STOCHASTIC

302 REVIEW OF ECONOMIC STUDIES
III. Variations in risk aversion
If the investor's attitude towards risk alters then this will alter the form of (3.10).
A decrease in risk aversion can either be reflected by a decrease in a (dotted line (3.16)
in Figure 9) or a decrease in the value of B (dotted line (3.15) in Figure 9).
(3.10)
The discussion so far has assumed that there exists at least one asset which is riskless
and which the investor holds. As a rule this riskless asset is assumed to be money. Considering
the variations in purchasing power it may currently be a doubtful assumption
to suppose that an investor considers money riskless. It might therefore be of some
interest to study the case when all assets are risky.
(b) All assets risky
Assume that a choice is to be made between two investment projects. We assume
that the amounts of money invested in these two projects are xs and x2. The returns,
arising in the next period are r, and r2 per unit of x, and x2 respectively. As in the previous
discussion we assume that ry and r2 are normally distributed with mean iil and \x2 and
variance a\ and a\ respectively.
We maximize the expected return in the second period while requiring that the probability
that the return is equal to or higher than a certain value B is at least a. The proportional
tax is 1(0 i ( S 1) and the lump sum tax is K. Thus we have the problem
subject to
max EixsiQ -i)+x2r2(l - /) - K),
Xy+x2 £ M,
P(xiri(\-t)+x2r2(l-t)-K g B) S a,
xu x2 i 0.
PART III. STATIC PORTFOLIO SELECTION MODELS