Asset Pricing

In the previous sections we used the Euler equation to derive optimal consumption and investment decisions. Now note that the Euler equation defines a relation between consumption (or other real variables) and prices.

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Previously we took the prices as given and derived the optimal path of consumption. We may, however, use the Euler relation in the other direction. Take the path of real variables, e.g., consumption as given and derive what the prices have to be. A straightforward way to do this is to assume that output is exogenous, like manna from heaven, and cannot be stored. In that environment we may introduce markets for capital and production facilities. This is the setup in the seminal Lucas (Econometrica, 1978) paper.

We will later relax the assumption about exogenous and non-storable output. This will give us the basic stochastic growth (or RBC) model.

Asset Pricing in the Lucas Tree Model and other CAPM

Asset Pricing in the Lucas Tree ModelLarge number of identical agents.

Equal number of trees with stochastic crop

dtThe distribution ofdtis Markov. Distribution is. The process known by all agents.Purpose: find

pt– the price of a tree as a function of the state of the economy (dt).The gross return on a tree is (per capita)

No safe asset.

Perfect market in ownership of trees. All equal so no trade in equilibrium.

No storage or foreign trade so consumption

ct=dt

Substitute for from the Euler equation noting that *w *=0 gives

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By 7)

For any type of expectations in based on *dt *we
can compute a price today as a function of *dt *Let the
individuals subjective expectations of *dt+1* be described
by and the
expectations about the relation between *pt+1 *and *dt+1*
be given by the function *ps*(*dt+1*). We then have

Now Lucas defined the very powerful concept of *rational
expectations*. Let’s require that and . Lucas proves that this together with defines a *unique*
and* *constant pricing function .

Use recursions on

A discounted sum of dividends. Stochastic discount rates unless marginal utility is constant.

A simple example with log utility

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With *dt *i.i.d. so

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Note that the price increases in *dt *(if *U
*is concave).

Assume an autocorrelation in *dt *then depends on *dt *Both
income and substitution effect, with log utility they cancel.

The Consumption and market CAPM

Previously we found that

The second equation in implies

.

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Substituting into the first equation in yields

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Consider a similar problem but with *n*
risky assets each with a stochastic net return of * *(º -1). FOC for each risk
asset *i *yields after the same substitution as above

So

*This is the Consumption CAPM.*

So a risky asset can have an expected rate of return that is larger or smaller than the safe return. Note that a positive covariance between consumption and the risky return implies a negative covariance between marginal utility and so that asset will have a risk premium.

Note that if we have CRRA () then we have

where I have used that .

Now assume there exists an asset m which return is perfectly negatively correlated with marginal utility. So and thus .

Then from

Substitute into

This is the *market or traditional CAPM*.
Note that *b**i *is the (true) regression coefficient in a
regression of asset *i *on *m. *The term can be interpreted as
the price of aggregate or systematic risk.

Empirics

The Mehra – Prescott Puzzle

Consider a representative household that "maximizes"

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*l* is
a stochastic growth rate that can take *n *different values all >0. The
probability of a specific growth rate depends only on last
periods growth rate.

.

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Assume there is a share that entitles the owner to the entire output the next period. From we have that the price of this share is

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where *i* Î {1,...,*n*}. So the
price is H(1) in *c *and *y.*

We can also use to get

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This is a linear equation system in *n *unknowns
so we can solve it for the price of the share in all states of
the world. Now we can calculate the net return on the asset

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and expected return is

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We can also compute the price of a safe asset in this economy.

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with a return of .

Now we want to find the unconditional (average)
returns on the assets. First we need the unconditional
probabilities of the states *p.*

Assume ergodic growth rates

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Then

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The risk premium is then defined as .

Results

Now simplify and assumed there is two states and a common transition probability.

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Now we calibrate the model using the mean growth rate of GDP, its variance and autocorrelation.

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Using these value and some reasonable values
for *b* and *a* <10 we find that the risk premium should be
something in the order of 0 to 0.4%. But on average the US stock
market has yielded 6% more on average than government.

Consumption versus marketb(Mankiw & Shapiro 1986)

Mankiw and Shapiro first estimates from 464
different stocks. They first estimate the market *b*, i.e., assuming it is
constant over time. Note that the coefficient on *b *should be
equal to the equity premium (around 6%) and constant for all
periods and assets. Then they use this variable to predict the
return on the corresponding asset.

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The coefficient is significant and around 6 for most estimation methods.

Now rewrite

Assume that all relevant moments in are
constant over time. We then *b**ci** *as sample moments

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Then we can run the regression

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Now the estimate of *g*1 is insignificant and unstable. M&S also run a
regression with both *b,
*then only the coefficient on *b**mi *comes
out significant.