Lecture 8: Portfolio Choice and Consumption-based Asset...

Lecture 8: Portfolio Choice and Consumption-basedAsset Pricing

Yulei Luo

Econ HKU

November 20, 2017

Luo, Y. (Econ HKU) Macro Theory November 20, 2017 1 / 67

The Portfolio Choice Problem

In the permanent income model we have discussed, we assumed thatindividuals can only invest in one risk free asset. However, individualscan invest in many assets including the risk free asset and risky assetshaving uncertain returns.

In the presence of multiple assets, individuals face two optimaldecisions: consumption-saving decision and portfolio choice.

We start from a simple static portfolio choice problem. Consider aninvestor with wealth level w0, who need to decide how to allocate hisor her wealth in two assets: one is a risky asset with uncertain rater̃ and the other is a risk free asset that pays a certain return rf .

The investor’s wealth at the end of the period is

w1 = a (1+ r̃) + (w0 − a) (1+ rf ) = w0 (1+ rf ) + a (r̃ − rf ) ,

where a is the amount of wealth invested in the risky asset.


(Conti.) Thus the portfolio choice problem can be written as

maxaE [u(w1)] = max

aE [u(w0(1+ rf ) + a(r̃ − rf ))] (1)

where E is the expectation operator, u(·) is the utility functiondefined on final wealth (Note that in the static setup, it is equivalentwith consumption), and then E [u(·)] the expected utility.Under risk aversion (u′′ < 0), the optimal solution for the aboveportfolio choice problem must satisfy:

E [u′(w0(1+ rf ) + a(r̃ − rf )) · (r̃ − rf )] = 0, (2)

which describes the relationship between risk aversion and optimalasset allocation.

Note that if u′ > 0, u′′ < 0, and a∗ is the solution to (2), then

a∗ R 0⇐⇒ E (r̃) R rf


Proof.Define W (a) = E [u(w0(1+ rf ) + a(r̃ − rf ))]. (2) means that

W ′(a) = E [u′(w0(1+ rf ) + a(r̃ − rf ))(r̃ − rf )] = 0,W ′′(a) = E [u′′(w0(1+ rf ) + a(r̃ − rf ))(r̃ − rf )2] < 0

because u′′(·) < 0. It follows that a∗ > 0 if and only if

W ′(0) = E [u′(w0(1+ rf )) · (r̃ − rf )] = u′(w0(1+ rf ))E (r̃ − rf ) > 0

because a will have to increase from 0 to reach the equality in (2).

This result means that a risk averse investor will invest in the riskyasset only if the expected return on the risky asset is greater than therisk free rate.


Optimal Asset Allocation: CRRA Case

Start with u(x) = ln(x) and for simplicity assume that the return onthe risky asset has two states (“down”or “up” in the stock market)

r̃ ={r1, with probability 1− π,r2, with probability π

.

and r2 > rf > r1. Hence, the FOC can be rewritten as

E[

r̃ − rfw0(1+ rf ) + a(r̃ − rf )

]= 0.

Using the distribution of the returns, we may eliminate theexpectation operator:

π(r2 − rf )w0(1+ rf ) + a(r2 − rf )

+(1− π)(r1 − rf )

w0(1+ rf ) + a(r1 − rf )= 0, (3)

which can be solved directly

aw0=(1+ rf )E [r̃ − rf ](rf − r1)(r2 − rf )

, (4)


Implications

The fraction of wealth invested in the risky asset, aw0, increases with

the risky premium, E [r̃ − rf ], and decreases with the dispersion ofthe return of the risky asset, (rf − r1)(r2 − rf ), (Note that in thistwo-state case, this dispersion measures the volatility of asset returns).

It is independent of the wealth level (w0) in the CRRA case.

A numerical example: suppose that r2 = 0.4, rf = 0.05, r1 = −0.2,and π = 0.5, substituting these values into the optimal ratio abovegives

aw0= 0.6.


The Effects of Risk Aversion on Optimal Asset Allocation

Consider u(x) = x 1−γ−11−γ , γ > 1. Following the same procedure above,

we have

π(r2 − rf )[w0(1+ rf ) + a(r2 − rf )]γ

+(1− π)(r1 − rf )

[w0(1+ rf ) + a(r1 − rf )]γ= 0, (5)

which implies that

aw0=

(1+ rf ){[(1− π) (rf − r1)]1/γ − [π (r2 − rf )]1/γ

}(r1 − rf ) [π (r2 − rf )]1/γ − (r2 − rf ) [(1− π) (rf − r1)]1/γ

(6)A numerical example: suppose that γ = 3,

aw0= 0.24 < 0.6, (7)

which means that risk aversion reduces the optimal allocation in therisky asset.


Optimal Asset Allocation: CARA

Consider u(x) = − exp(−αx) (that is, RA(x) = α), the maximizationproblem becomes

maxaE [− exp (−α [w0(1+ rf ) + a(r̃ − rf )])]. (8)

The FOC is thus

E [α(r̃ − rf ) exp (−α [w0(1+ rf ) + a(r̃ − rf )])] = 0. (9)

For simplicity, we also consider two-state distribution, the optimalamount invested in the risky asset is

a =1α

1r1 − r2

log(1− π

π

rf − r1r2 − rf

), (10)

which is independent of the wealth level w0, i.e.,

dadw0

= 0. (11)

Note that a > 0⇐⇒ 1−ππ

rf −r1r2−rf ∈ (0, 1)⇐= π > 1/2.


