Download Ch. 2. Asset Pricing Theory (721383S)

Ch. 2. Asset Pricing Theory (721383S)
Juha Joenväärä
University of Oulu
March 2014
Abstract
This chapter introduces the modern asset pricing theory based on the stochastic discount factor
approach. The main idea is that asset prices should be equal to discounted expected payo¤.
I start reviewing the main concepts related to expected utility and risk aversion. Indeed, the
expected utility provides a convenient way to rank risky investments between each other.
Next, I turn on state pricing. I introduce a basic state price rule to price assets. It bases heavily to
so called primitive securities, which can be used to price other assets. I also present conditions when
state prices exists. These conditions include the absence of arbitrage and the law of one price.
Finally, I focus on stochastic discount approach to price …nancial assets. First, I solve the maximization problem of representative investor. Using the …rst order conditions of the problem, one can
…nd the stochastic discount factor that can be used to price asset. In the modern …nance, the systemic risk of an asset is captured by the covariance with the stochastic discount factor.
Indeed, assets with a positive (negative) covariance with the stochastic discount factor has low (high)
expected return.
1
1
Asset Pricing Theory
The expected utility:
– Provides a convenient way to rank risky investments between each other.
– Risk aversion, risk premium and certainty equivalence.
State pricing.
– Bases heavily to so called primitive securities, which can be used to price other assets.
– State prices exists if conditions such as the absence of arbitrage and the law of one price holds
The stochastic discount factor approach to asset pricing.
– Solving a representative investor’s maximization problem can be obtained stochastic discount
factor.
– Using the …rst order conditions, one can …nd the stochastic discount factor that can be used to
price asset.
– In the modern …nance, the systemic risk of an asset is captured by the covariance with
the stochastic discount factor.
– Indeed, assets with a positive (negative) covariance with the stochastic discount factor has low
(high) expected return.
Does Consumption based Asset Pricing work in practice?
– Derivation of CCAPM
– Empirical evidence
– Equity Premium Puzzle
– Potential solutions for Equity Premium Puzzle
2
2
2.1
State-pricing
Utility theory and risk aversion
Time and Risk dimensions
Prefer for smooth consumption stream.
– It’s often assumed that individuals have a desire for a smooth stream of consumption.
– Plan A: Investor consumes 2 units in year 1 and 10 units in year two.
– Plan B: Investor consumes 6 units both years.
– Most investors choose plan B.
Time and Risk dimensions:
– Time dimension:
It’s di¢ cult to alter standard of living from period to period in response to a highly variable
consumption stream.
– Risk dimension:
Individuals would prefer to have the same standard of living in the next period no matter
what events will take place.
The role of …nancial markets:
– If investor generally prefer a smooth consumption stream, we can assume that they would be
better o¤ if they could diversify away some of their consumption risk.
– Financial market o¤ers instruments that allow for this kind of diversi…cation.
– Time dimension:
1. Borrow money to consume today. Pay back to debt in a future period from your consumption in that period.
2. Invest today in a asset that o¤ers you a payo¤ in a future period.
– Risk dimension:
1. An asset that has a variable payo¤ can also diversify away some the risks in the investor’s
consumption ‡ows. (Think this issue in SDF framework; What kind of assets have a positive
correlation with SDF i.e., marginal rate of substitution? How about their expected return,
low or high?)
3
Example 1.
– Assume that investor follows Plan 1 with current consumption (c0 ) is 9 units.
State
State
State
State
1. (s=1)
2. (s=2)
3. (s=3)
4 (s=4)
c0
c0
c0
c0
c0
Plan 1:
c1;s
= 9 c1;1 = 20
= 9 c1;2 = 12
= 9 c1;3 = 8
= 9 c1;4 = 4
Asset 1
x1;s
x1;1 = 1
x1;2 = 2
x1;3 = 3
x1;4 = 4
Asset 2
x2;s
x2;1 = 4
x2;2 = 3
x2;3 = 2
x2;4 = 1
– She does not know for sure the state of nature of next period.
– However, there are two assets having state-contingent payo¤s presented in the table above.
– Which one she chooses if she does not have to pay anything to obtain of on these assets?
– Both assets have a same expected payo¤ equaling to 2.5 units
– Of course, generally investors prefer Asset 1, because then her consumption will be less variable.
– Thus, investors can diversify consumption risks away.
– However, we do not have yet tools to value these assets, i.e., to assign a price and expected
return for them!
Expected Utility Theory:
– Provides a criterion for comparing and ranking di¤erent investments.
1. Specify preferences (utility function) for an investor.
2. Obtain a single number that serves a ranking between di¤erent investments.
The utility function de…ned in a this way is a von Neumann-Morgenstern (VNM) utility function.
– The utilities, V ( ) ; that are assigned to di¤erent gambles, portfolios, or consumption levels can
then used to rank these objects.
– The ranking will be in relation to investors preferences that are described by the utility function
U ( ) of which we are are taking expectation over.
– Speci…cally, if we have a consumption plan fc1;1;:::; c1;s g de…ned over S mutually exclusive states
of nature, the utility that the investor would obtain in each of these states is
V (c1 ) = E [U (c1 )] =
S
X
SU
(c1;S ) :
s=1
– A VNM utility-maximizing investor will then choose the consumption plan that maximizes his
expected utility.
We will later maximize an investor’s expected utility in order to …nd SDF!
4
Risk aversion
How can we understand risk aversion?
– We say that an investor is risk averse if she would not accept a fair gamble, where a fair gamble
is de…ned as one that has an expected value of zero.
