Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Asset PricesEcon 497F Lecture

Barry W. Ickes

The Pennsylvania State University

Spring 2007

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We consider a simple case that turns out to be powerful.

Why do people hold assets? To smooth consumption inorder to maximize utility. We have a basic choice problem.

Start by asking how an individual values a stream ofuncertain cash �ows. Let

U(ct , ct+1) = u(ct ) + βEt [u(ct+1)] (11)

be the utility function de�ned over present and futureconsumption.

E is expectations operator �uncertainty.future is not the present, we discount it by β � 1.

With concave utility we have diminishing marginal utilitywhich gives the desire to smooth consumption.

We have not just impatience (β), but risk aversionInvestors prefer a consumption stream that is steady overtime and across states of nature.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We consider a simple case that turns out to be powerful.Why do people hold assets? To smooth consumption inorder to maximize utility. We have a basic choice problem.

Start by asking how an individual values a stream ofuncertain cash �ows. Let

U(ct , ct+1) = u(ct ) + βEt [u(ct+1)] (11)

be the utility function de�ned over present and futureconsumption.

E is expectations operator �uncertainty.future is not the present, we discount it by β � 1.

With concave utility we have diminishing marginal utilitywhich gives the desire to smooth consumption.

We have not just impatience (β), but risk aversionInvestors prefer a consumption stream that is steady overtime and across states of nature.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We consider a simple case that turns out to be powerful.Why do people hold assets? To smooth consumption inorder to maximize utility. We have a basic choice problem.

Start by asking how an individual values a stream ofuncertain cash �ows. Let

U(ct , ct+1) = u(ct ) + βEt [u(ct+1)] (11)

be the utility function de�ned over present and futureconsumption.

E is expectations operator �uncertainty.future is not the present, we discount it by β � 1.

With concave utility we have diminishing marginal utilitywhich gives the desire to smooth consumption.

We have not just impatience (β), but risk aversionInvestors prefer a consumption stream that is steady overtime and across states of nature.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We consider a simple case that turns out to be powerful.Why do people hold assets? To smooth consumption inorder to maximize utility. We have a basic choice problem.

Start by asking how an individual values a stream ofuncertain cash �ows. Let

U(ct , ct+1) = u(ct ) + βEt [u(ct+1)] (11)

be the utility function de�ned over present and futureconsumption.

E is expectations operator �uncertainty.

future is not the present, we discount it by β � 1.With concave utility we have diminishing marginal utilitywhich gives the desire to smooth consumption.

We have not just impatience (β), but risk aversionInvestors prefer a consumption stream that is steady overtime and across states of nature.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We consider a simple case that turns out to be powerful.Why do people hold assets? To smooth consumption inorder to maximize utility. We have a basic choice problem.

Start by asking how an individual values a stream ofuncertain cash �ows. Let

U(ct , ct+1) = u(ct ) + βEt [u(ct+1)] (11)

be the utility function de�ned over present and futureconsumption.

E is expectations operator �uncertainty.future is not the present, we discount it by β � 1.

With concave utility we have diminishing marginal utilitywhich gives the desire to smooth consumption.

We have not just impatience (β), but risk aversionInvestors prefer a consumption stream that is steady overtime and across states of nature.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We consider a simple case that turns out to be powerful.Why do people hold assets? To smooth consumption inorder to maximize utility. We have a basic choice problem.

Start by asking how an individual values a stream ofuncertain cash �ows. Let

U(ct , ct+1) = u(ct ) + βEt [u(ct+1)] (11)

be the utility function de�ned over present and futureconsumption.

E is expectations operator �uncertainty.future is not the present, we discount it by β � 1.

With concave utility we have diminishing marginal utilitywhich gives the desire to smooth consumption.

We have not just impatience (β), but risk aversionInvestors prefer a consumption stream that is steady overtime and across states of nature.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We consider a simple case that turns out to be powerful.Why do people hold assets? To smooth consumption inorder to maximize utility. We have a basic choice problem.

Start by asking how an individual values a stream ofuncertain cash �ows. Let

U(ct , ct+1) = u(ct ) + βEt [u(ct+1)] (11)

be the utility function de�ned over present and futureconsumption.

E is expectations operator �uncertainty.future is not the present, we discount it by β � 1.

With concave utility we have diminishing marginal utilitywhich gives the desire to smooth consumption.

We have not just impatience (β), but risk aversion

Investors prefer a consumption stream that is steady overtime and across states of nature.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We consider a simple case that turns out to be powerful.Why do people hold assets? To smooth consumption inorder to maximize utility. We have a basic choice problem.

Start by asking how an individual values a stream ofuncertain cash �ows. Let

U(ct , ct+1) = u(ct ) + βEt [u(ct+1)] (11)

be the utility function de�ned over present and futureconsumption.

E is expectations operator �uncertainty.future is not the present, we discount it by β � 1.

With concave utility we have diminishing marginal utilitywhich gives the desire to smooth consumption.

We have not just impatience (β), but risk aversionInvestors prefer a consumption stream that is steady overtime and across states of nature.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

De�nition

De�ne xt+1 as the payo¤ of an investment.

E.G.,equity, the payo¤ tomorrow: xt+1 = pt+1 + dt+1;(not rate of return; but value of investment in t + 1)

Agent is a price taker in the asset. Let et , et+1 be thecurrent and future endowments, and φ be the amount ofthe asset he chooses to buyThen the consumer�s problem is to

maxφu(ct ) + Et [βu(ct+1)] s.t.

ct = et � ptφ,ct+1 = et+1 + xt+1φ

Future consumption depends on the payo¤ in t + 1, whichyou do not know today.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

De�nition

De�ne xt+1 as the payo¤ of an investment.

E.G.,equity, the payo¤ tomorrow: xt+1 = pt+1 + dt+1;(not rate of return; but value of investment in t + 1)Agent is a price taker in the asset. Let et , et+1 be thecurrent and future endowments, and φ be the amount ofthe asset he chooses to buy

Then the consumer�s problem is to

maxφu(ct ) + Et [βu(ct+1)] s.t.

ct = et � ptφ,ct+1 = et+1 + xt+1φ

Future consumption depends on the payo¤ in t + 1, whichyou do not know today.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

De�nition

De�ne xt+1 as the payo¤ of an investment.

E.G.,equity, the payo¤ tomorrow: xt+1 = pt+1 + dt+1;(not rate of return; but value of investment in t + 1)Agent is a price taker in the asset. Let et , et+1 be thecurrent and future endowments, and φ be the amount ofthe asset he chooses to buyThen the consumer�s problem is to

maxφu(ct ) + Et [βu(ct+1)] s.t.

ct = et � ptφ,ct+1 = et+1 + xt+1φ

Future consumption depends on the payo¤ in t + 1, whichyou do not know today.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

De�nition

De�ne xt+1 as the payo¤ of an investment.

