Lee Ohanian – Bunche Hall 8391 – ohanian@econ.ucla.edu
Macroeconomics
UCLA
1 A Dynamic Model of Asset Pricing
These notes describe a simple model we can use to price a variety of assets. An
asset is a claim to consumption in at least one state of the world. We can use
this model to value stocks, bonds, real estate, etc. The basic model we will use
is an endowment economy. The basic setup is the following. There will be a
single type of consumer in the economy. Consumption will be provided from an
infinitely-lived asset, which simply generates the consumption good costlessly
(think of fruit on a tree), but the amount is random and will fluctuate over
time. The household will own a proportional share of the asset. Also, consider
the “fruit” (endowment) that comes from the asset (the tree) as a dividend.
Now, the basic economic force that will be determining the price of the asset
will be as follows. There will be a competitive market for the asset in which you
can buy additional shares of the asset, or you can sell shares. Since everyone is
identical, the equilibrium price will be the value of shares today such that given
the price of the asset, no household has a preference for buying or selling. That
is, the price must be a value so that the household maximizes utility by simply
holding their existing number of shares. Note that this is the only equilibrium!
To see this, suppose the asset price was at a level that you wanted to buy
additional shares. Since everyone is identical, then if you want to buy, then
everybody wants to buy, and this will drive the price up to the point where no
one wants to buy. Similarly, suppose the asset price was at a level that you
wanted to sell some shares. Since everyone is identical, then if you want to sell,
then everybody wants to sell, and this will drive the price down to the point
where no one wants to sell. The details of all of this are below.
We will first determine optimal allocations as the solution to a planning
problem. Then we will decentralize the problem as a competitive equilibrium,
which will include a price for the asset.
There is a representative household, with preferences over an infinite stream
of consumption. The size of the total population is fixed and is given by N.
maxE
∞∑
t=0
βtu(ct)
The term ”E” is called the expectations operator. This just says that the con-
sumer will be maximizing expected utility. Think of the expectations operator
as you making your best possible forecast of the future.
Each period, there is an endowment of a non-storable consumption good,
denoted as X. This endowment is random. Think of it as fruit coming from a
tree. Sometimes this is called a “fruit tree economy”.
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This endowment is random. There are i different possible values for the en-
dowment. The endowment will thus randomly transit from one value to another
over time. In addition, the value of the endowment today may affect the prob-
ability of a particular realization of the endowment next period. (For example,
consider the temperature on any given day. Temperature tomorrow tends to be
related to temperature today.)
The notation for these conditional probablities is given as follows, in which
the realization of X today is given by Xi and the realization of X tomorrow is
given by Xj :
Pij = Pr(Xt+1 = Xj p Xt = Xi)
We will also require that these probabilities are well defined, in that they
sum to 1. That is for all possible realizations of X today, which we denote
as Xi,we have that the transition probabilities from the value today, Xi to all
possible values tomorrow satisfy the following property:
∑
j
Pij = 1
This is called a Markov process. A simple example of a Markov process is
the weather, and the outcomes are rain and sum. Here is the Markov transition
probability matrix for this example
Rain Sun
Rain Prr Prs
Sun Psr Pss
Here is the way to read this matrix. The upper left element, Prr is the
probability that it rains tomorrow, given that it is raining today. The upper
right element, Prs is the probability that it is sunny tomorrow, given that it is
raining today.
The lower left element is the probability that it rains tomorrow, given that
it is suuny today, and the lower right element is the probability that it is sunny
tomorrow, given that it is sunny today. Note that Prr + Prs = 1, and that
Psr + Pss = 1. This just says that if it is rainy today, then for certain it will
either be rainy or sunny tomorrow, and that if it is sunny today, then for certain
it will be rainy or sunny tomorrow.
