ∞ X max Et [ β j U (Ct +j )] j =0 where β is the time discount factor and U is a time-separable utility function. ▶ This investor can trade freely in some asset i and obtain a gross simple rate of return 1 + Ri,t +1 on the asset held from time t to time t + 1. ▶ First order condition or Euler equation describing the investor’s optimal consumption and and portfolio plan is: U ′ (Ct ) = Et [βU ′ (Ct +1 )(1 + Ri,t +1 )] Euler Equation ▶ How to interpret the Euler equation? U ′ (Ct ) = Et [βU ′ (Ct +1 )(1 + Ri,t +1 )] (5) ▶ The left hand side of equation (5) is the marginal utility cost of consuming one dollar less at time t. ▶ The right hand side is the expected marginal utility benefit from investing the dollar in asset i at time t, selling it at time t + 1 for 1 + Ri,t +1 dollars, and consuming the proceeds. ▶ The marginal cost equals the marginal benefit. The Fundamental Asset Pricing Equation ▶ The first order condition of the optimal consumption and portfolio choice problem gives the fundamental equation in asset pricing. ▶ Dividing equation (5) by U ′ (Ct ) yields: U ′ ( Ct + 1 ) 1 = Et [ β (1 + Ri,t +1 )] U ′ ( Ct ) ′ ▶ Denoting Mt +1 = β UU(′C(Ct +)1 ) , we obtain the fundamental t equation in asset pricing: 1 = Et [Mt +1 (1 + Ri,t +1 )] ▶ Mt +1 is the intertemporal marginal rate of substitution of the investor, also known as the stochastic discount factor (SDF), pricing kernel, or simply marginal utility. The Fundamental Asset Pricing Equation ▶ Mt +1 is a random variable and always positive. ▶ High Mt +1 corresponds to low consumption. Marginal utility is high when the level of the consumption is low. ▶ While the expected returns (e.g., Et [1 + Ri,t +1 ]) can vary across time and assets, the expected discounted return should always be the same, 1. 1 = Et [Mt +1 (1 + Ri,t +1 )] The Fundamental Asset Pricing Equation ▶ The above derivation for 1 = Et [Mt +1 (1 + Ri,t +1 )] assumes the existence of an investor maximizing a time-separable utility function, but in fact the equation holds more generally. ▶ The existence of a positive stochastic discount factor is guaranteed by the absence of arbitrage opportunities in markets. ▶ Different asset pricing models use different Mt +1. Think of Mt +1 as an index for bad times. The Fundamental Asset Pricing Equation: Implications ▶ We just derived the fundamental equation in asset pricing: 1 = Et [Mt +1 (1 + Ri,t +1 )] (6) ▶ One can manipulate this equation to get a pricing formula. Note that Xi,t +1 1 + Ri,t +1 = Pi,t where Xi,t +1 is the payoff of the asset at time t + 1. ▶ This implies: Xi,t +1 1 = Et [ M t + 1 ] ⇒ Pi,t = Et [Mt +1 Xi,t +1 ] Pi,t ▶ The value of an asset is equal to the expected value of discounted future payoffs (by Mt +1 ). The Fundamental Asset Pricing Equation: Implications ▶ Explore the implications of the fundamental equation in the return space. ▶ Using E (XY ) = E (X )E (Y ) + Cov (X , Y ), Et [Mt +1 (1 + Ri,t +1 )] = Et [Mt +1 ]Et [(1 + Ri,t +1 )] + Covt [Ri,t +1 , Mt +1 ] ▶ Substituting into (6) yields: 1 = Et [Mt +1 ]Et [(1 + Ri,t +1 )] + Covt [Ri,t +1 , Mt +1 ] ▶ Rearranging gives: 1 − Covt [Ri,t +1 , Mt +1 ] Et [(1 + Ri,t +1 )] = (7) Et [Mt +1 ] ▶ An asset with a high expected return must have low covariance with the stochastic discount factor. Such an asset tends to have low returns when investors have high marginal utility. It is risky in that it fails to deliver wealth in bad times when wealth is most valuable to investors. Investors therefore demand a higher expected return to hold it. The Fundamental Asset Pricing Equation: Implications ▶ Equation (7) must hold for any asset, including a riskless asset whose return has zero covariance with the stochastic discount factor (or any other random variable). 