Lecture 9: Stochastic Discount Factor - Modern

Readings:

Cochrane – Chapters 1 – 4 & 9 (Preparation for the empirical asset pricing class)
1
Classic Issues in Finance from the Stochastic Discount Factor (“SDF”) Perspective
The basic pricing equation v  E[mY ] or 1  E[mZ ] allows us to illustrate many of the classic
lessons of finance in a simple way. (For today’s discussion assume there exists a riskless asset.)
(1)
The time value of money or the risk free rate
1
Demonstrating this is simple.
E[m]
If a risk free asset exists – m must price it.
1
since Rf is a constant.
1  E[mR f ]  R f 
E[m]
Rf 
Let’s use this relation to consider the economics behind risk free rates:
1 1
Assume u (c) 
c power utility for illustration (log utility as   1) so u (c)  c  .
1 
From earlier discussions we know if there is no uncertainty about ct+1, then
1
=
Rf 
E[m]
So:
(A)
1

u (ct 1 )
u (ct )
=
1
ct1
 
ct
1 c 
=   t 1 
  ct 

Real interest rates are higher when people are impatient (  is low). It takes a high
real rate to get impatient people to save rather than consume. The impatience
parameter can reasonably be treated as an exogenous parameter of the economy.
(B)
Real rates are high when consumption growth is high. If consumption growth is high,
it requires a high rate of interest to induce investors to consume less now in return for
more consumption later. This interpretation considers Rf to be determined by the
consumption pattern which follows production (as in Lucas).
Alternatively: If Rf is high, consumers save more, increasing consumption growth.
In this interpretation Rf is looked at as being determined by the production technology
and that consumption adjusts to it. Actually, all three are endogenously determined.
(C)
Real rates are more sensitive to changes in consumption growth when  is large:
R f

 
ct 1
ct



c 
  t 1 
 ct 
 1
 Increasing in 
If  is large, then utility is more concave (for this utility function,  measures both
relative risk aversion – aversion to consumption changing across states of nature and
intertemporal substitution – aversion to consumption changing across time periods).
With more concave utility, the investor wants very badly to maintain a smooth
2
consumption stream across time and across states of nature. This implies investors
are less willing to change the consumption stream across time in response to interest
rate incentives. Thus, it takes a bigger change in Rf to induce the investor to a given
consumption growth. Said differently, consumption is less sensitive to interest rates
when the desire for a smooth consumption path is high.
Now introduce uncertainty: Let r f  Ln( R f ),   e  , Ln(ct 1 )  Ln(ct 1 )  Ln(ct )
Assume consumption growth is lognormally distributed.






1
R f  E


  ct 1  
 
  
  ct  
The combination of power utility and a lognormal distribution implies this expectation can be
written:
r f    Et (Ln(ct 1 )) 
2
 t2 (Ln(ct 1 ))
2
2
1
This comes from: if z is normal, E (e z )  e E ( z ) 2  ( z ) . Note that
 ct 1 


 ct 

e
c 
Ln  t 1 
 ct 

e
c 
Ln  t 1 
 ct 
and that since
c 
ct 1
is lognormal Ln t 1  is normal.
ct
 ct 
So, Rf is high when:
(A) Impatience is high (high  or low )
(B) Ln(ct 1 ) - consumption growth is expected to be high
(C) higher  makes Rf more sensitive to expected consumption growth; Ln(ct 1 )
(D)  2 (Ln(ct 1 )) captures precautionary savings (it’s negative), the impact of uncertainty in
this model. When consumption growth is very volatile, people with power utility are more
concerned with low consumption states than they are pleased by high consumption states so
they save to avoid the dips at sacrifice of current consumption. This is the concavity in
power utility and it implies that people want to save more, driving the risk free rate down.
(2)
Risk Corrections
E[Y ]
 Cov(m, Y )
Rf
Current price equals the expected payoff discounted by Rf, plus a risk correction.
v(Y )  E[mY ]  E[m]E[Y ]  Cov(m, Y ) 
Substituting for m from consumption/investment problem:
E[ y i ,t 1 ] Cov(  u (ct 1 ), y i ,t 1 )
vt ( y i ,t 1 ) 

