LINKING PSYCHOMETRIC RISK TOLERANCE WITH CHOICE BEHAVIOUR

FUR Conference – July 2008

Peter Brooks, Greg B. Davies and Daniel P. Egan

2

Presentation Aims

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?

3

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

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Example Utility Measure

5 Year Bond/Equity Mixes

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.21

0.23

0.25

0.27

0.29

0.31

0.33

0.35

0.37

0.39

100B

20E-80B

40E-60B

50E-50B

60E-40B

80E-20B

100E

Expect

ed U

tilit

y

Low values of imply risk tolerant behaviour – Optimal portfolio is 100% equities Higher values of imply

risk averse behaviour – Optimal portfolio is now a mix of equities and bonds

xeEEU1

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Optimal Portfolio Mixes with Cash

We have modelled a range of values of the risk parameter for 5 year returns

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.

0

10

20

30

40

50

60

70

80

90

100

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Ass

et

Allo

cati

on %

Equities

Bonds

Cash

5 Year Investment Horizon

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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

Responses given on a 5-point Likert scale

Produces a score between 8 and 40

Higher scores signal higher risk tolerance

Scores bucketed into 5 risk profiles from low up to high.

Risk Tolerance Scale Distribution

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Risk Tolerance Score

Fre

qu

ency

Higher=More Risk Tolerant

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Experiment Aims

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?

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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.

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Example Distribution

Mean = £103,000

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Example Distribution 2

Mean = £105,000

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Distribution Comparisons – Example Using Exponential Risk Measures

Utility Parameter ()

Expect

ed

Uti

lity

Mean = 106

Mean = 105

Mean = 104

Mean = 103Mean = 102

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Utility Parameter ()

Expect

ed

Uti

lity

Mean = 106

Mean = 105

Mean = 104Mean = 103

Mean = 102

Distribution Comparisons – Example Using Exponential Risk Measures

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Utility Parameter ()

Expect

ed

Uti

lity

Mean = 106

Mean = 105

Mean = 104

Mean = 103Mean = 102

Distribution Comparisons – Example Using Exponential Risk Measures

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Experiment Procedures

Participants recruited through iPoints

Participants paid in iPoints All participants reported either

gross annual income above £50k or investable wealth above £100k

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

6 section experiment1. Demographics

2. Psychometric Risk Tolerance

3. Training stage

4. 9 Pairwise choice tasks between distributions

5. Filler Task – maze

6. 9 Pairwise choice tasks between distributions

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Experimental Results

108 Participants completed all parts of the survey

1 participant removed for inconsistent responses

Over-sampling of the end points successful

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

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Low Med-Low Medium Med-High High

Fre

qu

ency

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MLE Fit Results

-0.5

0

0.5

1

1.5

2

5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00

Mean Risk Tolerance Score

Es

tim

ate

d U

tilit

y P

ara

me

ter

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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