STOCHASTIC

and
CONSUMPTION AND PORTFOLIO RULES 385
\ = hk-^-gk, k = 1,
where v, i) are arbitrary constants (v ^=0).
Proof. (I) (30) is a parametric representation of a line in the hyperplane
defined by £" wk* = l. 18 Hence, there exist two linearly independent
vectors (namely, the vectors of asset proportions held by the two mutual
funds) which form a basis for all optimal portfolios chosen by the individuals.
Therefore, each individual would be indifferent between choosing
a linear combination of the mutual fund shares or a linear combination
of the original n assets.
(2) Let V = NfP, = the total value of (either) fund where
Nf = number of shares of the fund outstanding. Let Nk — number of
shares of asset k held by the fund and fik = NkPk/V = percentage of
total value invested in the k-th asset. Then V = £" NkPk and
But
dV = Y.Nk dPk + £ i>*