E.G.,equity, the payo¤ tomorrow: xt+1 = pt+1 + dt+1;(not rate of return; but value of investment in t + 1)Agent is a price taker in the asset. Let et , et+1 be thecurrent and future endowments, and φ be the amount ofthe asset he chooses to buyThen the consumer�s problem is to

maxφu(ct ) + Et [βu(ct+1)] s.t.

ct = et � ptφ,ct+1 = et+1 + xt+1φ

Future consumption depends on the payo¤ in t + 1, whichyou do not know today.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

The solution to this problem yields the familiar �rst-ordercondition:

ptu0(ct ) = Et [βu0(ct+1)xt+1] (12)

or

pt = Et

�βu0(ct+1)u0(ct )

xt+1

�. (13)

Expression (12) is the standard �rst-order condition.

LHS is the utility loss to the consumer of purchasing onemore unit of the asset this period.What do you gain? u0(ct+1)xt+1, but we must discountthis future utility to equate it with the present.If the LHS was less than the RHS the consumer wouldpurchase more of the asset until the condition is satis�ed.

Expression (13) is the central asset pricing formula.

Notice that consumption is on the RHS and that isendogenous.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

The solution to this problem yields the familiar �rst-ordercondition:

ptu0(ct ) = Et [βu0(ct+1)xt+1] (12)

or

pt = Et

�βu0(ct+1)u0(ct )

xt+1

�. (13)

Expression (12) is the standard �rst-order condition.

LHS is the utility loss to the consumer of purchasing onemore unit of the asset this period.What do you gain? u0(ct+1)xt+1, but we must discountthis future utility to equate it with the present.If the LHS was less than the RHS the consumer wouldpurchase more of the asset until the condition is satis�ed.

Expression (13) is the central asset pricing formula.

Notice that consumption is on the RHS and that isendogenous.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

The solution to this problem yields the familiar �rst-ordercondition:

ptu0(ct ) = Et [βu0(ct+1)xt+1] (12)

or

pt = Et

�βu0(ct+1)u0(ct )

xt+1

�. (13)

Expression (12) is the standard �rst-order condition.LHS is the utility loss to the consumer of purchasing onemore unit of the asset this period.

What do you gain? u0(ct+1)xt+1, but we must discountthis future utility to equate it with the present.If the LHS was less than the RHS the consumer wouldpurchase more of the asset until the condition is satis�ed.

Expression (13) is the central asset pricing formula.

Notice that consumption is on the RHS and that isendogenous.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

The solution to this problem yields the familiar �rst-ordercondition:

ptu0(ct ) = Et [βu0(ct+1)xt+1] (12)

or

pt = Et

�βu0(ct+1)u0(ct )

xt+1

�. (13)

Expression (12) is the standard �rst-order condition.LHS is the utility loss to the consumer of purchasing onemore unit of the asset this period.What do you gain? u0(ct+1)xt+1, but we must discountthis future utility to equate it with the present.

If the LHS was less than the RHS the consumer wouldpurchase more of the asset until the condition is satis�ed.

Expression (13) is the central asset pricing formula.

Notice that consumption is on the RHS and that isendogenous.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

The solution to this problem yields the familiar �rst-ordercondition:

ptu0(ct ) = Et [βu0(ct+1)xt+1] (12)

or

pt = Et

�βu0(ct+1)u0(ct )

xt+1

�. (13)

Expression (12) is the standard �rst-order condition.LHS is the utility loss to the consumer of purchasing onemore unit of the asset this period.What do you gain? u0(ct+1)xt+1, but we must discountthis future utility to equate it with the present.If the LHS was less than the RHS the consumer wouldpurchase more of the asset until the condition is satis�ed.

Expression (13) is the central asset pricing formula.

Notice that consumption is on the RHS and that isendogenous.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

The solution to this problem yields the familiar �rst-ordercondition:

ptu0(ct ) = Et [βu0(ct+1)xt+1] (12)

or

pt = Et

�βu0(ct+1)u0(ct )

xt+1

�. (13)

Expression (12) is the standard �rst-order condition.LHS is the utility loss to the consumer of purchasing onemore unit of the asset this period.What do you gain? u0(ct+1)xt+1, but we must discountthis future utility to equate it with the present.If the LHS was less than the RHS the consumer wouldpurchase more of the asset until the condition is satis�ed.

Expression (13) is the central asset pricing formula.

Notice that consumption is on the RHS and that isendogenous.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

The solution to this problem yields the familiar �rst-ordercondition:

ptu0(ct ) = Et [βu0(ct+1)xt+1] (12)

or

pt = Et

�βu0(ct+1)u0(ct )

xt+1

�. (13)

Expression (12) is the standard �rst-order condition.LHS is the utility loss to the consumer of purchasing onemore unit of the asset this period.What do you gain? u0(ct+1)xt+1, but we must discountthis future utility to equate it with the present.If the LHS was less than the RHS the consumer wouldpurchase more of the asset until the condition is satis�ed.

Expression (13) is the central asset pricing formula.Notice that consumption is on the RHS and that isendogenous.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We can see the problem graphically in �gure 3

the endowments (et , et+1) at E : note that this cannot bean optimumif we move on the budget line to the northwest we moveon a higher indi¤erence curveoptimum is at C �; if we move from there utility must fall

Key di¤erence, we don�t know ct+1 or xt+1, just haveexpectations

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We can see the problem graphically in �gure 3

the endowments (et , et+1) at E : note that this cannot bean optimum

if we move on the budget line to the northwest we moveon a higher indi¤erence curveoptimum is at C �; if we move from there utility must fall

Key di¤erence, we don�t know ct+1 or xt+1, just haveexpectations

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We can see the problem graphically in �gure 3

the endowments (et , et+1) at E : note that this cannot bean optimumif we move on the budget line to the northwest we moveon a higher indi¤erence curve

optimum is at C �; if we move from there utility must fall

Key di¤erence, we don�t know ct+1 or xt+1, just haveexpectations

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We can see the problem graphically in �gure 3

the endowments (et , et+1) at E : note that this cannot bean optimumif we move on the budget line to the northwest we moveon a higher indi¤erence curveoptimum is at C �; if we move from there utility must fall

Key di¤erence, we don�t know ct+1 or xt+1, just haveexpectations

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Basic Framework

We can see the problem graphically in �gure 3

the endowments (et , et+1) at E : note that this cannot bean optimumif we move on the budget line to the northwest we moveon a higher indi¤erence curveoptimum is at C �; if we move from there utility must fall

Key di¤erence, we don�t know ct+1 or xt+1, just haveexpectations

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Consumer�s Problem

          C*

             E

tc

1+tc

te

1+te

Figure: Consumer�s problem

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

De�nition

We can de�ne the stochastic discount factor, mt+1 = β u0(ct+1)u 0(ct )

.

Clearly m is the marginal rate of substitution.

Why it is also called the sdf? Re-write (13) as:

pt = Et [mt+1xt+1 ] . (14)

With no uncertainty, pt = 1R f xt+1, where R

f = 1+ r r is

the gross risk-free rate of return: R f "discounts" x .With uncertainty for each asset i

pit =1R iEt (x it+1).

R i , is used to discount the uncertain payo¤, x it+1.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

De�nition

We can de�ne the stochastic discount factor, mt+1 = β u0(ct+1)u 0(ct )

.

Clearly m is the marginal rate of substitution.

Why it is also called the sdf? Re-write (13) as:

pt = Et [mt+1xt+1 ] . (14)

With no uncertainty, pt = 1R f xt+1, where R

f = 1+ r r is

the gross risk-free rate of return: R f "discounts" x .With uncertainty for each asset i

pit =1R iEt (x it+1).

