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Consumption-Based Asset Pricing CONSUMPTION-BASED ASSET PRICING Emi Nakamura and Jon Steinsson UC Berkeley Fall 2020 Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 1 / 48 BIG ASSET PRICING QUESTIONS Why is the return on the stock market so high? (Relative to the "risk-free rate") Why is the stock market so volatile? What does this tell us about the risk and risk aversion? Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 2 / 48 CONSUMPTION-BASED ASSET PRICING Consumption-based asset pricing starts from the Consumption Euler equation: 0 0 U (Ct ) = Et [βU (Ct+1)Ri;t+1] Where does this equation come from? Consume $1 less today Invest in asset i Use proceeds to consume $ Rit+1 tomorrow Two perspectives: Consumption Theory: Conditional on Rit+1, determine path for Ct Asset Pricing: Conditional on path for Ct , determine Rit+1 Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 3 / 48 CONSUMPTION-BASED ASSET PRICING 0 0 U (Ct ) = βEt [U (Ct+1)Ri;t+1] A little manipulation yields: 0 βU (Ct+1) 1 = Et 0 Ri;t+1 U (Ct ) 1 = Et [Mt+1Ri;t+1] Stochastic discount factor: 0 U (Ct+1) Mt+1 = β 0 U (Ct ) Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 4 / 48 STOCHASTIC DISCOUNT FACTOR Fundamental equation of consumption-based asset pricing: 1 = Et [Mt+1Ri;t+1] Stochastic discount factor prices all assets!! This (conceptually) simple view holds under the rather strong assumption that there exists a complete set of competitive markets Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 5 / 48 PRICES,PAYOFFS, AND RETURNS Return is defined as payoff divided by price: Xi;t+1 Ri;t+1 = Pi;t where Xi;t+1 is (state contingent) payoff from asset i in period t + 1 and Pi;t is price of asset i at time t Fundamental equation can be rewritten as: Pi;t = Et [Mt+1Xi;t+1] Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 6 / 48 MULTI-PERIOD ASSETS Assets can have payoffs in multiple periods: Pi;t = Et [Mt+1(Di;t+1 + Pi;t+1)] where Di;t+1 is the dividend, and Pi;t+1 is (ex dividend) price Works for stocks, bonds, options, everything. Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 7 / 48 STATE-PRICES Suppose Ps;t+1 is the price of the Arrow security that pays $ 1 if state s occurs at time t + 1 and zero otherwise The price of any security can by written two ways: X Pi;t = Ps;t+1Xs;t+1; Pi;t = Et [Mt+1Xt+1] s which implies Ps;t+1 Ms;t+1 = πs;t+1 where πs;t+1 is the probability of state s in period t + 1. This is why you sometimes see Et [Mt+1Xt+1] type terms in budget constraints Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 8 / 48 MODIGLIANI-MILLER THEOREM Value of an asset is the sum of its parts: X Pi;t = Ps;t+1Xs;t+1 s Why? Arbitrage! Doesn’t matter how the asset is sliced up! (as long as the total payoff is not changed) Capital structure of a firm doesn’t matter for its value! Dividend policy of a firm doesn’t matter for its value! Whether a firm buys insurance (hedges a risk) doesn’t matter! Holds when: Complete set of competitive markets exists (no bankruptcy costs, no agency costs, etc.) No taxes Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 9 / 48 MODIGLIANI-MILLER THEOREM Does hedging a risk raise the value of a firm? Let’s adopt vector notation: S state of the world in the future Xt+1 is an S × 1 vector of payoffs in these states Pt is an S × 1 vector of state prices Value of Firm A before hedging risk: A A Pt = Pt · Xt+1 A where Xt+1 denotes the payoffs of firm A over future states Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 10 / 48 MODIGLIANI-MILLER THEOREM B Consider some other cashflow Xt+1 Price of that cashflow: B B Pt = Pt · Xt+1 Suppose the firm were to purchase this cashflow At that point the firm’s value would be the value of the combined cashflow minus the price of the cashflow: A B B A B B A B B A Pt · [Xt+1 + Xt+1] − Pt = Pt · Xt+1 + Pt · Xt+1 − Pt = Pt + Pt − Pt = Pt True of any cashflow!! (Hedge, Bond, Dividend, etc.) Flows from the linearity of the pricing: By arbitrage, assets are