State Preference Theory
1.
Advanced economies facilitate individuals’
savings/consumption decisions and firms’
investing/financing decisions through securities trading.
2.
In equilibrium, securities supply equals demand and
firms maximize profits while consumers maximize
expected utility.
3.
This and the next two lectures consider issues necessary
to resolve the problem of how individuals select among
risky securities to maximize their expected utility over
time.
4.
Today’s Topic: How are security prices determined when
they offer given payoffs in particular states of the world,
but where the state that will be realized at a future point
in time is uncertain?
5.
What is a security? A vector of payoffs associated with
different states of the world at some future date.
6.
The investor’s portfolio can then be characterized as a
matrix of possible payoffs on his securities.
State Specific Securities
1. Simple Model
-Two states of nature, 1 and 2, with associated
probabilities 1 and 2 .
- Assume that the states are exclusive and exhaustive so that
the probabilities sum to 1. Here this means that
2 = (1 - 1).
- Pure state security 1 (2) pays off $1 if state 1 (2) is realized
and nothing otherwise. If both securities exist then
the securities market is said to be complete.
- Assume investors are able to associate payoffs with states
and that utility is not a direct function of the realized
state but depends only on how much wealth they
receive each state.
2. Under these conditions, investors can buy pure securities to
obtain their desired future wealth given the
constraint defined by their current wealth and the
prices of the pure securities p1 and p2. The prices
of the pure securities will reflect their supply from
firms and their demand from investors/consumers.
A Complete Capital Market
of Complex Securities
1. Markets consist of many complex securities rather than
pure securities.
2. Complex securities are just linear combinations of the pure
securities. For example, a security paying $3 in
state 1 and $2 in state 2 is equivalent to a portfolio
of 3 shares of pure security 1, and 2 shares of pure
security 2.
3. When there are S states, and there are at least S complex
securities that have linearly independent payoffs,
then the complex securities market is complete.
That is, the market can operate as if there are S
pure securities. In a complete market, all risk is
insurable.
Example: Suppose there are three states and three securities
have the following payoff vectors XS = [x1, x2, x3].
Assume you can buy or sell fractions of a share. Is
this market complete?
X1 =[6, 6, 2], X2=[3, 0, 0], X3 =[0, 3, 1].
Hint: Combine the vectors into a matrix and see if the matrix
rank is 3. Alternatively, see if the determinant is nonzero.
4. Options on complex securities allow an incomplete market
to be completed (see Ross 1976).
If a state can be described by some price for the complex
security, then we can write options on the security with a given
strike price to synthetically create a pure security that pays off
only in that state.
5. Long-lived securities represent portfolios on pure securities
that allow us to have effectively complete markets with a
relatively small number of securities. With many time periods
and many states in each time period, the number of pure
securities needed to complete a market seems very large. But
in fact, if there are enough long-lived complex securities to
cover the full range of states in any period, then we may get
by with a much smaller number of securities. Since
uncertainty (the state) is revealed one period at a time, if we
know how the state revealed this period affects all future
period payoffs, then we can use a relatively small number of
long-lived securities to effectively complete the market periodby-period. We buy some long-lived securities, not just
because the payoff they offer next period suits our needs, but
also because their payoffs for many future periods suit our
expected future needs as well.
1.
Example of how to find pure securities prices given a
complete market and securities payoffs.
-
ps = prices of pure securities
-
Pj = prices of complex securities
-
s = state probabilities
-
Qs = number of pure securities
2. Consider two complex securities with the following payoffs
in two states of the world X1 =[10, 20], X2=[30, 10]. The
price of the two securities is P1 = 8 and P2 = 9. We can
find the pure securities prices as
P1 = 8 = 10p1 + 20 p2
P2 = 9 = 30p1 + 10p2
Solving two equations in two unknowns gives
p1 = .20 and p2 = .30.
We pay 20 cents today for a security that pays off $1 if state 1
occurs in the future and 30 cents today for a security that
pays off $1 if state 2 occurs in the future.
3. Cramer’s rule can be used to solve a system of equations.
P1 = 8 = 10p1 + 20 p2
P2 = 9 = 30p1 + 10p2
We can get the pi as the ratio of determinants, pi = |Ai|/|A|
where A = is the matrix of coefficients on the pi in the system
and Ai is the same matrix with the ith column replaced by the
vector of complex securities prices.
A
10 20
30 10
8
9
p1 
10
30
A1 
20
10
 0.20
20
10
8 20
9 10
10
30
p2 
10
30
A2 
10 8
30 9
8
9
 0.30
20
10
Law of One Price
1.
Equilibrium in the securities market means that supply
equals demand for all securities.
2.
Equilibrium implies that securities with the same payoffs
carry the same price. If they did not have the same price,
supply would not equal demand – individuals would sell
the high-priced one and buy the low-priced one and earn
a risk free return on the difference between the prices.
3.
For a complete market, we can construct a risk-free
security by buying one of each of the pure securities,
which guarantees a $1 payoff. For the risk-free return r,
in the previous example we have
p1 + p2 = .20 + .30 = .50 = 1/(1 + r) => r = 1 = 100%
4.
The risk-free rate reflects the time value of money and
productivity of capital.
4.
Other securities reflect time value and risk and offer a
risk-premium, i.e., a larger rate of return.
5.
Assuming homogeneous expectations for s, (all
investors use the same state probabilities in their
maximization problem), and that the price of an expected
$1 payoff contingent on state S occurring is s, then
1 + E(Rs) = [πs1 + (1 - πs)0]/ps = πs/ps
so that
ps = s s = s
$1
1  E ( Rs )
Where E(Rs) is the expected return for a dollar payoff in
state s. When investors highly value a dollar payoff in a
particular state, they will accept a smaller return and pay
a higher price (s) today for it.
Question: Why would investors value a dollar in state 1 more
than a dollar in state 2? Doesn’t the difference in
probability of the two states occurring already account for
this?
Diversifiable versus
Undiversifiable Risk
1.
The variation in aggregate wealth is undiversifiable.
Because total wealth will be lower during recession and
higher during expansion, someone must bear the risk of
realizing a low return (low wealth) during a recession.
2.
Those that accept the risk do so by purchasing securities
that pay unusually low returns in recessions and
unusually high returns in expansions [positive payoff
covariance with aggregate wealth (market portfolio of
securities)]. Their reward for doing this is that the
average returns for their securities over the full cycle of
recession and expansion is larger than that of others.
3.
If states 1 and 2 offer the same aggregate wealth (I.e.,
same security payout) and you hold more shares of pure
security 1 than 2, you are taking on diversifiable risk.
If state 1 occurs, you get a larger piece of total wealth
but if state 2 occurs you get a smaller piece. Had you
simply “diversified” and held the same number of shares
of each, you would have received the same wealth in
each state. Your expected wealth is the same but you
have introduced variance in the outcome.
Risk aversion implies that you should not accept
additional variance in your wealth unless you are offered
a larger expected wealth. But others in the securities
market will not offer a larger return to you because it is
costless for them to simply diversify to eliminate their
risk. They do not need you to bear it for them.
Decomposition of Pure
Securities Prices
We can rewrite the previous equation as follows
ps = s s = s
= s
$1
1  E ( Rs )
$1  1  r 
1  r 1  E ( Rs ) 
= s $1 1  E ( Rs )  r 