R i , is used to discount the uncertain payo¤, x it+1.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

De�nition

We can de�ne the stochastic discount factor, mt+1 = β u0(ct+1)u 0(ct )

.

Clearly m is the marginal rate of substitution.

Why it is also called the sdf? Re-write (13) as:

pt = Et [mt+1xt+1 ] . (14)

With no uncertainty, pt = 1R f xt+1, where R

f = 1+ r r is

the gross risk-free rate of return: R f "discounts" x .

With uncertainty for each asset i

pit =1R iEt (x it+1).

R i , is used to discount the uncertain payo¤, x it+1.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

De�nition

We can de�ne the stochastic discount factor, mt+1 = β u0(ct+1)u 0(ct )

.

Clearly m is the marginal rate of substitution.

Why it is also called the sdf? Re-write (13) as:

pt = Et [mt+1xt+1 ] . (14)

With no uncertainty, pt = 1R f xt+1, where R

f = 1+ r r is

the gross risk-free rate of return: R f "discounts" x .With uncertainty for each asset i

pit =1R iEt (x it+1).

R i , is used to discount the uncertain payo¤, x it+1.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

De�nition

We can de�ne the stochastic discount factor, mt+1 = β u0(ct+1)u 0(ct )

.

Clearly m is the marginal rate of substitution.

Why it is also called the sdf? Re-write (13) as:

pt = Et [mt+1xt+1 ] . (14)

With no uncertainty, pt = 1R f xt+1, where R

f = 1+ r r is

the gross risk-free rate of return: R f "discounts" x .With uncertainty for each asset i

pit =1R iEt (x it+1).

R i , is used to discount the uncertain payo¤, x it+1.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The sdf thus generalizes the normal approach.

One factor, mt+1, the same one for each asset, can beused to discount any asset.

What does pt = Et [mt+1xt+1] imply?

what matters for asset prices is how payo¤s are correlatedwith the value of a dollar in a given statean asset that pays o¤ when a dollar is valuable is worthmore

When is a dollar more valuable?

m is high when when consumption is expected to be low(so the marginal utility of consumption is high)

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The sdf thus generalizes the normal approach.

One factor, mt+1, the same one for each asset, can beused to discount any asset.

What does pt = Et [mt+1xt+1] imply?

what matters for asset prices is how payo¤s are correlatedwith the value of a dollar in a given statean asset that pays o¤ when a dollar is valuable is worthmore

When is a dollar more valuable?

m is high when when consumption is expected to be low(so the marginal utility of consumption is high)

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The sdf thus generalizes the normal approach.

One factor, mt+1, the same one for each asset, can beused to discount any asset.

What does pt = Et [mt+1xt+1] imply?

what matters for asset prices is how payo¤s are correlatedwith the value of a dollar in a given statean asset that pays o¤ when a dollar is valuable is worthmore

When is a dollar more valuable?

m is high when when consumption is expected to be low(so the marginal utility of consumption is high)

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The sdf thus generalizes the normal approach.

One factor, mt+1, the same one for each asset, can beused to discount any asset.

What does pt = Et [mt+1xt+1] imply?

what matters for asset prices is how payo¤s are correlatedwith the value of a dollar in a given state

an asset that pays o¤ when a dollar is valuable is worthmore

When is a dollar more valuable?

m is high when when consumption is expected to be low(so the marginal utility of consumption is high)

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The sdf thus generalizes the normal approach.

One factor, mt+1, the same one for each asset, can beused to discount any asset.

What does pt = Et [mt+1xt+1] imply?

what matters for asset prices is how payo¤s are correlatedwith the value of a dollar in a given statean asset that pays o¤ when a dollar is valuable is worthmore

When is a dollar more valuable?

m is high when when consumption is expected to be low(so the marginal utility of consumption is high)

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The sdf thus generalizes the normal approach.

One factor, mt+1, the same one for each asset, can beused to discount any asset.

What does pt = Et [mt+1xt+1] imply?

what matters for asset prices is how payo¤s are correlatedwith the value of a dollar in a given statean asset that pays o¤ when a dollar is valuable is worthmore

When is a dollar more valuable?

m is high when when consumption is expected to be low(so the marginal utility of consumption is high)

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The sdf thus generalizes the normal approach.

One factor, mt+1, the same one for each asset, can beused to discount any asset.

What does pt = Et [mt+1xt+1] imply?

what matters for asset prices is how payo¤s are correlatedwith the value of a dollar in a given statean asset that pays o¤ when a dollar is valuable is worthmore

When is a dollar more valuable?

m is high when when consumption is expected to be low(so the marginal utility of consumption is high)

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

Example

Two-State derivation of SDF.

Recall relation between pi and A-D state price:

pi = q1x1i + q2x2i

where the q0s are the A-D state prices.

Clearly we can write

pi = π1

�q1π1

�x1i + π2

�q2π2

�x2i

� π1m1x1i + π2m2x2i

where the m0s are the ratio of state prices to theprobability of the state, and are thus sdf�s.Then, clearly,

pi = Ej [mjxji ]

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

Example

Two-State derivation of SDF.

Recall relation between pi and A-D state price:

pi = q1x1i + q2x2i

where the q0s are the A-D state prices.Clearly we can write

pi = π1

�q1π1

�x1i + π2

�q2π2

�x2i

� π1m1x1i + π2m2x2i

where the m0s are the ratio of state prices to theprobability of the state, and are thus sdf�s.Then, clearly,

pi = Ej [mjxji ]

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

Example

Two-State derivation of SDF.

Recall relation between pi and A-D state price:

pi = q1x1i + q2x2i

where the q0s are the A-D state prices.Clearly we can write

pi = π1

�q1π1

�x1i + π2

�q2π2

�x2i

� π1m1x1i + π2m2x2i

where the m0s are the ratio of state prices to theprobability of the state, and are thus sdf�s.

Then, clearly,pi = Ej [mjxji ]

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

Example

Two-State derivation of SDF.

Recall relation between pi and A-D state price:

pi = q1x1i + q2x2i

where the q0s are the A-D state prices.Clearly we can write

pi = π1

�q1π1

�x1i + π2

�q2π2

�x2i

� π1m1x1i + π2m2x2i

where the m0s are the ratio of state prices to theprobability of the state, and are thus sdf�s.Then, clearly,

pi = Ej [mjxji ]

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The stochastic discount factor is thus a random variablethat di¤ers across states, but is the same for all assets.

It is the ratio of state price to probability; thus, it isproportional to marginal utility of wealth in each state.The price of any asset is the average of its payo¤s,weighted by state probabilities and marginal utilities.If an asset pays o¤ in states that occur with highprobability it will be more valuable, other things equal. Ifan asset pays o¤ in states where marginal utility is high itwill be more valuable, other things equal.Equivalently, the price of any asset is the expected productof the SDF and the payo¤.This seems only notational. But involved here is a deepand useful separation. All asset pricing models amount toalternative ways of connecting the stochastic discountfactor to the data.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The stochastic discount factor is thus a random variablethat di¤ers across states, but is the same for all assets.

It is the ratio of state price to probability; thus, it isproportional to marginal utility of wealth in each state.

