Chapter 5 Macroeconomics and Finance

Transcription

1 Macro II Chapter 5 Macro and Finance 1 Chapter 5 Macroeconomics and Finance Main references : - L. Ljundqvist and T. Sargent, Chapter 7 - Mehra and Prescott 1985 JME paper - Jerman 1998 JME paper - J. Greenwood and B. Jovanovic, The Information-Technology Revolution and the Stock Market", AER, Kocherlakota 1996 survey in the JEL

2 Macro II Chapter 5 Macro and Finance 2 1 Introduction In this chapter, I want to 1. show how to compute asset prices in general equilibrium 2. discuss of the some quantitative properties of asset prices in (simple) GE models 3. show an application to the US stock market in the 70s (if I have time)

3 Macro II Chapter 5 Macro and Finance 3 2 Asset Prices in General Equilibrium I describe here the competitive equilibrium of a pure exchange infinite horizon economy with stochastic Markov endowments. This is a basic setting for studying risk sharing, asset pricing, consumption. 2 different market structures 1. Arrow-Debreu structure with complete markets in dated contingent claims all traded at period 0 2. recursive structure with complete one-period Arrow securities. The 2 have different asset structures but identical consumption allocations

4 Macro II Chapter 5 Macro and Finance The physical setting Preferences and endowments π(s s) is a Markov chain with initial distribution π 0 (s) P rob(s t+1 = s s t = s) = π(s s) and P rob(s 0 = s) = π o (s) a sequence of probability measures π(s t ) on histories s t = [s t, s t 1,..., s 0 ] is given by π(s t ) = π(s t s t 1 )π(s t 1 s t 2 )...π(s 1 s 0 )π 0 (s 0 ) (1) and conditional probability is given by π(s t s 0 ) = π(s t s t 1 )π(s t 1 s t 2 )...π(s 1 s 0 ) (2)

5 Macro II Chapter 5 Macro and Finance 5 Trading occurs after s 0 has been observed. the probability of state (history) s t conditional on being in state (history) s τ at date τ is π(s t s τ ) = π(s t s t 1 )π(s t 1 s t 2 )...π(s τ+1 s τ ) (3) (because of Markov property, π(s t s τ ) does not depend on history s τ 1

6 Macro II Chapter 5 Macro and Finance 6 Households: i = 1,..., I. Each owns a stochastic endowment of one good y i t = y i (s t ), and s t is publicly observable Each household purchase a history-dependant consumption plan c i = {c i t(s t )} t=0 Household objective U(c i ) = t=0 s t β t u[c i t(s t )]π(s t s 0 ) = E 0 u has all nice properties, including lim c 0 u (c) = + β t u[c i t(s t )] (4) t=0

7 Macro II Chapter 5 Macro and Finance Complete markets Household trade dated state-contingent claims to consumption qt 0 (s t ) = price of a claim on time-t consumption, contingent on history s t, in terms of a numéraire not specified the BC is qt 0 (s t )c i t(s t ) = qt 0 (s t )y i (s t ) (5) t=0 s t t=0 s t Hh problem: choose c i to maximize (4) s.t. (5) Notice that one can collapse the problem into a problem with a single budget constraint because of complete markets

8 Macro II Chapter 5 Macro and Finance 8 let µ i be the Lagrange multiplier of this constraint, FOC: and with the specification (4) of preferences, one gets U(c i ) c i t(s t ) = µi q 0 t (s t ) (6) U(c i ) c i t(s t ) = βt u [c i t(s t )]π(s t s 0 ) (7) β t u [c i t(s t )]π(s t s 0 ) = µ i q 0 t (s t ) (8)

9 Macro II Chapter 5 Macro and Finance 9 Definition 1 A price system is a sequence of functions {qt 0 (s t )} t=0. An allocation is a list of sequences of functions {c i t(s t )} t=0, one for each i. A feasible allocation satisfies y i (s t ) c i t(s t ) (9) i i Definition 2 A competitive equilibrium is a feasible allocation and price system such that the allocation solves each household problem

10 Macro II Chapter 5 Macro and Finance 10 Notice that (8) implies u [c i t(s t )] u [c j t(s t )] = µi (10) µ j which means thats ratios of marginal utilities between pairs of agents are constant across all sates and dates. An equilibrium allocation solves (10), (9),and (5). Note that (10) implies { } c i t(s t ) = u 1 u [c 1 t(s t )] µi µ 1 and substituting into feasibility condition (9) at equality gives { } u 1 u [c 1 t(s t )] µi = y i (s µ 1 t ) (12) i i the RHS of (12) does not depend on the entire history s t, but only on current state s t, therefore the LHS, therefore c 1 t(s t ). Then, from (11), it is also the case for all c i t(s t ). One then has the following proposition Proposition 1 The competitive equilibrium allocation is not history dependent; c i t(s t ) = c i (s t ) (11)