1 r 
1  E ( Rs ) 
This shows that the pure security price is determined by the
probability that the state occurs, the present value of a riskfree future payment of one dollar, and a risk adjustment factor.
The product of the first and third terms can be called the riskneutral probability.
For a security with much undiversifiable risk, its expected
return will be large, the risk adjustment term in square
brackets will be small, and the pure security price will be small
(holding s fixed).
Optimal Portfolio Choice
1.
Assume a perfect and complete market of pure state
securities exists. How do investors choose
shareholdings?
-
ps = prices of pure securities
-
s = state probabilities
-
Qs = number of pure securities
-
C0 = consumption at time 0
-
W0 = wealth at time 0
2. Investors maximize the utility of current consumption and
future wealth (which will be consumed), subject to the
constraint that current consumption and the value of
securities purchased does not exceed present wealth.
Max[U (C 0)  sU (Qs)]
C 0 ,Qs
s.t.
W 0  C 0   psQs
s
s
The Lagrangian is
L  U (C 0)   sU (Qs )   (W 0  C 0   psQs )
s
s
Note: There is no explicit time discount here but this could be
done explicitly or within the utility function. Also, the pure
security prices include an implicit market discount rate.
The first order conditions for C0, each pure security S, and 
L
U

 0
C 0 C 0
L
U
 s
 ps  0
Qs
Qs
for each S
L
 W 0  C 0   psQs  0
s

3. An interesting result is that
U
Qs
 ps
U
C 0
s
This says that optimization requires that I set the expected
marginal rate of substitution of consumption for each security
S, equal to the price of S, for all securities. That is, the utility
value I expect to give up now by reducing consumption now
and buying security S, should equal the amount I expect to
get in the future if state S occurs, the security pays off $1 and
I then consume that dollar (in the two period case).
It is clear that pure security S’s price reflects both the
probability that state S occurs and the utility value of a dollar
payoff in state S.
4. A related result gives the expected MRS between states
U
s
Qs ps
E[ MRS ] 