The price of any asset is the average of its payo¤s,weighted by state probabilities and marginal utilities.If an asset pays o¤ in states that occur with highprobability it will be more valuable, other things equal. Ifan asset pays o¤ in states where marginal utility is high itwill be more valuable, other things equal.Equivalently, the price of any asset is the expected productof the SDF and the payo¤.This seems only notational. But involved here is a deepand useful separation. All asset pricing models amount toalternative ways of connecting the stochastic discountfactor to the data.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The stochastic discount factor is thus a random variablethat di¤ers across states, but is the same for all assets.

It is the ratio of state price to probability; thus, it isproportional to marginal utility of wealth in each state.The price of any asset is the average of its payo¤s,weighted by state probabilities and marginal utilities.

If an asset pays o¤ in states that occur with highprobability it will be more valuable, other things equal. Ifan asset pays o¤ in states where marginal utility is high itwill be more valuable, other things equal.Equivalently, the price of any asset is the expected productof the SDF and the payo¤.This seems only notational. But involved here is a deepand useful separation. All asset pricing models amount toalternative ways of connecting the stochastic discountfactor to the data.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The stochastic discount factor is thus a random variablethat di¤ers across states, but is the same for all assets.

It is the ratio of state price to probability; thus, it isproportional to marginal utility of wealth in each state.The price of any asset is the average of its payo¤s,weighted by state probabilities and marginal utilities.If an asset pays o¤ in states that occur with highprobability it will be more valuable, other things equal. Ifan asset pays o¤ in states where marginal utility is high itwill be more valuable, other things equal.

Equivalently, the price of any asset is the expected productof the SDF and the payo¤.This seems only notational. But involved here is a deepand useful separation. All asset pricing models amount toalternative ways of connecting the stochastic discountfactor to the data.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The stochastic discount factor is thus a random variablethat di¤ers across states, but is the same for all assets.

It is the ratio of state price to probability; thus, it isproportional to marginal utility of wealth in each state.The price of any asset is the average of its payo¤s,weighted by state probabilities and marginal utilities.If an asset pays o¤ in states that occur with highprobability it will be more valuable, other things equal. Ifan asset pays o¤ in states where marginal utility is high itwill be more valuable, other things equal.Equivalently, the price of any asset is the expected productof the SDF and the payo¤.

This seems only notational. But involved here is a deepand useful separation. All asset pricing models amount toalternative ways of connecting the stochastic discountfactor to the data.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Stochastic Discount Factor

The stochastic discount factor is thus a random variablethat di¤ers across states, but is the same for all assets.

It is the ratio of state price to probability; thus, it isproportional to marginal utility of wealth in each state.The price of any asset is the average of its payo¤s,weighted by state probabilities and marginal utilities.If an asset pays o¤ in states that occur with highprobability it will be more valuable, other things equal. Ifan asset pays o¤ in states where marginal utility is high itwill be more valuable, other things equal.Equivalently, the price of any asset is the expected productof the SDF and the payo¤.This seems only notational. But involved here is a deepand useful separation. All asset pricing models amount toalternative ways of connecting the stochastic discountfactor to the data.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Payo¤s

De�ne the gross return as Rt+1 � xt+1pt.

Then1 = Et [mt+1Rt+1 ]

The gross risk free rate is Rf � 1+ rf .Thus, 1 = Et [mt+1Rf ] = E (m)Rf ,

as the payo¤ on a risk-free asset is know with certainty.Thus,

Rf =1

E (m)(15)

notice that we can use (15) to de�ne a shadow risk-freerate even if such a security did not actually exist

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Payo¤s

De�ne the gross return as Rt+1 � xt+1pt.

Then1 = Et [mt+1Rt+1 ]

The gross risk free rate is Rf � 1+ rf .Thus, 1 = Et [mt+1Rf ] = E (m)Rf ,

as the payo¤ on a risk-free asset is know with certainty.Thus,

Rf =1

E (m)(15)

notice that we can use (15) to de�ne a shadow risk-freerate even if such a security did not actually exist

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Payo¤s

De�ne the gross return as Rt+1 � xt+1pt.

Then1 = Et [mt+1Rt+1 ]

The gross risk free rate is Rf � 1+ rf .

Thus, 1 = Et [mt+1Rf ] = E (m)Rf ,

as the payo¤ on a risk-free asset is know with certainty.Thus,

Rf =1

E (m)(15)

notice that we can use (15) to de�ne a shadow risk-freerate even if such a security did not actually exist

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Payo¤s

De�ne the gross return as Rt+1 � xt+1pt.

Then1 = Et [mt+1Rt+1 ]

The gross risk free rate is Rf � 1+ rf .Thus, 1 = Et [mt+1Rf ] = E (m)Rf ,

as the payo¤ on a risk-free asset is know with certainty.Thus,

Rf =1

E (m)(15)

notice that we can use (15) to de�ne a shadow risk-freerate even if such a security did not actually exist

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Payo¤s

De�ne the gross return as Rt+1 � xt+1pt.

Then1 = Et [mt+1Rt+1 ]

The gross risk free rate is Rf � 1+ rf .Thus, 1 = Et [mt+1Rf ] = E (m)Rf ,

as the payo¤ on a risk-free asset is know with certainty.

Thus,

Rf =1

E (m)(15)

notice that we can use (15) to de�ne a shadow risk-freerate even if such a security did not actually exist

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Payo¤s

De�ne the gross return as Rt+1 � xt+1pt.

Then1 = Et [mt+1Rt+1 ]

The gross risk free rate is Rf � 1+ rf .Thus, 1 = Et [mt+1Rf ] = E (m)Rf ,

as the payo¤ on a risk-free asset is know with certainty.Thus,

Rf =1

E (m)(15)

notice that we can use (15) to de�ne a shadow risk-freerate even if such a security did not actually exist

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Payo¤s

De�ne the gross return as Rt+1 � xt+1pt.

Then1 = Et [mt+1Rt+1 ]

The gross risk free rate is Rf � 1+ rf .Thus, 1 = Et [mt+1Rf ] = E (m)Rf ,

as the payo¤ on a risk-free asset is know with certainty.Thus,

Rf =1

E (m)(15)

notice that we can use (15) to de�ne a shadow risk-freerate even if such a security did not actually exist

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Real interest rates

We can use (15) to think about the economics of realinterest rates

Suppose that u0(c) = c�γ, power utility

marginal utility declines with consumption (see �gure)

Suppose no uncertainty. Then,

Rf =1β

�ct+1ct

�γ

(16)

Three e¤ects follow:

1 real interest rates are high when β is low2 real interest rates are high when consumption growth ishigh

3 real interest rates are more sensitive to consumptiongrowth when γ is high

1 when utility is highly curved, investor cares more about asmooth consumption pro�le

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Real interest rates

We can use (15) to think about the economics of realinterest ratesSuppose that u0(c) = c�γ, power utility

marginal utility declines with consumption (see �gure)

Suppose no uncertainty. Then,

Rf =1β

�ct+1ct

�γ

(16)

Three e¤ects follow:

1 real interest rates are high when β is low2 real interest rates are high when consumption growth ishigh

