Week 6: Asset Pricing
Guanglian Hu
FINC3017
Overview
▶ State Space Model
▶ Consumption-Based Asset Pricing: Theory
State Space Model
Suppose next period there are S states of the world (state of nature),
which are labeled by s, s = 1, 2, 3, ...S, representing different economic
conditions.
One of the most important frameworks for modeling investments under
uncertainty, also known as the time-state model of Arrow (1964) and
Debreu (1959)
State Space Model
▶ State s occurs with probability πs , where 0 < πs < 1 and∑S
1 πs = 1. [π1, π2, π3,...,πS ] are physical probabilities.
▶ Consider Arrow-Debreu securities that pay off $1 if state of nature s
occurs next period and zero otherwise. Suppose we have one AD
security for each state of the world.
▶ For example, the AD security for state 1 has the following payoff
vector 
1
0
0
.
.
.
0

▶ Let ws be today’s price of such claim. Basic arbitrage arguments
dictate that 0 < ws < 1. The prices of AD securities [w1, w2,
w3,...,wS ] are also referred to as state prices.
State Space Model
▶ Once we know the prices of Arrow Debreu securities, we will be able
to price any risky asset with a random payoff Y :
Y =

y1
y2
.
.
yS

▶ Note that the random payoff Y can be replicated by buying y1 units
of AD claim for state 1, y2 units of AD claim for state 2, y3 units of
AD claim for state 3.......
y1

1
0
0
.
.
.
0

+ y2

0
1
0
.
.
.
0

+ .....+ yS

0
0
0
.
.
.
1

State Space Model
▶ The current price of Y , by arbitrage, must be given by adding up
the values of its component pieces:
p =
S∑
1
wsys (1)
▶ We can also use AD securities to infer the price of a risk-less bond
(B) that pay off $1 regardless of which state occurs.
B =
S∑
1
ws
State Prices and SDF
▶ Rewrite equation (1),
p =
S∑
1
πs
ws
πs
ys (2)
▶ Denote wsπs as Ms , we have
p =
S∑
1
πsMsys = E (Msys) (3)
▶ Ms is called stochastic discount factor (SDF) or pricing kernel and
equal to state prices divided by physical probabilities. The value of
an asset is equal to the expected value of discounted future cash
flow.
State Prices and Risk Neutral Probabilities
▶ Let
π˜s =
ws
B (4)
▶ Note that π˜s have all of the properties of probabilities: 0 < π˜s < 1
and
∑S
1 π˜s = 1.
▶ Using equation (4) and equation (1) on the previous slide, we can
rewrite the price of Y as
p = B
S∑
1
ws
B ys = B
S∑
1
π˜sys
State Prices and Risk Neutral Probabilities
▶ Lastly, using B = 11+rf , we have
p = 11+ rf
S∑
1
π˜sys
▶ The value of a random cash flow Y is calculated as the expected
value of cash flow under the risk neutral probabilities, discounted by
the risk-free rate.
▶ Note that
p ̸= 11+ rf
S∑
1
πsys
Summary
▶ We consider the state space model of Arrow (1964) and
Debreu (1959) in which there exists a set of elementary
contingent claims (AD securities), each paying one dollar in
one specific state of nature and nothing in any other states.
▶ In this set-up, we can break the payoffs or cash flows of any
risky asset down ’state-by-state’, and then pricing it as a
bundle of AD securities.
▶ Valuation can be done under both physical and risk neutral
probability measure.
▶ Stochastic discount factor equals state prices divided by
physical probabilities
▶ Risk neutral probabilities are state prices (the prices of AD
securities) divided by the price of the risk-free asset.
Inferring State Prices From Option Prices
▶ State prices can be inferred from prices of options with different
strikes.
▶ Suppose the state of nature is summarized by the value of the
market portfolio which has a discrete probability distribution with
possible values of: M = $1.00, $2.00, . . . , $N.
▶ Denote the vector of payoffs of a European call option on the
market with a strike price of X as C(X); its price today will be
denoted as c(X)
Market Portfolio C(0) C(1) C(2)
M=1 1 0 0
M=2 2 1 0
M=3 3 2 1
. . . .
. . . .
. . . .
M=N N N-1 N-2
Inferring State Prices from Option Prices
▶ The payoffs of AD securities can be replicated by combining call
options on the market portfolio with various strike prices.
▶ For example, the claim having a payoff of $1.00 only if M(T) = $1
may be constructed as [C(0) - C(1)] - [C(1) - C(2)]
1
1
1
.
.
.
1