U
pt
t
Qt
This says that optimization requires that I set the expected
marginal rate of substitution of security S for each security t
equal to the ratio of the securities’ prices.
This result is simply a reflection of the fact that each security’s
value is measured in consumption terms. Once we have the
first result, the second follows from the maximization,
otherwise, we could buy securities that are “cheap” in terms of
the expected utility of consumption and sell the “expensive”
ones to improve our total utility.
Results for an Economy of
Many Consumer/Investors
1.
Pareto Optimality - if all consumers perceive the state
probabilities the same way (homogeneous
expectations), we can see from the previous result that
the actual MRS (not just the expected) between any two
states will be the same for all investors. We know the
actual MRS’s between states are equalized because
each investor knows what he will get in each state
because he knows his portfolio of state securities.
U
Qs ps / s
MRS 

U
pt / t
Qt
This is Pareto Efficient – no one can benefit from further
security trading (risk sharing).
Here again, the information transmission of the price
system is at work. Because everyone faces the same
prices, in general equilibrium, everyone’s MRS must be
equal or else trading occurs, prices change, and at least
one person ends up better off at the new prices and no
one else is worse off.
2.
Risk separation - with Pareto optimality, individual risk
preferences are equalized at the margin (there is one
price for risk) so the specific risk preferences of any one
investor should not affect a firm’s investment decisions.
Managers maximize expected NPV.
Once risk is traded through the securities markets, there
is one price for risk. Both managers and investors use it
to make their investment decisions.
3. If we assume everyone has a utility function with the same
constant relative risk aversion coefficient (strong assumption)
and the same rate of time discount, then growth rates of
consumption will be equalized across states and time
(CCAPM).
4. From the previous result, we can rearrange and for all
investors I and j we have:
Ui Ui
Qs Qt

Uj Uj
Qs Qt
This says that the ratio of marginal utilities of consumption
across investors I and j is equal for all states (I.e.,
independent of the state).
From the above equation, if aggregate consumption is larger
in state s than state t, then everyone must consume more in
state s than state t to keep the ratios across individuals equal.
Thus, any two states yielding the same aggregate
consumption are identical (consumers make the same
consumption choices in both states).
5. The Consumption Capital Asset Pricing Model (CCAPM)
prices assets using consumption as a primitive to replace the
market portfolio used in the usual CAPM. The intuition behind
the CCAPM is that the amount of aggregate consumption in a
state can be used to define the outcome of the state.
To see this more clearly, use a previous result and assume
that s = t , then
U
U
s
Qs Qs ps


U
U
pt
t
Qt Qt
This holds for all consumer/investors. When aggregate
consumption is larger in state s than in state t, then this
implies that the marginal utility of consumption is smaller in
state s than in state t, so that the price of a pure security for
state s is smaller than that for state t.
Thus aggregate consumption is said to be a sufficient statistic
for state outcomes. This assumes that utility is not state
dependent. That is, only the amount of consumption matters.
For example, if the only two states are sunny-consume-20
and rainy-consume-21, you must be better off in the rainy
state because you get to consume more. The fact that it is
rainy should have no effect on your utility.
Maximizing the Value of the
Firm
1.
How do firms decide which investments to make and
how many pure securities to issue to finance their
investments?
2.
Assume complete and perfect markets so that firm’s
production decisions don’t affect market prices for
securities or the completeness of the market.
- Qjs = j(Ij, s) = a production function for firm j.
Transforms current investment into future state
contingent consumer goods.
- Ij = investment by firm j
- Yj = value of the firm
Max Yj   psQjs  Ij   psj ( Ij, s )  Ij
Ij
s
s
The first order condition is
dYj   psj ' ( Ij, s)  1  0
dIj s
This says that the firm should continue to invest an extra $1
(increase I) as long as the summation over all states, of
the output in each state times the price of output in each
state, exceeds the $1 investment. ps = πs/(1 + E(Rs))
from earlier slide so discounting is in price.
Example: Consider two firms with the following data.
Firm A => stock price = 62, Investment cost = 10
Firm B => stock price = 56, Investment cost = 8
States
Payoffs
on Stock
Firm A
Firm B
Payoffs
on Investment
Firm A
Firm B
1
100
40
10
12
2
30
90
12
6
A.
First find the pure securities prices as before.
100p1 + 30p2 = 62
40p1 + 90p2 = 56
=> p1 = 0.5 and p2 = 0.4
B.
Use these to find the NPVs for each investment using
the first order condition given above.
NPVA = 10p1 + 12p2 – I = 10(0.5) + 12(0.4) – 10 = -0.2
NPVB = 12p1 + 6p2 – I = 12(0.5) + 6(0.4) – 8 = 0.4
Firm A should reject its investment and firm B should accept.
Question: If the pure security prices are p1 = 0.3 and p2 = 0.6,
(util max. set these), what should the firms do? How
about if the investment payoffs increased by 10%?
These results illustrate how the market price of a firm’s
securities signals investor preference for it’s payoffs. Firms
make more or less investments depending upon their
technology’s ability to produce payoff’s in states that
consumers consider valuable.