3 real interest rates are more sensitive to consumptiongrowth when γ is high

1 when utility is highly curved, investor cares more about asmooth consumption pro�le

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Real interest rates

We can use (15) to think about the economics of realinterest ratesSuppose that u0(c) = c�γ, power utility

marginal utility declines with consumption (see �gure)

Suppose no uncertainty. Then,

Rf =1β

�ct+1ct

�γ

(16)

Three e¤ects follow:

1 real interest rates are high when β is low2 real interest rates are high when consumption growth ishigh

3 real interest rates are more sensitive to consumptiongrowth when γ is high

1 when utility is highly curved, investor cares more about asmooth consumption pro�le

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Real interest rates

We can use (15) to think about the economics of realinterest ratesSuppose that u0(c) = c�γ, power utility

marginal utility declines with consumption (see �gure)

Suppose no uncertainty. Then,

Rf =1β

�ct+1ct

�γ

(16)

Three e¤ects follow:

1 real interest rates are high when β is low2 real interest rates are high when consumption growth ishigh

3 real interest rates are more sensitive to consumptiongrowth when γ is high

1 when utility is highly curved, investor cares more about asmooth consumption pro�le

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Real interest rates

We can use (15) to think about the economics of realinterest ratesSuppose that u0(c) = c�γ, power utility

marginal utility declines with consumption (see �gure)

Suppose no uncertainty. Then,

Rf =1β

�ct+1ct

�γ

(16)

Three e¤ects follow:

1 real interest rates are high when β is low2 real interest rates are high when consumption growth ishigh

3 real interest rates are more sensitive to consumptiongrowth when γ is high

1 when utility is highly curved, investor cares more about asmooth consumption pro�le

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Real interest rates

We can use (15) to think about the economics of realinterest ratesSuppose that u0(c) = c�γ, power utility

marginal utility declines with consumption (see �gure)

Suppose no uncertainty. Then,

Rf =1β

�ct+1ct

�γ

(16)

Three e¤ects follow:1 real interest rates are high when β is low

2 real interest rates are high when consumption growth ishigh

3 real interest rates are more sensitive to consumptiongrowth when γ is high

1 when utility is highly curved, investor cares more about asmooth consumption pro�le

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Real interest rates

We can use (15) to think about the economics of realinterest ratesSuppose that u0(c) = c�γ, power utility

marginal utility declines with consumption (see �gure)

Suppose no uncertainty. Then,

Rf =1β

�ct+1ct

�γ

(16)

Three e¤ects follow:1 real interest rates are high when β is low2 real interest rates are high when consumption growth ishigh

3 real interest rates are more sensitive to consumptiongrowth when γ is high

1 when utility is highly curved, investor cares more about asmooth consumption pro�le

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Real interest rates

We can use (15) to think about the economics of realinterest ratesSuppose that u0(c) = c�γ, power utility

marginal utility declines with consumption (see �gure)

Suppose no uncertainty. Then,

Rf =1β

�ct+1ct

�γ

(16)

Three e¤ects follow:1 real interest rates are high when β is low2 real interest rates are high when consumption growth ishigh

3 real interest rates are more sensitive to consumptiongrowth when γ is high

1 when utility is highly curved, investor cares more about asmooth consumption pro�le

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Real interest rates

We can use (15) to think about the economics of realinterest ratesSuppose that u0(c) = c�γ, power utility

marginal utility declines with consumption (see �gure)

Suppose no uncertainty. Then,

Rf =1β

�ct+1ct

�γ

(16)

Three e¤ects follow:1 real interest rates are high when β is low2 real interest rates are high when consumption growth ishigh

3 real interest rates are more sensitive to consumptiongrowth when γ is high1 when utility is highly curved, investor cares more about asmooth consumption pro�le

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Marginal utility and gamma

0

0.2

0.4

0.6

0.8

1

1.2

c 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

u'gamma = 2

u'gamma= 5

u'gamma = 1.1

u'gamma = .5

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk-free rate

Thus, when γ is higher it takes a larger ∆rf to get peopleto change their consumption path

Now add uncertainty. We need to say how c varies.Suppose that c is lognormally distributed

this means that the log of c , written as ln c is normallydistributed

Then

lnRf = δ+ γEt (∆ ln ct+1)�γ2

2σ2t (∆ ln ct+1) (17)

where β = e�δ, ∆ ln ct+1 is the growth rate ofconsumption, and σ2t (∆ ln ct+1) is the variance of thegrowth rate of consumptionExpression (17) can be decomposed into three terms

lnRf = impatience term+ growth of c term

� variance of growth of c term

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk-free rate

Thus, when γ is higher it takes a larger ∆rf to get peopleto change their consumption pathNow add uncertainty. We need to say how c varies.Suppose that c is lognormally distributed

this means that the log of c , written as ln c is normallydistributed

Then

lnRf = δ+ γEt (∆ ln ct+1)�γ2

2σ2t (∆ ln ct+1) (17)

where β = e�δ, ∆ ln ct+1 is the growth rate ofconsumption, and σ2t (∆ ln ct+1) is the variance of thegrowth rate of consumptionExpression (17) can be decomposed into three terms

lnRf = impatience term+ growth of c term

� variance of growth of c term

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk-free rate

Thus, when γ is higher it takes a larger ∆rf to get peopleto change their consumption pathNow add uncertainty. We need to say how c varies.Suppose that c is lognormally distributed

this means that the log of c , written as ln c is normallydistributed

Then

lnRf = δ+ γEt (∆ ln ct+1)�γ2

2σ2t (∆ ln ct+1) (17)

where β = e�δ, ∆ ln ct+1 is the growth rate ofconsumption, and σ2t (∆ ln ct+1) is the variance of thegrowth rate of consumptionExpression (17) can be decomposed into three terms

lnRf = impatience term+ growth of c term

� variance of growth of c term

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk-free rate

Thus, when γ is higher it takes a larger ∆rf to get peopleto change their consumption pathNow add uncertainty. We need to say how c varies.Suppose that c is lognormally distributed

this means that the log of c , written as ln c is normallydistributed

Then

lnRf = δ+ γEt (∆ ln ct+1)�γ2

2σ2t (∆ ln ct+1) (17)

where β = e�δ, ∆ ln ct+1 is the growth rate ofconsumption, and σ2t (∆ ln ct+1) is the variance of thegrowth rate of consumption

Expression (17) can be decomposed into three terms

lnRf = impatience term+ growth of c term

� variance of growth of c term

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk-free rate

Thus, when γ is higher it takes a larger ∆rf to get peopleto change their consumption pathNow add uncertainty. We need to say how c varies.Suppose that c is lognormally distributed

this means that the log of c , written as ln c is normallydistributed

Then

lnRf = δ+ γEt (∆ ln ct+1)�γ2

2σ2t (∆ ln ct+1) (17)

where β = e�δ, ∆ ln ct+1 is the growth rate ofconsumption, and σ2t (∆ ln ct+1) is the variance of thegrowth rate of consumptionExpression (17) can be decomposed into three terms

lnRf = impatience term+ growth of c term

� variance of growth of c term

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Implications

Expression (17) implies previous 3 e¤ects, plus new one:

real interest rates are low when consumption growth ismore volatileprecautionary demand for savings

