The Consumption Capital AssetPricing Model

Instructor: Dmytro Hryshko

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Readings

Danthine, J. P., Donaldson, J. B. (2005), IntermediateFinancial Theory, 2nd Edition, Elsevier AcademicPress. Chapter 9.

Mehra and Prescott (2003), The Equity Premium inRetrospect, Handbook of the Economics of Finance,Elsevier. Chapter 14.

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The Representative Agent Hypothesis and its Notion ofEquilibrium

An infinitely-lived Representative Agent–Helps avoid terminal period problem–Equivalence with finite lives if operative bequestmotive–In complete markets, representative agent is the onewhose utility function is a weighted average of theutilities of all the economy’s participants

No Trade Equilibrium–Positive net supply: the representative agent willinglyholds total supply–Zero net supply: at the prevailing price, supply =demand =0

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An Exchange (Endowment) Economy. Lucas 1978

Recursive trading: many periods (one period inCAPM); investment decisions are made one period at atime, taking due account of their impact on the futurestate of the world

One perfectly divisible share

Dividend = economy’s total output, Yt

Output arises exogenously and stochastically (fruittree) and is perishable

Stationary stochastic process for Yt

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Transition probability matrix (TPM)

E.g., the 3-state TPM

T =

π11 π12 π13π21 π22 π23π31 π32 π33

with πij-th element denoting Prob(Yt+1 = Y j|Yt = Y i), andY j is output in state j.

Lucas fruit tree

Rational expectations economy: knowledge of theeconomy’s structure and the stochastic process

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The problem

max{zt+1}

E

(∞∑t=0

δtU(ct)

)s.t. ct + ptzt+1 ≤ ztYt + ptzt

zt ≤ 1, all t

F.O.C:

U ′(ct)pt = δEtU′(ct+1)(pt+1 + Yt+1) or

U ′(ct(Yi))pt(Y

i)︸ ︷︷ ︸utility loss

frompurchasing 1unit of sec.

at t

= δ∑j

πijU′(ct+1(Y

j))[pt+1(Y

j) + Yt+1(Yj)]

︸ ︷︷ ︸expected utility gain att+ 1 from liquidating theproceeeds and consuming

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Definition of an equilibrium. U ′(ct)pt = δEtU′(ct+1)(pt+1 + Yt+1)

For the entire economy to be in equilibrium, it must,therefore, be true that:

1 zt = zt+1 = zt+2 = . . . = 1—the representative agentowns the entire security

2 ct = Yt—ownership of the entire security entitles theagent to all the economy’s output

3 FOC is satisfied—the agents’ holdings of the securityare optimal given the prevailing prices

(1)–(3) imply

U ′(Yt)pt = δEtU′(Yt+1)(pt+1 + Yt+1) (*)

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ExampleAssume utility is U(c) = log c, δ = 0.96, the 3-state economy,(Y 1, Y 2, Y 3) = (3/2, 1, 1/2). The TPM is

T =

1/2 1/4 1/41/4 1/2 1/41/4 1/4 1/2

Denote p(Y i) = p(i), then

2

3︸︷︷︸=U ′[c(Y 1)]

p(1) = 0.96(1

2+

1

4+

1

4︸ ︷︷ ︸=∑j π1jU

′[c(j)]Y (j)

) + 0.96

1

3p(1) +

1

4p(2) +

1

2p(3)︸ ︷︷ ︸

=∑j π1jU

′[c(j)]p(j)

1p(2) = 0.96 + 0.96

[1

6p(1) +

1

2p(2) +

1

2p(3)

]2p(3) = 0.96 + 0.96

[1

6p(1) +

1

4p(2) + 1p(3)

]Solving the system yields p(1) = 36, p(2) = 24, p(3) = 12.

