STOCHASTIC

300 REVIEW OF ECONOMIC STUDIES
subject to
P(fcrTgfl)aa, ...(3.9)
where
rT = return from risky assets if (the whole) total fund is invested in risky assets
(assumed normally distributed),
k = proportion of total fund that is invested in risky assets,
HT = mean of rT,
aT = standard deviation of rT,
and B = risk level which the risky return must exceed at least with probability a.
Taking the deterministic equivalent of (3.9) we obtain
where
so that
In Figure 7 we have shown (3.10), (3.11) and (3.12).
It-F^o^B, ...(3.10)
c = kaT, and [t = kfiT, ...(3.11)
/.-^ff. ...(3.12)
According to (3.10) the solution must lie above the line (3.10). Since the solution
must lie on the line given by (3.11) and also on the boundary of the constraint set determined
by (3.10), the solution will be at A which corresponds to investing the proportion
fc, of the funds in risky assets. We can now use this formulation to study the effect of some
economic factors on the solution.
I. The effect of a proportional tax
A proportional tax with the possibility of offsetting losses will not affect (3.12), but
(3.11) will change to a — k(\ — t)oT and (3.10) will remain unaltered, since none of the
FIGURE 7
(3.11)
> o
324 PART III. STATIC PORTFOLIO SELECTION MODELS