We can also read the expression backwards

consumption growth is high when rf is highconsumption is less sensitive to rf when γ is high becausethey desire smoother consumption

Notice that with power utility γ governs intertemporalsubstitution, risk aversion and precautionary savings. Moregeneral utility functions allow this to be separated

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Implications

Expression (17) implies previous 3 e¤ects, plus new one:

real interest rates are low when consumption growth ismore volatile

precautionary demand for savings

We can also read the expression backwards

consumption growth is high when rf is highconsumption is less sensitive to rf when γ is high becausethey desire smoother consumption

Notice that with power utility γ governs intertemporalsubstitution, risk aversion and precautionary savings. Moregeneral utility functions allow this to be separated

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Implications

Expression (17) implies previous 3 e¤ects, plus new one:

real interest rates are low when consumption growth ismore volatileprecautionary demand for savings

We can also read the expression backwards

consumption growth is high when rf is highconsumption is less sensitive to rf when γ is high becausethey desire smoother consumption

Notice that with power utility γ governs intertemporalsubstitution, risk aversion and precautionary savings. Moregeneral utility functions allow this to be separated

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Implications

Expression (17) implies previous 3 e¤ects, plus new one:

real interest rates are low when consumption growth ismore volatileprecautionary demand for savings

We can also read the expression backwards

consumption growth is high when rf is highconsumption is less sensitive to rf when γ is high becausethey desire smoother consumption

Notice that with power utility γ governs intertemporalsubstitution, risk aversion and precautionary savings. Moregeneral utility functions allow this to be separated

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Implications

Expression (17) implies previous 3 e¤ects, plus new one:

real interest rates are low when consumption growth ismore volatileprecautionary demand for savings

We can also read the expression backwards

consumption growth is high when rf is high

consumption is less sensitive to rf when γ is high becausethey desire smoother consumption

Notice that with power utility γ governs intertemporalsubstitution, risk aversion and precautionary savings. Moregeneral utility functions allow this to be separated

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Implications

Expression (17) implies previous 3 e¤ects, plus new one:

real interest rates are low when consumption growth ismore volatileprecautionary demand for savings

We can also read the expression backwards

consumption growth is high when rf is highconsumption is less sensitive to rf when γ is high becausethey desire smoother consumption

Notice that with power utility γ governs intertemporalsubstitution, risk aversion and precautionary savings. Moregeneral utility functions allow this to be separated

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Implications

Expression (17) implies previous 3 e¤ects, plus new one:

real interest rates are low when consumption growth ismore volatileprecautionary demand for savings

We can also read the expression backwards

consumption growth is high when rf is highconsumption is less sensitive to rf when γ is high becausethey desire smoother consumption

Notice that with power utility γ governs intertemporalsubstitution, risk aversion and precautionary savings. Moregeneral utility functions allow this to be separated

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk Corrections

Use the basic price equation again, p = E (mx)

Recall the de�nition of covariance:cov(m, x) = E (mx)� E (m)E (x)So

p = E (m)E (x) + cov(m, x)

or, using (15) to write:

p =E (x)Rf

+ cov(m, x) (18)

the �rst term on RHS is discounted present value, secondterm is risk adjustment

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk Corrections

Use the basic price equation again, p = E (mx)

Recall the de�nition of covariance:cov(m, x) = E (mx)� E (m)E (x)

Sop = E (m)E (x) + cov(m, x)

or, using (15) to write:

p =E (x)Rf

+ cov(m, x) (18)

the �rst term on RHS is discounted present value, secondterm is risk adjustment

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk Corrections

Use the basic price equation again, p = E (mx)

Recall the de�nition of covariance:cov(m, x) = E (mx)� E (m)E (x)So

p = E (m)E (x) + cov(m, x)

or, using (15) to write:

p =E (x)Rf

+ cov(m, x) (18)

the �rst term on RHS is discounted present value, secondterm is risk adjustment

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk Corrections

Use the basic price equation again, p = E (mx)

Recall the de�nition of covariance:cov(m, x) = E (mx)� E (m)E (x)So

p = E (m)E (x) + cov(m, x)

or, using (15) to write:

p =E (x)Rf

+ cov(m, x) (18)

the �rst term on RHS is discounted present value, secondterm is risk adjustment

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk Corrections

Use the basic price equation again, p = E (mx)

Recall the de�nition of covariance:cov(m, x) = E (mx)� E (m)E (x)So

p = E (m)E (x) + cov(m, x)

or, using (15) to write:

p =E (x)Rf

+ cov(m, x) (18)

the �rst term on RHS is discounted present value, secondterm is risk adjustment

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk Corrections

It is useful to substitute for m in (18)

p =E (x)Rf

+cov [βu0(ct+1), xt+1]

u0(ct )(19)

Since u0 declines as c rises, an asset�s price is lower if thepayo¤ positively covaries with consumption

Conversely, an asset that pays o¤ when consumption is lowhas a higher priceInsuranceriskier securities must o¤er higher returns to get investorsto hold them

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk Corrections

It is useful to substitute for m in (18)

p =E (x)Rf

+cov [βu0(ct+1), xt+1]

u0(ct )(19)

Since u0 declines as c rises, an asset�s price is lower if thepayo¤ positively covaries with consumption

Conversely, an asset that pays o¤ when consumption is lowhas a higher priceInsuranceriskier securities must o¤er higher returns to get investorsto hold them

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk Corrections

It is useful to substitute for m in (18)

p =E (x)Rf

+cov [βu0(ct+1), xt+1]

u0(ct )(19)

Since u0 declines as c rises, an asset�s price is lower if thepayo¤ positively covaries with consumption

Conversely, an asset that pays o¤ when consumption is lowhas a higher price

Insuranceriskier securities must o¤er higher returns to get investorsto hold them

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk Corrections

It is useful to substitute for m in (18)

p =E (x)Rf

+cov [βu0(ct+1), xt+1]

u0(ct )(19)

Since u0 declines as c rises, an asset�s price is lower if thepayo¤ positively covaries with consumption

Conversely, an asset that pays o¤ when consumption is lowhas a higher priceInsurance

riskier securities must o¤er higher returns to get investorsto hold them

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Risk Corrections

It is useful to substitute for m in (18)

p =E (x)Rf

+cov [βu0(ct+1), xt+1]

u0(ct )(19)

Since u0 declines as c rises, an asset�s price is lower if thepayo¤ positively covaries with consumption

Conversely, an asset that pays o¤ when consumption is lowhas a higher priceInsuranceriskier securities must o¤er higher returns to get investorsto hold them

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Idiosyncratic risk

It is not risk itself that is compensated with return, onlyrisk correlated with m matters.

We can use (18) to see this. Suppose that some asset hasa payo¤ that has zero covariance with m: cov(m, x) = 0.

Then from (18) p = E (x )Rf,so that

E (x)p

= Rf (20)

thus the rate of return on this asset equals the risk-freerate, no matter how large is the variance of its payo¤,σ2(x).

We could let σ2(x) �! a billion, but if cov(m, x) = 0, itwill still yield only the risk-free rate. Even if people aretotally risk averse.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Idiosyncratic risk

It is not risk itself that is compensated with return, onlyrisk correlated with m matters.

