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Other Portfolio Selection Models Adapted by Team 2 Introduction Up to this point we have used the traditional mean-variance approach to portfolio management; – Investors are utility maximizers – Investors are risk averse – Security returns are normally distributed or utility functions are quadratic We will now look at other approaches to the portfolio problem Other Portfolio Selection Models Other models make less stringent assumptions about: – The investors choice framework – The form of the utility function – The form of the distribution of security returns Other Portfolio Selection Models Other models include: – The geometric mean return – Safety first – Stochastic dominance – Skewness analysis Geometric Mean Return Model Select the portfolio that has the highest expected geometric mean return Proponents of the GMR portfolio argue; – Has the highest probability of reaching, or exceeding, any given wealth level in the shortest period of time – Has the highest probability of exceeding any given wealth level over any period of time Opponents argue that expected value of terminal wealth is not the same as maximizing the utility of terminal wealth Properties of the GMR Portfolio A diversified portfolio usually has the highest geometric mean return A strategy that has a possibility of bankruptcy would never be selected The GMR portfolio will generally not be mean- variance efficient unless; – Investors have a log utility function and returns are normally or log-normally distributed Safety First Models A second alternative to expected utility theorem: safety first Says decision makers are unable/unwilling to go through the mathematics of the expected utility theorem and will use a simpler decision model that concentrates on bad outcomes Three Safety Criteria 1. Roy: The best portfolio has the smallest probability of producing a return below some specified level Minimize (Rp < RL) Where Rp = return on portfolio RL = minimum level to which returns can fall Three Safety Criteria If returns are normally distributed, the optimum portfolio exists when RL is the maximum number of standard deviations away from the mean. To determine how many standard deviations RL lies below the mean (if returns are normally distributed), minimize RL – Rp σp The portfolio that maximizes Roy’s criterion must lie along the efficient frontier in mean standard deviation space. Three Safety Criteria The use of Tchebyshev’s inequality produces similar results, same maximization problem – Gives an expression that allows the determination of the maximum odds of obtaining a return less than some number – Makes very weak assumptions about the underlying distribution Three Safety Criteria 2. Kataoka: Maximize the lower limit subject to the constraint that the probability of a return less than or equal to the lower limit is not greater than some predetermined value Maximize RL Subject to Prob(Rp < RL) ≤ α or RL ≤ Rp – (constant)σp Tchebyshev’s inequality produces same results as normally distributed returns Three Safety Criteria 3. Telser: An investor maximizes expected return, subject to the constraint that the probability of a return less than or equal to some predetermined limit is not greater than some predetermined number Maximize Rp Subject to Prob(Rp ≤ RL) ≤ α or RL ≤ Rp – (constant)σ The optimum portfolio lies in the efficient frontier in mean standard deviation space or does not exist Tchebyshev’s inequality produces same results Three Safety Criteria Under reasonable assumptions, safety criteria lead to mean-variance analysis and to the selection of a portfolio in the efficient set With unlimited lending and borrowing at risk-free rate, the analysis may lead to infinite borrowing – Possible problem with assumption that investors can borrow unlimited amounts at riskfree rate Stochastic Dominance Another alternative to mean variance analysis Define efficient sets under alternative assumptions about general characteristics of investor’s utility function – Three stronger assumptions • First Order: non-satiation • Second Order: risk averse (includes first) • Third Order: decreasing absolute risk aversion (includes previous two) Idea behind First-Order Dominance Formal Theorem: “If investors prefer more to less, and if the cumulative probability of A is never greater than the cumulative probability of B and sometimes less, then A is preferred to B.” Second-Order If the two curves cross, a choice is not possible Therefore, we have to make the second assumption (Risk Aversion) – Investor must be compensated for bearing risk • Decreasing marginal utility Idea behind Second-Order Formal Thm: If investors prefer more to less, are risk averse, and the sum of the cumulative probabilities for all returns are never more with A than B and sometimes less, then A dominates B with second order stochastic dominance. How does this relate to meanvariance analysis? If returns are normally distributed and short sales are allowed – Preferring higher mean for any standard deviation leads to efficient frontier No short sales – First order produces a set of portfolios that lie on upper half of outer boundary of the feasible set • This includes efficient set produced by meanvariance analysis Relating Con’t Second-order assumptions also lead to the efficient set theorem – Thus, with normal returns, the only set of portfolios that is not dominated, using second order, is the mean-variance efficient set Advantage of Stochastic Dominance Used to derive sets of desirable portfolios when returns are not normal or when the investor is unwilling to assume specific utility functions But direct use is infeasible – Infinite set of alternatives in portfolio selection Third-Order Assumes decreasing absolute risk aversion – Function exhibiting this is positive third derivative Thm: If the theorem for second order holds true, plus if the third derivative of the investor’s utility function is positive and the mean of A is greater than the mean of B, then A dominates B. Skewness and Portfolio Analysis The third moment SKEWNESS What is it? – Skewness measures the asymmetry of a distribution – A Normal Distribution has a skewness of??? ZERO – Empirical evidence indicates that investors actually have positive skewness…they prefer a higher probability of larger payoffs Skewness and Portfolio Analysis Value at Risk Semivariance – Measures downside risk relative to a benchmark VaR is the standard measure of downside risk – Measures the least expected loss that will be obtained at a given probability – Dependent on an accurate measure of tail area probability Alternative: Tail Conditional Expectation – Measures downside risk by a measure of the expected return less than the benchmark. This corresponds to the first lower partial moment of returns…gives rise to a first-order stochastic dominance criterion for choosing among portfolios VaR The use of VaR and other downside risk measures is motivated by the inadequacy of variance as a risk measure Unfortunately, downside risk measures are difficult to compute and work with in cases where the distribution of returns is asymmetrical VaR will be an underestimate Roy Criteria Investment alternatives leading to 100% investment in the riskless asset. Investment alternatives leading to infinite borrowing.
