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7
Random Rate of Return
When the return (or rate of return) is random, its exact value is unknown in advance though its probability
distribution is known. If even the probability distribution is unknown, we call it
uncertain
return.
In previous example, if the return
R
is random, then
is random, and hence the utility level
is also random. How do we make the savings decision? If there are more than one investment choice, how
do we allocate the savings among different investment instruments?
To answer these questions, we have to find the opportunity set (choice set) and have to know preferences of
the investor over the choices. The portfolio theory presented in the textbook focuses on
 the allocation of savings among different investment opportunities, assuming that the savings decision
has already been made (i.e., the first period consumption
is already determined). This excludes the
case in which one reduces the current consumption when he expects an exceptionally higher return from
the investment.
 the opportunity set defined by mean and standard deviation (risk) of the random return of the portfolio.
The second point implicitly assumes that investors care only about the mean and risk. The textbook does not
have much discussion about why investors are concerned only about the mean and risk, or how they choose
the best portfolio from the opportunity set. More fundamentally, what kind of investors are concerned only
about mean and risk of the outcomes? Why is the standard deviation of returns a reasonable measure of the
risk?
We will first discuss the preference over the random outcome
W
, which is the level of wealth at the end of
the investment period. This part includes expected utility theory, definition of risk aversion, risk premium,
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 Fall '08
 Staff

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