STOCHASTIC

(i) Suppose that V0 is positive definite. Show that the aspiration model has as its
deterministic equivalent
mm(c'x-d0)l(xV0xy n , s.t. Ax ^ b, x 6 0,
assuming x = 0 is not optimal.
(j) Show that x = 0 is optimal for the aspiration model if and only if d0 & 0.
27. Suppose investments X and Y have a common mean and finite positive variances.
Suppose an investor has initial wealth of one dollar and has a strictly increasing and strictly
concave utility function over wealth w. To maximize expected utility, one must maximize
{(X) = Eu[XX+(\ — A)y]|0S X g 1}. It is of interest to investigate how diversification is
affected by possible negative dependence between X and Y.
(a) Suppose X has a uniform distribution on [0,1] and that Y =— \2X 1 +\QX—\.
Show that cov (X, Y) = - i < 0 and EX = EY = i.
(b) Suppose that
{X if X£ TV,
u(X) = U- + 0 is sufficiently small. Show that « is strictly increasing, strictly concave, and
differentiable.
(c) Show that X* = 1 so that it is not optimal to diversify. [Hint: Show that d(\)ldX =
£[«'(*)(*-r)]>0.]
(d) Suppose X arxA u are as defined in (a) and (b), and that Y= MX 1 — lOA'+i. Show
that EX= EY= i, cov(X,Y) > 0, and that 0 < X* < 1. Hence, negative correlation is
neither necessary nor sufficient for diversification.
(e) Show that X* < 1 if and only if E[{X- Y) «'(*)] < 0.
(f) Show that the hypotheses with regard to the result in (e) may be weakened so that
u is concave and not necessarily differentiable or u is pseudo-concave. Show that the
monotonicity assumption is crucial, however.
(g) Show that 0 < X* if and only if E[(Y- X) «'(T)] < 0.
(h) Show that there is diversification, that is, 0 < X* < 1, if and only if
/•CO 1*2 j*GO /»2
(x-y)dF(x,y) SO and (y-x)dF(x,y) ^ 0
Jy= — oo Jx= — co Jx= — 00 Jy= — oo
for all z, provided that E[(X-Y)u'(X)] ^ 0 # E[(Y-X) u'(Y)l
(i) Use (h) to show that if all risk-averse investors should diversify between X and Y,
then EX = EY. [Hint: Let z -> oo.]
28. Refer to Exercise 27. It is of interest to develop a concept of negative dependence that
always leads to diversification in the two-asset case.
(a) Suppose that E(Y | X = x) is a strictly decreasing function of x. Show that it is optimal
to hold some Y that is X* < 1. [Hint: Show that
E[(X-Y)u\X)] = cov[A-,«'W] - cov[E(Y\X),u'(X)] < 0.
(b) Suppose that E(X\ Y = y) is a strictly decreasing function of y. Show that it is optimal
to hold some X. Hence if both conditional means are decreasing functions, the investor
will always diversify.
(c) Show that if P[YiS y | X= x] is strictly increasing in x for each y, then E(Y\X= x)
is strictly decreasing in x, and that the converse is false.
(d) Show that if E(Y\X=x) is strictly decreasing in x, then co\(X,Y) < 0 and the
converse is false.
360 PART III STATIC PORTFOLIO SELECTION MODELS