Design of Efficient Investment Portfolios with a Shortfall Probability as a Measure of Risk

Abstract

The paper presents a constructive description of the set of all efficient (Pareto-optimal) investment portfolios in a new setting, where the risk measure named “shortfall probability” (SP) is understood as the probability of a shortfall of investor’s capital below a prescribed level. Under a normality assumption, it is shown that SP has a generalized convexity property, the set efficient portfolios is constructed. Relations between the set of mean-SP and the set of mean-variance efficient portfolios as well as between mean-SP and mean-Value-at-Risk (VaR) sets of efficient portfolios are studied. It turns out that mean-SP efficient set is a proper subset of the mean-variance efficient set; interrelation with the mean-VaR efficient set is more complicated, however, mean-SP efficient set is proved to be a proper subset of mean-VaR efficient set under a sufficiently high confidence level. Besides a normal distribution, elliptic distributions are considered as an alternative for modeling the investor’s total return distribution. The obtained results provides the investor with a risk measure, that is more vivid than the variance and Value-at-Risk, and with determination of the corresponding set of effective portfolios.

APPENDIX

The proof of Theorem 1. Since Φ(x) is an increasing function, a necessary condition of efficiency of a fixed portfolio a* ∈ A^SP is that it must solve the problem

$$\left\{ \begin{gathered} \min (\alpha - \mu (a)){\text{/}}\sigma (a), \hfill \\ \mu (a) = M, \hfill \\ a \in A = \left\{ {a \in {{R}^{n}}\,:\sum\limits_{i = 1}^n {{{a}_{i}}} = 1} \right\}, \hfill \\ \end{gathered} \right.$$

(A.1)

where M = μ(a*).

Suppose, at first, that α $\geqslant$ M. If α = M then a* is not efficient since any portfolio a¹ : μ(a¹) > M dominates a* in the sense that (α – μ(a¹))/σ(a¹) < (α – μ(a*))/σ(a*) = 0 and μ(a¹) > μ(a*). If α > M then problem (A.1) reduces to maximizing σ(a). It is easy to construct a portfolio sequence {a^m} such that μ(a^m) = M and σ(a^m) → ∞ as m → ∞. Then, for sufficiently large m, σ(a^m) > σ(a*) and, hence, a^m dominates a*. We have shown that a condition α < M is necessary for efficiency of a*. Under this condition, problem (A.1) reduces to

$$\min {{\sigma }^{2}}(a),\quad {\text{s}}{\text{.t}}{\text{.}}\;\;\mu (a) = M,\;\;\sum\limits_{i = 1}^n {{{a}_{i}}} = 1.$$

(A.2)

Problem (A.2) is already solved by a standard method of Lagrange multipliers (see, e.g., [14, 17]). It is shown that (A.2) has a unique optimal point

$$a \text{*} ( = a(M)) = \frac{1}{\Delta }\left[ {{\mathbf{1}}\left\| m \right\|_{C}^{2} - m{{{\left\langle {{\mathbf{1}},m} \right\rangle }}_{C}} + M\left( {m\left\| {\mathbf{1}} \right\|_{C}^{2} - {\mathbf{1}}{{{\left\langle {{\mathbf{1}},m} \right\rangle }}_{C}}} \right)} \right]{{C}^{{ - 1}}}.$$

(A.3)

Now, we will investigate the intervals of monotonicity of the function SP[α, X_a(M)] = Φ((α – M)/σ(a(M)). Taking into account that

$${{\sigma }^{2}}(a(M)) = \left\langle {a(M),a(M)C} \right\rangle = \left( {{{M}^{2}}\left\| {\mathbf{1}} \right\|_{C}^{2} - 2M{{{\left\langle {{\mathbf{1}},m} \right\rangle }}_{C}} + \left\| m \right\|_{C}^{2}} \right){\text{/}}{{\Delta }^{2}},$$

the derivative

$$\frac{d}{{dM}}\Phi ((\alpha - M){\text{/}}\sigma (a(M))) = \frac{{\phi ((\alpha - M){\text{/}}\sigma (a(M))}}{{{{\Delta }^{2}}{{\sigma }^{3}}(a(M))}}\left[ { - {{\Delta }^{2}}{{\sigma }^{2}}(a(M))} \right.\left. { - (\alpha - M)\left( {M\left\| {\mathbf{1}} \right\|_{C}^{2} - {{{\left\langle {{\mathbf{1}},m} \right\rangle }}_{C}}} \right)} \right],$$

where ϕ(x) > 0 denotes the density of standard normal distribution. Consider the function in the square brackets,

$$r(M) = M\left( {{{{\left\langle {{\mathbf{1}},m} \right\rangle }}_{C}} - \alpha \left\| {\mathbf{1}} \right\|_{C}^{2}} \right) - \left\| m \right\|_{C}^{2} + \alpha {{\left\langle {{\mathbf{1}},m} \right\rangle }_{C}}.$$

(A.4)

(1) Let α $\geqslant$ α₀ = 〈1, m〉_C/$\left\| {\mathbf{1}} \right\|_{C}^{2}$. If α > α₀ then, as M > α > 0, r(M) < r(α) = $- {{\alpha }^{2}}\left\| {\mathbf{1}} \right\|_{C}^{2}$ – $\left\| m \right\|_{C}^{2}$ + 2α〈1, m〉_C. By Cauchy–Schwartz–Bunyakovskii inequality, we have r(α) < –(α||1||_C – ||m||_C)² $\leqslant$ 0. If α = α₀ then r(M) ≡ $- \left\| m \right\|_{C}^{2}$ + α₀〈1, m〉_C < 0.

(2) Let α < α₀. It follows from (A.4) that r(M) > 0 (=0) if and only if

$$M > ( = ){{M}^{{SP}}} = \frac{{\left\| m \right\|_{C}^{2} - \alpha {{{\left\langle {{\mathbf{1}},m} \right\rangle }}_{C}}}}{{{{{\left\langle {{\mathbf{1}},m} \right\rangle }}_{C}} - \alpha \left\| {\mathbf{1}} \right\|_{C}^{2}}},$$

(A.5)

i.e., the function SP[α, X_a(M)] increases only on the interval [M^SP, ∞). To sum up, (i) the condition α < 〈1, m〉_C/$\left\| {\mathbf{1}} \right\|_{C}^{2}$ is necessary and sufficient for existence of an efficient mean-SP portfolio, (ii) the set of efficient mean-SP portfolios is defined as A^SP = {a(M), M ∈ [M^SP, ∞)}, the expressions for a(M) and M^SP are given by (A.3) and (A.5), correspondingly.

$\square$