Note that we may also examine the property of a (i.e., dadw0) from the

FOC directly instead of solving for the explicit solution:Differentiating the FOC w.r.t. w0 gives

E

[α2 (r̃ − rf ) exp (−α [w0(1+ rf ) + a(r̃ − rf )]) ·[

(1+ rf ) + dadw0(r̃ − rf )

] ]= 0, (12)

which implies that

(1+ rf )E [α2(r̃ − rf ) exp(−α(w0(1+ rf ) + a(r̃ − rf )))]︸︷︷︸

=0 (Implied by the FOC (9))

+E [α2(r̃ − rf )2︸︷︷︸>0

exp (−α [w0(1+ rf ) + a (r̃ − rf )])︸︷︷︸>0

dadw0

] = 0,(13)

which also means that dadw0

= 0.


Optimal Asset Allocation: Quadratic utility

Consider u(x) = x − 12x2 (note that we impose that x < 1), the

maximization problem becomes

maxaE[w0(1+ rf ) + a(r̃ − rf )−

12[w0(1+ rf ) + a(r̃ − rf )]2

].

(14)The FOC is

E [(r̃ − rf ) {1− [w0(1+ rf ) + a(r̃ − rf )]}] = 0. (15)

Differentiating it w.r.t. w0 gives

E[(r̃ − rf )

[−(1+ rf )−

dadw0

(r̃ − rf )]]

= 0 =⇒

−(1+ rf )E [r̃ − rf ]︸︷︷︸>0

− E[(r̃ − rf )2

dadw0

]= 0 =⇒

dadw0

= − (1+ rf )E [r̃ − rf ]E [(r̃ − rf )2]

< 0. (16)


Given the two-state distribution, the optimal a is

π(r2 − rf ) (1− [w0(1+ rf ) + a(r2 − rf )])+(1− π)(r1 − rf ) (1− [w0(1+ rf ) + a(r1 − rf )]) = 0 (17)

which implies that

a =[π(r2 − rf )− (1− π)(r1 − rf )] [1− w0(1+ rf )]

π(r2 − rf )2 − (1− π)(r1 − rf )2. (18)


A More General Static Case

We abstract from consumption and saving decisions here. Theoptimizing portfolio choice problem is:

maxαt

Et[A1−γt+1

]1− γ

, subject to: At+1 = Rpt+1At , (19)

where the market portfolio includes two assets: one risky asset andone risk free asset, and

Rpt+1 = αtRet+1 + (1− αt )R f ,

which means

Et[Rpt+1

]= R f + αt

(Ret+1 − R f

)and var

[Rpt+1

]= α2t var [Ret+1]

(20)


Using the fact for the lognormal variable:

log Et [Xt+1] = Et [logXt+1] +12

var t [logXt+1] , (21)

the above optimizing problem is equivalent to

maxαtlog Et

[A1−γt+1

]= max

αt(1− γ)Et [at+1] +

12(1− γ)2 var t [at+1] ,

(22)where at+1 = logAt+1.

Taking log on both sides of the constraint gives

at+1 = rpt+1 + at , (23)

where rpt+1 = logRpt+1. Rewriting the optimizing problem:

maxαtEt[rpt+1

]+12(1− γ) var t

[rpt+1

](24)


Next, we need to approximate the return to the market portfolio:

rpt+1 − r f = αt (r et+1 − r f ) +12

αt (1− αt )ω2. (25)

because:

Rpt+1 = αtRet+1 + (1− αt )R f =⇒Rpt+1R f

= 1+ αt

(Ret+1R f− 1)=⇒

rpt+1 − r f = log[1+ αt

(exp

(r et+1 − r f

)− 1)],

which gives a nonlinear relation between rpt+1 − r f and r et+1 − r f .


This relation can be approximated using a second-order Taylorexpansion around the point r et+1 − r f = 0. The functionf(r et+1 − r f

)= log

[1+ αt

(exp

(r et+1 − r f

)− 1)]is approximated

as:

f(r et+1 − r f

)' f (0) + f ′(0)

(r et+1 − r f

)+12f ′′(0)

(r et+1 − r f

)2,

(26)where f ′(0) = αt and f ′′(0) = αt (1− αt ). In addition, replace(r et+1 − r f

)2by its expectation ω2.

Substituting this approximation back into the optimizing problemgives

maxαtEt

[r f + αt (r et+1 − r f ) +

12

αt (1− αt )ω2]+12(1− γ) α2tω

2,

(27)The FOC is

αt =E[r et+1 − r f

]+ω2/2

γω2 , (28)

which has the same implications as the CRRA case with the two-statedistribution we discussed above.Luo, Y. (Econ HKU) Macro Theory November 20, 2017 15 / 67

The Joint Saving-Portfolio Choice Problem

We have so far distinguished the consumption-saving decision and theportfolio allocation decisions. The two decisions, however, should beconsidered jointly. We now formalize the consumption-savings andportfolio choice problem:

max{a,s}

E [u(w0 − s) + βu(s(1+ rf ) + a(r̃ − rf ))] (29)

s.t.w0 ≥ s ≥ 0, (30)

where s denotes the total amount of saving and a is the amount investedin the risky asset.

When the utility function is CRRA, the FOCs are:

−(w0 − s)−γ + βE [(s(1+ rf ) + a(r̃ − rf ))−γ(1+ rf )] = 0,(31)

E [(s(1+ rf ) + a(r̃ − rf ))−γ(r̃ − rf )] = 0.(32)


Implications

The first FOC compares the marginal utility today with the expectedmarginal utility tomorrow.

For the second FOC,

E [(s(1+ rf ) + a(r̃ − rf ))−γ(r̃ − rf )] = 0 =⇒ (33)E [(s(1+ rf ) + a(r̃ − rf ))−γ r̃ ] = E [(s(1+ rf ) + a(r̃ − rf ))−γrf ], (34)

which means that if investors are behaving optimally, a marginalinvestment at t in any asset should yield the same expected marginalincrease in utility at t + 1.