Rf
u (ct )
Since u is decreasing in ct+1, an asset sells for a lower price if its payoff covaries positively with
future consumption or consumption growth (negatively with marginal utility of consumption).
3
Why? Investors don’t like uncertainty about consumption. Assets that payoff more in states you
are wealthy and less in those you are poor add to the volatility of consumption. Requires a lower
price on this asset for you to hold it – i.e. it provides compensation for the risk or a risk premium.
In terms of returns:
1  E[mzi ]  E[m]E[ z i ]  Cov(m, z i )
so,
E[ z i ]
E[ z i ]  R f  R f Cov(m, z i )
1
 Cov(m, z i )
or,
Rf
E[ z i ]  R f 
Cov(u (ct 1 ), z i )
E[u (ct 1 )]
All assets have an expected return that is the risk free rate plus a premium that is positive for
assets whose returns covary positively with consumption.
Why covariance matters rather than variance…
Remember that an investor really cares about volatility of consumption, not the volatility of
individual assets or even the volatility of the return on her portfolio. Consider what happens to
the volatility of consumption if an investor buys a little more, w, of a payoff y:
 2 (c  wy)   2 (c)  w 2 2 ( y)  2wCov(c, y)
At the margin, i.e. for small w, the change in volatility of consumption comes from the Cov(c, y)
term so the way in which y contributes to what matters is via its covariance with consumption,
not via its own variance.
(3)
Idiosyncratic risk is not priced
One interpretation of what we just saw is that assets with very volatile payoffs or returns need
not have large risk corrections. It is only the portion of payoff that is correlated with the
stochastic discount factor that implies a risk correction is required. The portion of payoff
uncorrelated with m receives no such risk correction, even if this volatility is large. (Where have
we seen that before?)
If Cov(m,y) = 0, then v( y ) 
E[ y ]
, no matter how volatile is y (how large is  2 ( y ) ). Recall
Rf
 2 ( y) has no first order effect on consumption volatility.
For any random payoff y, consider the decomposition: y  proj ( y m)   where proj ( y | m) is
the projection of y on m. This is the portion of y’s volatility that is perfectly correlated with m
(it’s like  m where  is a regression coefficient from a regression of y on m with no intercept).
proj ( y m) 
E[my]
m , further we know that E[ ]  E[ m]  0 by construction
E[ m 2 ]
4
Then, the value of the projection of y on m is equal to the value of y itself.
 m 2 E[my] 
 E[my] 
  E[my]  v( y)


v( proj ( y | m))  v
m   E
2
2
E
[
m
]
E
[
m
]




So,  must have a price of zero: v( )  0 its expectation is zero and it is orthogonal to m.
Note that here the  ’s can be uncorrelated across assets (as is assumed in the APT) or not (as
allowed in the CAPM). But, since this is based on the absence of arbitrage and the APT is as
well, and the CAPM is an equilibrium model, we knew that would follow.
(4)
Expected Return/Beta Representations
Assume there exists a riskless asset:
1  E[mzi ] i
1  E[m]E[ z i ]  Cov(m, z i )

E[ z i ]  R f  R f Cov(m, z i )

Cov(m, z i )   Var (m) 


Var (m)  E[m] 
 R f   i ,m m
 Rf 
where m is the price per unit risk (and a function of Var(m)) and  i,m measures the quantity of
risk for asset i.

c 
To relate this to the underlying variables of interest, recall m    t 1  assuming power
 ct 
Cov(m, z i )   Var (m) 
 with

utility. Performing a Taylor’s series expansion of E[ z i ]  R f 
Var (m)  E[m] 
c 
m    t 1 
 ct 

around consumption growth,
ct 1
, gives:
ct
Cov( z i , c)
c
, c  Var (c) , and c  t 1
Var (c)
ct
In the continuous time limit this approximation becomes precise.
E[ z i ]  R f   i ,c c
where  i ,c 

This says that expected returns increase linearly with an asset’s beta with consumption
growth. This is the consumption CAPM relation. This falls directly out of the power utility
framework.