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Equilibrium pricing

U ′(Yt)pt = δEtU′(Yt+1)(pt+1 + Yt+1)

Et[U′(Yt+1)pt+1] = Et

[Et+1δU

′(Yt+2)Yt+2 + Et+1δU′(Yt+2)pt+2

]︸ ︷︷ ︸=EtδU

′(Yt+2)Yt+2 + EtδU′(Yt+2)pt+2

by law of iterated expectations

EtδU′(Yt+1)pt+1 = Etδ

2U ′(Yt+2)Yt+2 + Etδ2U ′(Yt+2)pt+2

U ′(Yt)pt = EtδU′(Yt+1)Yt+1 + Etδ

2U ′(Yt+2)Yt+2 + Etδ2U ′(Yt+2)pt+2

...

U ′(Yt)pt = Et

∞∑j=1

δjU ′(Yt+j)Yt+j + Et limk→∞

δkU ′(Yt+k)pt+k︸ ︷︷ ︸must be=0

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Equilibrium pricing

pt = Et

∞∑j=1

δjU ′(Yt+j)

U ′(Yt)Yt+j

asset price is the sum of all expected discounted futuredividends (on the RHS, summing starts at t+ 1)

discounting is at the intertemporal marginal rate ofsubstitution (IMRS) of the representative agent!

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Interpreting the Exchange EquilibriumDefine the period t to t+ 1 return for security j as

1 + rjt+1 =pjt+1 + Yjt+1

pjt

Using equation (*),

1 = δEt

[U ′(ct+1)

U ′(ct)(1 + rjt+1)

]Let qbt denote the t-period price of a one-period riskless discountbond paying 1 unit of consumption at t+ 1 in every state.

qbt = δEt

[U ′(ct+1)

U ′(ct)· 1]

Since the riskfree rate is defined from qbt (1 + rft+1) = 1,

1

1 + rft+1= δEt

[U ′(ct+1)

U ′(ct)

]Under risk-neutrality, U(ct) is linear, rft+1 is constant. Linkbetween discount factor and risk-free rate in a risk neutralworld. 11 / 30

Interpreting the Exchange Equilibrium

1 = δEt

[U ′(ct+1)

U ′(ct)(1 + rjt+1)

]

1 = δEt

[U ′(ct+1)

U ′(ct)

]︸ ︷︷ ︸

=1/(1+rft+1)

Et[1 + rjt+1] + δcovt

(U ′(ct+1)

U ′(ct), rjt+1

)

(since cov(X,Y ) = E[XY ]− E[X]E[Y ]. Define Et[1 + rjt+1] = 1 + rjt+1.)

1 + rjt+1

1 + rft+1

= 1− δcovt

(U ′(ct+1)

U ′(ct), rjt+1

)

rjt+1 − rft+1 = −δ(1 + rft+1)covt

(U ′(ct+1)

U ′(ct), rjt+1

).

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CAPM and CCAPM

CCAPM: rjt+1 − rft+1 = −covt

(U ′(ct+1)

U ′(ct), rjt+1

)δ(1 + rft+1)

CAPM: rjt+1 − rft+1 =cov(rm, rj)

var(rm)︸ ︷︷ ︸=βj

(rmt+1 − rft+1),

CCAPM: asset j pays high on average, relative to theriskfree rate, if the covariance between U ′(ct+1) and thereturn on asset j is negative and large—i.e. consumption islow (= U ′(ct+1) is high) when the return is low. Such anasset will have low price pt, and high expected return

CAPM: asset j pays high on average if asset pays low whenmarket pays low (i.e. asset is risky).

CCAPM: Key to an asset’s value is its covariation with themarginal utility of consumption! Consumption-smoothingperspective

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Towards a CAPM equation in the CCAPM frameworkLet U(ct) = act − b

2c2t , a, b > 0, a− bct > 0 for all ct.

rjt+1 − rft+1 = −δ(1 + rft+1)covt

(a− bct+1

a− bct, rjt+1

)

rjt+1 − rft+1 = −δ(1 + rft+1)−b

a− bctcovt (ct+1, rjt+1)

rjt+1 − rft+1 =δb(1 + rft+1)

a− bct︸ ︷︷ ︸>0

covt (ct+1, rjt+1)

Asset j has a high return on average if it pays low whenconsumption is low, i.e. when covt (ct+1, rjt+1) > 0.

