Lecture 2: Stochastic Discount Factor
Simon Gilchrist
Boston Univerity and NBER
EC 745
Fall, 2013
Stochastic Discount Factor (SDF)
A stochastic discount factor is a stochastic process {Mt,t+s }
such that for any security with payoff xt+1 at time t + 1 the price
of that security at time t is
Pt = Et Mt,t+1 xt+1
Equivalently
Et Mt,t+1 Rt+1
for
Rt+1 =
xt+1
Pt
Ex: representative agent
Mt+1 = β
u0 (Ct+1 )
u0 (Ct )
Issues to think about:
Heterogeneous agenst with complete vs incomplete markets
Hansen-Jagannathan volatility bounds (JPE, 1991).
SDF must have (roughly) permanent innovations (Alvarez and
Jermann, Econometrica 2004)
Basic properties of SDF:
Risk free rate:
f
Rt+1
= Et Mt+1
CAPM – SDF is linear function of market return:
m
log Mt+1 = a − bRt+1
CCAPM – SDF is a function of consumption growth:
log Mt+1 = log β − γ∆ log Ct+1
Compensation for risk:
f
e
Et Rt+1
− Rt+1
=
where
σt (Mt+1 )
Et Mt+1
σt (Mt+1 )
e
e
corrt Mt+1 , Rt+1
σt Rt+1
Et Mt+1
may be interpreted as the market price of risk.
Constructing SDF under complete markets:
One period world, s = 1 : S possible states with probability πs .
Let qs be the state-contingent price – price of an Arrow-Debreu
security that payoffs one unit in state s. Then the price of a
security with payoffs {ds }Ss=1 is
S
X
p ({d}) =
s=1
S
X
=
s=1
qs ds
πs
qs
ds
πs
then we have defined the SDF as
ms =
qs
πs
so that
p ({d}) =
S
X
πs ms ds
s=1
= Ems ds
Incomplete markets: an example.
One period economy with N agents
There is uncertainty about the state s
Trade occurs before uncertainty is resolved.
Securities: L securities with prices ql and payoffs dls in state s.
Agents problem:
Max
max U (ci1 , ..ciS )
subject to
cis = yis +
L
X
ql θil ≤ wi
i=1
where yis is the endowment in state s, θil is number of shares in
security l purchased by agent i and wi is initial wealth of agent i.
FOC:
S
X
∂ui
s=1
∂cs
dls = ql λi
where λi is agent i’s Lagrange multiplier on the wealth
constraint.
Equilibrium:
A price vector q = {ql } and shares {θil }i,l such that each agent
maximizes utility and markets clear:
L
X
θil = 1,
l = 1..L
i=1
SDF: Consider any agent with λi > 0 then the Euler equation is
ql =
S
X
mis dls
s=1
where
mis =
∂ui
∂cs
λi
We can use the marginal rate of substitution of any consumer
who is unconstrained as an SDF. Agents may not be
unconstrained in each state but as long as there are no arbitrage
opportunities there exists an SDF.
Incomplete markets: no arbitrage pricing.
The system of securities characterized by {dls } and {ql } is
arbitrage free if there is no vector of portfolio choices {θl } such
that both
L
X
ql θ l ≤ 0
l=1
and for all s = 1..S
L
X
dls θl ≥ 0
l=1
> 0 for some state s.
In word: there does not exist a portfolio with a zero (or negative)
price that gives positive payoffs in all states of the world.
Implications
If prices {ql } result from a competitive equilibrium then there
will be no arbitrage – otherwise demand an infinite amount of
the security.
More generally we can use no arbitrage theory to place
restrictions on prices. Example if asset x pays more dividends
than asset y in all states then the price of asset x must be
(weakly) greater than the price of asset y.
Assets that are redundant, i.e. can be replicated or “spanned” by
other assets can be priced even if markets are incomplete.
Example
Example: S=6,l=3


121000
D= 131000 
010000
Since the third asset is the first minus the second it can be priced.
Black and Scholes – under NAO, options can be priced as linear
combinations of equity and debt payouts.
Some general results on SDF:
The law of one price holds iff there is (at least) one stochastic
discount factor.
LOOP implies prices are linear functions of payoffs:
P (x + y) = P (x) + P (y) . This implies that for a security that
payoffs x
X
P (x) =
ws xs
Define
ms =
then
P (x) =
ws
πs
X
πs ms xs
s
so that ms is an SDF. Converse is obvious.
Some general results on SDF:
There are no arbitrage opportunities iff there exists (at least) one
strictly positive SDF.
NAO: if x ≥ 0, p(x) ≥ 0 and if xs > 0 for some s then p (x) > 0,
use separating-hyperplane argument to argue that SDF exists.
Some general results on SDF:
The stochastic discount factor is unique iff markets are complete.
Suppose markets are complete then there is an asset that pays off
one unit in state s where the price of that asset ps satisfies:
ps = ms πs
This is true for any SDF:
ps = m∗s πs
therefore
m∗s = ms
Suppose the SDF ms is unique, assume that the market is
incomplete. Then there is a state j which is non-tradeable. Define
m∗s
= ms if s 6= j
= ms + 1 if s = j
The SDF m∗s will price all tradeable securities hence ms is not
unique.
Hansen-Jagannathan Bounds
Consider any risky return R and risk-free return Rf then
E (mR) = 1
E (m) Rf
= 1
E (mR) = E (m) Rf
This implies:
cov (m, R) = E (mR) − E (m) E (R)
= E (m) Rf − E (m) E (R)
Hansen-Jagannathan Bounds
Now use bound (−1 ≤ corr(x, y) ≤ 1):
−σ (x) σ (y) ≤ cov (x, y) ≤ σ (x) σ (y)
to obtain
−σ (m) σ (R) ≤ E (m) Rf − E (m) E (R) ≤ σ (m) σ (R)
Reverse inequality to obtain:
−
σ (m)
E (R) − Rf
σ (m)
≤
≤
E (m)
σ (R)
E (m)
Comments:
E(R)−Rf
σ(R)
σ(m)
is the Sharpe ratio and E(m)
is the market price of risk.
The Sharpe ratio for the market is 0.06/0.17=35%.
This applies to any asset – Sharpe ratios can easily be on the
order of 100% so the price of risk must be very high.
To obtain tight bounds, consider a portfolio with very high sharp
ratio.
Under complete markets, the price of risk is the slope of the
mean-variance frontier – this defines parabola in E(m), σ(m)
space.
Price of risk under CRRA
For CRRA and random-walk consumption:
∆ ln ct = µ + εt
where
εt ∼ N (0, σε2 )
Results from log-normal:
Ct+1
1 2
E(
= exp µ + σε
Ct
2
2
Ct+1
2
var(
) =
eσε − 1 e2µ+σε
Ct
Price of risk under CRRA
These results imply:
σε2
E (m) = β exp γ −µ + γ
2
and
1/2
σ (m)
= exp σε2 γ 2 − 1
E (m)
where γ is CRRA coefficient.
Quarterly calibration: σvarepsilon = 0.036/4
If γ
If γ
If γ
If γ
= 1 price of risk = 0.01
= 10 price of risk = 0.09
= 20 price of risk = 0.18
= 50 price of risk = 0.47