Transcript Other Portfolio Selection Models

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.