−

0
1
1
.
.
.
1

=

1
0
0
.
.
.
0

▶ In general, replicating the claim giving $1 only if the market is M
consists of one long call with X =M-1,one long call with X=M+1,
and two short calls with X=M (if you recall, this is butterfly spread).
Inferring State Prices from Option Prices: Example
▶ For example, if N = 3 (only three states)
C(0) C(1) C(2)
M=1 1 0 0
M=2 2 1 0
M=3 3 2 1
▶ Suppose the prices of calls are c(0) = $1.7, c(1) = $0.8, and c(2) = $0.1
▶ Then the respective state prices are:
▶ P(M = 1) = $0.2, c(0)+c(2)-2c(1)
▶ P(M = 2)=$0.6, c(1)-2c(2)
▶ P(M = 3) = $0.1, c(2)
▶ Also, the price of a riskless discount bond paying $1.00 would be $0.2 +
$0.6 + $0.1 = $0.9
Consumption Based Asset Pricing
Consumption-Based Asset Pricing
▶ Equity Risk Premium Puzzle
▶ Risk-Free Rate Puzzle
▶ Hansen-Jagannathan Bound
▶ Recent Development
Intertemporal Choice Problem
▶ Consumption-based asset pricing models start with the
intertemporal choice problem of an investor, who wants to maximize
the expected value of the life time utility:
max Et [
∞∑
j=0
βjU(Ct+j )]
where β is the time discount factor and U is a time-separable utility
function.
▶ This investor can trade freely in some asset i and obtain a gross
simple rate of return 1+ Ri ,t+1 on the asset held from time t to
time t + 1.
▶ First order condition or Euler equation describing the investor’s
optimal consumption and and portfolio plan is:
U ′(Ct) = Et [βU ′(Ct+1)(1+ Ri ,t+1)]
Euler Equation
▶ How to interpret the Euler equation?
U ′(Ct) = Et [βU ′(Ct+1)(1+ Ri ,t+1)] (5)
▶ The left hand side of equation (5) is the marginal utility cost
of consuming one dollar less at time t.
▶ The right hand side is the expected marginal utility benefit
from investing the dollar in asset i at time t, selling it at time
t + 1 for 1+ Ri ,t+1 dollars, and consuming the proceeds.
▶ The marginal cost equals the marginal benefit.
The Fundamental Asset Pricing Equation
▶ The first order condition of the optimal consumption and
portfolio choice problem gives the fundamental equation in
asset pricing.
▶ Dividing equation (5) by U ′(Ct) yields:
1 = Et [β
U ′(Ct+1)
U ′(Ct)
(1+ Ri ,t+1)]
▶ Denoting Mt+1 = β U
′(Ct+1)
U′(Ct ) , we obtain the fundamental
equation in asset pricing:
1 = Et [Mt+1(1+ Ri ,t+1)]
▶ Mt+1 is the intertemporal marginal rate of substitution of the
investor, also known as the stochastic discount factor (SDF),
pricing kernel, or simply marginal utility.
The Fundamental Asset Pricing Equation
▶ Mt+1 is a random variable and always positive.
▶ High Mt+1 corresponds to low consumption. Marginal utility
is high when the level of the consumption is low.
▶ While the expected returns (e.g., Et [1+ Ri ,t+1]) can vary
across time and assets, the expected discounted return should
always be the same, 1.
1 = Et [Mt+1(1+ Ri ,t+1)]
The Fundamental Asset Pricing Equation
▶ The above derivation for 1 = Et [Mt+1(1+ Ri ,t+1)] assumes
the existence of an investor maximizing a time-separable
utility function, but in fact the equation holds more generally.
▶ The existence of a positive stochastic discount factor is
guaranteed by the absence of arbitrage opportunities in
markets.
▶ Different asset pricing models use different Mt+1. Think of
Mt+1 as an index for bad times.
The Fundamental Asset Pricing Equation: Implications
