SIMON FRASER UNIVERSITY

BURNABY BRITISH COLUMBIA

Paul Klein

Office: WMC 3635

Phone: TBA

Email: paul klein 2@sfu.ca

URL: http://paulklein.ca/newsite/teaching/809.php

Economics 809

Advanced macroeconomic theory

Spring 2012

Lecture 8: Finance

1 No-arbitrage pricing

How far can we get in determining the “correct” price of assets by just as-

suming that there are no arbitrage opportunities? Notice that this is a weaker

assumption than (competitive) equilibrium. Surprisingly far, as it turns out. In

particular, we can work out how to price derivatives — claims that are defined

in terms of other securities, e.g. options. Arbitrage–free pricing is a lot like

expressing a vector as the linear combination of basis vectors.

1

1.1 A two–period model

Let t = 0, 1 (today and tomorrow). There areN securities. The price of security

n at time t is denoted by Snt . We write

St =

S1t

S2t...

SNt

.

S0 is deterministic, but S1 is stochastic. We write

S1(ω) =

S11(ω)

S21(ω)...

SN1 (ω)

where ω ∈ Ω = ω1, ω2, . . . , ωM.

Now define the matrix D via

DN×M

=

S11(ω1) S1

1(ω2) . . . S11(ωM)

S21(ω1)

. . . . . . ...... . . . . . . ...

SN1 (ω1) . . . . . . SN

1 (ωM)

=[d1 d2 . . . dM

].

Definition. A portfolio is a vector h ∈ RN . Interpretation: hn is the number

of securities of type n purchased at t = 0.

Remark: Fractional holdings as well as short positions (hn < 0) are allowed.

2

The value of a portfolio h at time t is given by

Vt(h) =N∑n=1

hnSnt = hTSt.

Definition. A vector h ∈ RN is called an arbitrage portfolio if

V0(h) < 0

and

V1(h) ≥ 0

for all ω ∈ Ω.

Remark. We can weaken the condition “for all ω ∈ Ω” to “with probability

one” if we want — but in this context that wouldn’t add much.

Theorem. Let securities prices S be as above. Then there exists no arbitrage

portfolio iff there exists a z ∈ RM+ such that

S0 = Dz.

Remark. This means that today’s (period 0’s) price vector has to lie in the

convex cone spanned by tomorrow’s (period 1’s) possible prices vectors. (A

convex cone is a subset C of a vector space X such that for any x, y ∈ C and

α ≥ 0 we have (αx) ∈ C and (x+ y) ∈ C. )

Proof. Absence of arbitrage opportunities means that the following system of

inequalities has no solution for h.hTS0 < 0

hTdj ≥ 0 for each j = 1, 2, . . . ,M.

Geometric interpretation: there is no hyperplane that separates S0 from the

columns of D. Such a hyperplane would have an arbitrage portfolio as a normal

3

vector. Now according to Farkas’ lemma, the non–existence of such a normal

vector is equivalent to the existence of non–negative numbers z1, z2, ..., zM such

that

S0 =M∑j=1

zjdj

or, equivalently,

S0 = Dz

where z ∈ RM+ .

We will prove Farkas’ lemma as a corollary of the Separating Hyperplane The-

orem.

Proposition (Farkas’ lemma). If Am×n

is a real matrix and if b ∈ Rm, then

exactly one of the following statements is true.

1. Ax = b for some x ∈ Rn+

2. (yTA) ∈ Rn+ and yT b < 0 for some y ∈ Rm.

Proof. In what follows we will sometimes say that x ≥ 0 when x is a vector.

This means that all the components are non–negative. To prove that (1) implies

that not (2), suppose that Ax = b for some x ≥ 0. Then yTAx = yT b. But

then (2) is not true. If it were, then yTA ≥ 0 and hence yTAx ≥ 0. But then

yT b ≥ 0. Hence (1) implies not (2). Next we show that not (1) implies (2). Let

X be the convex cone spanned by the columns a1, a2, ..., an of A, i.e.

X =

a ∈ Rn : a =

n∑i=1

λiai; λi ≥ 0, i = 1, 2, . . . , n

.

