Transcript Risk Measurement in Insurance

Risk Measurement in Insurance
Paul Kaye
CAS 2005 Spring Meeting
Phoenix, Arizona
17 May 2005
Company ABC
• Made up of 3 similar risk portfolios A, B and C
- Premium
- Expenses
- Losses
100
25
mean 65, std dev 20 (LogNormal)
• Loss behaviour of A and B highly correlated, C uncorrelated
• One year simulation with 100,000 trials
• Net underwriting result captured for
-
ABC (mean +30)
A
B
C
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Company ABC – how risky?
ABC Financial Result - frequency distribution
-200
-150
-100
-50
Distribution
0
Mean
50
100
150
Median
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Aims
Bring clarity to the risk measurement jungle
• Illustrate how risk can be measured
• Illustrate how risk can be allocated across sub-portfolios
• Highlight the benefits and pitfalls of different approaches
- Theory
- Practice
• Avoid getting unnecessarily technical
• Economic value focus
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BIG ASSUMPTIONS!
• All the risks relevant to the question in hand are in the model
- Or at least a way of making an allowance for omissions has been
established
• The individual model assumptions are valid
• The key risk inter-dependencies are incorporated
• The design and execution of the model are robust
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Coherence
A risk should measure…err…risk!
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Evaluation of methods
• Does the chosen risk measure adequately measure risk?
• Does the allocation methodology satisfy all relevant
stakeholders?
• Concept of Coherence formalises common sense behaviour
criteria
- Coherence of risk measure (Artzner et al)
- Coherence of allocation method (Denault)
An allocation method will not be Coherent if the Risk
Measure chosen is not at least Coherent
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Properties of Coherent Risk Measures
• Sub-additivity
Combining two portfolios should not create more risk
• Monotonicity
If a portfolio is always worth more than another (i.e. for all return
periods), it cannot be riskier
• Positive homogeneity
Scaling a portfolio by a constant will change the risk by the same
proportion
• Translation invariance
Adding a risk free portfolio to an existing portfolio creates no change
in risk
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Properties of Coherent Allocation Methods
Later…
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Category 1
Point measures
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Point measures
• The value of a distribution of outcomes at a single point
ABC Financial Result - frequency distribution
-200
-150
-100
-50
Distribution
0
Mean
50
100
150
1 in 100
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Point measures
Either
• Value at a specified percentile
- e.g. 1st/99th %ile (= 1 in 100)
or
• Probability less than / more than specified value
- e.g. 1% chance of a loss of 98.4 or worse for ABC
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Point measures - technical
Not coherent - fails sub-additivity test!
For example, consider 2 similar (non correlating) portfolios:
•
Up to and including 99.1th percentile value +10
•
Beyond 99.1th percentile value -100
•
1 in 100 result:
- Sum of parts: +20
- Combined: -90
Problem areas include:
•
Risk margin when pricing excess of loss reinsurance contracts
•
Observed in practice with cat modelling results
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Point measures - practice
Despite technical limitations we love them!
• Everyone understands them (don’t they?)
• Intuitively easy
• Only need to know (or estimate) one point on a distribution to
use - the easiest ‘risk’ measure
- Popular with regulators
… but beware of the limitations
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Category 2
Standard deviation and higher moments
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Standard deviation and higher moments
• Probability weighted deviation from mean
ABC Financial Result - frequency distribution
-200
-150
-100
Distribution
-50
Mean
0
1 * Std Dev
50
100
150
2.87 * Std Dev
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Std deviation and higher moments - theory
• Standard deviation limited to giving measure of spread
- But does take into account entire distribution
• Full description of distributions requires reference to higher
moments
- E.g. Skewness and Kurtosis
- Immediate elegance of a single metric lost
• Algebraically cumbersome
• Not coherent – fails Monotonicity test
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Std deviation and higher moments - practice
• Standard deviation is popular but often abused
- The school text book measure of volatility
- An abstract concept – do users really understand its values?
• Insurance distributions are never Normal!
- Neither are many financial risk distributions
‘Quick and dirty’ merits should not be ignored but be careful!
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Category 3
Expected exceedence measures
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Expected exceedence measures
•
Measures based on the expected result given the result is beyond a
given threshold
- i.e. the average of all the values beyond a given point of the distribution.
Measures include:
•
Tail Conditional Expectation (TCE)
•
Tail Value at Risk (TVaR) – same as TCE
•
Excess Tail Value at Risk (XTVaR) – threshold is the mean
•
Expected Shortfall (the shortfall beyond the threshold)
•
Expected Policyholder Deficit (the shortfall beyond the available
capital
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Expected exceedence measures
• TCE2.728% / TVAR2.728%
ABC Financial Result - frequency distribution
-200
-150
Distribution
-100
Mean
-50
0
Threshold (1 in 37)
50
100
150
Expected Exceedence
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Expected exceedence - theory
• Strong technical properties - coherent
• Focuses on one tail of the distribution
- Much richer than a simple point measure
- But what about the other tail?