– Consider a gamble where an investor can either gain an amount h with probability 1/2 or lose
an amount h with probability 1/2.
– Because this is a gamble with a zero expected payo¤, a risk averse investor with a level of
personal wealth W would not participate it.
– This implies that the expected utility that he gets fro the two possible outcomes of the gamble
is less that the utility he receives from keeping his personal wealth safe in his hand
U (W ) >
1
2
U (W + h) +
1
2
U (W
h) = E (U ) :
To capture this implication, we make two assumptions on the utility function U (W ) :
1. U 0 (W ) > 0:
– The …rst derivative of the utility function is positive.
– This implies that investor prefers more to less.
– His utility function increase whenever his wealth, consumption or whatever we use as an
argument in the utility function increases.
2. U 00 (W ) < 0:
– The second derivative of the utility function is negative.
– The utility increases at a decreasing speed.
– In other words, the utility that the investor receives from a …xed increase in his wealth gets
smaller the larger his current wealth is (the utility increase for a poor man is higher than
the utility increase for a rich man when wealth increase by, say, 100 units).
– This captures risk aversion in the investor’s preferences, and it also implies that risk averse
investor would rather have a smooth level of consumption rather than a variable one around
a …xed mean.
The two assumptions imply that the utility function is strictly increasing and strictly concave.
Absolute risk aversion:
RA (w)
U 00 (W )
:
U 0 (W )
– Measures risk aversion for a given level of wealth.
– For example, when the wealth level increases, will more or less be invested in risky assets?
– Often we assume decreasing ARA.
5
Relative risk aversion:
RR (w)
W
U 00 (W )
:
U 0 (W )
– How does the relative amount of the wealth invested in risk assets change when the wealth level
increases?
– Often we assume constant RRA.
Risk Premium and certainty equivalence
Consider a risk averse investor, with current wealth W , evaluating an uncertain risky project payo¤
xt+1 :
Then for any distribution function FW , we obtain using Jensen’s inequality
U (W + E [xt+1 ]) > E [U (W + xt+1 )] :
– This implies that uncertain payo¤ is available for sale, a risk averse investor will only be willing
to buy it at a price less that its expected payo¤.
Certainty equivalent:
– The maximal certain sum of money a person is willing to pay to acquire an uncertain opportunity.
Risk premium:
– The di¤erence between the certainty equivalent and expected value of the payo¤.
Think these de…nitions in a SDF framework, pt = E (mt+1 xt+1 ) :
– What pt really means in this equation?
– How risk premium is related to mt+1 ?
– We try to understand later!
6
2.2
State pricing
Next, we construct a simple one-period state-pricing model.
The …nancial asset is characterized by its price (p) at time 0 and its payo¤ (x) at time time 1.
– The payo¤ of an asset can be viewed as the sum of the ex-dividend price and the end of the
period and the dividend
xt+1 = pt+1 + dt+1 :
– This means that if you pay a stock today for a price of pt ; the payo¤ at the end of the period
is the ex-dividend price plus the dividend that you will receive.
– In a one-period model we assume that the whole value of the asset is paid as a dividend to the
investor at the end of the period.
I drop out time notations t in order to make notations easier; However remember that
price p is always “today’s” price and payo¤ x is always “tomorrow’s” payo¤.
To incorporate uncertainty (risk) in the analysis, we assume that there are S di¤erent and mutually
exclusive states of nature that can occur during the period.
– The concepts of risk the arises because the investor, making his investment decisions at time 0,
does not know with certainty the state of nature that will occur.
– Each state of nature is associated with probability
S
X
S
S,
which sum up to one
= 1:
s=1
The state-contingent payo¤ of an asset, that is indexed by i, can then be presented by a payo¤ vector
2
3
x1;i
6 x2;i 7
6
7
xi = 6 .. 7
4 . 5
xS;i
Let’s assume that we have the earlier example 1 with equal state probabilities
s
State
State
State
State
1
2
3
4
1
2
3
4
= 1=4
= 1=4
= 1=4
= 1=4
7
Asset 1
x1;s
x1;1 = 1
x1;2 = 2
x1;3 = 3
x1;4 = 4
Asset 2
x2;s
x2;1 = 4
x2;2 = 3
x2;3 = 2
x2;4 = 1
If there are N assets, each will have a payo¤ vector xi ; i = f1; 2; : : : ; N g, then the payo¤ vectors can
be stacked in a payo¤ matrix x that contains all the state-contingent payo¤s of the individual assets.
– This matrix will have a dimension of S N (S rows and N columns).
3
2
x1;1
x1;N
6
.. 7 :
...
x = 4 ...
. 5
xS;1
xS;N
– Let p represent the beginning period prices of theses N assets (an N
2
3
p1
6
7
p = 4 ... 5 :
pN
1 vector)
Consider an asset that provides a payo¤ 1 if and only if state s occurs, and a payo¤ of 0 in all other
states
– Denote the price of such an asset by qs :
– The payo¤ vector, denoted by es , of such an asset will be
2
3 2 3
x1
0
.
6 . 7 6 .. 7
6 . 7 6 . 7
6
7 6 7
es = 6 xs 7 = 6 1 7 :
6 . 7 6 . 7
4 .. 5 4 .. 5
xS
0
– If there exists such assets for each state of economy, i.e., there exists es for s = f1; 2; : : : ; Sg, we
can stack the payo¤ vectors of these assets in an S S identity matrix
2
3
1 0
0 0
. . 7
6
6 0 1 0 .. .. 7
6 .
.. 7
..
.
e =6
.
0
. 7
.
0
6
7:
6 . .