▶ We just derived the fundamental equation in asset pricing:
1 = Et [Mt+1(1+ Ri ,t+1)] (6)
▶ One can manipulate this equation to get a pricing formula.
Note that
1+ Ri ,t+1 =
Xi ,t+1
Pi ,t
where Xi ,t+1 is the payoff of the asset at time t + 1.
▶ This implies:
1 = Et [Mt+1
Xi ,t+1
Pi ,t
]⇒ Pi ,t = Et [Mt+1Xi ,t+1]
▶ The value of an asset is equal to the expected value of
discounted future payoffs (by Mt+1).
The Fundamental Asset Pricing Equation: Implications
▶ Explore the implications of the fundamental equation in the return
space.
▶ Using E (XY ) = E (X )E (Y ) + Cov(X ,Y ),
Et [Mt+1(1+Ri ,t+1)] = Et [Mt+1]Et [(1+Ri ,t+1)]+Covt [Ri ,t+1,Mt+1]
▶ Substituting into (6) yields:
1 = Et [Mt+1]Et [(1+ Ri ,t+1)] + Covt [Ri ,t+1,Mt+1]
▶ Rearranging gives:
Et [(1+ Ri ,t+1)] =
1− Covt [Ri ,t+1,Mt+1]
Et [Mt+1]
(7)
▶ An asset with a high expected return must have low covariance with
the stochastic discount factor. Such an asset tends to have low
returns when investors have high marginal utility. It is risky in that
it fails to deliver wealth in bad times when wealth is most valuable
to investors. Investors therefore demand a higher expected return to
hold it.
The Fundamental Asset Pricing Equation: Implications
▶ Equation (7) must hold for any asset, including a riskless asset
whose return has zero covariance with the stochastic discount
factor (or any other random variable).
1+ Rf ,t+1 =
1
Et [Mt+1]
(8)
▶ The above equation can be used to rewrite equation (7):
Et [(1+ Ri ,t+1)] = (1+ Rf ,t+1)(1− Covt [Ri ,t+1,Mt+1]) (9)
▶ An asset with a negative (positive) covariance with the
stochastic discount factor will earn an expected return that is
higher (lower) than the risk-free rate
Expected Return - Beta Representation
▶ The fundamental equation can be rewritten as a CAPM-type
expression.
▶ Start with
1 = Et [Mt+1]Et [(1+ Ri ,t+1)] + Covt [Ri ,t+1,Mt+1]
▶ Dividing both sides by Et [Mt+1] and using 1+ Rf ,t+1 = 1Et [Mt+1] :
Et [(1+ Ri ,t+1)]− (1+ Rf ,t+1) = −Covt [Ri ,t+1,Mt+1]Et [Mt+1]
▶ It follows
Et [Ri ,t+1]− Rf ,t+1 = Covt [Ri ,t+1,Mt+1]Vart [Mt+1] (−
Vart(Mt+1)
Et [Mt+1]
)
Expected Return - Beta Representation
▶ Equivalently, we have
Et [Ri ,t+1]− Rf ,t+1 = βi ,tλt
where
βi ,t =
Covt [Ri ,t+1,Mt+1]
Vart [Mt+1]
; λt = −Vart(Mt+1)Et [Mt+1]
▶ The expected return should be proportional to beta in a regression
of that returns on the SDF M.
▶ λt is price of risk and βi ,t is quantity of risk
▶ Note that λt is the same for all assets while the βi ,t varies from
asset to asset
Imposing Log-normality
▶ Assume that the joint conditional distribution of asset returns and
the stochastic discount factor is lognormal and homoskedastic
▶ When a random variable X is conditionally lognormally distributed,
it has the following convenient property that
log EtX = Et logX +
1
2Vart logX
▶ With joint conditional lognormality and homoskedasticity of asset
returns and consumption, we can take logs of equation (6):
0 = Et ri ,t+1 + Etmt+1 +
1
2 (σ
2
i + σ
2
m + 2σi ,m) (10)
where mt+1 = log(Mt+1), ri ,t+1 = log(1+ Ri ,t+1), σ2i is the
unconditional variance of log returns, σ2m is the unconditional
variance of the log stochastic discount factor, and σi ,m is the
unconditional covariance between log returns and stochastic
discount factor