The price of risk in this case depends upon the risk aversion coefficient of the investor and
the variance of consumption growth (the fundamental risk facing the investor).

The more risk averse are agents or the more risky their consumption, the larger is the
expected return required to induce them to hold risky assets (assets that covary positively
with consumption).
5
Now consider the following: If instead of the SDF (m) being a function of consumption growth
we assume m  a  bz Mkt , then we know that for any asset i:
1  E[m  z i ]  E[az i  bz Mkt z i ]
or,
1  az i  bz i z Mkt  bCov( z i , z Mkt )
or,
1
b

Cov( z i , z Mkt )  R f  bR f Cov( z i , z Mkt )
a  bz Mkt a  bz Mkt
recognizing that R f  1 E [ m ]
zi 
This must hold for all assets i and also for zMkt. So,
z Mkt  R f  bR f Var ( z Mkt )
and,
b 
( z Mkt  R f )
R f Var ( z Mkt )
So,
zi  R f 
Cov( z i , z Mkt )
( z Mkt  R f )
Var ( z Mkt )
So if the SDF is a linear function of the returns on the market portfolio we get our familiar
CAPM pricing relation.
One comment: Cochrane likes to use returns in specifying m since then it has a neat
interpretation. But, if m=a +bzMkt it may not always be positive. He’s a little loose with this – a
stochastic discount factor that is not strictly positive can correctly price the assets it is just not the
strictly positive SDF guaranteed by the absence of arbitrage. If we instead rely on the law of one
price we know there is a SDF, but it is not restricted to be positive. In the absence of arbitrage
there must be a strictly positive SDF ( m( s)  ke azMkt ( s )  0 ) giving a beta pricing representation.
(5)
Mean-Variance Frontier
All assets priced by a SDF must obey:
 (m)
E[ z i ]  R f 
 ( zi )
E[m]
This follows again from:
1  E[mzi ]  E[m]E[ z i ]   m, zi  ( z i ) (m)
 E[ z i ]  R f    m, zi
(using correlation instead of covariance)
 (m)
 ( zi )
E[m]
And since  1    1, the result follows.
6
We can see several things based on this simple relation:
(A)
Since  1    1, means and variances of all assets must lie in the wedge shaped
region bounded by the minimum variance frontier derived in the development of the
CAPM. Thus, the frontier is of general interest without assuming M-V preferences.
E[z]
Slope  E[( mm])
Systematic Risk