11 Macro II Chapter 5 Macro and Finance Equilibrium pricing function Let c i, i = 1,...I b an equilibrium allocation. Then (6) or (8) gives the price system qt 0 (s t ) as a function of the allocation to Hh i, for any i. The price system is a stochastic process Because the units of the price system are arbitrary, one can normalized one of the multipliers at any positive value. I set µ 1 = u [c 1 (s 0 )], so that q0(s 0 0 ) = 1, i.e. the price system is in units of time-0 goods. (one has therefore µ i = u [c i (s 0 )] for all i)

12 Macro II Chapter 5 Macro and Finance Examples: Risk sharing suppose u(c) = (1 γ) 1 c 1 γ, γ > 0 (CRRA). Then (10) implies ( ) µ c i t = c j i 1 γ t (13) time-t elements of consumption allocations to distinct agents are constant fractions of one another. The individual consumption is perfectly correlated with the aggregate endowment or aggregate consumption. The fractions assigned to each individual are independent of the realization of s t. There is extensive cross-time cross-state consumption smoothing. µ j

13 Macro II Chapter 5 Macro and Finance Asset pricing Pricing Redundant Assets Let {d(s t )} t=0 be a stream of claims on time t, state s t consumption, where d(s t ) is a measurable function of s t. The price of an asset entitling the owner to this stream must be a 0 0 = qt 0 (s t )d(s t ) (14) t=0 s t (this can be understood as an arbitrage equation) Riskless Consol A riskless consol offers for sure one unit of consumption at each period, i.e. d t (s t ) = 1 for all t and s t. The price is a 0 0 = t=0 s t q 0 t (s t )

14 Macro II Chapter 5 Macro and Finance Riskless strips Consider a sequence of strips of returns on the riskless consol. The time-t strip is the return process d τ = 1 if τ = t 0, and 0 otherwise. The price of time-t strip at 0 is s t q 0 t (s t ) Tail assets Consider the stream of dividends {d(s t )} t 0 For τ 1, suppose that we strip off the first τ 1 periods of the dividend and want to get the time-0 value of the dividend stream {d(s t )} t τ. Let a 0 τ(s τ ) be the time-0 price of an asset that entitles the dividend stream {d(s t )} t τ if history s τ is realized: a 0 τ(s τ ) = t τ { s t : s τ =s τ } q 0 t ( s t )d( s t ) (15)

15 Macro II Chapter 5 Macro and Finance 15 Let us convert this price into units of time τ, state s τ by dividing by qτ(s 0 τ ): a τ τ(s τ ) = a0 τ(s τ ) qτ(s 0 τ ) = qt 0 ( s t ) q 0 t τ τ(s τ ) d( st ) (16) Notice that for all consumers i { s t : s τ =s τ } q τ t (s t ) = q0 t (st ) q 0 τ (s τ ) = βt u [c i t (st )]π(s t ) β τ u [c i τ (s τ )]π(s τ ) = β t τ u [c i t (st )] u [c i τ (s τ )] π(st s τ ) Here q τ t (s t ) is the price of one unit of consumption delivered at time t, state s t in terms of the date-τ, state-s τ consumption good. The price at t for the tail asset is a τ τ(s τ ) = t τ { s t : s τ =s τ } (17) q τ t ( s t )d( s t ) (18) This tail asset formula is useful if one wants to create in a model a time series of equity prices: an equity purchased at time τ entitles the owner to the dividends from time τ forward, and the price is given by (18).

16 Macro II Chapter 5 Macro and Finance 16 Note: The relative price is (17) is not history dependent, given Proposition 1. This is stated in the following proposition: Proposition 2 The equilibrium price of date-t 0, state-s t consumption good expressed in terms of date τ (0 τ t), state s τ consumption good is not history dependent: qt τ (s t ) = q j t ( s k ) for j, k 0 such that t τ = k j and [s t, s t 1,..., s τ ] = [ s k, s k 1,..., s j ].

