Portfolio Optimisation for the Anxious

Greg B Davies, PhDHead of Behavioural Finance

greg.davies2@barclayswealth.comJune 2010

2

“The market can stay irrational, longer than you can stay solvent” John Maynard Keynes

Harry Markowitz – Nobel Prize 1990Daniel Kahneman – Nobel Prize 2002

Value function in descriptive theories: CPT

Losses (£)

Utility

Loss aversion: Steeper for losses

Reference Point

Gains (£)

4

And with risk=variance, expected utility theory is equivalent to mean-variance optimisation

Portfolio Efficient Frontier

r

Risk free rate

Market Portfolio

Optimal Portfolio

2,, rfRiskrfxuE

Risk/Return Trade-off

(Indifference Curve)

5

Only works under specific, and largely incorrect assumptions

Only true if one of two assumptions holds:

1.Log Returns are normally distributed (no fat tails; no black swans; no skewness)

OR

2.Individuals have a rational utility function that is quadratic

Neither assumption is valid

2,rfxuEEU Example Quadratic Utility Function

Return

Utility

6

This means both the risk measure and the risk-return trade-off are flawed

r

Risk free rate

2,rfxuEEU

The exponential function shows aversion to left tail events and preference for positive skewness in log returns Low Risk

Tolerance

Medium Risk

ToleranceHigh Risk Tolerance

Utility

Log returns

All display CRRA

throughout domain

8

These result in a remarkably simple rational trade-off between adjusted risk and expected returns

T

rrD Bf

2

Desirability

Expected Excess Returns

Total return – risk-free return

Compensation for risk

Risk / Risk Tolerance

9

Illustrating in risk-return space

Risk free return

r

frAll components

measured in % log returns

B

RejectAccept

Risk compensation

Trr Bf

2

Desirability

Maximum Desirability

Desirability

10

Because risk is mis-specified, the mean-variance ‘efficient’ frontier is not truly risk-return efficient

Desirability

r

fr(In)efficient Frontier

True Efficient Frontier

B

11

Traditional portfolio theory trades off risk and return of the portfolio in the long run

Portfolio Risk

Expected

Portfolio

Return

Efficient Frontier

Low risk tolerance, low portfolio risk, low return

High risk tolerance, high portfolio risk, high return

1212

The emotional experience with the investment journey has potentially greater influence on the final result Which investor is happier?

(Green, black or red)

Port

foli

o V

alu

e

Time

Danger of selling low

Danger of buying high

Ulysses

Self-control Dual self model Two systems of reasoning Methods for self-control Differences in short-term and long-term distributions

13

The short term investor has a further emotional transformation of returns

Composure value function

Rational linear

function

crrrv sgn

Investors with short-term reactions will attribute utility to returns differently to long-term rational investors

Loss aversion

c

rvT

rrrv

eEEU

sgn

12

Investors with short-term reactions will attribute utility to returns differently to long-term rational investors

Loss aversion

c

rvT

rrrv

eEEU

sgn

12

17

These result in a remarkably simple rational trade-off between adjusted risk and expected returns

T

rrD Bf

2

Desirability

Expected Excess Returns

Total return – risk-free return

Compensation for risk

Risk / Risk Tolerance

18

Effect can be completely reflected through a separate Anxiety score for any investment

AT

rrD adjf

2

Compensation for risk

Compensation for Anxiety

(relative to risk free)

Excess Returns

Desirability

19

Introducing the Anxiety measure

ffadjrrv

T rrvT

eET

A

22

ln2

Total psychological compensation for returns variability

Compensation for

rational risk

Reduction in anxiety from existence of

risk free investment

20

Illustrating in risk-return space

Risk free return

r

frAll components

measured in % log returns

B

RejectAccept

Risk compensation

Trr Bf

2

Desirability

Maximum Desirability

Desirability

21

Representing Anxiety graphically

Risk free return

r

fr

B

RejectAccept

Risk compensation

Trr Bf

2

Maximum Desirability

Desirability

ANXIETY

B

22

Portfolio optimisation for the anxious

Desirability

r

fr

True Efficient Frontier

Maximum Desirability

B

23

Portfolio optimisation for the anxious

Desirability

r

fr

True Efficient Frontier

B

24

Portfolio optimisation for the anxious

Desirability

r

fr

True Efficient Frontier

Anxiety

B

25

Portfolio optimisation for the anxious

r

fr

True Efficient Frontier

Anxiety Efficient Frontier

B

26

Portfolio optimisation for the anxious

r

fr

True Efficient Frontier

Anxiety Efficient Frontier

Desirability

Why would we use this?

Pander to short term self Understand costs of Anxiety Bargaining between planner and doer

Use small degree of short-term preferences to ‘take off the edge’

Use different time horizons to overcome myopia

27

28

1 Month 2 Month 3 Month 6 Month 1 Year 2 Year 5 Year

Excess Returns Risk Compensation Rational Desirability

The effect of time horizon on risk and desirability

29

1 Month 2 Month 3 Month 6 Month 1 Year 2 Year 5 Year

Excess Returns Risk Compensation Anxiety Compensation

The effect of time horizon on anxiety

30

1 Month 2 Month 3 Month 6 Month 1 Year 2 Year 5 Year

Excess Returns Risk Compensation Anxiety Compensation Desirability

The effect of time horizon on anxiety

31

1 Month 2 Month 3 Month 6 Month 1 Year 2 Year 5 Year

Rational Desirability Desirability

Desirability for the rational and the anxious