Asset Pricing Ljungqvist / Sargent Chapter 8, 13 Kjetil Storesletten University of Oslo
1 Motivation Facts: Real Returns 50 40 30 20 10 0 -10 -20 -30 -40 Stock T Bill -50 1930 1940 1950 1960 1970 1980 1990 2000 Real stock and bond returns 1927-2002 Bond (Stock-Bond) Mean annual % return 1.1 7.5 Standard Deviation 4.4 20.8 1 MOTIVATION 2
Real value of a dollar invested in 1927 3 10 stock bond 2 10 1 10 0 10 -1 10 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 ¨ 1 MOTIVATION 3
• Assets are used to shift consumption between periods. • Some assets are not perfectly safe . Hence, the agent experiences some uncer- tainty about future payoffs, which results in uncertainty about future levels of consumption . • Based on this thought, agents will price assets not only depending on its return compared to the risk-free rate, but also depending on each agent’s willingness to take risk (willingness to forego some consumption smoothing for a higher absolute level of consumption). 1 MOTIVATION 4
2 Asset Euler Equations Agent’s Maximization Problem: • Agents maximize expected lifetime utility: ∞ � β j u ( c t + j ) , E t 0 < β < 1 . (2.1) j =0 • Individuals can hold bonds and equity. Assets can be sold short , but borrowing constraints work on both equity and fixed-income holdings. • Notation: – L t : Bond holdings – N t : Equity holdings – Borrowing constraints: L t ≥ − b L , N t ≥ − b N . – Risk-free real gross interest rate: R t , measured in units of time t + 1 consumption good per time t consumption good. 2 ASSET EULER EQUATIONS 5
• Budget constraint: − 1 L t + p t N t ≤ A t . c t + R t (2.2) • Next period’s wealth: A t +1 = L t + ( p t +1 + y t +1 ) N t (2.3) y t +1 : stochastic dividend income • Hence, the agent’s maximization problem is a dynamic programming problem. 2 ASSET EULER EQUATIONS 6
Euler Equations: • At interior solutions, the Euler equations are − 1 = E t βu ′ ( c t +1 ) , u ′ ( c t ) R t u ′ ( c t ) p t = E t β ( y t +1 + p t +1 ) u ′ ( c t +1 ) . (2.4) • Transversality conditions: k →∞ E t β k u ′ ( c t + k ) R − 1 k →∞ E t β k u ′ ( c t + k ) p t + k N t + k = 0 . lim t + k L t + k = 0 , lim (2.5) • If any of the transversality conditions (2.5) would be positive, the agent would be overaccumulating assets. This is the analogous condition to a finite horizon model where the agent will not die with positive asset holdings. • Were they negative, the agent would be given credit he cannot pay up - a contract that no one would enter. 2 ASSET EULER EQUATIONS 7
3 Martingale Theories of Consumption and Stock Prices • Assume that the risk-free interest rate is constant ( R t = R > 1). Thus: E t u ′ ( c t +1 ) = ( βR ) − 1 u ′ ( c t ) . Hence, u ′ ( c ) follows a univariate linear first-order Markov process. • Rearranging the Euler equation yields: E t β ( y t +1 + p t +1 ) u ′ ( c t +1 ) u ′ ( c t ) = p t . (3.1) • The left-hand side of (3.1) is the expected value of a product of two random variables. Apply E t ( xz ) = E t xE t z + cov ( x, z ): � � u ′ ( c t +1 ) ( y t +1 + p t +1 ) , u ′ ( c t +1 ) βE t ( y t +1 + p t +1 ) E t u ′ ( c t ) + βcov t = p t (3.2) u ′ ( c t ) 3 MARTINGALE THEORIES OF CONSUMPTION AND STOCK PRICES 8
• Can it be the case that p t = βE t ( y t +1 + p t +1 )? Unlikely to be satisfied for general utility functions! Then u ′ ( c t ) • One obvious example where it fits is risk-neutrality of agents. becomes independent of c t , E t u ′ ( c t +1 ) /u ′ ( c t ) = 1, and � � ( y t +1 + p t +1 ) , u ′ ( c t +1 ) cov t = 0 . u ′ ( c t ) • Stock prices then suffice E t β ( y t +1 + p t +1 ) = p t . (3.3) • Adjusted for dividends and discounting, the share price follows a first-order univariate Markov process and no other variables Granger cause the share price. 3 MARTINGALE THEORIES OF CONSUMPTION AND STOCK PRICES 9
• The stochastic difference equation (3.3) in that case has the class of solutions ∞ � 1 � t � β j y t + j p t = E t + ξ t , (3.4) β j =1 � �� � � �� � bubbleterm expecteddiscounted futuredividends where ξ t is any random process that obeys E t ξ t +1 = ξ t . The ‘bubble term’ must be zero in equilibrium. 3 MARTINGALE THEORIES OF CONSUMPTION AND STOCK PRICES 10