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The Formal Consumption CAPM

rct+1 − rft+1 =δb(1 + rft+1)

a− bctcovt (ct+1, rct+1)

“c” denotes portfolio most correlated with consumption

rjt+1 − rft+1

rct+1 − rft+1

=covt (ct+1, rjt+1) /vart(ct+1)

covt (ct+1, rct+1) /vart(ct+1)=βj,ctβc,ct

rjt+1 − rft+1 = βj,ct (rct+1 − rft+1) if βc,ct = 1

Compare this to the CAPM equation:

rjt+1 − rft+1 = βj,m (rmt+1 − rft+1)

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Testing the Consumption CAPM: The Equity PremiumPuzzle

mean, % s.d., %

r 6.98 16.54rf 0.80 5.67r − rf 6.18 16.67

Source: Data from Mehra and Prescott (1985): 1889–1978.

A reasonably parameterized CCAPM is unable to explainr − rf (=the equity premium), the difference in ex-postreturn on a diversified portfolio of U.S. stocks (S&P 500),and the return on one-year T-bills.

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Equity premium, annual time seriesCh 14: The Equity Premium in Retrospect

CD

E

.Cr

a'n U

Year-end

Fig 1 Realized equity risk premium per year, 1926-2000 Source: Ibbotson ( 2001).

18

a0

P

waE

.2Ec

20-Year Period Ending

Fig 2 Equity risk premium over 20-year periods, 1926-2000 Source: Ibbotson ( 2001).

2.4 Variation in the equity premium over time

The equity premium has varied considerably over time, as illustrated in Figures 1 and 2.Furthermore, the variation depends on the time horizon over which it is measured.There have even been periods when the equity premium has been negative.

897

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Equity premium, 20-year average

Ch 14: The Equity Premium in Retrospect

CD

E

.Cr

a'n U

Year-end

Fig 1 Realized equity risk premium per year, 1926-2000 Source: Ibbotson ( 2001).

18

a0

P

waE

.2Ec

20-Year Period Ending

Fig 2 Equity risk premium over 20-year periods, 1926-2000 Source: Ibbotson ( 2001).

2.4 Variation in the equity premium over time

The equity premium has varied considerably over time, as illustrated in Figures 1 and 2.Furthermore, the variation depends on the time horizon over which it is measured.There have even been periods when the equity premium has been negative.

897

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Ch 14: The Equity Premium in Retrospect

Table 2Equity premium in different countriesa

Country % real return on % real return % equity premiuma market index (mean) on a relatively riskless (mean)

security (mean)

UK 1947-1999 5 7 1 1 4 6

Japan 1970-1999 4 7 1 4 3 3

Germany 1978-1997 9 8 3 2 6 6

France 1973-1998 9 0 2 7 6 3

a Source: UK from Siegel ( 1998), the rest are from Campbell ( 2001).

Table 3Terminal value of $ 1 invested in stocks and bondsa

Investment period Stocks T-billsReal Nominal Real Nominal

1802-1997 $ 558,945 $ 7,470,000 $ 276 $ 3,679

1926-2000 $ 266 47 $ 2,586 52 $ 1 71 $ 16 56

a Source: Ibbotson ( 2001) and Siegel ( 1998).

The annual return on the British stock market was 5 7 % over the post-war period,an impressive 4 6 % premium over the average bond return of 1 1 % Similar statisticaldifferentials are documented for France, Germany and Japan Table 2 illustrates theequity premium in the post-war period for these countries.

The dramatic investment implications of this differential rate of return can be seenin Table 3, which maps the capital appreciation of $ 1 invested in different assets from1802 to 1997 and from 1926 to 2000.

As Table 3 illustrates, $ 1 invested in a diversified stock index yields an ending wealthof $ 558,945 versus a value of $ 276, in real terms, for $ 1 invested in a portfolio ofT-bills for the period 1802-1997 The corresponding values for the 75-year period,1926-2000, are $ 266 47 and $ 1 71 We assume that all payments to the underlyingasset, such as dividend payments to stock and interest payments to bonds are reinvestedand that there are no taxes paid.