A More Realistic Infinite-horizon Consumption-Saving andPortfolio Choice Model

The infinite-horizon optimizing problem:

maxE0

[∞

∑t=0

βjc1−γt − 11− γ

](35)

subject toat+1 = R

pt+1(at + yt − ct ), (36)

where Rpt+1 = αRet+1 + (1− α)R f . It is diffi cult to solve this problem.

You may refer to Campbell and Viceira (Strategic Asset Allocation2002, Cambridge U. Press) for the details about how to solve thisproblem and some interesting discussions on long-term assetallocation, labor income risk, and optimal consumption and savings.


Consumption-based Asset Pricing

Even if we cannot easily solve the full-fledged optimal consumptionand portfolio choice model, we can still gain many interesting insightsabout the joint dynamics of the asset return and consumptiondynamics by inspecting the Euler equation.

We now assume that there are n risky assets such that

Rpt+1 =n

∑j=1

αjR jt+1 +

(1−

n

∑j=1

αj

)R f . (37)

In this case, the Euler equations for all assets are

u′ (ct ) =1

1+ ρEt[R jt+1u

′ (ct+1)],

where j = f , 1, · · ·, n.


Model Implications

Note that the Euler equations can be rewritten as

1 =1

1+ ρEt

[R jt+1

u′ (ct+1)u′ (ct )

]≡ Et

[R jt+1Mt+1

], (38)

where Mt+1 is the stochastic discount factor applied at t toconsumption in the following period. It is the intertemporal marginalrate of substitution, i.e., the discounted ratio of marginal utilities ofconsumption in any two subsequent periods.

Using (38), we have the key result of consumption-based asset pricing:

Et[R jt+1Mt+1

]= Et

[R jt+1

]Et [Mt+1] + cov t

(R jt+1,Mt+1

)=⇒

Et[R jt+1

]=

1Et [Mt+1]

[1− cov t

(R jt+1,Mt+1

)]. (39)


(conti.) In the case of the risk free asset, we have

R f =1

Et [Mt+1]. (40)

Combining (39) with (40), we have

Et[R jt+1

]− R f = −R f cov t

(R jt+1,Mt+1

), (41)

which means that: In equilibrium, the risky asset j whose return has anegative correlation with the SDF yields an expected return higherthan R f .

This asset is risky for the investor because it yields lower returns whenthe marginal utility of consumption relatively high due to a relativelylow level of consumption. In equilibrium investors are still willing tohold this asset only if such risk can be compensated by a premiumdetermined by an expected return higher than the risk free rate R f .


Implications of CRRA Utility

(Conti.) Assume the CRRA utility c1−γt −11−γ , we have

Et[R jt+1

]− R f = −R f cov t

(R jt+1, β

(ct+1ct

)−γ). (42)

Using the facts that (1) If ∆ct+1ct

is small,

ct+1ct' 1+ ∆ log ct+1. (43)

(2) If x is small,(1+ x)n ' 1+ nx , (44)

(42) can be written as

Et[R jt+1

]− R f ' −βR f cov t

(R jt+1, 1− γ∆ log ct+1

)' cov t

(R jt+1,γ∆ log ct+1

)(45)

Note that βR f is close to 1 if ∆ct+1ct

is small.


First Look: The Equity Premium Puzzle

Taking unconditional expectations on both sides of (45) gives

E[R jt+1

]− R f ' cov

(R jt+1,γ∆ log ct+1

)(46)

= γ corr(R jt+1,∆ log ct+1

)sd(R jt+1

)sd (∆ log ct+1)

Mehra and Prescott (1985) show that it is diffi cult to reconcileobserved returns on stocks and bonds with equation (45). Asdocumented in Campbell (2003), in the U.S. data from1947.2− 1998.4:

corr(R jt+1,∆ log ct+1

)= 0.34, (47)

sd(R jt+1

)= 15.6%, (48)

sd (∆ log ct+1) = 1.1%, (49)

E[R jt+1

]− R f = 7%, (50)

which means that γ = 120! It is highly unrealistic.Luo, Y. (Econ HKU) Macro Theory November 20, 2017 23 / 67

To gain some idea about what plausible values of γ are, consider thefollowing gamble: You must choose between a gamble in which youconsume $50000 in the rest of your life with probability 0.5 and$100000 with probability 0.5, or consuming some amount X withcertainty. The CRRA, γ, determines the value of X which wouldmake you indifferent between consuming X or being exposed to therisky gamble.

E.g., if γ = 0, then you have no risk aversion at all and will will beindifferent between $75000 with certainty and the 50/50 gamble withexpected value of $75000. Here are the values of X associated with

different γ:

γ X1 707115 5856510 5399130 51209∞ 50000


Consumption-based Capital Asset Pricing Model(C-CAPM)

In reality, assets are traded every period, as new information becomesavailable, and decisions are made sequentially. Consequently, today’sdecisions affect tomorrow’s opportunities. CCAPM can be used tocapture these dynamic features and to price assets in such anenvironment.

Another advantage: This theory provides a way to link the realeconomy (aggregate output and consumption) and financial markets(asset prices and returns).

Lucas (1978) first developed this CCAPM theory. It is an endowmenteconomy (i.e., no production decisions) and allows recursive securitytrading.