Suppose there is no x ≥ 0 such that Ax = b. Then b /∈ X. Since X is closed

and convex, the Separating Hyperplane Theorem says that there is a y ∈ Rm

4

such that yTa > yT b for all a ∈ X. Since 0 ∈ X, yT0 > yT b and consequently

yT b < 0. Now suppose (to yield a contradiction) that not yTA ≥ 0. Then there

is a column in A, say ak, such that yTak < 0. Since X is a convex cone and

ak ∈ X, we have (αak) ∈ X for all α ≥ 0. But by the (absurd) supposition,

for sufficiently large α, we have yT (αak) = α(yTak) < yT b, and this contradicts

separation so the supposition cannot be true. Thus yTai ≥ 0 for all columns ai

of A, i.e. yTA ≥ 0.

Separating Hyperplane Theorem. Let X ⊂ Rn be closed and convex, and

suppose y /∈ X. Then there is an a ∈ R and an h ∈ Rn such that hTx > a > hTy

for all x ∈ X.

Proof. Omitted.

A popular interpretation of the result S0 = Dz is the following (but beware of

over–interpretation). Define

qi =ziβ

where

β =M∑i=1

zi.

Then q can be thought of as a probability distribution on Ω and we may conclude

the following.

Theorem. The market S is arbitrage–free iff there is a scalar β > 0 and a

probability measure Q such that

S0 = βEQ[S1] = β[q1S1(ω1) + q2S1(ω2) + . . .+ qMS1(ωM)]

and we call this a martingale measure (for reasons that are not immediately

obvious in this context). Economically speaking, the qis are “state prices”.

Relative prices of $1 in state i. Arrow–Debreu prices.

5

Yet another approach is to define a so–called pricing kernel. For an arbitrage–

free market S, there exists a (scalar) random variable m : Ω → R such that

such that

S0 = E[m · S1]

where the expectation is now taken under the “objective” measure P . Provided

the outcomes ωi all obtain with strictly positive probabilities, we have

m(ωi) = βQ(ωi)/P (ωi) = βqi/pi.

where the pis are the objective probabilities. So the pricing kernel takes care

both of discounting and the change of measure.

1.1.1 Pricing contingent claims

Definition. A contingent claim is a mapping X : Ω → R. We represent it as a

vector x ∈ RM . Interpretation: the contract x entitles the owner to $xi in state

ωi.

Theorem. Let S be an arbitrage–free market. Then there exist β, q such that

if each contingent claim x is priced according to

π0[X] = βqTx = βEQ[X]

then the market consisting of all contingent claims is arbitrage free.

Example. Let Ω = ω1, ω2. Let

S10 = b S2

0 = s

S11(ω1) = (1 + r)b S1

1(ω2) = (1 + r)b

S21(ω1) = x S2

1(ω2) = y

where r > 0 and, without loss of generality, y ≤ x. Then

D =

[(1 + r)b (1 + r)b

x y

].

6

Abusing the notation somewhat, define q = q1. No arbitrage impliesb = β[q(1 + r)b+ (1− q)(1 + r)b]

s = β[qx+ (1− q)y]

0 ≤ q ≤ 1.

We may conclude that

β =1

1 + rand

q =s(1 + r)− y

x− y, 1− q =

x− s(1 + r)

x− y.

The condition 0 ≤ q ≤ 1 then becomes

0 ≤ s(1 + r)− y

x− y≤ 1.

That is to say, x = y and

y ≤ s(1 + r) ≤ x.

This is not a surprising no–arbitrage condition. In this case, it turns out that

there is a unique martingale measure, so all contingent claims can be uniquely

priced. Below we give a more general treatment of this phenomenon, and it

turns out that this uniqueness is intimately related to the notion of market

completeness.

1.1.2 Completeness and uniqueness of martingale measure

Can contingent claims be prices in a unique way? One class of contingent claims

certainly can: the hedgeable ones.