This long-term perspective underscores the remarkable wealth building potential ofthe equity premium It should come as no surprise therefore, that the equity premium isof central importance in portfolio allocation decisions, estimates of the cost of capitaland is front and center in the current debate about the advantages of investing SocialSecurity funds in the stock market.

In Table 4 we report the premium for some interesting sub-periods: 1889-1933,when the USA was on a gold standard; 1934-2000, when it was off the gold standard;

895

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Ch 14: The Equity Premium in Retrospect

Table 2Equity premium in different countriesa

Country % real return on % real return % equity premiuma market index (mean) on a relatively riskless (mean)

security (mean)

UK 1947-1999 5 7 1 1 4 6

Japan 1970-1999 4 7 1 4 3 3

Germany 1978-1997 9 8 3 2 6 6

France 1973-1998 9 0 2 7 6 3

a Source: UK from Siegel ( 1998), the rest are from Campbell ( 2001).

Table 3Terminal value of $ 1 invested in stocks and bondsa

Investment period Stocks T-billsReal Nominal Real Nominal

1802-1997 $ 558,945 $ 7,470,000 $ 276 $ 3,679

1926-2000 $ 266 47 $ 2,586 52 $ 1 71 $ 16 56

a Source: Ibbotson ( 2001) and Siegel ( 1998).

The annual return on the British stock market was 5 7 % over the post-war period,an impressive 4 6 % premium over the average bond return of 1 1 % Similar statisticaldifferentials are documented for France, Germany and Japan Table 2 illustrates theequity premium in the post-war period for these countries.

The dramatic investment implications of this differential rate of return can be seenin Table 3, which maps the capital appreciation of $ 1 invested in different assets from1802 to 1997 and from 1926 to 2000.

As Table 3 illustrates, $ 1 invested in a diversified stock index yields an ending wealthof $ 558,945 versus a value of $ 276, in real terms, for $ 1 invested in a portfolio ofT-bills for the period 1802-1997 The corresponding values for the 75-year period,1926-2000, are $ 266 47 and $ 1 71 We assume that all payments to the underlyingasset, such as dividend payments to stock and interest payments to bonds are reinvestedand that there are no taxes paid.

This long-term perspective underscores the remarkable wealth building potential ofthe equity premium It should come as no surprise therefore, that the equity premium isof central importance in portfolio allocation decisions, estimates of the cost of capitaland is front and center in the current debate about the advantages of investing SocialSecurity funds in the stock market.

In Table 4 we report the premium for some interesting sub-periods: 1889-1933,when the USA was on a gold standard; 1934-2000, when it was off the gold standard;

895

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The equity premium puzzle

U(c) =c1−γ

1− γAssume xt+1 ≡ ct+1

ct∼ iid lognormal(1.0183, 0.03572), and

so log xt+1 ∼ iidN(µx, σ2x). Then,

1.0183 = exp(µx +1

2σ2x)

0.03572 = exp(2µx + σ2x)[exp(σ2

x − 1)]

Solving, µx = 0.01752, σ2x = 0.00123.

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The equity premium puzzleAssume

pt = vYt

The stock price at t is proportional to the dividend paid att.Verify that it qualifies for a solution:

vYt = δEt

[(vYt+1 + Yt+1)

U ′(ct+1)

U ′(ct)

]

v = δEt

(v + 1)Yt+1

Yt︸︷︷︸=xt+1: Lucas tree

x−γt+1

v =

δE[x1−γ]

1− E[x1−γ]indeed a constant

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The equity premium puzzle

Rt+1 = 1 + rt+1 =pt+1 + Yt+1

pt+1

=v + 1

v

Yt+1

Yt=v + 1

vxt+1

EtRt+1 =v + 1

vEtxt+1 =

E[x]

δE[x1−γ]

Rft+1 =1

qbt=

[δEt

U ′(ct+1)

U ′(ct)

]−1=

1

δ

1

E[x−γ]

EtRt+1

Rft+1

=E[x]E[x−γ]

E[x1−γ]=

exp(µx + 0.5σ2x) exp(−γµx + 0.5γ2σ2

x)

exp((1− γ)µx + 0.5(1− γ)2σ2x)

= exp(γσ2x)

log(E[R])− log(Rf ) ≈ r − rf = γσ2x

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The equity premium puzzle

r − rf = γσ2x

γ =r − rfσ2x

=0.0698− 0.008

0.03572≈ 50

If a more realistic γ = 2 is assumed, r − rf ≈ 0.2%.