There are also unconditional probabilities, which are just how frequently it
rains, and how frequently it is sunny. These probabilities, which we will denote
as π. This is also called the invariant distribution, and are given by the following:
πj =
∑
i
Pijπi
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The resource constraint requires that consumption, C cannot exceed the
endowment. Since we assume consumers prefer more of the good to less, then
it will be the case that all of the endowment is consumed:
Xt − Ct = 0 (1)
The planner’s objective is to maximize weighted discounted expected utility,
and the planner values each household equally. The optimal solution is to allo-
cate the endowment equally across households so that each household consumes
the endowment per-capita each period:
ct = Xt/N = xt
Now let’s set this up as a solution to a competitive equilibrium which will
allow us to form market prices.. Here is the idea. We want to construct a price
for the asset, which in this case is the fruit tree. We will see how the value of
the asset (the fruit tree) is influenced by preferences, the current endowment,
and the probability of future endowments.
As conventional, the numeraire will be the endowment/consumption good.
We assume that the household can buy or sell “shares” to the endowment. At
time 0, each household is endowed with an equal number of shares, s0, which
we normalize to 1, and at any date t, the number of shares that the household
begins the period with is denoted as st. Each share is entitled to a per-share
“dividend”, which is denoted as dt. Shares can be bought or sold, and the price
of shares at date t is denoted as qt The number of shares the household leaves
the period with is denoted as st+1.We will treat these shares as shares of equity,
or common stock. That is, like common stock, the share in our model yields a
random dividend, and the shares can be purchased or sold.
The household’s maximization problem is:
maxE
∞∑
t=0
βtu(ct)
subject to the budget constraint
st(qt + dt) ≥ ct + qtst+1
Recall that the term E is the expectations operator, which means that the
household will be forecasting the future, given the information that they have
available today.
We are now are dealing with a household operating under uncertainty. Re-
call that we have assumed that a household that is dealing with uncertainty
maximizes expected utility. To see how this works, consider the easiest case,
which is two possible outcomes, a high endowment, and a low endowment. De-
fine the transition probabilities as Pll, Plh, Phl, Phh. Suppose that the current
realization is low. Then the expected utility for next period is given by the sum
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of utility in state l, multiplied by the probability of state l occurring, and the
utility in state h, multiplied by the probability of state h occurring:
β{Pllu(cl) + Plhu(ch)}
Note that when the consumer maximizes expected utility, then the opti-
mization of this problem will result in expected marginal utilities in the future.
These marginal utilities will just be a probability weighted sum of the marginal
utilities in all the possible states. In this simple case, there are just two states.
To interpret this, note that all we are doing is taking the probability-weighted
average of utilities, in which we are multiplying the utility of the low state
tomorrow by the probability that occurs, and we are multiplying the utility of
the high state tomorrow by the probability that occurs, and then we just add
them together. In particular, the expected maringal utility at date t+1, given
today’s state is low, is given by
β{Pllu′(cl) + Plhu′(ch)}
The household chooses consumption (ct) and the number of shares for the
following period (st+1).
The Lagrangian is given by maximizing expected, discounted utility subject
to the budget constraint every period, and is given by:
L = maxE{
∑
βtu(ct) + λt[st(qt + dt)− ct − qtst+1]}
The household chooses consumption today (ct) and the number of shares for
tomorrow (st+1). Recall we have the expectations operator, which tells us that
we are making a forecast of all the terms within the brackets. Note that the
The first order maximization condistions are given by:
βtE(u′(ct)) = E(λt)
Today, however, note that the realization of the random variable has already
happened, so we don’t need to make forecasts for today. Thus, there is no need
for an expectations operator (the forecast for today is just what has already
happened today), and the first order condition becomes:
βtu′(ct) = λt
The first order condition for choosing how many shares to own does involve
an expectations operator for next period, and is given by:
qtλt = E(λt+1(qt+1 + dt+1))
Combining equations, we get:
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uctqt = βE(uct+1(qt+1 + dt+1))
The economics in this condition is very intuitive. Like all first order conditions,
it equates the marginal benefit of a choice with its marginal cost. The left hand
side can be interpreted as the marginal cost of purchasing an additional share of
the endowment. The purchase requires that it gives up qt units of consumption
for that share, and the household values those qt units at marginal utility:
uctqt. Acquiring an extra share will pay dividends next period, and moreover,
the share can be sold next period. Thus, the expected marginal benefit is equal
to βE(uct+1(qt+1 + dt+1)).
