ECOM 009 Macroeconomics B Lecture 7 Giulio Fella c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 187/231
Plan for the rest of this lecture Introducing the general asset pricing equation Consumption-based asset pricing. Consumption-based asset pricing in general equilibrium: Lucas s tree model Some applications Readings: Ljungqvist and Sargent, Ch. 13.1-13.3, 13.5-13.8 c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 188/231
The basic pricing equation Consider the following equation p j t (z t) = E t [m t+1 x j t+1 ] = n π(z t+1 z t )m(z t+1 z t )x j (z t+1 ) (142) i=1 z t {z 1, z 2,..., z n } is the value of the state variable at time t. We assume it is first-order Markov. p j t (z t) is the price of asset j at time t in state z t x j t+1 is the payoff of asset j at time t + 1 m t+1 = m(z t+1 z t ) is the price (in units of the numeraire) in the current state z t and time t of one (certain) unit of numeraire in state z t+1 at time t + 1. c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 189/231
The basic pricing equation II The price of an asset is the expected value of the future state-contingent asset payoffs times the state-contingent prices. The state-contingent pricing function m(z t+1 z t ) is called the stochastic discount factor or pricing-kernel. The stochastic discount factor is the same for all assets! Relative price of the numeraire across time and states. Different asset pricing theories involve different assumptions on what determines the stochastic discount factor. c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 190/231
The basic pricing equation: an example Consider a deterministic economy. z t = z 1 for all t. The basic pricing equation becomes p j t = m(z 1)x j t+1 p j t is the discounted present value of the future payoff x j t+1 m(z 1 ) is the price of one unit of numeraire tomorrow in terms of numeraire today (the market discount rate) m( ) is called the (stochastic) discount factor because it generalizes the above notion to a stochastic environment. c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 191/231
Breaking down the basic pricing equation One can rewrite the pricing equation as p j t (z t) = E t [m t+1 x j t+1 ] = E t(m t+1 )E t x j t+1 + cov (m t+1, x j t+1 ) If x j t+1 is uncorrelated with m t+1 (e.g. x j t+1 is deterministic; i.e. risk-free) the covariance term is zero. Same as in the deterministic case with the only difference that m(z 1 ) is replaced by E t (m t+1 ). (Note: z t+1 and, consequently, m t+1 are NOT deterministic). p j t (z t) is higher (lower) if the covariance term is +ve (-ve). An asset for which x j t+1 is high in states with high price m t+1 has a higher price today. c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 192/231
Consumption-based asset pricing Effectively, the macroeconomic theory of asset pricing. With two assets the optimization problem for consumer k is W (L k t 1,Nt 1, k z t ) = max c k t,lk t,n t k u(c k t ) + βew (L k t, Nt k, z t+1 ) (143) s.t. c k t + Rt 1 L k t + p t Nt k = L k t 1 + (p t + y t )Nt 1 k (144) L k t, Nt k 0, L k t 1, Nt 1 k given (145) L k t and N k t choice of stock of risk-free and risky asset. Future return on risky asset (y t+1 + p t+1 )/p t is stochastic. No labour income (for simplicity) c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 193/231
Euler equations Replacing for c k t, maximizing with respect to L k t and Nt k using the envelope condition, yields the Euler equations [ Rt 1 = E t β u (c k t+1 ) ] u (c k t [ ) p t = E t β u (c k t+1 ) ] u (c k t ) (p t+1 + y t+1 ) and (146) (147) Those Euler equations are basic asset pricing equations with m k t+1 = β u (c k t+1 ) u (c k t ) ] In consumption-based asset pricing theories the stochastic discount factor is the MRS between consumption today and tomorrow. Unique prices (no arbitrage) only if m k t+1 = m t+1 for all k. c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 194/231
Risky assets p t = E t [m t+1 (p t+1 + y t+1 )] = E t [ β u (c k t+1 ) ] u (c k t ) (p t+1 + y t+1 ) The price of a risky asset is higher (its expected return lower) the more it pays in states in which m t+1 is high; i.e. in which c t+1 is low. Such an asset provides more insurance in worse state. Desirable, hence p t c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 195/231
Only non-diversifiable risk matters The pricing equation can also be written as p t =E t [m t+1 (p t+1 + y t+1 )] =E t [m t+1 ]E t [(p t+1 + y t+1 )] + cov[m t+1, (p t+1 + y t+1 )] Idiosyncratic risk (covariance term is zero) does not affect prices No need for compensation as risk is fully diversified. Only risk correlated with future consumption affects prices (and returns). c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 196/231