4 Equilibrium Asset Pricing Lucas’ breakthrough on asset pricing (Econometrica 1978) • So-called ‘Lucas’ Tree Model’. Trees are durable, dividends (fruits) are not. Each agent owns one tree at time 0. • Dividend follows a Markov process and is the only state variable of the economy. • An asset-pricing model has the following features: 1. It is a dynamic economy. 2. Specified preferences, technology and endowments. 3. Equilibrium is competitive 4 EQUILIBRIUM ASSET PRICING 11
• Equilibrium intertemporal consumption allocation then satisfies: E t u ′ ( c t +1 ) = ( βR ) − 1 u ′ ( c t ) . � � βu ′ ( c t +1 ) E t u ′ ( c t ) · ( y t +1 + p t +1 ) = p t . (4.1) • Equate consumption that appears in the Euler equation to the equilibrium consumption derived in the first step. This procedure will give the asset price at t as a function of the state of the economy at t . u ′ ( y t ) R − 1 = E t βu ′ ( y t +1 ) , t u ′ ( y t ) p t = E t β ( y t +1 + p t +1 ) u ′ ( y t +1 ) . 4 EQUILIBRIUM ASSET PRICING 12
5 Stock Prices without Bubbles • Using recursions on the previous equation and the law of iterated expectations ( E t E t +1 ( · ) = E t ( · )), we arrive at the equilibrium share price: ∞ � u ′ ( y t ) p t = E t β j u ′ ( y t + j ) y t + j + E t lim k →∞ β k u ′ ( y t + k ) p t + k . (5.1) j =1 • Market clearing implies that the last term must be zero (agents must be willing to hold their trees forever). • Would it be positive, then the marginal utility from selling shares, u ′ ( y t ) p t , exceeds the marginal utility loss of holding the asset forever and consuming the � ∞ j =1 β j u ′ ( y t + j ) y t + j . Then, everybody would future stream of dividends, E t like to sell shares and asset prices would immediately be driven down. • The same arguments holds vice versa: If the term is negative, all agents would like to purchase more shares and the price would soar. 5 STOCK PRICES WITHOUT BUBBLES 13
• Hence, in equilibrium, the price must satisfy ∞ β j u ′ ( y t + j ) � p t = E t u ′ ( y t ) y t + j , (5.2) j =1 which is a generalization of equation (3.4). 5 STOCK PRICES WITHOUT BUBBLES 14
Consider a simple example: • u ( c ) = log ( c ) • y t is a random walk with constant drfit g : log y t = log y t − 1 + g + ε t • Consider equation (5.2); ∞ ∞ β j y t β � � β j = p t = E t y t + j = y t 1 − βy t y t + j j =1 j =1 – Constant price-dividend ratio p t /y t – Asset price is independent of expected growth rate g 5 STOCK PRICES WITHOUT BUBBLES 15
6 The Term Structure of Interest Rates Consider there are two zero-bonds with maturity of 1 and 2 years, respectively. The law of motion for wealth is then c t + R − 1 1 t L 1 t + R − 1 2 t L 2 t + p t N t ≤ A t , (6.1) A t +1 = L 1 t + R − 1 1 t +1 L 2 t + ( p t +1 + y t +1 ) N t . (6.2) Hence the Bellman equation for this problem is: � u [ A t − R − 1 1 t L 1 t − R − 1 v ( A t , s t ) = max 2 t L 2 t − p ( s t ) N t ] L 1 t ,L 2 t ,N t � + βE t v ( L 1 t + R − 1 t +1 L 2 t + [ p ( s t +1 ) + y t +1 ] N t , s t +1 ) 6 THE TERM STRUCTURE OF INTEREST RATES 16
The first-order conditions are: u ′ ( c t ) R − 1 1 t = βE t v 1 ( A t +1 , s t +1 ) , (6.3) � � u ′ ( c t ) R − 1 v 1 ( A t +1 , s t +1 ) R − 1 2 t = βE t . (6.4) 1 t +1 Applying the envelope theorem and setting c t = y t = s t , the equilibrium rates of return equal � u ′ ( s t +1 ) � R − 1 ≡ R 1 ( s t ) − 1 1 t = βE t (6.5) u ′ ( s t ) � u ′ ( s t +1 ) � � u ′ ( s t +2 ) � R − 1 u ′ ( s t ) R − 1 ≡ β 2 E t ≡ R 2 ( s t ) − 1 2 t = βE t (6.6) 1+1 u ′ ( s t ) 6 THE TERM STRUCTURE OF INTEREST RATES 17
Because of the Markov assumptions, interest rates can be written as time-invariant functions of the economy’s current state s t . The general expression of (6.6) is as follows: � u ′ ( s t + j ) � R − 1 jt = β j E t . (6.7) u ′ ( s t ) This rate is not annualized and smaller than 1. If we invert and exponentiate with ( · ) 1 /j , we get the yield to maturity of a zero coupon bond: jt = β − 1 � u ′ ( s t )[ E t u ′ ( s t + j )] − 1 � 1 /j R jt = R 1 /j ˜ (6.8) If dividends are i.i.d. over time, the yields to maturity for a j-period and a k-period bond are then related as follows: u ′ ( s t )[ Eu ′ ( s )] − 1 � k − j � R jt = ˜ ˜ kj R kt 6 THE TERM STRUCTURE OF INTEREST RATES 18
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