Definition. A contingent claim X is said to be hedgeable if there is an h ∈ RN

such that

V1(h) ≡ X

7

i.e.

hTS1(ω) = X(ω)

for all ω ∈ Ω or

hTdi = X(ωi)

for all i = 1, 2, . . . ,M or

hTD = xT

i.e. hedgeability of X boils down to the corresponding x being in the row space

of D.

Proposition. Suppose the contingent claim X is hedgeable via h. Then the

only price of X at 0 that is consistent with no arbitrage is

π0[X] = hTS0.

Proof. Exercise.

Proposition. Suppose X is a contingent claim that is hedgeable via h and that

it is also hedgeable via g. Suppose also that there are no arbitrage opportunities.

Then

hTS0 = gTS0.

Proof. Exercise.

Thus any hedgeable contingent claim can be uniquely priced. Under what

circumstances can all contingent claims be uniquely priced?

Definition. A market S is said to be complete if all contingent claims are

hedgeable.

Proposition. A market S is complete iff rank(D) ≥ M . A necessary condition

is N ≥ M .

Proof. Exercise.

8

Meta–theorem. A market S is generically arbitrage–free and complete if

N = M . If N > M it is generically not arbitrage–free. If N < M it is

incomplete.

Proposition. Suppose a market is arbitrage–free and complete. Then there

is a unique probability measure Q such that every contingent claim is priced

according to

π0[X] = βEQ[X]

where

β = π[1]

where 1 is the “unit” contingent claim that delivers $1 no matter what.

Proof.

π0[X] = hTS0 = hTβEQ[S1] =

= βEQ[hTS1] = βEQ[X].

Meanwhile,

π0[1] = βEQ[1] = β.

2 Equilibrium pricing

In general equilibrium, relative prices are given by marginal rates of substitu-

tion. This gives a way of determining prices even if they are not given exoge-

nously. Lucas (1978) pioneered the pricing of assets in dynamic equilibrium.

We will assume here that preferences are represented by

E

[ ∞∑t=0

βt c1−αt

1− α

].

9

Now let’s suppose that a single asset with stochastic return rt is available, so

that the consumer’s period-by-period budget constraint is

at+1 = wt + rtat − ct

Suppose that the information flow is given by the filtration Ft∞t=0 generated

by the processes rt and wt. Applying the principles of stochastic dynamic

optimization, we form the Hamiltonian:

H = βt c1−αt

1− α+ λt+1[wt + rtat − ct]

and the optimality conditions are

E[λt+1|Ft]rt = λt

and

βtc−αt = E[λt+1|Ft].

Apparently—though it is not immediately obvious why this is relevant—

E[λt+2|Ft+1]rt+1 = λt+1

and

βt+1c−αt+1 = E[λt+2|Ft+1]

Thus

βt+1c−αt+1rt+1 = λt+1 (1)

and you may recall that

βtc−αt = E[λt+1|Ft].

Thus, taking expectations with respect to Ft on each side of Equation (1), we

get

βt+1E[c−αt+1rt+1|Ft

]= βtc−α

t

10

or, simplifying and rearranging,

1 = βE[c−α

t+1rt+1|Ft]

c−αt

establishing that

mt+1 = βc−αt+1rt+1

c−αt

is a pricing kernel for this economy.

2.1 What kind of assets yield high average returns?

It is tempting to conclude that, in general, if people are risk averse, then risky

assets have to yield a high return to compensate for risk. That would be wrong,

however. What matters is how good an asset is in providing insurance against

consumption variance. An risky asset that pays more when consumption is low

and less when it is high is actually better than a safe asset and will still be held

even if the average return is lower than that of a safe asset.

To establish this result, we will assume that consumption growth ln ct+1 − ln ct

is i.i.d. normally distributed with mean µ and variance σ2. The normal distri-

bution is convenient because the linear combination of two random variables is

also normally distributed.

We also need the following result. If X is normally distributed with mean µX

and variance σ2X , then

E[eX ] = exp

µX +

1

2σ2X

.

It follows that if X and Y are joint normal with means µX and µX , variances

σ2X and σ2

Y and covariance σXY then

E[eX+Y ] = exp

µX + µY +

1

2σ2X +

1

2σ2Y + σXY

.