1 1 + Rf ,t +1 = (8) Et [Mt +1 ] ▶ The above equation can be used to rewrite equation (7): Et [(1 + Ri,t +1 )] = (1 + Rf ,t +1 )(1 − Covt [Ri,t +1 , Mt +1 ]) (9) ▶ An asset with a negative (positive) covariance with the stochastic discount factor will earn an expected return that is higher (lower) than the risk-free rate Expected Return - Beta Representation ▶ The fundamental equation can be rewritten as a CAPM-type expression. ▶ Start with 1 = Et [Mt +1 ]Et [(1 + Ri,t +1 )] + Covt [Ri,t +1 , Mt +1 ] ▶ Dividing both sides by Et [Mt +1 ] and using 1 + Rf ,t +1 = 1 : Et [Mt +1 ] Covt [Ri,t +1 , Mt +1 ] Et [(1 + Ri,t +1 )] − (1 + Rf ,t +1 ) = − Et [Mt +1 ] ▶ It follows Covt [Ri,t +1 , Mt +1 ] Vart (Mt +1 ) Et [Ri,t +1 ] − Rf ,t +1 = (− ) Vart [Mt +1 ] Et [Mt +1 ] Expected Return - Beta Representation ▶ Equivalently, we have Et [Ri,t +1 ] − Rf ,t +1 = βi,t λt where Covt [Ri,t +1 , Mt +1 ] Vart (Mt +1 ) βi,t = ; λt = − Vart [Mt +1 ] Et [Mt +1 ] ▶ The expected return should be proportional to beta in a regression of that returns on the SDF M. ▶ λt is price of risk and βi,t is quantity of risk ▶ Note that λt is the same for all assets while the βi,t varies from asset to asset Imposing Log-normality ▶ Assume that the joint conditional distribution of asset returns and the stochastic discount factor is lognormal and homoskedastic ▶ When a random variable X is conditionally lognormally distributed, it has the following convenient property that 1 log Et X = Et log X + Vart log X 2 ▶ With joint conditional lognormality and homoskedasticity of asset returns and consumption, we can take logs of equation (6): 1 0 = Et ri,t +1 + Et mt +1 + (σi2 + σm 2 + 2σi,m ) (10) 2 where mt +1 = log(Mt +1 ), ri,t +1 = log(1 + Ri,t +1 ), σi2 is the unconditional variance of log returns, σm 2 is the unconditional variance of the log stochastic discount factor, and σi,m is the unconditional covariance between log returns and stochastic discount factor Imposing Log-normality ▶ Consider the log return on the risk-free asset: 2 σm rf ,t +1 = −Et mt +1 − (11) 2 which is the log counterpart of equation (8) ▶ Subtracting equation (11) from equation (10) yields an expression for the expected excess return of a risky asset over the riskless rate: σi2 Et [ri,t +1 ] − rf ,t +1 + = −σi,m (12) 2 which is the log counterpart of equation (9) ▶ The right hand side of equation (12) says that the risk premium is determined by the negative of the covariance of the asset with the stochastic discount factor. ▶ An asset with a negative covariance with the stochastic discount factor must have higher expected returns. Jensen’s Inequality σi2 Et [ri,t +1 ] − rf ,t +1 + = −σi,m 2 σi2 ▶ What is with the 2 term? σi2 ▶ 2 arises from the fact that we are describing expectations of log return. σ2 log Et [1 + Ri,t +1 ] = Et [ri,t +1 ] + i 2 ▶ Two ways to write equity risk premium equation: σi2 Et [ri,t +1 ] − rf ,t +1 + = −σi,m 2 log Et [1 + Ri,t +1 ] − log(1 + Rf ,t +1 ) = −σi,m International Stock Market Data International Consumption and Dividend Data Data Summary ▶ Stock markets have delivered high average returns with high standard deviations ▶ Short term debt (risk-free asset) has delivered low returns with low standard deviation ▶ Consumption growth is smooth with low standard deviation Consumption Based Asset Pricing with Power Utility ▶ Assume that there is a representative agent who maximizes a time-separable power utility function defined over aggregate consumption Ct : C 1−γ − 1 U (Ct ) = t 1−γ where γ is the coefficient of relative risk aversion. With constant relative risk aversion(CRRA), fraction allocated to risky asset is independent of wealth. ▶ Power utility implies: U ′ (Ct +1 ) Ct +1 −γ U ′ (Ct ) = Ct−γ ; Mt +1 = β ′ = β( ) ; U ( Ct ) Ct ▶ The log of the stochastic discount factor is mt +1 = log β − γ∆ct +1 where ∆ct +1 = ct +1 − ct and ct = log(Ct ). Consumption Based Asset Pricing with Power Utility ▶ Recall from equation (10) 1 0 = Et ri,t +1 + Et mt +1 + (σi2 + σm 2 + 2σi,m ) 2 ▶ With power utility, mt +1 = log β − γ∆ct +1 and the above equation becomes: 1 0 = Et ri,t +1 + log β − γEt ∆ct +1 + (σi2 + γ 2 σc2 − 2γσi,c ) 2 ▶ σc2 is the unconditional variance of log consumption growth (∆ct +1 ), and σi,c is the unconditional covariance between log stock returns and consumption growth Cov (ri,t +1 , ∆ct +1 ) Consumption Based Asset Pricing with Power Utility ▶ The risk-free rate in equation (11) now becomes: γ 2 σc2 rf ,t +1 = − log β + γEt ∆ct +1 − (13) 2 ▶ Risk-free rate is high when β is low (investors are impatient) ▶ Risk-free rate is linear in expected consumption growth, with slope coefficient equal to the coefficient of relative risk aversion. ▶ The conditional variance of consumption growth has a negative effect on the riskfree rate which can be interpreted as a precautionary savings effect. Consumption Based Asset Pricing with Power Utility ▶ The equity risk premium in equation (12) now becomes: σi2 Et [ri,t +1 ] − rf ,t +1 + = γσi,c (14) 2 ▶ Equation (14) says that the risk premium on any asset is the coefficient of relative risk aversion times the covariance of the asset returns with consumption growth ▶ Intuitively, an asset with a high consumption covariance (e..g, σi,c ) tends to have low returns when consumption is low, that is, when the marginal utility of consumption is high. Such an asset is risky and commands a large risk premium. The Equity Risk Premium Puzzle The Equity Risk Premium Puzzle ▶ RRA(1) uses equation (14) to estimate the risk aversion, dividing the adjusted average excess return (aere ) by the estimated covariance cov (ere , ∆c ). ▶ Using US as an example, RRA(1)= 0.08071/0.0003354=240. Please note that returns and standard deviations are reported in percent so cov (ere , ∆c ) = σ (ere ) ∗ σ (∆C ) ∗ ρ(ere , ∆c ) = 0.15271 ∗ 0.01071 ∗ 0.205 ▶ RRA(2) assumes that the correlation is equal to 1 to calculate the implied risk aversion (RRA(2)= 0.15271∗0.01071∗1 0.08071 = 49). It indicates the extent to which the equity premium puzzle arises from the smoothness of consumption rather than the low correlation between consumption and stock returns. ▶ Most economists believe risk aversion γ should be less than 10. ▶ Power utility model can only fit the equity premium if the coefficient of relative risk aversion is very large. The Risk-Free Rate Puzzle ▶ One response to the equity premium puzzle is to consider larger values for the coefficient of relative risk aversion. ▶ Recall equation (13) shows that the riskless interest rate is γ 2 σc2 rf ,t +1 = − log β + γEt ∆ct +1 − 2 ▶ High values of γ would imply high values of γEt ∆ct +1. This can be reconciled with low interest rates only if the time discount factor β is close to or even greater than one, corresponding to a low or even negative rate of time preference. Hansen-Jagannathan Bound ▶ Recall from equation (12) the expected excess return on any risky asset is given by σi2 Et [ri,t +1 ] − rf ,t +1 + = −σi,m 2 ▶ Expanding the covariance term, σi,m = σi σm ρi,m where ρi,m is the correlation between asset return and stochastic discount factor. ▶ Since ρi,m ≥ −1, −σi σm ρi,m ≤ σi σm , which then implies −σi,m ≤ σi σm. Hansen-Jagannathan Bound ▶ A little manipulation yields the Hansen-Jagannathan Bound σi2 Et [ri,t +1 ] − rf ,t +1 + σm ≥ 2 (15) σi ▶ Equation (15) says that the standard deviation of the log stochastic discount factor must be greater than the Sharpe ratio for any arbitrary asset i, that is, it must be greater than the maximum possible Sharpe ratio obtainable in asset markets. Hansen-Jagannathan Bound ▶ The Sharpe ratio of the market portfolio is approximately 50% on an annual basis. ▶ This means that the standard deviation of the stochastic discount factor must be equal to or greater than 50%. ▶ Recall with power utility, the (log) of the pricing kernel is given by mt +1 = log β − γ∆ct +1 ⇒ σ (m) = γσ (∆ct +1 ) ▶ In the data, σ (∆ct +1 ) is about 1%, which means we need a very large risk aversion coefficient. Revisiting the Equity Risk Premium Puzzle Another way to look at the equity risk premium puzzle is that the stochastic discount factor implied from the standard consumption-based asset pricing models is not volatile enough. Extensions The second generation of consumption based asset pricing models, which depart from log normality and power utility includes: ▶ Long-run risks model: Bansal and Yaron (2004) ▶ Habit model: Campbell and Cochrane (1999) ▶ Rare disaster model: Rietz (1988), Barro (2006), and Wachter (2013). ▶ The recent models have more free parameters and therefore fit the data better.