Imposing Log-normality
▶ Consider the log return on the risk-free asset:
rf ,t+1 = −Etmt+1 − σ
2
m
2 (11)
which is the log counterpart of equation (8)
▶ Subtracting equation (11) from equation (10) yields an expression
for the expected excess return of a risky asset over the riskless rate:
Et [ri ,t+1]− rf ,t+1 +
σ2i
2 = −σi ,m (12)
which is the log counterpart of equation (9)
▶ The right hand side of equation (12) says that the risk premium is
determined by the negative of the covariance of the asset with the
stochastic discount factor.
▶ An asset with a negative covariance with the stochastic discount
factor must have higher expected returns.
Jensen’s Inequality
Et [ri ,t+1]− rf ,t+1 + σ
2
i
2 = −σi ,m
▶ What is with the σ
2
i
2 term?
▶ σ2i2 arises from the fact that we are describing expectations of log
return.
log Et [1+ Ri ,t+1] = Et [ri ,t+1] +
σ2i
2
▶ Two ways to write equity risk premium equation:
Et [ri ,t+1]− rf ,t+1 +
σ2i
2 = −σi ,m
log Et [1+ Ri ,t+1]− log(1+ Rf ,t+1) = −σi ,m
International Stock Market Data
International Consumption and Dividend Data
Data Summary
▶ Stock markets have delivered high average returns with high
standard deviations
▶ Short term debt (risk-free asset) has delivered low returns
with low standard deviation
▶ Consumption growth is smooth with low standard deviation
Consumption Based Asset Pricing with Power Utility
▶ Assume that there is a representative agent who maximizes a
time-separable power utility function defined over aggregate
consumption Ct :
U(Ct) =
C1−γt − 1
1− γ
where γ is the coefficient of relative risk aversion. With constant
relative risk aversion(CRRA), fraction allocated to risky asset is
independent of wealth.
▶ Power utility implies:
U ′(Ct) = C−γt ; Mt+1 = β
U ′(Ct+1)
U ′(Ct)
= β(
Ct+1
Ct
)−γ ;
▶ The log of the stochastic discount factor is
mt+1 = log β − γ∆ct+1
where ∆ct+1 = ct+1 − ct and ct = log(Ct).
Consumption Based Asset Pricing with Power Utility
▶ Recall from equation (10)
0 = Etri ,t+1 + Etmt+1 +
1
2 (σ
2
i + σ
2
m + 2σi ,m)
▶ With power utility, mt+1 = log β − γ∆ct+1 and the above
equation becomes:
0 = Etri ,t+1 + log β − γEt∆ct+1 + 12 (σ
2
i + γ
2σ2c − 2γσi ,c)
▶ σ2c is the unconditional variance of log consumption growth
(∆ct+1), and σi ,c is the unconditional covariance between log
stock returns and consumption growth Cov(ri ,t+1,∆ct+1)
Consumption Based Asset Pricing with Power Utility
▶ The risk-free rate in equation (11) now becomes:
rf ,t+1 = − log β + γEt∆ct+1 − γ
2σ2c
2 (13)
▶ Risk-free rate is high when β is low (investors are impatient)
▶ Risk-free rate is linear in expected consumption growth, with
slope coefficient equal to the coefficient of relative risk
aversion.
▶ The conditional variance of consumption growth has a
negative effect on the riskfree rate which can be interpreted as
a precautionary savings effect.
Consumption Based Asset Pricing with Power Utility
▶ The equity risk premium in equation (12) now becomes:
Et [ri ,t+1]− rf ,t+1 + σ
2
i
2 = γσi ,c (14)
▶ Equation (14) says that the risk premium on any asset is the
coefficient of relative risk aversion times the covariance of the
asset returns with consumption growth
▶ Intuitively, an asset with a high consumption covariance (e..g,
σi ,c) tends to have low returns when consumption is low, that