7
4 .. .. 0 1 0 5
0 0
0 1
– The state price vector of these assets is denoted by q (an S
2
3
q1
6
7
q = 4 ... 5 :
1 vector)
qS
– These assets, whose payo¤ is denoted by es ; are called primitive securities, Arrow-Debreu securities, pure securities, state securities and state contingent claims in the literature
8
These primitive securities are the building blocks of modern asset pricing theory.
– Note that the state price, qs ; gives the current price of one unit of payo¤ that we get in
state s:
– Since individual assets are described by their state-contingent payo¤s, and if we know the prices,
given by qs ; of these state-contingent payo¤s, we can use the state prices to assign a price to
any other asset on the market.
– Prices of primitive securities are the state price vector ( the superscript T denotes a matrix
transpose)
2
p1
6 ..
4 .
pN
2
p1
6 ..
4 .
pN
p = xT q;
3
2
1
7
6
5 = 4 0
0
3
2
q1
7
6 ..
5 = 4 .
qS
x=e
32
3
0 0
q1
76 . 7
...
0 5 4 .. 5 ,
0 1
qS
3
7
5
The pricing formula is simply given by (the superscript T denotes a matrix transpose)
p = xT q;
Example 2:
– There are three states of the economy, i.e., S = 3:
– The payo¤ vectors of the assets are x1 = (8:5; 10; 4)T , x2 = (8:5; 8; 10)T and x3 = (7; 8; 6)T :
– There are also three primitive assets, one for each state of the economy, with the state price
vector q = (0:18; 0:45; 0:30)T :
– The prices of these three assets are set using p = xT q
2
3T 2
3
q1
x1;1 x1;2 x1;3
4 x2;1 x2;2 x2;3 5 4 q2 5
q3
x3;1 x3;2 x3;3
2
32
3
x1;1 x2;1 x3;1
q1
4 x1;2 x2;2 x3;2 5 4 q2 5
x1;3 x2;3 x3;3
q3
2
32
3 2
3
8:5 10 4
0:18
7:23
= 4 8:5 8 10 5 4 0:45 5 = 4 8:13 5
7 8 6
0:30
6:66
3
p1
4 p2 5 =
p3
2
3
p1
4 p2 5 =
p3
2
9
In actual …nancial markets, these primitive securities do not traded directly.
– However, under some conditions, we can infer their prices from the assets that are traded.
– Finding the state price vector q from prices and payo¤s of the traded assets is called a reverse
decomposition problem.
The condition that must hold if we are to …nd a unique state price vector q by using the observed
prices and payo¤s of individual assets are the following
Theorem 1: Market Completeness
– For unique state prices for each state to exists, the payo¤ matrix x must be of rank S, rank(x) =
S: This implies that there are as many linearly independent assets as there are states of nature.
Then, the market is said to be complete, the matrix x is invertible, and state prices can be
recovered by calculating
1
q = xT
p:
(1)
Example 3:
– Consider the same setting as in previous example, but there is also the forth asset with a payo¤
vector x4 = (4; 6; 2)T :
– In the line of the law of one price, the price of Asset 4 is p4 = 4:02:
– The stacked payo¤ matrix is now
2
3
8:5 8:5 7 4
x = 4 10 8 8 6 5
4 10 7 2
and the rank of x is 3, rank(x) = 3.
– This implies that there are 3 linearly independent securities on the market.
– Since there are three states of nature, the market is complete.
– In order to take inverse of the payo¤ matrix, we need to have a square matrix (as many rows
as columns).
– We can take out one of the assets to get rid of one column and check if the remaining matrix
still has a rank of three.
– Taking out, for example Asset 3, we are left with a matrix
2
3
8:5 8:5 4
x = 4 10 8 6 5
4 10 2
that still has a full rank.
10
– Then, states can be recovered using formula 1.
1
q = xT
2
3
2
q1
8:5
4 q2 5 = 4 8:5
q3
4
2
0:18
4
0:45
=
0:30
p
3
10 4
8 10 5
6 2
3
5
1
2
3
7:23
4 8:13 5
4:02
Using the prices for the individual assets and their payo¤s, we arrive at the same state prices
that we de…ned in the previous example.
11
2.3
The Law of One Price
The law of one price states that two assets having the same state contingent payo¤ must have the
same price.
– Formally,.let p (x1 ) be the current price of an asset with a payo¤ vector x1 ; then the Law of One
Price implies that we can …nd a linear pricing rule.
De…nition 1: The Law of one Price
– If the Law of One Price holds, then we have a linear pricing rule in the form of
p ( 1 x1 +
where
1
2 x2 )
=
is the amount invested in asset 1 and
1 p (x1 )
2
+
2 p (x2 ) ;
is the amount invested in asset 2.
The Law of One Price states that the price of a portfolio consisting individual securities, must be
given by the prices of these individual securities.
– It’s not possible to repackage two asset into one portfolio and sell the portfolio at a higher price
that what is implied by the prices on the individual assets.
– A violation of this law would give rise to an immediate kind of arbitrage pro…t, as you could
sell the expensive version and buy the cheap version of the same portfolio.
Theorem 2. State prices exists if and only if the Law of One Price holds.
– This implication goes for both directions:
The law of One Price implies the existence of state prices.
The existence of state prices implies that the Law of One Price holds.
12
2.4
Absence of arbitrage
Absence of arbitrage says that you cannot get for free a portfolio that might pay o¤ positively.