We can use (18) to see this. Suppose that some asset hasa payo¤ that has zero covariance with m: cov(m, x) = 0.

Then from (18) p = E (x )Rf,so that

E (x)p

= Rf (20)

thus the rate of return on this asset equals the risk-freerate, no matter how large is the variance of its payo¤,σ2(x).

We could let σ2(x) �! a billion, but if cov(m, x) = 0, itwill still yield only the risk-free rate. Even if people aretotally risk averse.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Idiosyncratic risk

It is not risk itself that is compensated with return, onlyrisk correlated with m matters.

We can use (18) to see this. Suppose that some asset hasa payo¤ that has zero covariance with m: cov(m, x) = 0.

Then from (18) p = E (x )Rf,so that

E (x)p

= Rf (20)

thus the rate of return on this asset equals the risk-freerate, no matter how large is the variance of its payo¤,σ2(x).

We could let σ2(x) �! a billion, but if cov(m, x) = 0, itwill still yield only the risk-free rate. Even if people aretotally risk averse.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Idiosyncratic risk

It is not risk itself that is compensated with return, onlyrisk correlated with m matters.

We can use (18) to see this. Suppose that some asset hasa payo¤ that has zero covariance with m: cov(m, x) = 0.

Then from (18) p = E (x )Rf,so that

E (x)p

= Rf (20)

thus the rate of return on this asset equals the risk-freerate, no matter how large is the variance of its payo¤,σ2(x).

We could let σ2(x) �! a billion, but if cov(m, x) = 0, itwill still yield only the risk-free rate. Even if people aretotally risk averse.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Idiosyncratic risk

It is not risk itself that is compensated with return, onlyrisk correlated with m matters.

We can use (18) to see this. Suppose that some asset hasa payo¤ that has zero covariance with m: cov(m, x) = 0.

Then from (18) p = E (x )Rf,so that

E (x)p

= Rf (20)

thus the rate of return on this asset equals the risk-freerate, no matter how large is the variance of its payo¤,σ2(x).

We could let σ2(x) �! a billion, but if cov(m, x) = 0, itwill still yield only the risk-free rate. Even if people aretotally risk averse.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Idiosyncratic risk

The reason why idiosyncratic risk does not earn a riskpremium should be obvious.

Buying this asset does not impact the variance of yourconsumption stream, so why should you need a riskpremium to hold it.

One way of saying this is that there is no compensation foridiosyncratic risk.Only systematic risk generates a risk correction.Systematic risk is that part of the variance that iscorrelated with m. Idiosyncratic risk is the residualvariance.This is the risk that can be diversi�ed away. Since it is notsystematically related to m, we can avoid this risk.Idiosyncratic risk is like the residual in a regressionequation.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Idiosyncratic risk

The reason why idiosyncratic risk does not earn a riskpremium should be obvious.

Buying this asset does not impact the variance of yourconsumption stream, so why should you need a riskpremium to hold it.

One way of saying this is that there is no compensation foridiosyncratic risk.Only systematic risk generates a risk correction.Systematic risk is that part of the variance that iscorrelated with m. Idiosyncratic risk is the residualvariance.This is the risk that can be diversi�ed away. Since it is notsystematically related to m, we can avoid this risk.Idiosyncratic risk is like the residual in a regressionequation.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Idiosyncratic risk

The reason why idiosyncratic risk does not earn a riskpremium should be obvious.

Buying this asset does not impact the variance of yourconsumption stream, so why should you need a riskpremium to hold it.

One way of saying this is that there is no compensation foridiosyncratic risk.

Only systematic risk generates a risk correction.Systematic risk is that part of the variance that iscorrelated with m. Idiosyncratic risk is the residualvariance.This is the risk that can be diversi�ed away. Since it is notsystematically related to m, we can avoid this risk.Idiosyncratic risk is like the residual in a regressionequation.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Idiosyncratic risk

The reason why idiosyncratic risk does not earn a riskpremium should be obvious.

Buying this asset does not impact the variance of yourconsumption stream, so why should you need a riskpremium to hold it.

One way of saying this is that there is no compensation foridiosyncratic risk.Only systematic risk generates a risk correction.

Systematic risk is that part of the variance that iscorrelated with m. Idiosyncratic risk is the residualvariance.This is the risk that can be diversi�ed away. Since it is notsystematically related to m, we can avoid this risk.Idiosyncratic risk is like the residual in a regressionequation.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Idiosyncratic risk

The reason why idiosyncratic risk does not earn a riskpremium should be obvious.

Buying this asset does not impact the variance of yourconsumption stream, so why should you need a riskpremium to hold it.

One way of saying this is that there is no compensation foridiosyncratic risk.Only systematic risk generates a risk correction.Systematic risk is that part of the variance that iscorrelated with m. Idiosyncratic risk is the residualvariance.

This is the risk that can be diversi�ed away. Since it is notsystematically related to m, we can avoid this risk.Idiosyncratic risk is like the residual in a regressionequation.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Idiosyncratic risk

The reason why idiosyncratic risk does not earn a riskpremium should be obvious.

Buying this asset does not impact the variance of yourconsumption stream, so why should you need a riskpremium to hold it.

One way of saying this is that there is no compensation foridiosyncratic risk.Only systematic risk generates a risk correction.Systematic risk is that part of the variance that iscorrelated with m. Idiosyncratic risk is the residualvariance.This is the risk that can be diversi�ed away. Since it is notsystematically related to m, we can avoid this risk.Idiosyncratic risk is like the residual in a regressionequation.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Systematic risk

Figure: Systematic Risk

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

We can also use the basic pricing equation setup to thinkabout the equity premium. That is, how large is theexpected return on stocks over a risk-free asset likeTreasury Bills. Turns out we have an interesting puzzle.

It is useful to start with a measure called the Sharpe ratio,the ratio of the mean excess return to standarddeviation:E (Ri )�Rf

σ(Ri ).

This is the slope of the capital allocation line.The maximum Sharpe ratio is thus given by the tangent tothe market portfolio: E (RP )�Rf

σ(RP ), as in �gure 5.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

We can also use the basic pricing equation setup to thinkabout the equity premium. That is, how large is theexpected return on stocks over a risk-free asset likeTreasury Bills. Turns out we have an interesting puzzle.

It is useful to start with a measure called the Sharpe ratio,the ratio of the mean excess return to standarddeviation:E (Ri )�Rf

σ(Ri ).

This is the slope of the capital allocation line.The maximum Sharpe ratio is thus given by the tangent tothe market portfolio: E (RP )�Rf

σ(RP ), as in �gure 5.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

We can also use the basic pricing equation setup to thinkabout the equity premium. That is, how large is theexpected return on stocks over a risk-free asset likeTreasury Bills. Turns out we have an interesting puzzle.

It is useful to start with a measure called the Sharpe ratio,the ratio of the mean excess return to standarddeviation:E (Ri )�Rf

σ(Ri ).

This is the slope of the capital allocation line.

The maximum Sharpe ratio is thus given by the tangent tothe market portfolio: E (RP )�Rf

σ(RP ), as in �gure 5.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

We can also use the basic pricing equation setup to thinkabout the equity premium. That is, how large is theexpected return on stocks over a risk-free asset likeTreasury Bills. Turns out we have an interesting puzzle.