Lucas (1978)’s Asset Pricing Model

Imagine the security as representing ownership of a fruit tree wherethe nonstorable output varies over time.Suppose that there is an economy with N identical agents with RE.That is, each agent’s expectations are conditional on all availableinformation (including the structure of the economy, the outputprocess, etc.)Suppose that all output is obtained from an asset which produces astochatic endowment of perishable consumption goods for each unitof the asset the agent owns at the beginning of t.If an agent owns zt units of the asset at the beginning of t, hereceives an endowment of ztyt of the consumption good and yt isidentical for each unit of the asset held by an agent and is anexogenous stochastic process.Assume that they have identical endowments. Since the agents haveidentical preferences, they will make the same decisions given thestate of the economy.Luo, Y. (Econ HKU) Macro Theory November 20, 2017 26 / 67

(Conti.) The typical agent chooses optimal holdings of securities andconsumes dividends:

max{ct ,zt+1}

E

[∞

∑t=0

βtu(ct )

], (51)

s.t. ct + ptzt+1 ≤ ztyt + ptzt , ∀t, (52)

where pt is the period t real price of the security in terms ofconsumption and zt is the agent’s beginning-of-period t holdings ofsecurity. The expectation operator applies across all possible states ofy .In the this economy:

Financial markets are in equilibrium iff at the prevailing price, supplyequals demand and the equilibrium price is that price at which theagents wishes to hold exactly the amount of the securities present inthe economy.Assume that the total supply of assets is N. With N agents wantin thesame number of assets, we must have zt = 1 for all t and for everyagent in equilibrium.The total net supply of I-owe-You (IOU) type of contract (insidebonds) must be zero; otherwise, there is no equilibrium.


Output is an exogenously stochastic process. We may assume thatthe output process follows a n-state probability transition. E.g., ytcould take three states:

[y1 y2 y3

]and it follows a probability

transition matrix to switch from one state to another state:Table: Three-state probability transition matrixoutput in t + 1

output in t Π =

π11 π12 π13π21 π22 π23π31 π32 π33

where πij = prob

(yt+1 = y j |yt = y i

)for any t.

Note this discrete-state distribution can be approximated from anAR(1) process.


Approximating an AR(1) Process

Given the following AR(1) process with continuous states,

yt+1 = ρyt + (1− ρ) y + εt+1, (53)

we can numerically approximate it with finite discrete states. TheMatlab codes posted on the course website can used to approximatethe AR process: discretizationAR1.m and Tauchen.m.

E.g., given that ρ = 0.9, ω = 0.1, y = 1, and n = 3, running thediscretization code yields:[

y1 y2 y3]=

[0.5412 1 1.4588

], (54)

Π =

0.9668 0.0332 0.00000.0109 0.9782 0.01090.0000 0.0332 0.9668

. (55)


Using the Bellman equation to Solve the Model

Define the value function as

v(ms ) = maxEs

[ ∞

∑t=s

βt−su(ct )]

(56)

s.t. ct + ptzt+1 ≤ ztyt + ptzt (57)

where ms is the beginning-of-period s wealth:

ms = (ys + ps ) zs , (58)

and rewrite the budget constraint as

cs + pszs+1 = ms (59)


The Bellman equation can be written as

v(ms ) = max{cs}{u(cs ) + βEs [v(ms+1)]} (60)

= max{cs}{u(cs ) + βEs [v((ys+1 + ps+1) zs+1)]} (61)

= max{cs}

{u(cs ) + βEs

[v((ys+1 + ps+1)

ms − csps

)

]}(62)

The FOC is:

u1(cs )− βEs

[V1(ms+1)

ys+1 + ps+1ps

]= 0 (63)

The envelop theorem is:

v1(ms ) = βEs

[v1(ms+1)

ys+1 + ps+1ps

], (64)

which means that u1(cs ) = v1(ms ) and u1(cs+1) = v1(ms+1).


Substituting this Envelop condition into the FOC gives the Eulerequation:

u1(cs )ps = βEs [u1(cs+1) (ys+1 + ps+1)] . (65)

Given the functional dependence on the output state variables, theEuler equation can be written as

u1(cs(y i))ps(y i)= β ∑

jπij[u1(cs+1

(y j)) (

y j + ps+1(y j ))], ∀i

(66)

Economic implications:

The LHS: the utility loss in period t associated with the purchase of anadditional unit of the security.The RHS: the expected discounted utility gain associated with sellingthe extra unit of the security.If this equality is not satisfied, the agent will try either to increase or todecrease his holdings of the security to increase his utility.


In Equilibrium

zt = zt+1 = · · · = 1, in other words, every agents owns the samenumber of trees, 1.

ct = yt , that is, ownership of the entire security entitles the agent toall the economy’s output.

u1(cs )ps = βEs [u1(cs+1) (ys+1 + ps+1)] are optimal given theprevailing prices. Substituting ct = yt into this Euler equation gives:

u1(ys )ps = βEs [u1(ys+1) (ys+1 + ps+1)] , (67)

which is the fundamental equation of the consumption-based CAPM.


Asset Prices

A recursive substitution of (67) into itself yields

ps = Es∞

∑j=1

[βju1(ys+j )u1(ys )

ys+j

], (68)

where Ms ,s+j = βju1(ys+j )u1(ys )

is the stochatic discount factor (i.e., theintertemporal marginal rate of substitution) and assume that the priceis bounded. (68) means that the stock price is the sum of allexpected discounted future dividends.Example: If the utility function displays risk neutrality (i.e., u1(·) isconstant),

ps = Es∞

∑j=1

[βjys+j

]= Es

∞

∑j=1

[ys+j

(1+ rf )j

], (69)

which means that the stock price is the sum of expected futuredividends discounted at the constant risk free rate.Luo, Y. (Econ HKU) Macro Theory November 20, 2017 34 / 67

The difference between (68) and (69) is the necessity of discountingthe flow of expected dividends at a rate higher than the risk free rate,so as to include a risk premium. Which factors affect risk premium isa central issue in finance.

Another examples: If the utility function is log, the price is

ps = Es∞

∑j=1

(βjysys+j

ys+j

)= Es

∞

∑j=1

(βjys

)(70)

=β

1− βys , (71)


Calculating the equilibrium price function.