De…nition 2: Arbitrage
– An arbitrage is a portfolio satisfying one of the following conditions
(i) p < 0 and x > 0
(ii) p 6 0 and x > 0 with Prob x > 0 > 0:
Condition (i) suggests that we get something today, without having any negative out‡ows from the
portfolio in the future.
Condition (ii) implies that we can construct a portfolio today that costs us nothing, but that has a
positive probability of producing a positive cash ‡ow in the future.
Indeed, you should not be able to construct a portfolio that will certainly not cost you anything, but
that might pay of positively.
This de…nition is di¤erent from the colloquial use of word “arbitrage”.
– Most people use “arbitrage” to mean a violation of the law of one price - a riskless way of
buying something cheap and selling it for a higher price.
– “Arbitrage”here might pay o¤, but they again might not.
Absence of arbitrage implies the following theorem.
Theorem 3: The positivity of state prices
– If there is no arbitrage opportunities on the market, then the state prices will be strictly positive.
– This implies both direction:
The positivity of state prices implies absence of arbitrage.
The absence of arbitrage implies the positivity of state prices
To understand this, assume that the state prices are strictly positive (qs > 0 for all states s) :
– Recall that both of the arbitrage conditions require that the payo¤s from portfolio must be
either zero or positive.
– The prices of all portfolios are give by the sum of the products between the state prices and the
payo¤s that portfolio produces.
– This implies that products between state prices and payo¤s must be either zero or positive, and
thus, the price of the portfolio cannot be negative.
– If the portfolio produces a positive payo¤, its price today must be positive if the state price is
positive.
13
– Thus, we cannot have any arbitrage if state prices are positive
T
x
| {z }
=
q
|{z}
all elements either all elements strictly
positive
positive or zero
p
|{z}
:
n e ve r n e g a tive
The Law of One Price and Absence of Arbitrage are two distinct concepts.
– The absence of arbitrage implies that the law of one price holds.
– If the law of one price did not hold, then we would have an arbitrage opportunity.
– On the other hand, the law of the one price does not imply absence of arbitrage opportunities.
– There can be arbitrage opportunities even if the law of one price holds.
Example 3:
– Suppose there are two assets and two possible states of nature. The payo¤s from the two assets
are x1 = (0; 1)T and x2 = (1; 2)T with prices p1 = 0:9 and p2 = 1:6: Because payo¤s are not
linearly dependent (Rank(x) = 2), the market is complete. Suppose that the law of one price
holds, so that price of any portfolio is simply given by the prices of the individual assets,
p =
1 p1
+
2 p2
=
1
(0:9) +
2
(1:6) :
Consider now a new portfolio that shorts two units of asset 1 and buys one unit of asset two.
That is, 1 = 2 and 2 = 1:
Because the law of one price holds, the price of such a portfolio
p =
2 (0:9) + 1 (1:6) =
0:2:
The state contingent payo¤s of the portfolio are
x =
1
0
:
Thus, the portfolio has a negative price today, and never produces a negative payo¤ and it has
a positive probability for a positive payo¤.
Hence, we clearly have arbitrage even thought the law of one price holds.
We know from the theorem above that positive state prices guarantee the absence of arbitrage
opportunities. Hence, because there is an arbitrage opportunity in this example, we cannot
have positive state prices. Using the formula that allows us to recover state prices in a complete
market, we get
q =
q =
xT
1
p
0 1
1 2
1
0:9
1:6
=
0:2
0:9
Thus, the …rst state price is negative, implying that there are arbitrage opportunities.
14
2.5
Summary and the most important ideas
We have studied how to assign asset prices given the existence of state prices.
In addition, we have examined the conditions under which these state prices, and consequently, an
asset pricing formula exists.
The existence of state prices:
– If Law of one price holds, there will always exists state prices.
– If the stronger condition, the absence of arbitrage holds, not only do these state prices exists,
but there will be strictly positive state prices.
– If markets are complete, there exists a unique state vector (only one).
I expect that you understand the most important concepts after this section:
– Expected utility and risk aversion.
– Basics of state pricing:
The role of primitive securities i.e., Arrow-Debreu assets.
Markets completeness.
How the pricing formula works.
– Meaning of the absence of arbitrage.
– Meaning of the law of one price.
– The conditions about existence of state prices.
Next, we turn on the economic determinants of state prices.
15
3
Stochastic discount factor approach
How state prices are determined?
What are economic sources determining why a unit of payo¤ is more valuable in some states of nature
and less valuable in some other states?
Because every asset can be priced with these state prices, the answer to the question posed above
should also help us to understand why some assets have lower prices than some other assets.
3.1
Maximization problem of representative investor
Assume that there exists a representative investor.
– A representative investor is somewhat abstract concept.
– When posing existence of such an investor, we assume that actions of all investors can be
aggregated in such a way that we need to study the actions of one investor who is aggregate of
the individual investors.
– Rubinstein (1974) shows that there are several technical conditions under such an aggregation
is possible to do.
– We assume that these conditions are ful…lled.
The representative investor much choose how much to consume today and how much to consume in
the s possible di¤erent states of nature tomorrow. The maximization problem of the representative
investor:
max E [U (C)] = U (c0 ) +
c0 ;c1 ;:::cs
S
X
sU
(cs ) ;
s=1
|
{z
Objective function
Subject to
c0 +
|
}
S
X
qs cs = W;
s=1
{z
Budget Constraint
}
where subjective time-preference factor, S probability of state s occurring, qS is the state price
for state s, W is the investor’s current wealth.
– The investor chooses his consumption levels today (t) and tomorrow (t + 1) in such a way that
his expected utility is maximized.
