asset pricing: the lucas tree model

7/29/2019 Asset Pricing: The Lucas Tree Model

1/40

Asset Pricing: The Lucas Tree Model

Kaiji ChenThe University of Hong Kong

October 21, 2009

1


2/40

Review of Last Class

Denition of Great Depression

A large negative deviation from trend (balanced) growth path.

On balanced growth path, capital output ratio is constant, and all per

capita variables grow at constant rate except hours per working age

person.

episode including not only sharp decline but also probably slow recovery.

2


3/40

Great Depression Methodology

Growth accounting: various shocks aect aggregate output during de-

pression through three channels: eciency that inputs are combined

together for production, capital input, and labor input (from both sup-

ply and demand sides).

identify the quantitative importance of these channels through dynamic

general equilibrium model.

Simulation tells us sharp declines in TFP are important for the output

drop during U.S. Great Depression, but not the slow recovery.

Diagnose depression by measuring deviation of the models rst-order

conditions.

3


4/40

Tell us at which margins the standard model misses so as to provide

directions to which types of frictions we need when constructing new

models?

4


5/40

Road map of this Class

Lucas Tree Model

Applications of Lucas Tree Model to risk free assets and risky assets.

5


6/40

1. LUCAS TREE MODEL

1 Lucas Tree Model

Idea

What is nancial asset? Contract that promises to deliver an amount ofgoods in some future periods and particular states.

Whats the roles of nancial asset? Agents would like to smooth consump-

tion across time and states. In a market economy, this is implemented bybuying and selling nancial assets, which transfer resources across time

and states.

6


7/40

1. LUCAS TREE MODEL

How the price of an asset is determined? In a competitive equilibrium,

prices are such that all markets clear (demand meets supply).

Previously, our focus is optimal resource allocation (across time), given

resources constraint (in a social planners problem) or individual agents

budget constraints (in a competitive equilibrium).

Alternatively, given equilibrium quantities and demand function, we can

back out equilibrium prices.

In particular, take consumption process as given, solve for the equilibrium

prices of given nancial assets that transfer resource across time and dif-

ferent states.

7


8/40

1. LUCAS TREE MODEL

The Economy

Consider an economy inhabited by a large number of identical agents.

Endowment: the only durable good in the economy is a set of trees, which

are equal in the number to the number of people in the economy.

Each agent starts life at time zero with one tree.

Each period, each tree yields fruit or dividends in the amount dt to its

owner at the beginning of period t.

8


9/40

1. LUCAS TREE MODEL

Technology: Fruits cannot be stored. Dividends are exogenous and follow

a stochastic process.

Preference: agents in this economy consume a single good, which is fruit.

E0

1Xt=0

t

u (ct)

where u () is concave, strictly increasing and twice continuously dieren-

tiable.

9


10/40

1. LUCAS TREE MODEL

Market structure: how can the agent transfer resource across time? Assume

a competitive market in trees.

Ownership of a tree at the beginning of period t entitles the owner to

receive the dividend in period t and to have the right to sell the tree at

price pt in terms of consumption good:

Since all agents are identical in terms of preference and endowment, we

can assume there is a representative agent in this economy.

In equilibrium, there is only 1 tree (supply). Our purpose is nd the

price of the tree that will make the aggregate demand of the tree equal

to 1.

10


11/40

1. LUCAS TREE MODEL

Solution Strategies

Find the competitive equilibrium allocation via the social planners problem.

Calculate the FOCs for individual agents with the opportunity to buy and

sell the share of trees (assets)

Find the equilibrium prices that support the competitive equilibrium allo-

cation.

11


12/40

1. LUCAS TREE MODEL

Step 1: Social Planners Problem

maxfctg

1t=0

E0

1Xt=0

tu (ct)

s:t: ct dt

Solution: ct = dt; 8t:

12


13/40

1. LUCAS TREE MODEL

Step 2: Representative Agents Problem in a Competitive Market

Economy

Let st be the share of tree held at the beginning of time t and pt the price

of one share of tree at time t:

The problem of the representative agent is

maxfct;st+1g

1t=0

E0

1Xt=0

tu (ct) (1)

subject to

ct + ptst+1 = (pt + dt) st

s0 = 1 given

13


14/40

1. LUCAS TREE MODEL

First-order condition

ptu0 (ct) = Etu

0 (ct+1) (pt+1 + dt+1)

The utility (shadow) cost of buying one tree today is ptu0 (ct) :

Tomorrow, each tree delivers payo pt+1 + dt+1, which is a random vari-

ables: an investor does not know exactly how much he will get from his

investment, but he can assess the probability of various possible outcomes.

The shadow value of the tree, discounted to today, is Etu0 (ct+1) (pt+1 + dt+1) :

At the margin, an investor (consumer) is indierent between buying an

additional unit of tree.

14


15/40

1. LUCAS TREE MODEL

Transversality condition

limk!1

Etku0

ct+k

pt+kst+k = 0 (2)

If the expressions in equation was positive, the agent would be over-

accumulating assets so that a higher expected lifetime utility could be

achieved by increasing consumption today.