This Euler equation will help us construct a competitive equilibrium for this
economy. It is competitive, because each consumer is a price-taker; that is, they
view the price of the share parametrically. Note that there are two requirements
for an equilibrium. One is that there can be no trade of shares – this is because
this is a representative agent economy. In particular, if the price was such that
any one household wanted to sell shares, then all households would want to sell
shares, and the price must fall. Analagously, if the price was such that any one
household wanted to buy shares, than all households would want to buy, and
the price must rise. The equilibrium price will be such that households will
be indifferent between buying and selling the asset. No other price can be an
equilibrium price. The second is that each household consumes its per-capita
share.
To construct the equilibrium, note that we already know the allocations –
each household consumes the per-capita endowment: ct = xt. Now we can solve
for the price using the equilibrium allocations, and we need prices to insure
that consuming the endowment is indeed an equilibrium. Return to equation
to substitute in using the endowment, and solve the difference equation forward
to eliminate the fact that the expected future price appears on the right hand
side. For example, note that we can write the asset pricing equation as
uxtqt = βE(uxt+1dt+1 + βuxt+1dt+1 + β
2uxt+3qt+3)
Continuing with forward substitution, we get
qt = (1/uxt)E
∞∑
j=t+1
βj−tucjdj
The rate of return (ror) between on the asset t− 1 and t is defined as
rort,t−1 =
dt + qt
qt−1
− 1
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Note that the price of the asset depends on 4 objects: (1) discounting (the
value of β), (2) the form of the utility function, and in particular, the degree
of risk aversion, (3), the current endowment, (4), expectations of future endow-
ments. Let’s consider each of these in turn. The higher the rate of discounting,
the lower is the price of the asset. This is because the asset delivers claims to
consumption in the future, and the more the household discounts the future
(that is, the lower is β), then the lower will be the equilibrium price of the
asset. The form of the utility function is important because risk aversion affects
the marginal utility of consumption. For example, suppose that households are
risk neutral. In this case, we obtain the classic “present value model” of stock
valuation, in which the price of a share is the expected present value of future
dividends
qt = βE(dt+1 + βdt+2 + β
2dt+3 + …)
(Note that here, we assume that today’s dividend has been already paid out.
Thus, the security you purchase will not pay its first dividend to you until the
following period.)
Now consider the case in which utility is of the CRRA variety: c
1−σ
1−σ :
qt = βE(
(
dt
dt+1
)σ
dt+1 + β
(
dt
dt+2
)σ
dt+2 + β
2
(
dt
dt+3
)σ
dt+3 + ……)
For the case of log utility (σ = 1), and noting that the infinite sum of
β+ β2+ β3+ … converges to β
1−β , we get the following simple pricing function:
qt =
β
1− β
dt
In this case, the today’s endowment raises the price of the share, as a high
endowment induces households to try to defer consumption to the future, and
this drives up the price of the asset. For σ > 1, note that expectations of future
endowments now affect the price. All things equal, the expectation of higher
future endowments lowers the price, as households would like to shift some of
that future consumption to today. This force results in the price of the asset,
all things equal, declining, as households try to raise consumption today.
Adding Other Assets to the Model
Above, we show how to construct the competitive price of the risky asset.
Note that we can price any asset to this model, even if the asset is in zero net
supply. (Zero net supply means that the asset is not issued).
