STOCHASTIC

7. Consider an investor having the quadratic utility function of Exercise 6. We are interested
in preference criteria for the dominance of Xi over X2 when the only information known is
that (i) 0 g Xi, X2 g k; (ii) the means and variances Hi,n2 and aua2 are known with
Afi = M2—ft £ 0; and (iii) Xt and X2 have symmetric distributions. Let i?, and R2 be the
length of the ranges of Xi,X2, i.e.,
Rt = inf [b-a\P(Xt >b) = P(X, < a) = 0], i = 1,2.
(a) Show that Rt^2(k-Vi)-
(b) Show that i?,S 2CT,.
Let A = max{/ii + ai, #2+a2}.
(c) Show that Xi is preferred to X2 if 2 A//L4 - p) - ACT 2 > 0.
(d) Given symmetric returns Xi, X2, X3 as above, choose
A = max{^i + EX2 and
3(EXi 2 -EX2 2 ) 2 < 4(EXi-EX2)(EXi 3 -EX2 3 ).
(b) Show that the condition in (a) is transitive, that is, if Xt is preferred to X2 and X2 is
preferred to X3, then Xi is also preferred to X3.
9. The Tobin separation theorem gives sufficient conditions for the optimal proportions
held in the risky assets to be independent of the investor's utility function and the proportion
of the investor's initial wealth that is invested in the risk-free asset. The theorem is proved
and discussed in Lintner's paper in a mean variance analysis setting. It is of interest to develop
similar separation theorems for wider classes of risk measures.
Suppose initial wealth is one dollar and that x = (x2, ...,*„) are the investments made in
the risky investments i = 2,..., n which have the known joint cumulative distribution function
F(p2, ...,p„). Let — x0 be the level of borrowing in a risk free asset at rate p0, and Xi be the
level of lending in a risk free asset at rate pi. The return is then £"= o Pi *i • Consider a risk
measure R(x) and suppose the R is homogeneous of degree m, differentiable, and concave
on the convex set K which represents constraints on the choice of x save the budget constraint.
Assume that {(x0, ...,x„)\xe K, x0 £ 0, Xi g 0, ^,1=0xi = 1} is compact.
(a) Suppose pa — Pi and that K = {x \ x S 0}. Show that the ratios of the optimal investments
in x2,...,x„ are independent of the optimal investment in the risk free asset as
long as this investment (x0 + Xi) is nonzero. {Hint: Develop the Kuhn-Tucker conditions
for
minR(x)
\
J^PiX, got, £ x, = 1, x, SO, i'= \,...,n, XoSO
1 = 0 1 = 0 J
and show that if x* satisfies them, then so does (/?i*o, /?i*i, Pix2,...,p2x^ for all
pi >0,p2> 0.]
(b) Show that the separation property holds if only lending is allowed and jt, > 0.
(c) Show that the separation property holds if only borrowing is allowed and x0 < 0.
Suppose p0 > pi and that K = {x | x g 0}.
(d) Show that it is never optimal to borrow at p0 and lend at pi.
(e) Show that the separation property holds unless x0 = Xi = 0.
COMPUTATIONAL AND REVIEW EXERCISES
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