17 Macro II Chapter 5 Macro and Finance Pricing One Period Returns The one-period version of equation (17) is q τ τ+1(s τ+1 ) = β u (c i τ+1) u (c i τ) π(s τ+1 s τ ) The RHS is the one-period pricing kernel at time τ. The price at time τ in state s τ of a claim to a random payoff ω(s τ+1 ) is given, using the pricing kernel, by p τ τ(s τ ) = s τ+1 qτ+1(s τ τ+1 )ω(s τ+1 ) [ ] = E τ β u (c τ+1 ) u (c τ ) ω(s τ+1) where superscripts i and dependence to s τ have been deleted. Let denote the one-period gross return on the asset by R τ+1 = ω(s τ+1 )/p τ τ(s τ ). Then, for any asset, equation (19) implies The term m τ+1 = β u (c τ+1 ) u (c τ ) [ ] 1 = E τ β u (c τ+1 ) u (c τ ) R τ+1 (19) (20) is a stochastic discount factor. Equation (20) can be understood

18 Macro II Chapter 5 Macro and Finance 18 as a restriction on the conditional moments of returns and m τ+1. Applying the law of iterated expectations to equation (20), one gets the unconditional moments restrection: 1 = E [ ] β u (c τ+1 ) u (c τ ) R τ+1 (21) 2.3 A Recursive Formulation: Arrow Securities One introduce another market structure that preserves the equilibrium allocation from our competitive equilibrium. This setting also preserves the one-period asset-pricing formula (19). Arrow (1964): one-period securities are enough to implement complete markets, provided that new one-period markets are reopened for trading each period See Ljundqvist and Sargent for a formal proof

19 Macro II Chapter 5 Macro and Finance 19 3 A Quantitative Model: Mehra & Prescott 3.1 Data See Table and Figures

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24 Macro II Chapter 5 Macro and Finance 24 The risk premium is high (6.18 %), as the s.d. of real returns is 5.67% for riskless asset and 16.54% for risky asset. Mehra and Prescott have proposed a relatively simple endowment economy to quantitatively reproduce this fact.

25 Macro II Chapter 5 Macro and Finance A Pure Exchange Economy Environment representative agent, E 0 t=0 βt u(c t ); 0 < β < 1, u(c; α) = c1 α 1 1 α One productive unit (a tree) gives y t units of a perishable good. This tree is an equity share that is competitively traded, and y t is its dividend. the growth rate of y t is stochastic: y t+1 = x t+1 y t x is markov: x t+1 {λ 1,..., λ n }, Prob(x t+1 = λ j x t = λ i ) = φ ij. Is is assumed that this markov chain is ergodic, and that λ i > 0, y 0 > 0. y t is observed at the beginning of the period and securities are traded ex-dividend.

26 Macro II Chapter 5 Macro and Finance 26 Equilibrium Proposition 3 Define A = [a ij ], a ij = βφ ij λ 1 α j a Debreu competitive equilibrium exists. and assume lim m A m = 0. Then,

27 Macro II Chapter 5 Macro and Finance 27 Pricing In this economy, the ex dividend price of a security with dividends {d t } is [ ] P t = E t β s tu (y s ) u (y t ) d s s=t+1 For the equity, given the functional forms and the fact that d = y, [ ] Pt e = P e (y t, x t ) = E t β s tyα t y ys α s s=t+1 (y t, x t ) is a sufficient description of the past history. It defines the state of the economy. { } Pt e = E t β u (y t+1) u (y t ) (P t+1 e + y t+1 ) Given that y s = y t x t+1 x t+2 x s, P e t is homogenous of degree 1 in y t, which is the current endowment of consumption good. Given that equilibrium values of the economy are time invariant functions of (y t, x t ), the subscript t can be dropped. The state can be written as (c, i), where y t = c and x t = λ i.

28 Macro II Chapter 5 Macro and Finance 28 With these notations, the price of an equity satisfies P e (c, i) = β n j=1 φ }{{} ij (λ } jc) {{ α } [P e (λ j c, j) + cλ j ] }{{} c α }{{} i ii iii iv with i: probability of state j knowing i ii: inverse of marginal utility of tomorrow consumption in state j iii: tomorrow price + dividend in state j iv: marginal utility of today consumption Given that P e is homogenous of degree 1 in c, we can write P e (c, i) = w i c where w i is a constant. Then the pricing equation becomes n w i = β φ ij λ (1 α) j (w j + 1) i = 1,..., n j=1 This is a system of n linear equations in n unknowns (the w i ) this has a unique positive solution when a competitive equilibrium exists we can derive prices

29 Macro II Chapter 5 Macro and Finance 29 Prices The return of an equity if current state is c, i and next state j is r e ij = P e (λ j c, j) + λ j c P e (c, i) P e (c, i) and the equity expected return is, conditional on state i: n Ri e = φ ij rij e j=1 = λ j(w j + 1) w i 1 Let us also consider a riskless security that pays 1 unit of good in each state, i.e. d i = 1 i. The price P f of this asset is and R f i = 1/P f i 1 P f i = P f (c, i) = β n j=1 φ ij u (λ j c) u (c) d j = β n j=1 φ ij λ α j Now we can compute expected returns w.r.t. the stationary distribution Let π R n be the vector of stationary probabilities of the markov chain: π is such that π = φ π