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To see this, consider adding a risk-free asset to the model. Without loss of
generality, consider a one-period asset. That is, if you purchase the asset today,
it pays off 1 unit of consumption tomorrow in any state of nature, and it has no
further payoffs after tomorrow. The budget constraint becomes:
bt + st(qt + dt) ≥ ct + qtst+1 + ptbt+1
where bt is the number of bonds, and pt is the price of acquiring a bond
today that pays off tomorrow. The first order condition for this bond is given
by:
uxtpt = βE(uxt+1)
pt = β
E(uxt+1)
uxt
and its rate of return is given by
uxt
βE(uxt+1)
Now let’s compare this risk-free asset to the risky asset. In particular, to
keep things simple imagine a one-period risky asset, so that the Euler equation
is
qt =
βE(uxt+1dt+1)
uxt
Suppose further that the unconditional mean of d is one. Now if we compare
the price of the risky asset and the price of the real asset, we see that the
difference in the price of the two assets will be determined by the covariance of
the realization of the endowment and the marginal utility:
qt =
β[E(uxt+1)E(dt+1) + cov(uxt+1, dt+1)]
uxt
If this covariance is positive, then the risky asset will sell for a higher price
than the risk-free asset. In contrast, if the covariance is negative, then the risky
asset will sell for a lower price than the risk-free asset. The intuition for this
is straightforward. If the covariance is positive, then the risky asset tends to
provide a relatively high endowment when the household needs it the most –
that is, when marginal utility of consumption is high (e.g. consumption is low).
On the other hand, if the covariance is negative, then the risky asset tends to
provide a relatively high endowment when the household does not particularly
need a high endowment (e.g. when marginal utility is low, which implies that
consumption is already high).
We can evaluate this covariance and see how this affects the pricing of the
asset. Again, let’s consider two assets, a safe asset with a sure payoff of one
unit, and the risky asset with an expected payoff of 1 unit.
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For the case of CRRA utility, we get the following:
cov(uxt+1, dt+1) = cov(d
−σ
t+1, dt+1) < 0
Note that the risky asset has a negative covariance between marginal utility
and the endowment. Since the covariance is negative, then it will sell for a lower
price, ceteris parabus, thaConditional on the average value of the dividend, then
a lower price will generate on average a higher rate of return than the safe asset.
In particular, we have the following prices for the safe asset (with a sure payoff
of 1 unit) and the risky asset (with an expected payoff of one 1 unit).
q
safe
t = βE(uxt+1)
q
risky
t = βE(uxt+1) + cov(d
−σ
t+1, dt+1)
Since the covariance term is negative, then the risk asset sells for a lower
price than the safe asset.
We can also assess how this impacts the rate of return on the two assets.
For the safe asset, we have:
E(rorsafe) = E
(
qsafet+1 + 1
q
safe
t
)
= E
(
(uxt+1 + 1
uxt+2
)
For the risky asset, (noting that E(d) = 1),we have:
E(rorrisky) = E
(
(uxt+1 + cov(d
−σ
t+1, dt+1) + 1
uxt+2 ++cov(d
−σ
t+2, dt+2)
)
We can also assess how this impacts the expected rate of return on the two
assets. For the safe asset, we have:
E(rorsafe) = E
(
qsafet+1 + 1
q
safe
t
)
= E
(
(uxt+1 + 1
uxt+2
)
For the risky asset, we have:
E(rorrisky) = E
(
(uxt+1 + cov(d
−σ
t+1, dt+1) + 1
uxt+2 ++cov(d
−σ
t+2, dt+2)
)
We can also see that E(rorrisky) > E(rorsafe). To see this, note that
cov(d−σt+1, dt+1) = cov(d
−σ
t+2, dt+2),, since the dividends at both points in time
are generated from the same statistical distribution. Since this covariance is
negative, then we are reducing the numerator and the denominator by the same
amount in the expression above. This mechanically means that E(rorrisky) >
E(rorsafe), given that the numerator is bigger than the denominator.
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