 Asset i
Rf
(B)
Idiosyncratic Risk (see note (F) below)
 (z )
Only if   1 does asset i lie on the minimum variance efficient frontier. Thus all
portfolios on the frontier are perfectly correlated with the SDF, m. Returns on the
upper limb have  m , z  1 , so are perfectly negatively correlated with m and thus,
perfectly positively correlated with consumption growth. They are maximally risky
and so demand the highest expected return per unit variance. The converse is true for
assets on the lower limb.
i
(C)
All frontier returns are also perfectly correlated with each other. They are all
perfectly correlated with m. Therefore, we know we can span the return of any
frontier portfolio using any two distinct frontier returns. For example, pick any single
frontier return z MV 1 (not Rf). Note that Rf is also on the frontier. Any other frontier
return zMV2 can be written z MV 2  R f  a( z MV1  R f ) for some constant a.
Show this is true as a homework problem.
(D)
Since each return on the frontier is perfectly correlated with m, we can find constants
a, b, d, e such that for any minimum variance return:
m  a  bz MV
z MV  d  em
and,
What does this mean? It means that any mean-variance efficient portfolio contains all
the pricing information in m. For example, in the CAPM the market contains all the
pricing information in m, which we knew already. Its return is a sufficient statistic
for m or marginal utility. Thus, zMkt can serve as m – requires zMkt is on the frontier.
Given any mean-variance efficient return and the risk free rate, we can find a SDF
that prices all assets, and vice versa. See problems 1-3 in Cochrane.
(E)
Given m, we can also construct a single-beta representation such that expected returns
are expressed in a single-beta model using the return on any mean-variance efficient
portfolio (except Rf):
E[ z i ]  R f   i , MV ( E[ z MV ]  R f )
7
(F)
All asset returns can be decomposed into a “priced” or systematic component and a
“non-priced” or idiosyncratic component. The priced component is perfectly
correlated with m and any frontier return and so this component would “plot on the
frontier” (see picture on previous page). The unpriced component is uncorrelated
with m and generates no expected return or risk adjustment. (Recall the
decomposition we did in our development of the CAPM.)
Note: Assets “inside” the frontier are not “worse” than assets on the frontier. The
frontier and its interior characterize equilibrium asset returns. Rational investors are
happy to hold all assets. You just don’t put all your wealth in an inefficient asset, but
you are happy to put small amounts of wealth in many such assets.
(6) The Slope of the Mean/Standard Deviation Frontier and “The Equity Premium Puzzle”
The ratio of the mean excess return to standard deviation is known as the Sharpe Ratio:
E[ z i ]  R f
Sharpe Ratio 
 ( zi )
This is more interesting and a better indication of performance than mean return alone. For
example, if you borrow at the risk free rate and invest the proceeds into some risky security, you
increase expected return but you don’t increase the Sharpe ratio since  p increases at the same
rate as E[zp].
The slope of the mean/standard deviation frontier is the maximal Sharpe ratio. It tells us how
much more mean return you can get by taking on added (priced) volatility.
Let zMV be a frontier return.
 (m)
From E[ z i ]  R f 
 ( zi )
E[m]
We can see that:
E[ z i ]  R f
 ( m)

  (m) R f for all assets i.
 ( zi )
E[m]
For zMV (that is, for frontier returns), since their correlation with m is one:
E[ z MV ]  R f
 ( m)

  ( m) R f .
MV
E[m]
 (z )
Thus the slope of the frontier is governed by the volatility of m and this slope we know
determines the risk premium.
Consider again the power utility framework: u (c)  c

c 
and so m    t 1 
 ct 

8
  c  
   t 1 
  ct 



MV

E[ z ]  R f


Then:
MV

 (z )
 c  
E  t 1  
 ct  
 (m ) increases if consumption is more volatile or if  is large.
If consumption growth is lognormal, “it can be shown,” using the transformation above, that:
E[ z MV ]  R f
 2 2 ( Ln ( ct 1 ))