– The budget constraint states that the price of the consumption pattern (c0 ; c1 ; : : : cs ) must equal
his current wealth (W ).
16
This means that he cannot today allocate more wealth to consumption than he owns.
However, the budget constrain makes sure that he consumes all the wealth during either
the …rst period (t) or second period (t + 1) :
– We further assume that the price of one unit of consumptions today is 1 unit, q0 = 1:
We can solve this constrained optimization problem using Lagrangian method:
L
:
U (c0 ) +
S
X
sU
(cs ) +
W
s=1
s=1
First order conditions for c0
@L
@c0
:
U 0 (c0 )
=0
, U 0 (c0 ) =
First order conditions for cs
@L
@cs
:
sU
0
(cs )
,
sU
0
(cs ) = qs
Using FOCs
:
sU
0
(cs ) = U 0 (c0 )qs
State Price s
:
qs =
c0 +
S
X
qs = 0
U 0 (cs )
;
s 0
U (c0 )
17
s = 1; : : : ; S:
qs c s
!!
3.2
Economic determinants of state prices
To understand what economic determinants drives state prices (qS ) i.e., the unit value of future
payo¤ in a given state s, let’s have look are the state-price equation
qs =
1.
s
U 0 (cs )
;
U 0 (c0 )
s = 1; : : : ; S:
(2)
is the investor’s subjective time-preference factor.
– The lower is the value of ; the more value the investor values consumption today relative
tomorrow.
– Low makes all the state prices smaller suggesting that one would rather consume wealth
today that tomorrow.
– One is not willing to pay much today for a unit of consumption (payo¤) that one will receive
tomorrow.
2.
s
is the probability that state s is occurring.
– One is not willing to pay much for a unit of consumption (payo¤) tomorrow if there is a
small probability that one will get the payo¤.
3. The third quantity,
U 0 (cs )
;
U 0 (c0 )
is the most important important for asset pricing purposes.1
– U 0 (cs ) is the marginal utility of the consumer/investor.
– The important economic questions is to understand U 0 (cs ) and the associated payo¤ for a
some speci…c state of nature s:
– Since investors are assumed to be risk averse:
(a) The …rst derivative of utility function is U 0 (cs ) is always positive.
(b) The second derivative of U 00 (cs ) is always negative.
– This implies that marginal utility U 0 (cs ) is high when consumption level cs is low.
– Hence, state prices qS will be high in states of aggregate consumptions is low.
State prices qS are high exactly in states of nature where investors value extra unit of payo¤ most.
– One is willing to pay much for an assets that gives you high payo¤s in states where you are
“hungry”and where one might otherwise be “starving”.
– Indeed, that kind of assets diversify away some of the risks in your consumption ‡ows and
increase one’s utility.
1
The denominator, U 0 (c0 ), is the same for all states, thus one can only concentrate on the nominator U 0 (cs ):
18
3.3
From state prices to stochastic discount factors
The equation (2) involves probabilities for each state of nature s.
– Thus, it’s di¢ cult to compare state prices with each other on a relative basis.
– Hence, it’s hard to compare the relative “goodness”and “badness”of states with each other.
To get rid of the probability measure
probability of state happening
s
qs
in equation (2) ; one can divide state prices qs with the
=
s
U 0 (cs )
U 0 (c0 )
ms :
where ms is called to the stochastic discount factor!
The name - stochastic discount factor - is was used by Hansen and Richard (1987).
– Other names for SDF that appear in literature are state price de‡ator, state price density,
pricing kernel or change of probability measure.
3.4
Asset Pricing with stochastic discount factors
Following pricing framework for state prices:
pi =
=
=
S
X
s=1
S
X
s=1
S
X
qs xs;i
qs
s
xs;i
s
s ms xs;i
s=1
pi = E [mxi ]
Example 4:
– Recall earlier example where an investors has to choose between to di¤erent assets. The table
shows investor’s consumption plan.
State
State
State
State
1.
2.
3.
4
c0
c0
c0
c0
c0
Plan 1:
c1;s
= 9 c1;1 = 20
= 9 c1;2 = 12
= 9 c1;3 = 8
= 9 c1;4 = 4
Asset 1
x1;s
x1;1 = 1
x1;2 = 2
x1;3 = 3
x1;4 = 4
Asset 2
x2;s
x2;1 = 4
x2;2 = 3
x2;3 = 2
x2;4 = 1
19
– We agreed earlier that Asset 1 looks like interesting, since it delivers high payo¤ when investor’s
consumption is low.
– Then, however, we didn’t had tools to verify our intuition.
– Using the stochastic discount factor
U 0 (cs )
;
U 0 (c0 )
ms =
we price these assets.
– However, we need to make couple of assumptions about investor preferences.
1. Investor has a logarithmic utility function
U (c) = ln (c)
with the …rst derivative
1
U 0 (c) = :
c
2. Investor’s subjective time discount parameter is
= 0:85:
3. There are equal state probabilities
s
=
1
4
– Using these facts, one can calculate the value of the stochastic discount factor for each state of
nature
ms =
1
cs
1
c0
c0
; s = 1; 2; 3; 4:
cs
=
m1 = 0:85
9
= 0:383
20
m2 = 0:638
m3 = 0:956
m4 = 1:913
– Using formula
pi = E [mxi ] =
S
X
s=1
one can calculate the prices of Assets 1 and 2.
20
s ms xs;i ;
– The price for Asset 1:
1
m1
4
1
m3
4
= 3:044
p1 =
x1;1
+
x1;3
+
x2;1
+
x2;3
+
1
4
1
4
m2
x1;2
m4
x1;4
m2
x2;2
m4
x2;4
+
– The price for Asset 2:
1
m1
4
1
m3
4
= 1:817:
p2 =
1
4
1
4
+
– Thus, investor has a “hunger”for Asset 1.