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Is γ = 50 realistic?

Assume current wealth 50, 000, utility function u(W ) = W 1−γ

1−γand a chance to play the gamble (0,1/2; 50000,1/2). What isthe certainty equivalent (CE) of the gamble? (Π is themaximum willing to pay to avoid playing the gamble(–25000,1/2; 25000,1/2) with the current wealth of 75,000.)

γ CE Π

0 25,000 01 20,711 4,2893 13,246 11,7545 8,566 16,43410 3,991 21,00920 1,858 23,14230 1,209 23,79150 712 24,288

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The risk-free rate puzzleSimilarly, if γ = 2,

δ =1

Rft+1E[x−γ]=

1

1.008 exp(−γµx + 0.5γ2σ2x)≈ 1.02

The difficulty in explaining the low rate of return on theriskfree asset is called the risk-free rate puzzle.

Rft+1 =1

δE[x−γ]=

1

δ exp(−γµx + 0.5γ2σ2x)

log(Rft+1) = − log(δ)︸ ︷︷ ︸impatience term

+ γµx︸︷︷︸consumption

smoothing

under certainty

− 0.5γ2σ2x︸ ︷︷ ︸

demand for

precautionary

savings

If γ = 3 and δ = 0.99, rft+1 ≈ 6%.26 / 30

Testing the Consumption CAPM: Hansen-JagannathanBounds

1 = Et

δU ′(ct+1)

U ′(ct)︸ ︷︷ ︸≡mt+1

(1 + rjt+1)

1 = Et

[mt+1Rt+1

]Should hold unconditionally:

1 = E[mt+1Rt+1

]m is called the equilibrium pricing kernel, or the stochasticdiscount factor.

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Hansen-Jagannathan bounds

E[mR

]= 1

E

m(Ri − Rj︸ ︷︷ ︸≡Ri−j

)

= 0

E[m]E[Ri−j] + cov(m, Ri−j) = 0

E[Ri−j]

σRi−j= −ρ(m, Ri−j)

σmE[m]

|E[Ri−j]|σRi−j

= |ρ(m, Ri−j)|σmE[m]

≤ σmE[m]

σmE[m]

≥ |E(Ri−j)|σRi−j

ρ(m, Ri−j) is the correlation between the pricing kernel and

Ri−j. 28 / 30

Hansen-Jagannathan bounds

Set i to m(arket) and j to f (riskfree asset) to obtain

σmE[m]

≥ |E(rm − rf )|σrm−rf

=0.062

0.167≈ 0.37

Assuming δ = 0.99 and γ = 2

E[m] = δx−γ = δ exp(−γµx + 0.5γ2σ2x) ≈ 0.96

Therefore, the lowest value for σm ≈ 0.36.

Using the chosen parameters, however, σm = 0.002, anorder of magnitude lower!

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Solutions?Habit formation—makes IMRS, the pricing kernel inthe CCAPM, more volatileHeterogeneity—if markets are incomplete, income riskis persistent, and the variance of consumption growthis countercyclical (higher in recessions than in normaltimes), the model-implied equity premium will behigher—“. . . the risk premium is highest in a recession since equities

are a poor hedge against the potential loss of employment. . . even

though per capita consumption growth is poorly correlated with

stocks returns, investors require a hefty premium to hold stocks

over short-term bonds because stocks perform poorly in

recessions, when an investor is more likely to be laid off.” Mehra

& Prescott (2003).

Limited stock-market participation—the relevantstochastic discount factor is IMRS of stockholderswhose consumption growth is more volatile 30 / 30