STOCHASTIC

Some properties of the expected value semivariance criterion are discussed in Exercise 8.
This exercise discusses a solution approach for such problems.
Let the investment allocations be x = {xu ...,xn)' and the random return vector be
p = (/?!, ...,p„)'. Hence end-of-period wealth is w = p'x. The semivariance of w is Sh(x) =
$min{0,p'x-h} 2 dF(p).
(a) Show that Sh is a convex function of x.
(b) Show that Sh is continuously differentiable and
8 (X) 2
" 2 I min{0,p'x-k} p,dF(p)
8x, 'Jas
long as the variance-covariance matrix of p has finite components. [Hint: Utilize
the dominated convergence theorem (Exercise 13) along with the mean-value theorem
and the Cauchy-Schwarz and triangle inequalities to show the existence of the partials.]
The efficiency problem is
^(a) = min5k(x), s.t. x e K, p'x £ a,
where Xis assumed to be convex and compact.
(c) Show that ^ is a convex function of a. Graph this function and interpret the meaning
of its shape.
(d) Show that the efficient boundary may be found by solving {minSh{x) — Xp'x\xe K)
for all A £ 0.
Suppose the pt have the joint discrete distribution p,[p = p' > 0,1= \,...,L < oo ].
(e) Develop the expressions for p'x, Sh{x), and 8S,(x)jdx,.
(f) Assume that K = {x \ Ax S b, x 2: 0). Illustrate the use of the Frank-Wolfe algorithm
(Exercise 14) to solve the efficiency problem for a given a.
(g) Discuss the advantages and disadvantages of the algorithm in (f). How could the
algorithm be efficiently modified to solve the problem when a takes on many monotonic
discrete values.
(h) Suppose that K takes the simple form K = {x|0g x, g#, £*f= 1}, for fixed
constants jff, £ 0. Show how the algorithm simplifies. Find a simple way to solve the
direction-finding problem (dfp). [Hint: Note that the dfp is a knapsack problem in
continuous variables.]
(i) Show that a lower bound on the optimal solution value is
max{Sh(x k ) - tyx" + V5„(x J )'(/-^)},
where / is the present iteration number, x" is the feasible solution at iteration k, and y k
solves the direction-finding problem in iteration k.
26. (Chance constraints and the aspiration and fractile objectives) Consider the linear
program
minc'x, s.t. Ax £ b, x £ 0,
where A is m x n, b is m x ], c is n x 1, and the decision vector x is n x 1. When some or all of
the coefficients of b and A are random variables one way to extend the formulation is by
introducing marginal chance constraints by requiring that each row of Ax & b be satisfied
most of the time, that is,
where the /?, e [0,1] are given constants.
Pr[^iX^6,] £ P,, i= \,...,m,
358 PART III STATIC PORTFOLIO SELECTION MODELS