The Euler equation, u1(ys )ps = βEs [u1(ys+1) (ys+1 + ps+1)],implicitly defines the equilibrium price. And we can calculate theactual equilibrium prices once specifying parameter values andfunction forms: select β, the utility function u(c), and the transitionmatrix Π.Specifically, we can solve for

{p(y j ), j = 1, · · ·,N

}as the solution to

a system of linear equations:

u1(y1)p(y1)= β

N

∑j=1

π1j[u1(y j )

(y j + p

(y j))]

(72)

· · · (73)

u1(yN )p(yN)= β

N

∑j=1

πNj[u1(y j )

(y j + p

(y j))]

(74)

with unknowns{p(y j ), j = 1, · · ·,N

}.


Numerica Example

Suppose that β = 0.96, u(c) = ln(c), (y1, y2, y3) = (1.5, 1, 0.5),and the transition matrix Π :.

Three-state probability transition matrixoutput in s + 1 (ys+1)

output in s (ys ) Π =

0.5 0.25 0.250.25 0.5 0.250.25 0.25 0.5

where πij = prob

(ys+1 = y j |ys = y i

)for any s.

Hence, we have three equations and three unknowns{p(y j ), j = 1, 2, 3

}and can solve for the equilibrium prices:

p(1) = 24; p(1.5) = 36; p(0.5) = 12. (75)


Link Asset Prices to Asset Returns

Define the return of security j from period t to t + 1 as

1+ rj ,t+1 =pj ,t+1 + yj ,t+1

pj ,t. (76)

Using this definition, the above Euler equation can be rewritten as

1 = βEt

[u1(ct+1)u1(ct )

(1+ rj ,t+1)]

(77)

Let qt denote the price in t of a one-period risk free bond in zero netsupply, which pays one unit of consumption in every state in t + 1.Hence,

qtu1(ct ) = βEt [u1(ct+1) · 1] , (78)

where qt is the equilibrium price at which the agent desires to holdzero units of the security, and thus supply equals demand.


Since (1+ rf ,t+1) qt = 1, we have

11+ rf ,t+1

= qt = βEt

[u1(ct+1)u1(ct )

], (79)

which means that in the risk neutrality case, the risk free rate mustbe a constant.

Combining the above two pricing equations, we have

1 = βEt

[u1(ct+1)u1(ct )

]Et [1+ rj ,t+1] + β covar t

[u1(ct+1)u1(ct )

, 1+ rj ,t+1

],

(80)where we use the fact that for two random variables:covar [x , y ] = E [xy ]− E [x ]E [y ] .Denote Et [1+ rj ,t+1] = 1+ r j ,t+1, we have

1 =1+ r j ,t+11+ rf ,t+1

+ β covar t

[u1(ct+1)u1(ct )

, rj ,t+1

](81)


Rearranging gives

r j ,t+1 − rf ,t+1 = −β (1+ rf ,t+1) covar t

[u1(ct+1)u1(ct )

, rj ,t+1

](82)

This equation is the central relationship of the CCAPM and is verysimilar to the pricing equation obtained in the last lecture. The LHSis the risk premium on security j .

It means that the risk premium will be large when

covar t[u1(ct+1)u1(ct )

, rj ,t+1]is large and negative, that is, for those

securities paying high returns when consumption is high (i.e., u1(·) islow), and low returns when consumption is high (i.e., u1(·) is high).These securities are not very desirable for reducing consumption riskbecause they pay high returns when investors don’t need them and lowreturns when they are most needed.Since they are not desirable, they have a low price and high expectedreturns for compensation.


A Discussion

The standard CAPM in finance tells us a security is relativelyundesirable and thus commands a high return when it covariespositively with the market portfolio. The Consumption CAPM addssome some further degree of precision: from the viewpoint ofconsumption smoothness and risk diversification, an asset is desirableif it has a high return when consumption is low and vice versa.

C-CAPM is more convincing in the multiperiod context because thevalue of an asset is to provide the investor intermediate consumptionover time; consequently, the key to an asset’s value is its covariationwith the marginal utility of consumption.

An unappealing feature of the above CCAPM is that the marginalutility of consumption is not observable. We can eliminate thisfeature by adopting a specific utility function.


Solving the CCAPM with growth

So far, we have assumed that output (=dividend=consumption inequilibrium) is stationary. In reality, consumption and output aregrowing over time. Here we assume that output growth rather thanoutput itself follows a distribution with possible finite states:(x1, · · ·, xN ) whose realizations are governed by a stochastic processwith transition matrix Π. Then for whatever xi is realized in periodt + 1:

dt+1 = xt+1yt = xt+1ct = xict . (83)

Using the CRRA utility, in equilibrium (yt = ct):

y−γt p(yt , xi ) = β

N

∑j=1

πij (xjyt )−γ [xjyt + p(xjyt , xj )] or (84)

p(yt , xi ) = βN

∑j=1

πij (xj )−γ [xjyt + p(xjyt , xj )] , (85)

which means that the SDF is determined exclusively by theconsumption growth rate xj .Luo, Y. (Econ HKU) Macro Theory November 20, 2017 42 / 67

Just like Mehra and Prescott (1985), we guess

p(yt , xi ) = viyt , (86)

for a set of constants {v1, · · ·, vN} , each should be identified with thecorresponding growth rate.

With this functional form, the asset pricing equation reduces to

viyt = βN

∑j=1

πij (xj )−γ [xjyt + vjxjyt ] or (87)

vi = βN

∑j=1

πij (xj )−γ [xj + vjxj ] = β

N

∑j=1

πij (xj )1−γ (1+ vj ) ,(88)

which is again a system of linear equations in the N unknowns{v1, · · ·, vN} .