is, when the marginal utility of consumption is high. Such an
asset is risky and commands a large risk premium.
The Equity Risk Premium Puzzle
The Equity Risk Premium Puzzle
▶ RRA(1) uses equation (14) to estimate the risk aversion, dividing
the adjusted average excess return (aere) by the estimated
covariance cov(ere ,∆c).
▶ Using US as an example, RRA(1)= 0.08071/0.0003354=240.
Please note that returns and standard deviations are reported in
percent so
cov(ere ,∆c) = σ(ere) ∗σ(∆C) ∗ρ(ere ,∆c) = 0.15271∗0.01071∗0.205
▶ RRA(2) assumes that the correlation is equal to 1 to calculate the
implied risk aversion (RRA(2)= 0.080710.15271∗0.01071∗1 = 49). It indicates
the extent to which the equity premium puzzle arises from the
smoothness of consumption rather than the low correlation between
consumption and stock returns.
▶ Most economists believe risk aversion γ should be less than 10.
▶ Power utility model can only fit the equity premium if the coefficient
of relative risk aversion is very large.
The Risk-Free Rate Puzzle
▶ One response to the equity premium puzzle is to consider
larger values for the coefficient of relative risk aversion.
▶ Recall equation (13) shows that the riskless interest rate is
rf ,t+1 = − log β + γEt∆ct+1 − γ
2σ2c
2
▶ High values of γ would imply high values of γEt∆ct+1. This
can be reconciled with low interest rates only if the time
discount factor β is close to or even greater than one,
corresponding to a low or even negative rate of time
preference.
Hansen-Jagannathan Bound
▶ Recall from equation (12) the expected excess return on any risky
asset is given by
Et [ri ,t+1]− rf ,t+1 +
σ2i
2 = −σi ,m
▶ Expanding the covariance term,
σi ,m = σiσmρi ,m
where ρi ,m is the correlation between asset return and stochastic
discount factor.
▶ Since ρi ,m ≥ −1, −σiσmρi ,m ≤ σiσm, which then implies
−σi ,m ≤ σiσm.
Hansen-Jagannathan Bound
▶ A little manipulation yields the Hansen-Jagannathan Bound
σm ≥
Et [ri ,t+1]− rf ,t+1 + σ
2
i
2
σi
(15)
▶ Equation (15) says that the standard deviation of the log
stochastic discount factor must be greater than the Sharpe
ratio for any arbitrary asset i , that is, it must be greater than
the maximum possible Sharpe ratio obtainable in asset
markets.
Hansen-Jagannathan Bound
▶ The Sharpe ratio of the market portfolio is approximately 50% on
an annual basis.
▶ This means that the standard deviation of the stochastic discount
factor must be equal to or greater than 50%.
▶ Recall with power utility, the (log) of the pricing kernel is given by
mt+1 = log β − γ∆ct+1 ⇒ σ(m) = γσ(∆ct+1)
▶ In the data, σ(∆ct+1) is about 1%, which means we need a very
large risk aversion coefficient.
Revisiting the Equity Risk Premium Puzzle
Another way to look at the equity risk premium puzzle is that the stochastic
discount factor implied from the standard consumption-based asset pricing
models is not volatile enough.
Extensions
The second generation of consumption based asset pricing models,
which depart from log normality and power utility includes:
▶ Long-run risks model: Bansal and Yaron (2004)
▶ Habit model: Campbell and Cochrane (1999)
▶ Rare disaster model: Rietz (1988), Barro (2006), and Wachter
(2013).
▶ The recent models have more free parameters and therefore
fit the data better.