Utility Analysis of Chance-Constrained Portfolio Selection: A Correction

Extract

In [1, p. 999] I wrongly stated that “the solution locus generated by the chance-constrained problem is efficient (for the class of utility function implied by the expected wealth-probability of ruin criterion) if the assets follow a multinomial distribution with means above the survival level.” In support of this statement footnote 6 of [1] attempted to establish the quasiconcavity of the expected utility function

in the (μ, σ) plane, where F is the normal distribution, z = (s-μ)/σ < 0, s is the survival level, μ is the mean of the portfolio, σ is the standard deviation, and c is a positive constant. The condition

was claimed to hold for any two assets (μ₁., σ₁) and (μ₂., σ₂) on any given indifference curve of (1), where z_i. = (s-μ_i.)/σ_i., i = 1,2, and z_γ = (s−μ_γ)/σγ corresponds to the convex combination μ_γ = γμ₁ + (1−γ) μ₂, σ_γ = γσ₁+ (1−γ) σ₂, in the (μ, σ) plane. (Note that this is not the same as asserting that F(z) is a convex function of μ and σ, which it is not as an examination of its Hessian determinant would readily show.)