Thus, for any state (y , xj ) = (c, xj ), the equilibrium asset price is

p(yt , xj ) = vjyt , (89)

if we assume that the current state is (y , xi ) while next period it is(xjy , xj ), then the one-period return is

rij =p(xjyt , xj ) + xjy − p(y , xi )

p(y , xi )(90)

=vjxjyt + xjy − viy

viyt=xj (1+ vj )

vi− 1, (91)

Hence, the mean or expected return, conditional on state i , isri = ∑N

j=1 πij rij ,and the unconditional equity return is given byE [r ] = ∑N

j=1 πj rj , where πj are long-run stationary probability ofeach state.The price of the risk free asset is

pf (c, xi ) = βN

∑j=1

πij (xj )−γ . (92)


The Empirical Validity of the CCAPM

A few key empirical observations regarding financial returns in USmarkets:

E[R jt+1

]− R f = 7%,

based on Campbell (2003)’s U.S. data from 1947.2− 1998.4.The equity premium puzzle found by Mehra and Prescott (1985): thestandard CCAPM is completely unable to replicate the high observedequity premium once reasonable parameter values are inserted in themodel.


The Reasoning of the Puzzle

According to the CCAPM theory, the only factors determining thecharacteristics of asset returns are the RA’s utility function, thesubjective discount factor, and the consumption process.

The utility function: CRRA c1−γ

1−γ and empirical studies have placed γ inthe range of [1, 5).The consumption process. In reality, consumption is growing over time.If there were no uncertainty in the model, and if the constant growthrate of consumption were to equal to its long-run historical average(around 1.0183), the asset pricing equation would reduce to

1 = βEt

[(ct+1ct

)−γ

Rt+1

]= βg−γR, (93)

where Rt+1 is the gross rate on capital, g and R are historical averagesof consumption growth and the gross rate.


For γ = 1, g = 1.0183, and R = 1.04, we can solve for the impliedβ ' 0.97. (Note that the economywide debt-to-equity ratios are notvery different from 1.) Since we use an annual estimate for g , theresulting β must be viewed as an annual or yearly subjective discountfactor. Similarly, on quarterly basis, β = 0.99.

If assume that γ = 2, the implied β = 0.99 annually and a quarterly βeven closer to 1. Specifically, assuming higher rates of risk aversionwould be incompatible with maintaining the hypothesis of a timediscount factor β < 1.At the root of this diffi culty is the low return on the risk free asset(1%). Highly risk averse individuals want to smooth consumption overtime (1/γ low if γ high), meaning they want to transfer consumptionfrom good times to bad times. Hence, when consumption is growingpredictably, the good times lies in the future. Agents want to borrownow against their future income.In the RA model, it is hard to reconcile with a low rate on borrowing:everyone is on the same side of the market and then inevitably forces ahigher rate. As a result, we need either an independent explanation forthe low average risk free rate or accept β > 1. Here, we just limitγ ≤ 2.


A Way to Test the CCAPM: Hansen-Jagannathan (HJ)Bounds

The bound proposed by HJ (1991) leads to a falsification of thestandard CCAPM and can also be applied in other asset pricingformulations.For all homogeneous agent economies, the equilibrium asset pricingcan be rewritten as:

p(st ) = Et [mt+1(st+1)X (st+1)] , (94)

where st is the state today, X (st+1) is the total return in next period,and mt+1(st+1) is the SDF:

mt+1(st+1) = βu1(ct+1(st+1))

u1(ct ). (95)

Suppress the state dependence, the pricing equation:

pt = Et [mt+1Xt+1]⇐⇒ 1 = Et [mt+1Rt+1] , (96)

where Rt+1 is the gross return.Luo, Y. (Econ HKU) Macro Theory November 20, 2017 48 / 67

Since the equation holds for each state st , it also holdsunconditionally:

1 = E [mR ] , (97)

where E denotes the unconditional expectation. For any two assets, iand j ,

E [m(Ri − Rj )] = 0, or E [mRi−j ] = 0, (98)

which implies that

E [m]E [Ri−j ] + covar [m,Ri−j ] = 0 =⇒ (99)

E [m]E [Ri−j ] + ρ [m,Ri−j ] sd [m] sd [Ri−j ] = 0 =⇒E [Ri−j ]sd [Ri−j ]

+ ρ [m,Ri−j ]sd [m]E [m]

= 0. (100)

Since ρ [m,Ri−j ] ≤ 1,

sd [m]E [m]

≥ |E [Ri−j ]|sd [Ri−j ]

(101)


The inequality is referred as the Hansen-Jagannathan lower bound onthe SDF. If i is the market portfolio and j is the risk free asset, wehave:

std [m]E [m]

≥ |E [RM − Rf ]|std [RM−f ]

=|E [RM − Rf ]|

std [RM ]=0.0620.167

= 0.37. (102)

We now can check whether this bound is satisfied for the standardCCAPM in which m = β(xt )−γ. Given

E [m] = β exp(−γµx +

12

γ2σ2x

)= 0.96 if γ = 2, (103)

for the HJ bound to be satisfied, the standard deviation of the SDFcannot be much lower than

0.37 · 0.96 = 0.355. (104)


Given the information about x (lognormal distribution), it isstraightforward to compute that

std [m] = 0.002! (105)

which is much lower than what is required for reaching the HJ bound(0.355).

Intuition: aggregate consumption is just too smooth and the MU ofconsumption doesn’t vary suffi ciently to satisfy the HJ bound impliedby asset data.


Summary

Reviewing the source of the failure of CCAPM in matching the data.Recall the original pricing equation

rM ,t+1 − rf ,t+1 = −β (1+ rf ,t+1) covar t

[u1(ct+1)u1(ct )

, rM ,t+1

]= − (1+ rf ,t+1) ρ

[βu1(ct+1)u1(ct )

, rM ,t+1

]std [m] std [RM ]

Implications: the equity premium depends on

The standard deviation of the SDFThe standard deviation of the market portfolioThe correlation between the two variables.

Hence, for the US and other industrial countries, the problem withthe CCAPMs is that aggregate consumption does not vary much atall. To make this model better fit the data, we must modify it in away that will increase the standard deviation of the relevant SDF.