15


16/40

1. LUCAS TREE MODEL

The Representative Agents Problem in Recursive Form

The states of the economy

aggregate state: d: p = p (d)

individual state: s:

Because d is time-invariant Markov process, the consumers problem is

time-invariant.

The controls: c; s0:

16


17/40

1. LUCAS TREE MODEL

Bellman Equation

V (d; s) = maxc;s0

nu (c) + E

hV

d0; s0

j dio

(3)

subject to

c + ps0 = (p + d) s

17


18/40

1. LUCAS TREE MODEL

First-order Conditions

u0 (c) =

p = EhVs0 d0; s0 j di

Envelop Condition

Vs (d; s) = (p + d)

Euler equation

u0 (c)p = Eh

u0

c0

p0 + d0

j di

18


19/40

1. LUCAS TREE MODEL

Step 3: Equilibrium

Denition of Sequential Market Equilibrium: Given the stochastic process

fdtg1t=0 ; and the initial endowment s0 = 1; a sequential market equilib-

rium consist of allocation fct; st+1g1t=0 ; and prices fptg

1t=0 such that

Given fptg1t=0 ; fct; st+1g

1t=0 solve the representative consumers prob-

lem (1).

fptg1t=0 is such that the tree market clears: st+1 = 1; 8t: (This implies

ct

= dt)

19


20/40

1. LUCAS TREE MODEL

Denition of Recursive Competitive Equilibrium: a recursive competitive

equilibrium for this economy consists of value function V (d; s) ; a set of

decision rules c (d; s) ; s0 (d; s) ; and price function p (d) such that

Given the price p (d) ; V (d; s) solve the representative consumers

problem (3) ; with the decision rules c (d; s) ; s0 (d; s) :

Market clear. p (d) is such that s0 (d; s) = 1:

20


21/40

1. LUCAS TREE MODEL

Equilibrium Price

Our goal is to obtain the price that will induce the consumer to consume

the equilibrium allocation.

We impose the equilibrium allocation on the FOCs: (ct = dt)

pt = Etu0 (dt+1)

u0 (dt)(pt+1 + dt+1) (4)

The current price of a tree is equal to the expectation of the product

of the future payo on that tree with the intertemporal marginal rate of

substitution.

21


22/40

1. LUCAS TREE MODEL

Equation (4) can be generalized as

pt = Et (mt+1xt+1)

where mt+1 = u0(dt+1)

u0(dt)is also called stochastic discount factor (SDF),

and xt+1 is a payo, the value of investment at time t + 1 (e.g. for a

stock, the payo is pt+1 + dt+1).

it says that one can incorporate all risk corrections by dening a single

SDFthe same one for each assetand putting it inside the expectation.

The correlation between the random components of the common dis-

count factor m and the asset-specic payo x generate asset-specic

risk corrections.

In fact, all asset pricing models amount to alternative ways of connect-

ing the stochastic discount factor to data.

22


23/40

1. LUCAS TREE MODEL

Forwarding (4) by one period and use the law of iterated expectation

(EtEt+1 () = Et ()), we arrives the following expression

ptu0 (dt) = Et

1Xj=1

ju0

dt+j

dt+j + limk!1

Etku0

ct+k

pt+k (5)

No-arbitrage condition implies that the last term in (5) must be zero.

If this term is strictly positive, then the marginal utility gain of selling

share exceeds the marginal utility loss of holding the asset forever and

consuming the future streams of dividend.

Asset bubble can also be ruled out by directly referring to transversality

condition (2) ; and market clearing condition (st = 1, and ct = dt).

23


24/40

1. LUCAS TREE MODEL

As a result, equation (5) becomes

pt = Et

1Xj=1

ju0

dt+j

u0 (dt)dt+j (6)

Equation (6) says that the price of a share of the tree (asset) is an ex-

pected discounted stream of future dividends, with the discount factors

intertemporal marginal rates of substitution. Here the discount factor is

time varying and stochastic, since consumption (or dividends) is time vary-

ing and stochastic.

ju0(ct+j)

u0(ct)is marginal rate of substitution between time t + j goods and

time t goods (the personal valuation (price) of one unit of time t + j

dividend in terms of time t dividend.)

24


25/40

1. LUCAS TREE MODEL

Special Case 1: log utility

pt = Et

1Xj=1

jdt

dt+jdt+j

=

1 dt

Price of a share of asset only depend on it current dividend payo, not the

future payo.

In particular, assume that the fruits only take two values dt 2 fd1; d2g,

with d1 > d2: Then p1 > p2, that is the price of the tree is high in the

state when aggregate output (fruit) is high:

25


26/40

1. LUCAS TREE MODEL

Intuition: when current output is high, consumer tends to transfer

resources from today to tomorrow by purchasing the tree (by saving).

As a result, the demand for the tree is high, pushing up the price of

the tree.

26


27/40

1. LUCAS TREE MODEL

Intuition for the impact of an increase in future dividend dt+j on

pt

When dt+j becomes larger, this have two eects.

Income eects: an increase in dt+j increase the agents life time income.

