State Preference Theory

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State Preference Theory

• Advanced economies facilitate individuals
  savings/consumption decisions and firms
  investing/financing decisions through securities
  trading.
• In equilibrium, securities supply equals demand
  and firms maximize profits while consumers
  maximize expected utility.
• 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.
• Todays 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?
• What is a security? A vector of payoffs
  associated with different states of the world at
  some future date.
• The investors portfolio can then be
  characterized as a matrix of possible payoffs on
  his securities.

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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.

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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, X23, 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.
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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
period-by-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.
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• 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, X230, 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.

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3. Cramers 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.
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Law of One Price

• Equilibrium in the securities market means that
  supply equals demand for all securities.
• 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.
• 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) gt r 1
  100
• The risk-free rate reflects the time value of
  money and productivity of capital.

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• Other securities reflect time value and risk and
  offer a risk-premium, i.e., a larger rate of
  return.
• 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) ps1 (1 - ps)0/ps ps/ps
• so that
• ps ?s ?s ?s
• 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? Doesnt
  the difference in probability of the two states
  occurring already account for this?

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Diversifiable versus Undiversifiable Risk

• 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.
• 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.
• 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.

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Decomposition of Pure Securities Prices
We can rewrite the previous equation as
follows ps ?s ?s ?s ?s ?s
This shows that the pure security price is
determined by the probability that the state
occurs, the present value of a risk-free future
payment of one dollar, and a risk adjustment
factor. The product of the first and third terms
can be called the risk-neutral 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).
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Optimal Portfolio Choice

• 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.
• The Lagrangian is
• 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.

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The first order conditions for C0, each pure
security S, and ? for each S 3. An
interesting result is that 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).
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It is clear that pure security Ss 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 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 securitys 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.
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Results for an Economy of Many Consumer/Investors

• 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 MRSs
  between states are equalized because each
  investor knows what he will get in each state
  because he knows his portfolio of state
  securities.
• 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,
  everyones 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.
• 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 firms 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.

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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 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).
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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 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.
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Maximizing the Value of the Firm

• How do firms decide which investments to make and
  how many pure securities to issue to finance
  their investments?
• Assume complete and perfect markets so that
  firms production decisions dont 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
• The first order condition is
• 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 ps/(1 E(Rs)) from
  earlier slide so discounting is in price.

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• Example Consider two firms with the following
  data.
• Firm A gt stock price 62, Investment cost 10
• Firm B gt stock price 56, Investment cost 8
• States Payoffs Payoffs on Stock on
  Investment Firm A Firm B Firm A Firm B
• 1 100 40 10 12
• 2 30 90 12 6
• First find the pure securities prices as before.
• 100p1 30p2 62
• 40p1 90p2 56
• gt p1 0.5 and p2 0.4
• 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?

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These results illustrate how the market price of
a firms securities signals investor preference
for its payoffs. Firms make more or less
investments depending upon their technologys
ability to produce payoffs in states that
consumers consider valuable.