Habit Formation

Main objective: admit utility functions that exhibit higher rates of riskaversion and thus can translate small variations in consumption into alarge variability of the SDF.One way to achieve this objective without causing the risk free ratepuzzle (which is exacerbated if we simply assume a higher RRA γ):introducing some form of habit formation.Habit formation: the agent’s utility today is determined not by herabsolute consumption level, but rather by the relative position of hercurrent consumption. The stock of habit can summarizes either herpast consumption history (with more or less weight placed on distantor old consumption levels) or the history of aggregate/averageconsumption (summarizing in a sense the consumption habits of herneighbors: a “keeping up with the Joneses” effect).Utility of consumption is primarily dependent on departures from priorconsumption history, either one’s own or that of a social referencegroup.


Habit Formation and The Equity Premium Puzzle

The RA’s preference takes the following form:

u(ct , ct−1) = u(ct − χct−1) =(ct − χct−1)1−γ

1− γ, (106)

where χ ≤ 1 is a parameter. When χ = 1, the utility depends only onthe deviation of current consumption ct from the previous period’sconsumption ct−1. Note that a general specification of habitformation can be written as

u(ct , xt ) =(ct − χxt )1−γ

1− γ, (107)

wherext = (1− θ) xt−1 + ct−1. (108)

Actual data indicate that aggregate consumption in the US and otherdeveloped countries is very smooth. This implies that (ct − ct−1) islikely to be very small most of the time.


For this specification, the agent’s effective relative risk aversionreduces to

RR (ct ) = −ctu′′(·)u′(·) =

γ

st, (109)

wherest = 1− χ (ct−1/ct ) . (110)

With ct−1 ≈ ct , the degree of effective risk aversion RR (ct ) couldthus be very high, even with a low γ, and the RA will appear asthough he is very risk averse. This opens the possibility for a veryhigh return on the risky asset.Note that when st ↓ 0, RR (ct ) ↑ ∞. With habit, the SDF can bewritten as

SDF = β

(ct+1ct

)−γ ( st+1st

)−γ

, (111)

where(st+1st

)−γcould be very volatile and correlated. If so, habit

formation might help explain the equity premium puzzle. SeeCampbell and Cochrane (JPE1999) for details.


Epstein-Zin Recursive Utility

Epstein and Zin (1989, 1991) propose a class of utility functions thatallows each dimension to be parameterized separately. Specifically,preferences are defined recursively over current (known) consumptionand a certainty equivalent of the next period’s total utility:

Ut = U(ct , ct+1, ct+2, · ··) = W (ct ,Rt (Ut+1)), (112)

where Rt (Ut+1) , CE t denotes the certainty equivalence in terms ofperiod-t consumption of the uncertain total utility in the futureperiods. Consider the CES aggregator function:

U(ct ,CE t ) =[(1− β)c1−1/ρ

t + β (CE t )1−1/ρ

] 11−1/ρ

. (113)

R t (Ut+1) = G−1 (Et [G (Ut+1)]) , (114)

where W and G are increasing and concave.


Asset Pricing Implications

Consider a CRRA specification for the certainty equivalent (CE). Fora random variable Ut+1, the period-t + 1 onward utility, the certaintyequivalent CE t (Ut+1) (= CE t ) is defined as follows:

[CEt (Ut+1)]1−γ = Et

[U1−γt+1

]when γ > 0 and γ 6= 1,(115)

=⇒ CEt (Ut+1) =(Et[U1−γt+1

])1/(1−γ)(116)

(113) can be rewritten as:

U(ct ,CEt ) =[(1− β)c

1−γθ

t + β (CEt )1−γ

θ

] θ1−γ

, (117)

where γ, ρ > 0,γ 6= 1, θ = 1−γ1−1/ρ .

If γ = 1/ρ (i.e., θ = 1) or if consumption is deterministic, we havethe usual standard time-separable expected utility setting with thediscount factor β and IES ρ (RRA γ = 1/ρ).


(Conti.) A proof. When γ = 1/ρ (θ = 1),

Ut (ct ,CEt )1−γ = (1− β)c1−γt + β (CEt )

1−γ

U1−γt = (1− β)c1−γ

t + β

((Et[U1−γt+1

])1/(1−γ))1−γ

U1−γt = (1− β)c1−γ

t + βEt[U1−γt+1

].

By unwinding the recursion, we get

U1−γt = (1− β)c1−γ

t + βEt[(1− β)c1−γ

t+1 + βEt[U1−γt+2

]]= · · · = (1− β)Et

[ ∞

∑s=t

βs−tc1−γs

],

which reduces to Ut =[(1− β)Et

[∑∞s=t βs−tc1−γ

s

]]1/(1−γ)and is

equivalent to the standard EU: Vt = Et

[∑∞j=0 βj

c1−γt+j1−γ

]. (Note that

Vt = 1(1−β)(1−γ)

U1−γt is an increasing function of Ut and therefore

represents the same preference as Ut .)Luo, Y. (Econ HKU) Macro Theory November 20, 2017 58 / 67

Asset Pricing Implications of RU

Epstein and Zin (1989,1991): derived the following asset pricingequation:

Et

[

β

(ct+1ct

)−1/ρ]θ [

1Rpt+1

]1−θ

R jt+1

= 1, (118)

where Rpt+1 is the period-(t + 1) return on the market portfolio, andR jt+1 is the return on some asset in it. Note that when θ = 1(γ = 1/ρ), the above pricing equation reduces to the standardtime-separable CCAPM case.

Note that in the EZ model, the SDF is a geometric average (withweights θ and 1− θ) of the SDF of the standard CCAPM

(β(ct+1ct

)−γ) and the SDF of the log case ( 1

R pt+1).