11

With the assumptions we have made so far we are able to derive an explicit

expression for the price of a bond—a riskless asset that yields precisely 1 unit

of consumption in period 1. Denote the period 0 price of a bond by q. Then

q = E[m · 1] = βE

[c−αt+1

c−αt

]=

= βE[exp−α(ln ct+1 − ln ct)] =

β exp

−αµ+

1

2α2σ2

.

On the other hand, the risk-free rate r∗ is

r∗ =1

q= β−1 exp

αµ− 1

2α2σ2

.

(The risk free rate might have depended on time. But it doesn’t in this case,

so we omit the time subscript.)

Now let’s introduce an arbitrary risky asset with stochastic rate of return rt.

Let’s assume ln rt+1 and ln ct+1 − ln ct are joint normal. Then Aiyagari (1993)

claims (rightly of course) that

E[rt+1]

r∗= exp αCOV(ln rt+1, ln ct+1 − ln ct) .

Let’s try to verify this! First of all, we have the following beautiful and very

general result which holds for all rates of return processes rt:

E[mt+1 · rt+1] = 1.

for some non-negative stochastic process mt (adapted to Ft∞t=0) With our

specification of preferences, as we have seen,

mt+1 = β exp−α(ln ct+1 − ln ct)

12

and the equation E[mt+1 · rt+1] = 1 becomes

βE[expln rt+1 − α[ln ct+1 − ln ct]] = 1

which implies

β exp

µln rt+1

+1

2σ2ln rt+1

− αµ+1

2α2σ2 − αCOV(ln rt+1, ln ct+1 − ln ct)

= 1.

This can be factorized to become

β exp

µln rt+1

+1

2σ2ln rt+1

exp

−αµ+

1

2σ2

exp −αCOV(ln rt+1, ln ct+1 − ln ct) = 1

and the result follows.

We take away from this that an asset earns a premium on average when its re-

turn is positively correlated with consumption growth—or negatively correlated

with future marginal utility.

2.2 The risky asset is a claim to consumption

Here we will assume, in the spirit of Mehra and Prescott (1985), that the risky

asset return is proportional to per-capita consumption growth, i.e.

rt+1 = A · ct+1

ct.

for some A > 0. But then the covariance between ln r and ln ct+1 − ln ct is just

the variance of ln ct+1 − ln ct! Plugging this into Aiyagari’s formula, we get

E[rt+1]

r∗= expασ2.

This is a very satisfying result, because it means that the ratio of returns

depends on the variance of consumption growth and α only.

13

According to my calculations, based on data from the U.S. Bureau of Economic

Analysis, the growth rate of annual per capita consumption in the United States

has averaged about 2.0 percent between 1929 and 2005, with a standard de-

viation of about 0.02. If we believe that rE = 1.06 and rB = 1.01 then we

get

α ≈ 0.05

0.022= 125.

Now if α = 125 and µ = 0.02 we can compute β from the formula

r∗ = β−1 exp

αµ− 1

2α2σ2

.

The result is

β ≈ 0.53.

So we can make the model fit the data. Is there really an equity premium

puzzle?

2.3 The more general case

Consider the approximate version of Aiyagari’s formula:

E[r]− r∗ ≈ αCOV(ln r, ln ct+1 − ln ct).

Now apply Cauchy-Schwarz’ inequality which says that, for any random vari-

ables X and Y

COV(X,Y ) ≤√

V(X) · V(Y ) = STD(X) · STD(Y ).

We get (approximately at least)

E[r]− r∗ ≤ αSTD(ln r) · STD(ln ct+1 − ln ct).

14

According to Aiyagari (1993), the standard deviation of the return on equity

is about 0.07. Thus the predicted equity premium is about 0.0014α. If the

equity premium is 0.05, we must have α ≥ 36. So do we really have an equity

premium puzzle?

Most economists believe there is. Indeed, the conventional wisdom is that none

of the proposed solutions have been successful. Kocherlakota (1996) concludes

that it is still a puzzle. This in spite of, for example,

• Constantinides (1990). Habit formation.

• Huggett (1993). Market incompleteness.