When are stock prices a martingale? p t =E t [m t+1 (p t+1 + y t+1 )] =E t [m t+1 ]E t [(p t+1 + y t+1 )] + cov[m t+1, (p t+1 + y t+1 )] Two necessary conditions for p t to be a martingale E t m t+1 = E t [βu (c t+1 )/u (c t )] is a constant The covariance term is zero If agents are risk-neutral E t m t+1 = β and p t = E t β(p t+1 + y t+1 ) = j=1 β j E t y t+j + lim k βk E t p t+k The last (bubble) term equals zero in the general equilibrium models we consider. c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 197/231
Consumption-based asset pricing in general equilibrium Our pricing equations are informative on prices only to the extent that we know the stochastic discount factor. To determine the SDF all models proceed along the following lines. 1. Postulate an economic environment. 2. Derive the equilibrium allocation. 3. Assume competitive markets for the assets of interests and solve for the agents Euler equations pricing equations. 4. Impose that the consumption allocation in 2. coincides with consumers demand in 3. c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 198/231
Lucas s tree model One good: coconuts (non-storable). Two assets: equity (ownership of one coconut trees) and bonds. All coconut trees yield the same payoff y t (z t ) in state z t (aggregate shocks). All agent are identical: same preferences and initial endowment of one tree Trivial equilibrium: each agent consumes the current flow of coconuts from her tree and net zero bond supply. c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 199/231
Equilibrium asset prices in Lucas model Same pricing equations but now consumption is determined. [ ] Rt 1 = E t β u (y t+1 ) u (y t ) [ ] p t = E t β u (y t+1 ) u (y t ) (p t+1 + y t+1 ) c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 200/231
Stock prices in Lucas model One can iterate on the second equation to obtain u (y t )p t = E t j=1 β j u (y t+j )y t+j + E t lim k βk u (y t+k )p t+k For agents to be willing to hold their tree forever in equilibrium the last term has to be zero (no bubble). Suppose not... The expression for the equilibrium stock price can be written as p t = E t j=1 β j u (y t+j ) u (y t ) y t+j c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 201/231
A special case Suppose that u(c) = log(c). The expression for the stock price becomes p t = E t j=1 β j y t y t+1 y t+j = β 1 β y t Remark: different choices of functional forms for u imply different asset pricing models. c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 202/231
The term structure of interest rates Let s now introduce a second (two-period) risk-free bond W (L 1,t 1, L 2,t 1 N t 1, y t ) = max u(c t ) + βew (L 1,t, L 2,t, N t, y t+1 ) c t,l 1,t,L 2,tN t s.t. c t + R 1 1,t L 1,t + R 1 2,t L 2,t + p t N t = L 1,t 1 + R 1 1,t L 2,t 1 + (p t + y t )N t 1 L 1,t, L 2,t, N t 0, L 1,t 1, L 2,t 1, N t 1 given and R 1 2,t are the current prices of a bond with respectively a one-period and two-period remaining maturity. R 1 1,t Absence of arbitrage requires the time-t price of a two-period bond issued last period to be R 1 1,t It follows that the one-period-ahead return of a newly-issued two-period bond is uncertain. c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 203/231
Bond pricing equations Imposing equilibrium (c t = y t ), the Euler (or pricing) equations for the two bonds can be written as R 1 1,t = βe t R 1 2,t = βe t [ u ] (y t+1 ) u (y t ) [ u (y t+1 ) u (y t ) R 1 1,t+1 ] [ u = β 2 ] (y t+2 ) E t u (y t ) The first equality in the second equation can be written as [ u R2,t 1 ] [ ] = βe (y t+1 ) t u E t R1,t+1 1 (y t ) + cov β u (y t+1 ) u (y t ), R 1 1,t+1 [ ] = R1,t 1 E tr1,t+1 1 + cov β u (y t+1 ) u (y t ), R 1 1,t+1 c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 204/231
The pure expectation theory of the term structure [ ] R2,t 1 = R 1 1,t E tr1,t+1 1 + cov β u (y t+1 ) u (y t ), R 1 1,t+1 The first addendum embodies the pure expectation theory of the term structure. Long rates are just a (geometric) average of expected future short rate. R 2,t > R 1,t if rates are expected to increase. The pure expectation theory holds exactly only if the covariance term is zero. E.g. Risk-neutral agents No uncertainty c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 205/231
A different look at the term structure The pricing equation [ R2,t 1 = Et u ] (y t+2 ) β2 u (y t ) generalizes to a j period bond [ Rj,t 1 = Et u ] (y t+j ) βj u (y t ) It can be written in terms of returns rather than prices as R j,t = β j [ u (y t )[E t u (y t+2 )] 1] c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 206/231
A different look at the term structure II The corresponding annual implied return is R j,t = R 1/j j,t = β 1 [ u (y t )[E t u (y t+j )] 1] 1/j If dividends are i.i.d. the expectation term E t u (y t+j ) = Eu (y t ) is constant for all j > 0 and we can write R j,t R k,t = [ u (y t )[Eu (y)] 1] 1 j 1 k = [ u (y t )[Eu (y)] 1] k j kj If k > j, R k,t > R j,t if u (y t ) < Eu (y) Shorter rates are below longer rates if consumption today is relatively high (people want to save for the future). c Giulio Fella, 2014 ECOM 009 Macroeconomics B - Lecture 7 207/231