Portfolio Optimisation for the Anxious
Greg B Davies, PhD
Head of Behavioural Finance
greg.davies2@barclayswealth.com
June 2010
“The market can stay irrational, longer than you can stay solvent”
John Maynard Keynes
Harry Markowitz – Nobel Prize 1990
2
Daniel Kahneman – Nobel Prize 2002
Value function in descriptive theories: CPT
Utility
Reference Point
Losses (£)
Gains (£)
Loss
aversion:
Steeper for
losses
And with risk=variance, expected utility theory is equivalent to
mean-variance optimisation
r
E u  x   f  r , Risk

f r , 
2

Risk/Return Trade-off
(Indifference Curve)
Optimal Portfolio
Risk
free
rate
Market Portfolio
Portfolio Efficient Frontier

4
Only works under specific, and largely incorrect assumptions
EU  E u  x   f r , 
2

 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
5
Example Quadratic Utility Function
Utility
Return
This means both the risk measure and the risk-return trade-off
are flawed
EU  E u  x   f r , 
2

r
Risk
free
rate

6
The exponential function shows aversion to left tail events and
preference for positive skewness in log returns
Utility
All display
CRRA
throughout
domain
Low Risk
Tolerance
Medium Risk
Tolerance
High Risk
Tolerance
Log returns
These result in a remarkably simple rational trade-off between
adjusted risk and expected returns
D  r  r f  
B
2
T
Desirability
Compensation for risk
Expected Excess Returns
Total return – risk-free return
8
Risk / Risk Tolerance
Illustrating in risk-return space
Maximum Desirability
r
B
2
r  rf 
Accept
Reject
Desirability
Desirability
rf
Risk compensation
All components measured
in % log returns
Risk free return
B
9
T
Because risk is mis-specified, the mean-variance ‘efficient’
frontier is not truly risk-return efficient
r
True Efficient Frontier
Desirability
rf
(In)efficient Frontier
B
10
Traditional portfolio theory trades off risk and return of the
portfolio in the long run
Expected
Portfolio
Return
Efficient
Frontier
High risk tolerance,
high portfolio risk,
high return
Low risk tolerance, low
portfolio risk, low return
Portfolio Risk
11
The emotional experience with the investment journey has
potentially greater influence on the final result
Portfolio Value
Which investor is happier?
(Green, black or red)
Danger of buying high
Danger of selling low
Time
12
Ulysses





13
Self-control
Dual self model
Two systems of reasoning
Methods for self-control
Differences in short-term and long-term distributions
The short term investor has a further emotional transformation
of returns
Composure
value
function
Rational
linear
function
v  r   sgn  r  r
c
Investors with short-term reactions will attribute utility to returns
differently to long-term rational investors
Loss
aversion
2
 v r  

EU  E 1  e T



v  r   sgn  r  r
c
Investors with short-term reactions will attribute utility to returns
differently to long-term rational investors
Loss
aversion
2
 v r  

EU  E 1  e T



v  r   sgn  r  r
c
These result in a remarkably simple rational trade-off between
adjusted risk and expected returns
D  r  r f  
B
2
T
Desirability
Compensation for risk
Expected Excess Returns
Total return – risk-free return
17
Risk / Risk Tolerance
Effect can be completely reflected through a separate Anxiety
score for any investment
D  r  r f  
 adj
2
A
T
Desirability
Excess Returns
18
Compensation for
risk
Compensation for
Anxiety
(relative to risk free)
Introducing the Anxiety measure
2

 v  r  r  
 adj

T
T
A  ln E  e
 v r f   r f

2
T


2
Total psychological
compensation for returns
variability
19
Compensation
for rational risk

Reduction in anxiety from
existence of risk free
investment
Illustrating in risk-return space
Maximum Desirability
r
B
2
r  rf 
Accept
Reject
Desirability
Desirability
rf
Risk compensation
All components measured
in % log returns
Risk free return
B
20
T
Representing Anxiety graphically
Maximum Desirability
r
B
2
r  rf 
Accept
ANXIETY
Reject
Desirability
Risk compensation
rf
Risk free return
B
21
T
Portfolio optimisation for the anxious
r
Maximum Desirability
True Efficient Frontier
rf
Desirability
B
22
Portfolio optimisation for the anxious
r
True Efficient Frontier
rf
Desirability
B
23
Portfolio optimisation for the anxious
r
Anxiety
True Efficient Frontier
rf
Desirability
B
24
Portfolio optimisation for the anxious
r
True Efficient Frontier
Anxiety Efficient Frontier
rf
B
25
Portfolio optimisation for the anxious
r
True Efficient Frontier
Anxiety Efficient Frontier
rf
Desirability
B
26
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
The effect of time horizon on risk and desirability
Excess Returns
1 Month
28
2 Month
Risk Compensation
3 Month
6 Month
Rational Desirability
1 Year
2 Year
5 Year
The effect of time horizon on anxiety
Excess Returns
1 Month
29
2 Month
Risk Compensation
3 Month
6 Month
Anxiety Compensation
1 Year
2 Year
5 Year
The effect of time horizon on anxiety
Excess Returns
1 Month
30
2 Month
Risk Compensation
3 Month
6 Month
Anxiety Compensation
1 Year
2 Year
Desirability
5 Year
Desirability for the rational and the anxious
Rational Desirability
1 Month
31
2 Month
3 Month
Desirability
6 Month
1 Year
2 Year
5 Year