3.5
Link SDF to returns and risk-free rate
Asset pricing’s sledgehammer m is ready to use!
Indeed, the price of any …nancial asset is given by the expectation of the product between the asset’s
payo¤ and stochastic discount factor
pt = E [mt+1 xt+1 ] :
(3)
Note that I take back t notations for time!
We usually divide the payo¤ xt+1 by the price pt to obtain a gross return:
Rt+1
xt+1
;
pt
where Rt+1 is the gross return on the asset (1 + rt+1 ) :
We can think of a return as a payo¤ that has price equal to one: if we pay one euro today, the return
is how many euros or units or consumption we get tomorrow.
Thus, returns obey
xt+1
pt
= E m
,
pt
pt
1 = E (mt+1 Rt+1 ) :
Returns are commonly used in empirical work, since they are typically stationary (the means, variances and autocovariances are independent of time) over time: they do not have trends.
21
If there is no uncertainty, we can express returns
1 = E (mt+1 Rt+1 ) = E (mt+1 ) Rf
(4)
where Rf is the gross risk-free rate.
The risk-free rate is related to the discount factor by
Rf = 1=E(mt+1 ):
(5)
Since Rf is typically greater than one, the payo¤ xt+1 sells ”at a discount.”
3.6
Understanding Risk Corrections
Using the de…nitions of covariance
cov(mt+1 ; xt+1 ) = E(mt+1 xt+1 )
E(mt+1 )E(xt+1 );
we can write the pricing equation pt = E(mt+1 xt+1 ) as
pt = E(mt+1 )E(xt+1 ) + cov(mt+1 ; xt+1 ):
Substituting the risk-free rate equation (5), we obtain
pt =
E(xt+1 )
+ cov(mt+1 ; xt+1 );
Rf
(6)
where
– the …rst term is the standard discounted present-value formula, giving the asset’s price in a
risk-neutral world (where consumption is constant or utility is linear)
– the second term is risk adjustment.
Using expected returns, we obtain risk premia or expected return
E(Rt+1 )
Rf =
Rf cov(mt+1 ; Rt+1 )
(7)
that is higher for assets having a large negative covariance with the discount factor.
This is an extremely important result! I hope that you understand the meaning of this
after taking this course!
This helps us to understand why some assets have higher expected returns than others.
– Indeed, equation (7) shows that assets with returns that have a positive covariance with a
stochastic discount factor have a low risk premium.
– The intuition should be clear from a previous example.
– The SDF mt+1 can be theoretically understood as a function of investor marginal utilities U 0 (c):
22
The SDF obtains high (low) values in states where marginal utility is high (low).
An asset with a positive (negative) covariance with the SDF yields high (low) returns in
states with high (low) values of the SDF.
– If covariance term cov(mt+1 ; Rt+1 ) in equation (7) is positive, this means that such an asset
have a lower expected return that a risk-free asset.
This due to fact that such an asset o¤ers insurance against bad states of nature.
– On the other hand, if covariance term cov(mt+1 ; Rt+1 ) in equation (7) is negative, this means
that such an asset have a higher expected return that a risk-free asset.
A higher negative covariance implies higher risk and expected return, thus investors are
also willing to include such an asset into their portfolios.
Intuitively, that kind of assets increase the volatility of an investor’s consumption stream.
Idiosyncratic (asset-speci…c) risk does not a¤ect prices.
– One important implication of equations (6) and (7) is that the variance of the asset’s
returns, or it’s payo¤s, does not a¤ect prices or expected returns.
– Only the covariance with the SDF matters, since if
cov(mt+1 ; Rt+1 ) = 0 )
Rf = 0
E(Rt+1 )
– This prediction is true no matter how large is the variance of the asset return (V ar Rt+1 ):
Indeed, one of main principles of modern …nance is that only the systematic risk of an asset’s
payo¤ should be priced or rewarded higher expected risk.
In the modern …nance, the systemic risk of an asset is captured by the covariance with the
stochastic discount factor.
Example 5:
– Consider the previous example.
– The cross risk-free rate is
Rf =
1
1
= S
P
E [m]
s ms
s=1
i.e., the risk-free rate is 2.86%.
23
=
1
= 1:0286
0:9722
– Using prices calculated earlier, one can …nd that the expected returns are
Asset 1
E [x1 ]
1=
E [r1 ] =
p1
17:9%
Asset 2
E [x2 ]
E [r2 ] =
1 = 37:6%:
p2
– Thus, expected return on Asset 1 is negative, where as the expected return on Asset 2 is positive.
– How can be understand the these somewhat puzzling return patterns?
– First, note that covariances of assets’returns with the stochastic discount factor
Cov [R1 ; m] = E [R1 m] E [R1 ] E [m]
= 1 (1 17:9) (0:9722)
Cov [R1 ; m] = 0:2016
Cov [R2 ; m] =
0:3377:
– Asset 1 has a positive covariance with the SDF.
Hence, it delivers high returns in bad states of economy when these high returns are needed.
Therefore investors are willing to take a sort of insurance against bad state of economy.
– Asset 2 has a negative covariance with the SDF.
Thus, Asset 2 increases the volatility of consumption stream.
It contains risk, therefore investors require a high premium in order to invest in this asset.
– Using equation (7), we can also retrieve the risk premiums on the assets.
