Mean-Variance Analysis
In this chapter, we leave the certain world and introduce risk into our analysis. We start with a review
of the expected utility hypothesis. We note the connection between risk-aversion and diminishing
marginal income. We then introduce the concept of a risky asset, which is practically any asset that is
available for investment (because the real world is always uncertain). Using EUH, we can compare
different risky assets. Our next step is to study portfolios, which are baskets of assets. Using EUH, we
can compare not only assets, but also portfolios. In this setting, the Markowitz theory of optimal
portfolio tells us how the investor can conceptually choose the best portfolio, i.e. the portfolio that
maximizes his/her EU. Finally, we compare the optimal portfolios of different investors. It is the
remarkable insight of Markowitz that these optimal portfolios are actually very similar to each other.
1. Expected Utility Hypothesis
Let
U
(
y
) be a utility function which is defined over income. The income that an asset gives is
equal to capital gain plus dividend. Both capital gain and dividend may be uncertain. To decide which
asset we prefer to hold depends on the relative size of these capital gains and dividends.
To enable to
make comparisons among assets, we use the expected utility hypothesis.
Suppose asset income (or equivalently, asset return) can take on
n
different values,
y
1
,...,
y
n
,
with probabilities
p
1
,...,
p
n
respectively. Then the expected utility of holding the asset is
EU =
p
1
U
(
y
1
)+.
..+
p
n
U
(
y
n
)
To apply EUH, we calculate the EU for each asset and rank them, the most preferred asset being the
one having the highest EU. If the random return can take on continuous values, the idea is the same,
but instead of summing, we need to take the following integral over the entire range of return,
EU =
f
(
y
)
U
(
y
)
dy
where
f
is the (probability) density function.
The function
U
reflects the investor's attitude towards risk. The shape of
U
-- whether linear,
concave, or convex -- corresponds to the three main types of investor preference -- risk-neutral,
risk-averse, and risk-loving.
Consider a chance game whose expected return is 0.(Also known as an actuarially fair bet).
An example is the toss of a fair coin: head, you win $1; tail, you lose $1. If an investor starts out
RN
RA
RL
y
U(y)
U(y)
y
y
U(y)