worth the sum of their parts Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 11 / 48 KETCHUP ECONOMICS Larry Summers (1985) critique of (then) finance Two kinds of research on ketchup market General Economists: Ask what the fundamental determinants of the price of ketchup is Analyze messy data on supply and demand Tough question, modest progress Ketchup Economists: Analyze hard data on transactions Two quart sized ketchup bottles invariably sell for twice as much as one No bargains from storing ketchup or mixing ketchup, etc. Conclude from this that ketchup market is efficient Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 12 / 48 EVERYTHING LOG-NORMAL 1 = Et [Mt+1Ri;t+1] Let’s assume for simplicity that everything is log-normal If Xt+1 is log-normal, then 1 log E X = E log X + Var log X t t+1 t t+1 2 t t+1 Now let’s take logs of asset pricing equation: 0 = log Et [Mt+1Ri;t+1] 1 0 = E r + E m + (σ2 + σ2 + 2σ ) t i;t+1 t t+1 2 i m im 2 where ri;t+1 = log Ri;t+1, mt = log Mt+1, σi = Vart log Ri;t+1, etc. Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 13 / 48 RISK FREE RATE 1 0 = E r + E m + (σ2 + σ2 + 2σ ) t i;t+1 t t+1 2 i m im This equation should hold for all assets What does it imply about the risk-free asset? 1 r = −E m − σ2 f ;t t t+1 2 m Or in levels: 1 1 = Et [Mt+1Rf ;t ] => Rf ;t = Et Mt+1 Since risk free return is risk free, it is determined at time t Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 14 / 48 EQUITY PREMIUM Combining: 1 0 = E r + E m + (σ2 + σ2 + 2σ ) t i;t+1 t t+1 2 i m im 1 r = −E m − σ2 f ;t t t+1 2 m we get 1 E r − r + σ2 = −σ t i;t+1 f ;t 2 i im Equity premium is negative of covariance of return on equity and the stochastic discount factor Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 15 / 48 1 log E R − log R = E r − r + σ2 t i;t+1 f ;t t i;t+1 f ;t 2 i (Log of) Geometric mean: Et ri;t+1 (Log of) Arithmetic mean: log Et Ri;t+1 1 log E R = E r + Var log R t i;t+1 t i;t+1 2 t i;t+1 Standard deviation annual real return on stocks is roughly 18% 1 1 Var log R = σ2 = 1:5% 2 t i;t+1 2 i EQUITY PREMIUM 1 E r − r + σ2 = −σ t i;t+1 f ;t 2 i im 1 2 What is with the 2 σi term? Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 16 / 48 EQUITY PREMIUM 1 E r − r + σ2 = −σ t i;t+1 f ;t 2 i im 1 2 What is with the 2 σi term? 1 log E R − log R = E r − r + σ2 t i;t+1 f ;t t i;t+1 f ;t 2 i (Log of) Geometric mean: Et ri;t+1 (Log of) Arithmetic mean: log Et Ri;t+1 1 log E R = E r + Var log R t i;t+1 t i;t+1 2 t i;t+1 Standard deviation annual real return on stocks is roughly 18% 1 1 Var log R = σ2 = 1:5% 2 t i;t+1 2 i Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 16 / 48 EQUITY PREMIUM Two ways to write equity premium equation: 1 E r − r + σ2 = −σ t i;t+1 f ;t 2 i im log Et Ri;t+1 − log Rf ;t = −σim Also recall that log of expected gross return is approximately equal to the expected net return: log(1 + x) ≈ x for small x Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 17 / 48 Consider agent with CRRA utility and wealth W facing portfolio choice between risky and risk-free asset. Fraction allocated to risky asset is independent of wealth. (CARA utility: Dollar amount invested in risky asset is independent of wealth) Consistent with stable interest rate and risk premia in the presence of long-run growth POWER UTILITY Now suppose that C1−γ − 1 U(C ) = t t 1 − γ This utility function is sometimes called CRRA utility for “constant relative risk aversion” Relative risk aversion: U00(C)C − = γ U0(C) Why do we think that this utility function is reasonable? Nakamura-Steinsson (UC Berkeley) Consumption-Based Asset Pricing 18 / 48 POWER UTILITY Now suppose that C1−γ − 1 U(C ) = t t 1 − γ This utility function is sometimes called CRRA utility for “constant relative risk aversion” Relative risk aversion: U00(C)C − = γ U0(C) Why do we think that this utility function is reasonable? Consider agent with CRRA utility and wealth W facing portfolio choice between risky and risk-free asset.
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