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FE II – Week 11 Tutorial Questions

Consider the following problem of an individual faced with two possible future statesof-nature, j 2 {1, 2}. In state-of-nature j = 1 the individual receives income y1 whereas

in state-of-nature j = 2, individual i receives income y2 , y1 6= y2 . Let cj denote the

amount of the consumption good enjoyed if the state-of-nature is j. State-of-nature j

occurs with probability ?j and ?1 + ?2 = 1.

Prior to learning the state-of-nature, individuals have the ability to purchase (or

sell) assets. There are two assets and the returns to both assets are paid in units of a

consumption good. One asset is risky and pays a rate of interest rjb in state-of-nature j

with r1b 6= r2b . Denote the amount of the risky asset purchased by the individual by b. The

other asset is a risk-free asset and pays an interest rate of rf in both states-of-nature.

The amount of the risk-free asset purchased by the individual is given by a. Thus the

gross return of the asset portfolio (a, b) is (1 + rjb )b + (1 + rf )a in state-of-nature j.

The timing of events is as follows: first, the individual enters an asset market and

purchases a portfolio of assets. The individual is endowed with an initial portfolio (a, b)

which can be sold. The relative price of the risky asset is q and the price of the risk-free

asset is normalized to one. Hence it costs q units of the risk-free asset to purchase one

unit of the risky asset. The individual uses the market value of the portfolio endowment

to finance his/her desired optimal portfolio. After purchasing the desired asset portfolio,

nature reveals the state-of-nature. Asset positions are settled and consumption occurs.

The individual chooses the asset his/her asset portfolio to maximize expected utility.

Utility is derived from consuming the consumption good and is represented by a utility

function with the properties that more is better, u0 (c) > 0, there is diminishing marginal

utility to consumption, u00 (c) < 0 and limc!0 u0 (c) = 1 so that having some consumption
in each state-of-nature is strictly preferred to having no consumption in any state-ofnature.
The individuals objective function is
max {?1 u(c1 ) + ?2 u(c2 )} .
c1 ,c2 ,a,b
1. Write down the individuals optimization problem.
2. Write down the Lagrangean for the individuals problem.
3. Solve for the individuals optimal trade-o? condition between the risk-free and the
risky asset.
4. Now suppose that there many individuals and they are all identical. As they all
face identical income risk and have the same endowments, they all choose the
same portfolio and enjoy the same consumption in each state-of-nature. With
this assumption derive the Consumption CAPM equation that relates the expected
return of the risky asset to the expected return of the risk-free asset. Here you can
define the expected return of the risk-free asset by Rf = 1 + rf while defining the
? (1+r1b )+?2 (1+r2b ))
expected return of the risky asset by Rb = 1
.
q
1
5. Suppose that ?1 , ?2 , rf , q, r1b and r2b as such that
mean for c1 and c2 ?
u0 (c2 )
u0 (c1 )
=
1. What does this
6. Using the individuals optimal trade-o? condition, show what this implies for the
risk-free rate and the expected return of the risky asset. Explain this result in
words with reference to the CCAPM equation previously derived.
2
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Tags:
Financial Economics
Lagrange Multipliers
optimization problem
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