e
 1   (Ln(ct 1 ))
 ( z MV )
This shows more directly that the slope of the mean/standard deviation frontier is higher if
consumption growth is more volatile or if risk aversion is higher.
In post-war data (50 years) for the U.S., E[ z Mkt ]  9% ,  ( z Mkt )  16% , and R f  1% (all in real
terms).
Aggregate consumption growth has had a mean about equal to 1% and a standard deviation of
about 1%. We can plug these values into the above equation to get:
9%  1%
 0.50   (.01)
16%
This implies a risk aversion coefficient of roughly 50!
This is an order of magnitude too high to be believable. The interpretation is that consumption is
not volatile enough to explain asset returns unless investors have risk aversion coefficients much
larger than we think they are. This is the point of the famous Mehra-Prescott paper.
Possible Explanations:
1. People are much more risk averse than we think.
2. Stock returns are largely a result of unexpected good fortune over the last 50 years and
are not indicative of expectations.
3. There may be real problems with measures of consumption.
4. Something is deeply wrong with the unconditional model.
This question/issue has been a central focus of the asset pricing literature for the last 15-20 years.
(7) Exact Factor Structure in the APT
If returns are given by a 2-factor model (for example):
z i  z i   i1 f1   i 2 f 2
The law of one price implies there is a discount factor SDF linear in the factors that price the
assets:
m  a  bf1  df 2 .
9
It is also true that if returns are given as above, we can use 1=E[mz] to derive the beta pricing
relation:
z i  z i   i1 f1   i 2 f 2
The law of one price implies we can use m to price both sides of this equation:
1  E[ z i m]  z i m   i1 E[mf1 ]   i 2 E[mf 2 ]
zi
 R f  R f  i1 E[mf1 ]  R f  i 2 E[mf 2 ]
 R f   i1 ( R f  E[mf1 ])   i 2 ( R f  E[mf 2 ])
 R f   i11   i 2 2
Now, form portfolios with returns:
z 1  z 1  1  f1  0  f 2
z 2  z 2  0  f1  1  f 2
The above pricing equation must hold for these portfolios, so:
so,
and,
z 1  R f  1
1  z 1  R f
so,
z 2  R f  2
2  z 2  R f
So, this representation takes us a ways towards developing the APT. See Cochrane Ch. 9.
(8) Random Walks and Time Varying Expected Returns
The predictability of future returns has become an issue of increasing interest. If we look across
time rather than across assets we can examine this issue. From the basic first-order condition:
pt u (ct )  Et [u (ct 1 )( pt 1  d t 1 )]
Now, if investors are risk neutral (or if consumption is constant), dt+1=0, and   1 (the time
horizon is very short), then the FOC becomes:
pt  Et [ pt 1 ]
or, we can write:
pt 1  pt   t 1
So, prices follow a martingale (or a random walk if we further assume that  2 ( t 1 ) is constant
across time) and expected returns should be constant across time.
The basic FOC really tells us that prices (adjusting for dividends) scaled by marginal utility
follow a martingale. For short horizons prices should be close to a martingale since consumption
and risk aversion don’t vary much over a day. Returns over longer horizons may be predictable.
For long horizons we write our expected returns relation:
Cov(mt 1 , z t 1 )
E[ z t 1 ]  R ft  
E[mt 1 ]
 (m )
  t t 1  t ( z t 1 )  (mt 1 , z t 1 )
Et [mt 1 ]
10
  t  t (Ln(ct 1 )) t ( z t 1 )  t (mt 1 , z t 1 )
Now, looking at:
E[ z t 1 ]  R ft   t  t (Ln(ct 1 )) t ( z t 1 )  t (mt 1 , z t 1 )
we see that variation in E[ z t 1 ]  R ft could come from variation in  t ( z t 1 ) . However, this is not
borne out in the data, as variables correlated with mean changes are not correlated with variance
changes (and vice versa).
Predictable excess returns can then come from changes in aggregate risk -  t (Ln(ct 1 )) - or risk
aversion -  t - or perhaps from  t (mt 1 , z t 1 )
The literature doesn’t consider  t (mt 1 , z t 1 ) much. But, it is natural to think that  t (Ln(ct 1 ))
and  t change over the business cycle which is the horizon over which returns are relatively
predictable (which is not short).
(9) Present Value Statement:
Price can be related to a stream of cash flows rather than just next period’s price/cash flow.
We can use a longer term objective function:
E t  j  0  j u (c t  j )

Now, suppose that you can purchase the stream {dt+j} for pt, the FOC gives us the pricing formula
directly (developed in the Lucas paper):
pt  Et  j 1 

j
u (ct  j )
u (ct )
d t  j  Et  j 1 mt ,t  j d t  j

Now by noticing that this holds at t and at t+1 and noting that multiplying pt+1 by 
u (ct 1 )
puts the
u (ct )
equation for pt+1 in terms of mt,t+j rather than mt+1,t+1+j. We can then get to:
pt  Et [mt 1 (d t 1  pt 1 )]
from the multi-period relation.
So, the two date solution and the multi-period solution are equivalent.
From the multi-period relation we can write:
pt   j 1

Et (d t  j )
R f t ,t  j
  j 1 Covt (d t  j , mt ,t  j )

and the same message about pricing risk comes out.
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