E(R)
Rf =
Rf cov(m; R)
– Taking Asset 2 as an example, one can easily …nd that
Rf cov(m; R) =
1:0286
( 0:33772) = 0:3474
being equal to
E(R)
Rf = 1:376
1:0286 = 0:3474:
– Finally, consider Asset 3 with payo¤s (2:624; 3:498; 0:972; 2:907)
This asset has a volatile payo¤ stream.
Hence, it must be a risky asset? Or must it?
One calculate its price
p3 = 2:4307:
24
The expected return on the Asset 3 is (the expected payo¤ is 2.5)
E [x3 ]
p3
2:5
=
2:4307
E [r3 ] =
1
1 = 2:851%;
which turns out to be equal to risk-free rate of return!
E [r3 ] = rf:
This can be understood by calculating covariance between asset return and the SDF
Cov [R3 ; m] = E [R3 m] E [R3 ] E [m]
= 1 E [Rf ] E [m]
1
= 1
E [m]
E [m]
= 1 1=0
Thus, Asset 3 is uncorrelated with the SDF.
Even though asset has a variable payo¤ stream, this variation is not connected to systemic
risk - all the risk involved in the asset’s payo¤ is unsystematic (idiosyncratic).
Therefore, one could conclude that asset risk-free and earns a risk-free rate of return.
What one can learn these example!
– One could erroneously conclude that Asset 2 is “better” that Asset 1, due to fact that Asset 1
has a negative return and Asset 2 has a positive return.
– However, such claims are completely unfounded if one simply looks at raw returns without
resorting asset pricing model.
– Asset 2 provides investors with high returns simply because it is a “bad” asset - its payo¤
contains more risk than the payo¤s of Asset 1.
– In market equilibrium, the fact that a company is good (excellent management, many investment
opportunities, new and promising product lines) is re‡ected in a high price of the company today.
– However, expected returns should, on average, only be a compensation for the systematic riskiness of stock’s payo¤s.
– Furthermore, in market equilibrium, there are no “good”and “bad”assets.
– Prices and expected returns are set in such a way that the badness of a given stock (measure
using systematic risk) is exactly balance by higher expected return that it must o¤er.
– Thus, investors will, at margin, be indi¤erent between di¤erent assets in their portfolio choices
when market reaches equilibrium.
25
3.7
Existence of stochastic discount factors
There is a close correspondence between the SDF and state prices, one can obtained very similar
existence theorems for the SDF.
Theorem 4: The fundamental theorem of asset pricing
– The following conditions on prices p and payo¤s x are equivalent
(i) Absence of arbitrage
(ii) Existence of a consistent positive linear pricing rule (positive state prices) (9q >> 0) p = xT q
(iii) Some agent with strictly increasing preference U has an optimum
Theorem is based on Cox and Ross (1976) and Ross (1977, 1978).
It’s also know as the fundamental theorem of …nancial economics.
The points (i) and (ii) familiar form case of state prices.
– The condition for the existence of any asset pricing model is that state prices are strictly positive
and that there are no arbitrage opportunities.
The point (iii) implies that if there were in fact arbitrage opportunities, then the representative
investor’s maximization problem could not react an optimum.
– Buying more and more consumption in a state that has a negative state price would increase
the investor’s wealth in…nitely, and his expected utility would not be bounded from above.
26
4
Consumption based Asset Pricing Model (CCAPM)
When we write the basic model
p = E (mx) ;
we do not assume
1. Markets are complete, or there is a representative investor
2. Asset returns or payo¤s are normally distributed, or independent of time
3. Two-period investors, quadratic utility, or separable utility
4. Investors have no human capital or labor income
5. The market has reached equilibrium, or individuals have bought all the securities they want to
These assumptions come in special cases. But, we do assume that the investor can consider a small
marginal investment or disinvestment.
The basic pricing equation should hold for any asset (stock, bond, option, real investment opportunity,
etc. ), and any monotone and concave utility function.
The consumption-based model is, in principle, a complete answer to all asset pricing questions, but
works poorly in practice.
– This observation motivates other asset pricing models.
All we need in CCAPM is a functional form for utility, numerical values for the parameters, and a
statistical model for the conditional distribution of consumption and payo¤s.
Consider the standard power utility function
u0 (c) = c :
Then excess returns should be
0 = Et
"
ct+1
ct
#
e
Rt+1
:
Taking unconditional expectations and applying the covariance decomposition, expected excess returns
follow
"
#
c
t+1
e
e
E Rt+1
= Rf cov
; Rt+1
:
ct
27
4.1
Empirical Findings
Theoretically, the CCAPM appears preferable to the traditional CAPM
– It takes into account the dynamic nature of portfolio decisions.
– It integrates the many forms of wealth beyond …nancial asset wealth.
– Consumption should deliver the purest measure of good and bad times as investors consume
less when their income prospects are low or if they think future returns will be bad.
However, empirically, the original version of the consumption-based model has not been a great
success.
Hansen and Singleton (1982, 1983) formulate a consumption-based model in which a representative
agent has time-separable power utility of consumption.
– They reject the model on U.S. data, …nding that it cannot simultaneously explain the timevariation of interest rates and the cross-sectional of average returns on stocks and bonds.
Wheatley (1988) rejects the model based on international data.
Mankiew and Shapiro (1986) show that the CCAPM performs no better, and in many respects even
worse than the CAPM.
– They regress the average returns of the 464 NYSE stocks that were continuously traded from
1959 to 1982 on their market betas, on consumption growth betas, and on both betas.
– They …nd that the market betas are more strongly and robustly associated with their cross
section of average returns, and that market beta drives out consumption beta in multiple regressions.