2.4 An infinitely-lived asset

Like Mehra and Prescott (1985), I now assume that the risky asset is a claim

to per-capita consumption in every period from the next one onwards. (Unlike

them, I assume log-normal and i.i.d. consumption growth.) The purpose of

this is to give an example of an asset whose rate of return is proportional to

consumption growth. Denote the price of the risky asset by pt. Might there be

a stationary equilibrium where pt = Act for some A > 0? If so,

Act = βEt

c−αt+1c

αt · [ct+1 + Act+1]

.

It follows that (since we assume ct to be known when expectations are evaluated)

A = β(1 + A)Et exp [(1− α)(ln ct+1 − ln ct] .

Hence

A = β(1 + A) exp

(1− α)µ+

1

2(1− α)2σ2

.

15

Rearrangement (for no obvious purpose right now) yields

1 + A

A= β−1 exp

−(1− α)µ− 1

2(1− α)2σ2.

Given what we know already,

Et[rt+1] =1 + A

Aexp

µ+

1

2σ2

.

Substituting in our expression for (1 +A)/A we get

E[rt+1] = Et[rt+1] = β−1 exp

αµ+ α

(1− 1

2α

)σ2

.

Summarizing our results so far, we haveE[rt+1] = β−1 exp

αµ+ α

(1− 1

2α)σ2

r∗ = β−1 expαµ− 1

2α2σ2

.

It follows from this thatE[rt+1]

r∗= expασ2.

2.5 The original Mehra-Prescott paper

Mehra and Prescott (1985) begin their paper by noting the following fact. Be-

tween 1889 and 1978, the average return on equity was 7 percent per year and

the average return on bonds was less than one percent. The difference (by def-

inition) is called the equity premium. Jagannathan et al. (2000) claim that this

premium has declined significantly, and that it was approximately zero 1970-

2000. But this new fact (if indeed it is a fact) was not known by Mehra and

Prescott in 1985.

Mehra and Prescott analyze an exchange (endowment) economy where a rep-

resentative consumer maximizes

E

[ ∞∑t=0

βtu(ct)

]16

where

u(c) =c1−σ − 1

1− σ.

Output follows an exogenous stochastic process satisfying

yt+1 = xt+1yt

and since all agents are alike and there is no trade, we have

ct = yt.

Meanwhile, xt is a finite–state Markov chain in discrete time. Specifically,

xt ∈ λ1, λ2, . . . , λn

and

P [xt+1 = λj|xt = λi] = φij.

It follows that the distribution of xt satisfies

µt+1 = ΦTµt.

Suppose there is a unique stationary distribution π such thatπ = ΦTπ

πT1 = 1.

and

limt→∞

µt = π

for all µ0 ∈ Rn such that µT0 1 = 1.

Definition. A share of equity at t is a claim to ys∞s=t+1. Denote the price of

this bundle of claims by pt.

The price of equity satisfies the following difference equation.

u′(yt)pt = βEt [u′(yt+1)(pt+1 + yt+1)] .

17

The presence of the conditional expectation is justified in my handout on

stochastic dynamic optimization.

Might there exist a pricing function

pt = p(y, i)

where y is current output and i is the current state of x? We can define it

implicitly via

p(y, i) = βn∑

j=1

φij(λjy)−σ[p(λjy, j) + λjy]y

σ.

It turns out that this function not only exists but is linear in y! So there exist

constants ω1, ω2, . . . , ωn such that

p(y, i) = ωi y.

A system of equations for these constants is given by

ωi = βn∑

j=1

φijλ1−σj (ωj + 1); i = 1, . . . , n.

We now want to derive the average return on equity. We begin by defining the

period return on equity when going from state i to state j via

Reij =

p(λjy, j) + λjy − p(y, i)

p(y, i)=

λj(ωj + 1)

ωi− 1.

The conditionally expected return on equity is

Rei =

n∑j=1

φijReij

and the unconditionally expected return (equal to the long–run average by the

LOLN) is

Re =n∑

i=1

πiRei .