It is useful to start with a measure called the Sharpe ratio,the ratio of the mean excess return to standarddeviation:E (Ri )�Rf

σ(Ri ).

This is the slope of the capital allocation line.The maximum Sharpe ratio is thus given by the tangent tothe market portfolio: E (RP )�Rf

σ(RP ), as in �gure 5.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Sharpe Ratio

Figure: Sharpe Ratio E¢ cient Portfolios

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

To derive the maximum Sharpe ratio, start from the basicequation

1 = E (mRi )

= E (m)E (Ri ) + ρm,Ri σ(Ri )σ(m)

where we used the de�nition of covariance.

we know that 1 = E (m)Rf , so we can subtract this fromboth sides and re-arrange:

E (Ri ) = Rf � ρm,Ri σ(Ri )σ(m)E (m)

(21)

notice that a correlation coe¢ cient cannot be greater thanunity, so this gives us a bound on asset prices:

jE (Ri ) = Rf j � σ(Ri )σ(m)E (m)

(22)

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

To derive the maximum Sharpe ratio, start from the basicequation

1 = E (mRi )

= E (m)E (Ri ) + ρm,Ri σ(Ri )σ(m)

where we used the de�nition of covariance.

we know that 1 = E (m)Rf , so we can subtract this fromboth sides and re-arrange:

E (Ri ) = Rf � ρm,Ri σ(Ri )σ(m)E (m)

(21)

notice that a correlation coe¢ cient cannot be greater thanunity, so this gives us a bound on asset prices:

jE (Ri ) = Rf j � σ(Ri )σ(m)E (m)

(22)

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

To derive the maximum Sharpe ratio, start from the basicequation

1 = E (mRi )

= E (m)E (Ri ) + ρm,Ri σ(Ri )σ(m)

where we used the de�nition of covariance.

we know that 1 = E (m)Rf , so we can subtract this fromboth sides and re-arrange:

E (Ri ) = Rf � ρm,Ri σ(Ri )σ(m)E (m)

(21)

notice that a correlation coe¢ cient cannot be greater thanunity, so this gives us a bound on asset prices:

jE (Ri ) = Rf j � σ(Ri )σ(m)E (m)

(22)

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

To derive the maximum Sharpe ratio, start from the basicequation

1 = E (mRi )

= E (m)E (Ri ) + ρm,Ri σ(Ri )σ(m)

where we used the de�nition of covariance.

we know that 1 = E (m)Rf , so we can subtract this fromboth sides and re-arrange:

E (Ri ) = Rf � ρm,Ri σ(Ri )σ(m)E (m)

(21)

notice that a correlation coe¢ cient cannot be greater thanunity, so this gives us a bound on asset prices:

jE (Ri ) = Rf j � σ(Ri )σ(m)E (m)

(22)

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

From (22) we derive the maximum Sharpe ratio, sinceρm,Rp = 1:

E (RP )� Rfσ(RP )

=σ(m)E (m)

= σ(m)Rf (23)

this is an important result: the maximum Sharpe ratio �the slope of the capital allocation line � is governed by thevolatility of m, the stochastic discount factor.

What does σ(m), depend on? We know that

m = βu 0(ct+1)u 0(ct )

, so it depends on the volatility ofconsumption, and on how curved is the utility function �that is how little we like irregular consumption.If consumption growth is volatile we have a higher equitypremium �a larger Sharpe ratio

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

From (22) we derive the maximum Sharpe ratio, sinceρm,Rp = 1:

E (RP )� Rfσ(RP )

=σ(m)E (m)

= σ(m)Rf (23)

this is an important result: the maximum Sharpe ratio �the slope of the capital allocation line � is governed by thevolatility of m, the stochastic discount factor.

What does σ(m), depend on? We know that

m = βu 0(ct+1)u 0(ct )

, so it depends on the volatility ofconsumption, and on how curved is the utility function �that is how little we like irregular consumption.

If consumption growth is volatile we have a higher equitypremium �a larger Sharpe ratio

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

From (22) we derive the maximum Sharpe ratio, sinceρm,Rp = 1:

E (RP )� Rfσ(RP )

=σ(m)E (m)

= σ(m)Rf (23)

this is an important result: the maximum Sharpe ratio �the slope of the capital allocation line � is governed by thevolatility of m, the stochastic discount factor.

What does σ(m), depend on? We know that

m = βu 0(ct+1)u 0(ct )

, so it depends on the volatility ofconsumption, and on how curved is the utility function �that is how little we like irregular consumption.If consumption growth is volatile we have a higher equitypremium �a larger Sharpe ratio

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

if we had power utility then

E (RP )� Rfσ(RP )

=

σ

��ct+1ct

��γ�

E��

ct+1ct

��γ�

and if consumption growth was lognormal

E (RP )� Rfσ(RP )

� γσ(∆ ln c) (24)

Over the last 50 years in the US, real stock returns haveaveraged � 9% with σ = 16%, while the real return on T-billsis about 1%.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

if we had power utility then

E (RP )� Rfσ(RP )

=

σ

��ct+1ct

��γ�

E��

ct+1ct

��γ�

and if consumption growth was lognormal

E (RP )� Rfσ(RP )

� γσ(∆ ln c) (24)

Over the last 50 years in the US, real stock returns haveaveraged � 9% with σ = 16%, while the real return on T-billsis about 1%.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

Using this in (24) we have

E (RP )� Rfσ(RP )

=.09� .01.16

= .5

But consumption growth is rather smooth, with σ � 1%.This implies that γ = 50 to reconcile (24).

This is the equity premium puzzle. Why are the returns toequity so high?

Consumption does not seem volatile enough to explain it,and people do not seem so risk averse (γ = 4 would seempretty high).Very hard to explain, many have tried.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

Using this in (24) we have

E (RP )� Rfσ(RP )

=.09� .01.16

= .5

But consumption growth is rather smooth, with σ � 1%.This implies that γ = 50 to reconcile (24).

This is the equity premium puzzle. Why are the returns toequity so high?

Consumption does not seem volatile enough to explain it,and people do not seem so risk averse (γ = 4 would seempretty high).Very hard to explain, many have tried.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

Using this in (24) we have

E (RP )� Rfσ(RP )

=.09� .01.16

= .5

But consumption growth is rather smooth, with σ � 1%.This implies that γ = 50 to reconcile (24).

This is the equity premium puzzle. Why are the returns toequity so high?

Consumption does not seem volatile enough to explain it,and people do not seem so risk averse (γ = 4 would seempretty high).

Very hard to explain, many have tried.

Lecture Note

Asset Prices

CAPM

CAPM andBetaBeta

BasicFramework

StochasticDiscountFactorSome uses of theBasic PriceEquationSystematic Risk

EquityPremium

Risk Aversionand AssetPrices

APT

Equity Premium

Using this in (24) we have

E (RP )� Rfσ(RP )

=.09� .01.16

= .5

But consumption growth is rather smooth, with σ � 1%.This implies that γ = 50 to reconcile (24).

This is the equity premium puzzle. Why are the returns toequity so high?

Consumption does not seem volatile enough to explain it,and people do not seem so risk averse (γ = 4 would seempretty high).Very hard to explain, many have tried.