STOCHASTIC

298 REVIEW OF ECONOMIC STUDIES
F ("y)
FIGURE 6
3. THE EFFECT OF TAXES IN CHANCE-CONSTRAINED MODELS
(a) One riskless and several risky assets
In this section the separation theorem of Tobin [20] plays a central role. This
theorem states, as was mentioned above, that (for risk averters in purely competitive
markets) the proportionate composition of the non-cash assets is independent of their
share of the total asset holdings when utility functions are quadratic, or the rates of
return are normally distributed.
This means that in the event that some of the riskless asset is held, the proportions
in which the risky assets are held in an optimal portfolio are independent of wealth, the
minimum return level and the risk level. If there existed a known distribution of stock
prices, then all investors would hold the same proportions of risky assets and the theorem
could be tested empirically. 1
Proof of the separation theorem using the E-formulation of chance-constrained programming
and assuming normally distributed asset returns with two constraints.
Assume that the investor can invest in one riskless asset, with return #•„ and in risky
assets whose rate of return r, are distributed with mean nt and variance erf, (i = 2, ..., N).
The amounts invested in the risky assets are denoted by x{, (i = 1, ..., N). We would then
try to solve the problem
subject to
maxl rlxi + Z /W
i