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) ≤ αor 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) ≤ αor 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 risk-free rate

Stochastic Dominance

Another alternative to mean variance analysisDefine 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 mean-variance analysis?

If returns are normally distributed and short sales are allowed– Preferring higher mean for any standard

deviation leads to efficient frontierNo 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 mean-variance 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

derivativeThm: 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 AnalysisThe third moment

SKEWNESSWhat 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.