Breeden, Gibbons, and Lizenberger (1989) …nd comparable performance of the CAPM and a model
that uses a mimicking portfolio for consumption growth as the single factor.
Recently, Lettau and Ludvigson (2001), Amir and Bansal (2004) and Savov (2011) among others
suggest that Consumption Strikes Back!
Savov (2011, JF) “Asset Pricing with Garbage”
– A new measure of consumption, garbage, is more volatile and more correlated with stocks than
the canonical measure, National Income and Product Accounts (NIPA) consumption expenditure.
– A garbage-based consumption capital asset pricing model matches the U.S. equity premium
with relative risk aversion of 17 versus 81 and evades the joint equity premium-risk-free rate
puzzle.
– These results carry through to European data. In a cross-section of size, value, and industry
portfolios, garbage growth is priced and drives out NIPA expenditure growth.
28
4.2
4.2.1
The Basic Equity Premium Puzzle
Equity Premium Puzzle
The Hansen-Jagannathan (1991) bounds are the characterizations of the discount factors that price
a given set of asset returns.
– Manipulating 0 = E(mRe ), we …nd
jE(Re )j
(Re )
| {z }
(m)
:
E(m)
Sharpe ratio
– Proof:
0 = E(mRe )
0 = E(m)E(Re ) + cov(m; Re )
0 = E(m)E(Re ) + mRe (m) (Re )
– Dividing both sides by E(m)
m
implies that
jE(Re )j
(Re )
(m)
=
E(m)
mR
f
– The highest Sharpe ratio is associated with portfolios lying on the mean-variance e¢ cient frontier.
– Notice that the slope of the frontier is governed by the volatility of the discount factor.
– Under the CCAPM it follows that
ct+1
ct
e
jE(R )j
=
(Re )
E
ct+1
ct
The postwar U.S. mean value weighted NYSE is about 8% per annum over the T-bill rate, with a
standard deviation of about 16%.
– Thus, the market Sharpe ratio E (Re ) = (Re ) is about 0.5 (8/16) for an annual investment
horizon.
– If there were a constant risk-free rate, E(m) = 1=Rf would nail down E(m).
– The T-bill rate is not very risky, hence E(m) is not far from the mean of the inverse of the mean
T-bill rate (1=(1 + rf )), or about E(m) 0:99.
– Thus, the facts about the mean return, 8%, and volatility, 16%, imply (m) > 0:5; which means
that the volatility of the discount factor must be about 50% of its level in annual data!
– Per capita consumption growth has standard deviation about 1% per year; and with log utility
that implies
ct+1
ct
= 0:01 = 1%, which is o¤ by a factor of 50.
– To match the equity premium we need
= 50, which seems a huge level of risk aversion!
29
4.2.2
Correlation Puzzle
The H-J bounds take the extreme possibility that consumption and stock returns are perfectly correlated.
– They are not. The correlation of annual stock returns and nondurable plus services consumption
growth in postwar U.S. data is no more than 0.2.
– If we use this information, the calculation becomes
(m)
E(m)
– With (m)
4.2.3
jE(Re )j
1
=
0:5 = 2:5:
e
j m;Re j (R )
0:2
1
( c); we now need a risk aversion coe¢ cient of 250!
Risk-Free Rate Puzzle
Average Interest Rates and Subjective Discount Factors:
– Traditionally, we consider risk aversion numbers
– What is wrong with
= [1; 5] or so.
= 50 to 250?
– The most basic piece of evidence for low comes from the relation between consumption growth
and interest rates:
"
#
1
ct+1
= E(mt+1 ) = Et
;
Rf
ct
that can be expressed in continuous time,
rtf = + Et [ c]
1
( + 1)
2
2
t
( c) :
– Real interest rates are typically quite low, about 1%. With a 1% mean and 1% standard
deviation of consumption growth, the predicted interest rate rises quickly as we raise .
– For example, with = 50 and a typical
50 51 0:012 = 0:38 or 38%!
– Using
= 0:01 (1%), we predict rf = 0:01 + 50
0:01
1
2
= 2, the risk-free rate should be around 5% to 6% per year.
– The actually observed rate is less than 1%.
– To get a reasonable 1% real interest rate, we have to use a subjective discount factor on negative
37%.
– That is not impossible –such economic model can be speci…ed where present values can converge
with negative discount rates - but it doesn’t seem very reasonable; since people prefer earlier
utility.
30
4.2.4
How Shall We Resolve the Equity Premium and Risk-Free Puzzles?
Perhaps investors are much more risk averse than we may have thought.
– This indeed resolves the equity premium puzzle.
– But higher risk aversion parameter implies higher risk-free rate. So, higher risk aversion reinforces the risk-free puzzle.
Perhaps the stock returns over the last 50 years are good luck rather than an equilibrium compensation for risk.
– If so, the equity premium will disappear in several decades as more reasonable returns are
realized.
Perhaps something is deeply wrong with the utility function speci…cation and/or the use of aggregate
consumption data.
– Indeed, the CCAPM assumes that agents’preferences are time additive von Neumann –Morgenstern expected utility representation (e.g., power utility).
– Standard power utility preferences impose tight restrictions on the relation between the equity
premium and the risk-free rate.
– In power utility, the elasticity of intertemporal substitution (EIS) and the relative risk aversion
parameters are reciprocals of each other - economically they should not be tightly linked.
– EIS is about deterministic consumption paths – it measures the willingness to exchange consumption today with consumption tomorrow for a given risk-free rate; whereas risk aversion is
about preferences over random variables (lotteries).
31