Expected return—expected loss approach to optimal portfolio investment

Abstract

Standard models of portfolio investment rely on various statistical measures of dispersion. Such measures favor returns smoothed over all states of the world and penalize abnormally low as well as abnormally high returns. A model of portfolio investment based on the tradeoff between expected return and expected loss considers only abnormally low returns as undesirable. Such a model has a comparative advantage over other existing models in that a first-order stochastically dominant portfolio always has a higher expected return and a lower expected loss. Expected return—expected loss model of portfolio investment can rationalize the equity premium puzzle. Two random variables are not comoving if there is at least one state of the world in which one random variable yields a positive return and the other—a negative return. Such random variables provide hedging benefits from diversification in portfolio investment according to the expected return—expected loss model. A special case of this model, when an investor linearly trade-offs expected returns and expected losses, is also a special case of the prospect theory when a decision-maker has a piece-wise linear value function without any probability weighting.

Appendix

1.1 Proof of proposition 1

For any portfolio α we denote the return of this portfolio in state of the world s_i as

$$ R_{i} \left( {\varvec{\alpha}} \right)\mathop = \limits^{{{\text{def}}}} \mathop \sum \limits_{j = 1}^{m} \alpha_{j} R_{j} \left( {s_{i} } \right) $$

For any two portfolios α and β let X denote the set of their possible returns:

$$ X\mathop = \limits^{{{\text{def}}}} \bigcup\limits_{i = 1}^{n} {\left\{ {R_{i} \left( {\varvec{\alpha}} \right) , R_{i} \left( {\varvec{\beta}} \right)} \right\}} $$

Let N be the number of elements in set X and let us number those elements by subscript k in the ascending order so that x₁ is the minimum element and x_N is the maximum element in X. Finally. We denote the cumulative distribution function of portfolio α as

$$ F_{k} \left( \alpha \right)\mathop = \limits^{{{\text{def}}}} \mathop \sum \limits_{{\begin{array}{*{20}c} {i = 1} \\ {R_{i} \left( \alpha \right) \le x_{k} } \\ \end{array} }}^{n} p\left( {s_{i} } \right) $$

If portfolio α first-order stochastically dominates portfolio β then $F_{k} \left( {\varvec{\alpha}} \right) \le F_{k} \left( {\varvec{\beta}} \right)$ for all $x_{k} \in X$.

Expected return (3) of portfolio α can be then written as

$$ \begin{aligned} ER\left( {\varvec{\alpha}} \right) & = \mathop \sum \limits_{j = 1}^{m} \alpha_{j} ER_{j} = \mathop \sum \limits_{i = 1}^{n} R_{i} \left( {\varvec{\alpha}} \right)p\left( {s_{i} } \right) \hfill \\ \;\;\;\;\;\;\;\;\;\;\; & = x_{N} + \mathop \sum \limits_{k = 1}^{N - 1} F_{k} \left( {\varvec{\alpha}} \right)\left[ {x_{k} - x_{k + 1} } \right] \ge x_{N} \hfill \\ \;\;\;\;\;\;\;\;\;\;\; & + \mathop \sum \limits_{k = 1}^{N - 1} F_{k} \left( {\varvec{\beta}} \right)\left[ {x_{k} - x_{k + 1} } \right] = ER\left( {\varvec{\beta}} \right) \hfill \\ \end{aligned} $$

Let X_- denote the set of all losses (negative returns) in X and zero:

$$ X_{ - } \mathop = \limits^{{{\text{def}}}} \left\{ {x \in X{|}x < 0} \right\} \cup \left\{ 0 \right\} $$

Let L be the number of elements in set X_-. As before. We keep the numbering in the ascending order so that x₁ is the minimum element (the biggest loss) in X_- and x_L = 0_. Expected loss of portfolio α can be then written as

$$ EL\left( \alpha \right) = - \sum _{{\mathop {i{\text{ }} = {\text{ }}1}\limits_{{R_{i} \left( \alpha \right) < 0}} }}^{n} R_{i} \left( \alpha \right)p\left( {s_{i} } \right) = \sum\limits_{{k = 1}}^{{L - 1}} {F_{k} } \left( \alpha \right)\left[ {x_{{k + 1}} - x_{k} } \right] \le \sum\limits_{{k = 1}}^{{L - 1}} {F_{k} } \left( \beta \right)\left[ {x_{{k + 1}} - x_{k} } \right] = ER\left( \beta \right) $$

Q.E.D.