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LINKING PSYCHOMETRIC RISK TOLERANCE WITH CHOICE BEHAVIOUR
Peter Brooks, Greg B. Davies and Daniel P. Egan
FUR Conference – July 2008
Presentation Aims
2

To introduce the Barclays Wealth Risk Tolerance Scale

To introduce the effects of an exponential utility function on asset
allocation

To describe an experiment that provides a link between risk tolerance
scores and risk parameters.

Are different risk profiles characterised by different risk/utility parameters
in choices?
Pre-experiment Analysis Overview
 Examine risk and utility measures using simulated portfolios involving
equities and bonds
 Mix the simulated portfolios with different proportions of cash
 Holding cash is assumed to be a riskless alternative
 Calculate the optimal portfolio for different values of the risk parameter
3
Example Utility Measure
5 Year Bond/Equity Mixes
Low values of  imply risk
tolerant behaviour – Optimal
portfolio is 100% equities
6
Higher values of  imply risk
averse behaviour – Optimal
portfolio is now a mix of
equities and bonds
2
-4
-6
100B
20E-80B
-8
-10
40E-60B
50E-50B
60E-40B
80E-20B
100E
-12
-14
4
1  e  x 
EU  E 




0.39
0.37
0.35
0.33
0.31
0.29
0.27
0.25
0.23
0.21
0.19
0.17

0.15
0.13
0.11
0.09
-2
0.07
0
0.05
Expected Utility
4
Optimal Portfolio Mixes with Cash
5 Year Investment Horizon
 For low  – optimal portfolio is 100%
Equities
 For  between 0.08 and 0.16, the
optimal portfolio is a mix of equities
and bonds
 For  greater than 0.17, the optimal
asset allocation includes cash.
100
90
Asset Allocation %
 We have modelled a range of values
of the risk parameter for 5 year
returns
80
Cash
70
60
50
40
Bonds
30
20
Equities
10

5
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0
Pre-experiment Analysis Overview 2
 The analysis suggests that the optimal portfolio is sensitive to the value
of a risk parameter.
 Assuming utility maximisation individual choices between portfolios make
it possible to calibrate a risk parameter.
 Choices constrain a risk parameter to a range of values where the
portfolio would be preferred by a utility maximising individual.
 Analysing a number of choices makes it possible to find a “best” value of
the risk parameter for each individual.
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Barclays Wealth Risk Tolerance Scale
 8 question psychometric
questionnaire
Risk Tolerance Scale Distribution
Higher=More Risk Tolerant
0.08
 Responses given on a 5-point Likert
scale
0.07
 Produces a score between 8 and 40
 Scores bucketed into 5 risk profiles
from low up to high.
0.05
Frequency
 Higher scores signal higher risk
tolerance
0.06
0.04
0.03
0.02
Risk Tolerance Score
7
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
0
8
0.01
Experiment Aims
8

To test various risk measures and utility functions using actual choices

To estimate risk/utility parameters for individual respondents.

To provide a link between the risk tolerance scores and risk parameters.

Are different risk profiles characterised by different risk/utility parameters
in choices?
Experimental Design
 Create stylised distributions of the final values of an investment.
 It is difficult to use distributions based upon real data. Increases in
volatility cause the tails of the distribution to become long.
 Long tailed distributions are difficult to display accurately to survey
respondents.
 Take log-normal distribution and set the mean and standard deviation.
 Generate 120 equally spaced observations across the distribution.
 Round each of these observations to the nearest integer.
 Plot the frequency table of the observations to create the distributions for
the experiment.
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Experimental Design
 Expected utility is increasing in the mean of the distribution.
 Expected utility is decreasing in the “risk” of the distribution.
 Create a preference order between two distributions by compensating for
an increase in “risk” by increasing the mean.
 The most risk averse will prefer lower mean and lower “risk” distributions.
 The least risk averse will prefer higher mean and higher “risk”
distributions.
10
Example Distribution
 Mean = £103,000
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Example Distribution 2
 Mean = £105,000
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Distribution Comparisons – Example Using Exponential
Risk Measures
1
0.8
0.6
Expected Utility
0.4
Mean = 103
Mean = 102
Mean = 104
0.2
Mean = 105
0
-0.2
Mean = 106
-0.4
-0.6
-0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Utility Parameter ()
13
0.8
0.9
1.0
Distribution Comparisons – Example Using Exponential
Risk Measures
1.2
1
Expected Utility
0.8
Mean = 104
0.6
Mean = 103
0.4
Mean = 102
Mean = 105
0.2
Mean = 106
0
-0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Utility Parameter ()
14
0.8
0.9
1.0
Distribution Comparisons – Example Using Exponential
Risk Measures
1
0.8
0.6
Expected Utility
Mean = 103
0.4
Mean = 102
0.2
Mean = 104
0
Mean = 105
-0.2
Mean = 106
-0.4
-0.6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Utility Parameter ()
15
0.8
0.9
1.0
Experiment Procedures
 Participants recruited through iPoints
6 section experiment
 Participants paid in iPoints
1. Demographics
 All participants reported either gross
annual income above £50k or
investable wealth above £100k
2. Psychometric Risk Tolerance
 Delivered through a non-branded
external website
 Respondents had participated in
previous surveys but had not
participated within the past 6 months
 Over-sampling of the extreme risk
profiles
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
3. Training stage
4. 9 Pairwise choice tasks between
distributions
5. Filler Task – maze
6. 9 Pairwise choice tasks between
distributions
Experimental Results
 108 Participants completed all parts
of the survey
 1 participant removed for inconsistent
responses
 Individuals in higher risk profiles tend
to choose higher variance
distributions more often
 Use MLE to estimate the utility risk
parameter for individuals - grouped
by risk tolerance score
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0.3
0.25
Frequency
 Over-sampling of the end points
successful
0.35
0.2
0.15
0.1
0.05
0
Low
Med-Low
Medium
Med-High
High
MLE Fit Results
2
Estimated Utility Parameter
1.5
1
0.5
0
5.00
10.00
15.00
20.00
25.00
-0.5
Mean Risk Tolerance Score
18
30.00
35.00
40.00
Conclusions and Extensions
 Our psychometric risk tolerance measure is consistent with risky choice
 There is potential for a behavioural calibration of a risk measure for
portfolio optimisation
 Separate work on whether utility measures are better than variance, VaR
or CVaR as risk measures for portfolio optimisation
 Geographical calibration exercise – current ongoing work
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