18

Now consider a riskless one–period bond. How are we to price it, in terms of

current consumption? Since it is a claim to 1 unit of consumption in each state,

the formula is

q(y, i) = βn∑

j=1

φij

u′(λjy)

u′(y)· 1.

With our specification of preferences, we get

q(y, i) = βn∑

j=1

φijλ−σj .

Incidentally,βu′(ct+1)

u′(ct)

constitutes a pricing kernel for this economy. We can now define the period

return on a bond in state i via

Rbi =

1

q(y, i)− 1

and the unconditionally expected return is

Rb =n∑

i=1

πiRbi .

The equity premium is then defined as Re −Rb.

But how are we to judge whether theory is consistent with Re = 0.07 and

Rb = 0.01? We calibrate! Here are the parameters we need to determine.

(1) The states λ1, λ2, . . . , λn and the transition probabilities φij; i, j = 1, 2, . . . , n.

(2) The preference parameters β and σ.

We begin with (1).

There is data on consumption growth rates and they have an average µ.

19

Impose symmetry so that Φ = ΦT .

Set n = 2.λ1 = 1 + µ+ δ

λ2 = 1 + µ− δ.

Φ =

[φ 1− φ

1− φ φ

]

Apparently π1 = π2 =1

2.

Finally, use δ and φ to match variance and autocorrelation! The (unconditional)

variance of consumption growth is

1

2[δ2] +

1

2[δ2] = δ2.

Meanwhile, the autocovariance is

1

2[φ · δ2 − (1− φ)δ2] +

1

2[φδ2 − (1− φ)δ2]

and the autocorrelation is the autocovariance divided by the variance, i.e.

1

2[φ− 1 + φ]

1

2[φ− 1 + φ] =

1

2[4φ− 2] = 2φ− 1.

Mehra and Prescott (1985) find that, with annual data,

µ = 0.018

δ = 0.036

and

φ = 0.43

resulting from an autocorrelation of consumption growth equal to −0.14.

(2) Determining β and σ. It would, in principle, be feasible to choose β and

σ so as to match Re and Rb. Strangely, Mehra and Prescott (1985) do not

20

consider this alternative. If they did, they would find that σ must be be very

large. Apparently Mehra and Prescott (1985) take as an axiom that σ < 10.

So they conclude that the model cannot account for the facts.

Apparently most economists are still uncomfortable with σ ≫ 1. It is not

altogether clear why. But if one were to estimate β and σ in the way that I

suggested, the parameter estimates would vary wildly with the sampling period.

This suggests that there might be something fishy with the model.

Many attempts have been made since Mehra and Prescott (1985) to alter pref-

erences in such a way as to solve the equity puzzle. Have any of these attempts

been successful and persuasive? Kocherlakota (1996) doesn’t think so.

Exercise 1 Suppose consumption ct satisfies

ln ct+1 = ρ ln ct + εt+1

where εt is i.i.d. normal with mean 0 and variance σ2. Assume the same pref-

erences as in these lecture notes. Suppose 0 < ρ < 1. Find an expression for

the period t riskless rate of return.

21

References

Aiyagari, S. R. (1993). Explaining financial market facts: The importance of incomplete marketsand transactions. Federal Reserve Bank of Minneapolis Quarterly Review 17 (1), 17–31.

Constantinides, G. (1990). Habit formation: A resolution of the equity premium puzzle. Journalof Political Economy 98 (3).

Huggett, M. (1993). The risk free rate in heterogeneous-agents, incomplete insurance economies.Journal of Economic Dynamics and Control 17 (5/6), 953–970.

Jagannathan, R., E. R. McGrattan, and A. Scherbina (2000). The declining U.S. equity pre-mium. Federal Reserve Bank of Minneapolis Quarterly Review 24 (4), 3–19.

Kocherlakota, N. R. (1996, March). The equity premium: It’s still a puzzle. Journal ofEconomic Literature 34 (1), 42–71.

Lucas, R. E. (1978, November). Asset prices in an exchange economy. Econometrica 46 (6),1429–1445.

Mehra, R. and E. Prescott (1985). The equity premium: A puzzle. Journal of MonetaryEconomics 15, 145–61.

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