Therefore agent would like to increases consumption at each period, in-

cluding time t:

Since dt is not changed, an increase in desirable ct induce agent to

reduce the holdings of shares of the tree (decrease in savings).

This tend to push the demand curve for asset to the left.

27


28/40

1. LUCAS TREE MODEL

Substitution eects: an increase in dt+j implies a higher future return for

investing in shares of the tree. Therefore, agent would like to postpone

consumption and increase savings.

This tend to push the demand curve for asset to the right.

When utility is log, income eect and substitution eect of an increase in

future dividend dt+j cancel out.

This leaves the demand curve for share of asset unchanged.

Hence, the price of a share of the tree is unchanged.

28


29/40

1. LUCAS TREE MODEL

Special Case 2: Risk Neutrality (u0 (c) = c)

From (4) ; we have

pt = Et (pt+1 + dt+1)

Or in terms of returns, we have

Et

"pt+1 + dt+1

pt

#=

1

1 (7)

Equation (7) states that the rate of return on each asset is unpre-dictable given current information, which is taken in Finance literature

as the implication of ecient market hypothesis.

29


30/40

1. LUCAS TREE MODEL

Special Case 3: A nite-state version

Assume dividend follows a time invariant Markov process dt 2n

d1;:::;dno

prob

dt+1 = di j dt = dj

= ji

The (n n) matrix with element ji is called a stochastic matrix.

The matrix satisesnX

l=1

kl = 1 for each k:

All elements in are nonnegative.

30


31/40

1. LUCAS TREE MODEL

Express equation (4) as

ptu0 (dt) = Etpt+1u

0 (dt+1) + Etdt+1u0 (dt+1) (8)

Express the price at t as a function of the state p

dk

= pk:

Dene ptu0 (dt) = p

ku0

dk

vk; k = 1; :::n:

Also dene k = Etpt+1u0 (dt+1) =

n

Xl=1

dlu0 dl kl:

31


32/40

1. LUCAS TREE MODEL

Then equation (8) can be expressed as

pku0

dk

= nX

l=1

plu0

dl

kl + nX

l=1

dlu0

dl

kl

or in matrix terms,

v = + v

This equation has a unique solution

v = (I )

1

(9)

The price of an asset at state k can be found from pk = vk=u0

dk

:

32


33/40

2. LUCAS TREE MODEL WITH RISK-FREE BOND

2 Lucas tree Model with Risk-free Bond

Now consider the same problem , but agent can also buy and sell a risk-freeone-period discounted bond for the price R1t (Rt can be seen as the rate

of return for a one period risk free bond); which deliver one unit of goodthe next period in any state.

Assume when agents were born, they are endowed with no-risk free bondb0 = 0.

The agents budget constraint is

ct + ptst+1 + R1t bt+1 = (pt + dt) st + bt

In recursive form, R = R (d) :

33


34/40


We follows the same steps in solving for the equilibrium bond

prices.

In Step 1, by solving the social planners problem, we still get ct = dt; 8t:

In Step 2, we solve for the representative agents problem and obtain

R1t = Etu0 (ct+1)

u0 (ct)

In Step 3, equilibrium condition gives bt+1 = 0: There is no one to buythese bonds or sell to, since all agents are the same in terms of preference

and endowments.

34


35/40


Our goal then is to nd the equilibrium price R1t that induce agents to

choose to buy 0 bond.

35


36/40


Again we impose the equilibrium allocation ct = dt to the rst-order

condition with respect to bond demand.

R1t = Etu0 (dt+1)

u0 (dt)

= 1

u0 (dt)Etu

0 (dt+1)

or

Rt =u0 (dt)

Etu0 (dt+1)

In general, we can price any asset in the same way.

36


37/40


Factors aecting the price of a risk-free bond, R1 (or real interest

rate, R)

The price of risk-free bond is low when people are impatient, i.e. when

is low. If everyone wants to consume now (thus having low demand

for bond), it takes a high interest rate (a low price of risk-free bond) toconvince to save (buy the bond).

The price of risk-free bond increases with Etu0 (dt+1) and decreases with

u0 (dt), that is, decreases with consumption growth).

The more abundant is todays resource relative to tomorrows, the more

valuable is tomorrows consumption relative to todays (the larger isEtu

0(dt+1)u0(dt)

).

37


38/40


The larger is the demand for this asset to transfer resource from today

to tomorrow.

Hence the higher price is risk-free bond (supply is always given).

38


39/40


assume u0 (c) = c (CRRA utility):

R1t =1

Et

dt

dt+1

!

The prices of risk-free bond (real interest rate) are more sensitive to con-

sumption growth if is large.

If utility is highly curved, then investors care more about maintaining

a consumption prole that is smooth over time.

hence they are less willing to rearrangement consumption over time in

response to changes in interest rate (change of price of bond).

Thus it takes a large interest rate change to induce him to a given

consumption growth.

39


40/40


Reinterpretation of the above pricing equation

Consumption growth is high when real interest rate is high (or the prices

of bond are low)

Consumption is less sensitive to real interest rate (the price of risk-free

bond ) as the desire for a smooth consumption stream, captured by ;

rise.

40

asset pricing: the lucas tree model

Documents