(Conti.) Hence, two covariances matter for an asset’s return pattern:1 the covariance of asset returns with consumption growth;2 the covariance of asset return with the return on the market portfolio.

The covariance with consumption growth captures its risk acrosssuccessive time periods (intertemporally, as in the standard CCAPM,while the covariance with the market portfolio captures its atemporalsystematic risk (as in the standard static CAPM).


Proof.We first rewrite the definition of the recursive utility together with thecertainty equivalent (115) as follows:

U1−γt =

[(1− β)c1−1/ρ

t + β(Et[U1−γt+1

])(1−1/ρ)/(1−γ)] 1−γ1−1/ρ

, (119)

which can be rewritten as

Ut =

[(1− β)c1−1/ρ

t + β(Et[U1−γt+1

])(1−1/ρ)/(1−γ)] 11−1/ρ

(120)

=[(1− β)c1−1/ρ

t + β (CEt )1−1/ρ

] 11−1/ρ ≡ f (ct ,CEt ) .

Further, the budget constraint is

At+1 = Rpt+1 (At − ct ) , for any t. (121)


Proof.(Conti.) With the RU, the SDF can be written as

SDFt ,t+1 =∂Ut/∂ct+1∂Ut/∂ct

= fCE (ct ,CEt )(Ut+1CEt

)−γ fc (ct+1,CEt+1)fc (ct ,CEt )

,

where the RHS is evaluated at the optimal consumption process,

∂Ut∂ct

= fc (ct ,CEt ) ,

∂Ut∂ct+1

= fCE (ct ,CEt )d (CEt )d (Ut+1)

∂Ut+1∂ct+1

= fCE (ct ,CEt )CEγt U−γt+1fc (ct+1,CEt+1)

where we use the fact that

d (CEt )d (Ut+1)

=(Et[U1−γt+1

]) 11−γ−1

U−γt+1 = CE

γt U−γt+1.


Proof.(Conti.) Note that given (120), the derivatives are

fc (ct ,CEt ) = (1− β)c−1/ρt U1/ρ

t ,

fCE (ct ,CEt ) = β (CEt )−1/ρ U1/ρ

t ,

fc (ct+1,CEt+1) = (1− β)c−1/ρt+1 U

1/ρt+1,

and the SDF can be reduced to

SDFt ,t+1 = β (CEt )−1/ρ U1/ρ

t

(Ut+1CEt

)−γ (1− β)c−1/ρt+1 U

1/ρt+1

(1− β)c−1/ρt U1/ρ

t

= β

(Ut+1CEt

)1/ρ−γ (ct+1ct

)−1/ρ

. (122)

In reality, however, we cannot observe the future utility index, Ut+1CEt. In

the next step, we will link the SDF to the return to the marketportfolio.Luo, Y. (Econ HKU) Macro Theory November 20, 2017 63 / 67

Link the SDF to the Return to the Market Portfolio

First, given At+1 = Rpt+1 (At − ct ). and R

pt+1 satisfying

1 = Et[SDFt ,t+1R

pt+1

], (123)

we haveAt = ct + Et [SDFt ,t+1At+1] . (124)

Second, we will show that at optimum, we have

At =Jt

fc (ct ,CEt ),

where the value function Jt satisfying

Jt = maxc ,α

Ut = maxc ,α

[(1− β)c1−1/ρ

t + β (CEt )1−1/ρ

] 11−1/ρ

. (125)

We guess that At = Jtfc (ct ,CEt )

holds, i.e.,

At =Jt

fc (ct ,CEt )=

Jt

(1− β)c−1/ρt J1/ρ

t

=1

1− βc1/ρt J1−1/ρ

t .


(Conti.) Substituting this into (123), we have

Et [SDFt ,t+1At+1]

= Et

[β

(Ut+1CEt

)1/ρ−γ (ct+1ct

)−1/ρ 11− β

c1/ρt+1J

1−1/ρt+1

]

=β

1− βc1/ρt Et

[(1CEt

)1/ρ−γ

J1−γt+1

]

=β

1− βc1/ρt (CEt )

1−1/ρ

Substituting this equation into (124):

11− β

c1/ρt J1−1/ρ

t = ct +β

1− βc1/ρt (CEt )

1−1/ρ =⇒

J1−1/ρt = (1− β) c1−1/ρ

t + β (CEt )1−1/ρ ,

which is just (125).


(Conti.) The return on wealth can be written as

Rpt+1 =At+1At − ct

=At+1

Et [SDFt ,t+1At+1]=

11−βc

1/ρt+1J

1−1/ρt+1

β1−βc

1/ρt (CEt )

1−1/ρ

=c1/ρt+1J

1−1/ρt+1

βc1/ρt (CEt )

1−1/ρ= β−1

(ct+1ct

)1/ρ (Jt+1CEt

)1−1/ρ

.

Therefore,

SDFt ,t+1 = β

(Jt+1CEt

)1/ρ−γ (ct+1ct

)−1/ρ

= β

[βRpt+1

(ct+1ct

)−1/ρ] 1/ρ−γ1−1/ρ (ct+1

ct

)−1/ρ

= β

[βRpt+1

(ct+1ct

)−1/ρ]θ−1 (

ct+1ct

)−1/ρ

= βθ(Rpt+1

)θ−1(ct+1ct

)−θ/ρ

where θ = 1−γ1−1/ρ .


Implications

If we assume that consumption growth and the return to the riskyasset are jointly normally distributed, (118) can be written in thefollowing log-linear form:

Et[r jt+1

]− r f +

ω2j

2=

θ

ρcov t

(r jt+1,∆ct+1

)+(1− θ) cov t

(r jt+1, r

pt+1

),

(126)which means that the expected excess return on the risky asset is aweighted average of the risky asset’s covariance with consumptiongrowth (divided by the IES ρ) and the asset’s covariance with themarket portfolio return.


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