30 Macro II Chapter 5 Macro and Finance 30 with n i=1 π i = 1 and φ = {φ ij } Then we define the expected returns as and the risk premium is given by R e R f. R e = n i=1 π ir e i R f = n i=1 π ir f i

31 Macro II Chapter 5 Macro and Finance Results preference parameters: α and β Technology parameters {φ ij }, {λ i } : it is assumed that λ takes two values: λ 1 = 1 + µ + δ and λ 2 = 1 + µ δ, and φ 11 = φ 22 = φ, φ 12 = φ 21 = 1 φ. For US data over , consumption growth =.018, consumption growth s.d. =.036, consumption growth serial correlation = -.14 µ =.018, δ =.036, φ =.43 Then, we search for (α, β) so that the average risk free rate and the equity risk premium are reproduced. α = people s willingness to substitute consumption between successive yearly time period not greater that 10. β ]0, 1[ The model cannot reproduce a equity premium of more that.35%, while it is 6 in the data (given that the risk free rate is.8%)

32 Macro II Chapter 5 Macro and Finance 32 There is therefore an Equity Premium Puzzle A large literature has been devoted to this question See Kocherlakota 1996 for a nice survey Here I present a quantitative solution of this (quantitative) puzzle, as proposed by Jerman A Possible Resolution of the Puzzle Jerman (1998, JME) proposed a model with production, capital, habit formation and K adjustment costs that is quantitatively satisfactory.

33 Macro II Chapter 5 Macro and Finance The Model Firms where β k Λ t+k Λ t max E t k=0 β kλ t+k Λ t [A t+k F (K t+k, X t+k N t+k ) w t+k N t+k I t+k ] is the MRS of the owners of the firm. K t+1 = (1 δ)k t + φ ( It K t ) K t φ( ) < 1 is a positive concave function that models adjustment costs on capital the shadow price of one installed unit of capital, q, differs from the price of one new unit of capital (Tobin s q) Firms are financed by retained earnings, and dividends are given by D t = A t F (K t, X t N t ) w t N t I t

34 Macro II Chapter 5 Macro and Finance 34 Hh s.t. w t N t + a t(v a t max E t k=0 + D a t ) C t + a t+1v a t (Λ t ) β k u(c t+k ) a t is a vector of financial assets held at t and chosen at t 1. this vector contains the representative firm, + possibly other assets. V a is the vector of asset prices and D a the vector of dividends payments. The Hh also face a time constraint : N t + L t = 1, and we assume habit persistence : u = u(c t αc t 1 ) Market equilibrium: A t F (K t, X t N t ) = C t + I t. Shocks are to the technology A

35 Macro II Chapter 5 Macro and Finance Model Solution If we use a log-linear approximation to solve the model, the expected returns will be the same for all agents no possibility to account for the risk premium The model is solved by log-linearization, but asset prices are computed in a second round using lognormal pricing formulas (see Hansen & Singleton, 1983) The model solution can be written s t = Ms t 1 + ε t (22) where ε could be a multivariate normal iid shock (In this model, it is univariate). Then we use basic asset pricing formula: a claim on future payment D t+k (s t+k ) has a value [ ] V t (s t ) = β k Λt+k (s t+k D t+k (s t+k ) E t Λ t (s t ) (23)

36 Macro II Chapter 5 Macro and Finance 36 If Λ and D are lognormal, with distribution given by (22), then the risk free rate ca be computed from (23) with k = 1 and D(s t+1 ) = 1 : where λ t = Λ t X t, X t = γx t 1, β = βγ t The return on equity is given by E(R t,t+1 (s t )) = γ β exp { 1 2 (var(e tλ t+1 λ t ) var(λ t+1 E t λ t+1 ) } R d t,t+1(s t, s t+1 ) = V t+1(s t+1 ) + D t+1 (s t+1 ) V t (s t ) Jerman uses simulations to find the unconditional mean.

37 Macro II Chapter 5 Macro and Finance Quantitative predictions Calibration easy part : 1. long run restrictions: Cobb-Douglas elasticity on labor =.64, γ = (per quarter), δ = Productivity shocks : A follows a AR(1) process, with persistence.95 or 1, with s.d. of the innovation which is such that postwar US gdp s.d. is reproduced 3. Risk aversion: c 1 τ 1 τ τ = 5

38 Macro II Chapter 5 Macro and Finance 38 Difficult part : 1. habit formation parameter α 2. K adjustment cost elasticity ξ 3. time preference β 4. shock persistence ρ Let θ 1 = [α, β, ξ, ρ]. θ 1 is chosen (estimated) to minimize F = (θ 2 f(θ 1 )) Ω(θ 2 f(θ 1 )) where θ 2 is a vector of moments to match, f(θ 1 ) is the vector of corresponding moments generated by the model and Ω a weighting matrix. (Simulated Method of Moments) θ 2 =(s.d. of c growth/s.d. of y growth, s.d. of i growth/s.d. of y, mean risk free rate, equity premium), Ω is identity

39 Macro II Chapter 5 Macro and Finance 39 The solution is that F = with α ξ β ρ

40 Macro II Chapter 5 Macro and Finance 40 The simulation results are Table 1: Simulation results, Jerman 1998 σ c σ y σ i σ y E(r f ) E(r e r f ) Data Standard RBC Standard RBC + τ = Habit persistence K adj. costs Benchmark

41 Macro II Chapter 5 Macro and Finance 41 4 The Information Technology Revolution and the Stock Market 4.1 Motivations Here we show that a simple model of asset pricing can account for the late 60 s early 70 s drop in U.S. (and OECD) market capitalization

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43 Macro II Chapter 5 Macro and Finance 43 Puzzling phenomenon: The market value of U.S. equity relative to GDP plunged in Greenwood & Jovanovic story: 1. The market declined in the late 1960 s because installed firms would have to give way to IT, while IT firms were not yet listed 2. IT innovators boosted the stock market s value only in the 1980 s. The main assumptions are 1. The IT revolution was heralded in 1973, or perhaps in stages during The IT revolution favored new firms.

44 Macro II Chapter 5 Macro and Finance 44 Reasons for which the IT revolution did favor new firms: 1. Awareness and skill: the manager of an old firm may not know what the new technology offers or may be unable to implement it. 2. Vintage capital: An old firm s human and physical capital is tied to its current practices, and may not easily convert to new technology. 3. Vested interests: management and workers in an older firm may resist new technology because it devalues their skills. In doing so, they harm the interests of the firms shareholders. Let s put down a formal model

45 Macro II Chapter 5 Macro and Finance A Simple Model Fundamentals Consider a Lucas 1978 economy: exchange economy, many infinitely-lived identical agents, equally many infinitely-lived trees. Preferences t=0 βt U(y t ), perfect foresight A tree promises a stream of dividends {d t }, its date-0 price is [ U P 0 = β t ] (y t ) d U t (y 0 ) t=0 We assume that a tree yield 1 unit of output in each period, forever, and that is the only source of income for a representative agent, so that [ U P 0 = β t ] (1) U (1) t=0 = 1 1 β

46 Macro II Chapter 5 Macro and Finance 46 Assume that some unexpected news arrives at t = 0 ( IT revolution ) The news is some of the existing trees will die in the future, say at period T, and they will be replaced by more productive trees. Formally: a fraction x of the existing trees will die at period T. They will be replaced by equally many new trees yielding forever 1 + z per period. The type of a tree (dying at period T or living forever) is announced at period 0, and new trees are not traded before period T. At T, the owner ship of those new trees will be allocated equally among agents. Per capita output is given by y t = { 1 for t T xz for t T

47 Macro II Chapter 5 Macro and Finance Asset Prices Before period T, two types of trees are traded on the stock market: 1. type-1 trees (that dies at T ) : price P 1t = 1 βt t 1 β 2. type-2 trees (that lasts forever) : price [ U P 2t = P 1t + β τ ] (1 + xz) U (1) τ=t t [ t βt U ] (1 + xz) P 2t = P 1t + 1 β U (1)

48 Macro II Chapter 5 Macro and Finance Stock Market Dynamics Before T, the stock market value is a weighted average of the two types of trees [ t βt U ] (1 + xz) P t = xp 1t + (1 x)p 2t = P 1t + (1 x) 1 β U (1) After T, the stock market value is P t = 1 + xz 1 β

49 Macro II Chapter 5 Macro and Finance 49 Comment : Both x and z act to lower P before T, while T raises P. Why? 1. P t is decreasing in x: some trees are expected to be replaced by trees which are not yet in the market portfolio. 2. A rise in x raises the interest rate, and then decreases P. 3. P decreases in z because of an interest rate effect. 4. A rise in T rises P 1t : trees live longer. In T, the stock price increases permanently to P t = 1+xz 1 β The dynamics of P/GDP is depicted on the figure. The empirical counterpart of this figure is the first figure.

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51 Macro II Chapter 5 Macro and Finance Some More Observations

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