• All points beyond threshold carry equal weight
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Expected exceedence - practice
• Intuitively appealing
• Consider a 1 in 100 threshold
- Point measure: result of X or worse every 100 years
- Expected exceedence: expected result Y every 100 years
• Need to understand behaviour of full tail of distribution
• Calculations easy using computer simulation
Popular – strong properties but accessible
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Category 4
Transform measures
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Transform measures
• Mean of transformed distribution (or the difference between
this and the original mean)
Two ways of going about the transform:
• Transform percentiles
- E.g. Proportional Hazard Transform (PHT) and Wang Transforms
• Transform results
- E.g. Concentration charge (Mango)
- Note exceedence measures are a very specific form of result
transform
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Proportional Hazard Transform
• PHT5.1375
ABC Financial Result - frequency distribution
-200
-150
Distribution
-100
-50
Mean
0
PHT
50
100
150
Transformed mean
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Proportional Hazard Transform
• PHT5.1375
ABC Financial Result - cumulative distribution
100%
80%
60%
40%
20%
0%
-200
-150
Distribution
-100
Mean
-50
0
PHT
50
100
150
Transformed mean
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Concentration Charge
• Possible weights: *8 if <0 and *4 if >100, otherwise *1
ABC Financial Result - cumulative distribution
100%
80%
60%
40%
20%
0%
-200
-150
-100
-50
0
Distribution
Mean
CC mean
Scaled CC mean
50
100
150
Conc Charge
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Transform measures - theory
• Strong technical properties - coherent
• Transforming percentiles is equivalent to transforming results
- In both cases giving different weights to different outcomes
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Transform measures - practice
• Where do the weights come from and what do they mean?
• Percentile transforms:
- Weights generated buy the transform
- But abstract. Why – other than as a means to an end?
• Concentration charge style weights:
- Rationale for setting weights?
- Mango’s ‘Capital Hotel’ analogy
• Relatively easy to calculate using scenario based simulation
Excellent measures but will the punters understand?
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Category 5
Performance ratios
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Performance ratios
• Performance rather than merely risk focused
• Risk Coverage Ratio (aka R2R – Reward to Risk)†
Expected Result
Probability of downside * Expected result given downside
† Downside: PV < 0
• Omega function‡
Probability of upside * Expected result given upside
Probability of downside * Expected result given downside
‡ For any threshold
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Omega function
• Omega value based on threshold of the mean (always 1)
ABC Financial Result - frequency distribution
100
10
1
0.1
0.01
Omega value (log scale)
1000
0.001
-200
-150
-100
Distribution
-50
Mean
0
50
1 in 100
100
150
Omega
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Omega function
• Omega values based on full range of thresholds
ABC Financial Result - frequency distribution
100
10
1
0.1
0.01
Omega value (log scale)
1000
0.001
-200
-150
-100
Distribution
-50
Mean
0
50
1 in 100
100
150
Omega
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Omega function
• Introduce a new distribution – more volatile but higher mean
ABC Financial Result - frequency distribution
-200
-150
-100
-50
0
50
100
Distribution(1)
Mean(1)
1 in 100(1)
Distribution(2)
Mean(2)
1 in 100(2)
150
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Omega function
• Introduce a new distribution – more volatile but higher mean
ABC Financial Result - frequency distribution
100
10
1
0.1
0.01
Omega value (log scale)
1000
0.001
-200
-150
-100
-50
0
50
100
Distribution(1)
Mean(1)
1 in 100(1)
Distribution(2)
Mean(2)
1 in 100(2)
Omega(1)
Omega(2)
150
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Performance measures – theory and practice
• The technical properties of any performance ratio will depend
on its construct and its purpose
- The Risk Coverage Ratio and Omega function are constructed
from measures which have strong technical properties
• Complex to explain despite relatively simple foundations
- The value isn’t a profit or loss, rather a semi-abstract hybrid
• Upside and downside characteristics brought together in an
elegant way
• Again, relatively easy to calculate using scenario based
simulation
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Risk / Capital allocation
between sub-portfolios
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Properties of Coherent Allocation Methods
• No undercut
The allocation for a sub-portfolio (or coalition of sub-portfolios) should
be no greater than if it was considered separately
- A sub-portfolio’s allocation <= standalone capital requirement
- A sub-portfolio’s allocation >= marginal (last in) allocation
• Symmetry
If the risk of two sub-portfolios is the same (as measured by the risk
measure), the allocation should be the same for each
• Riskless allocation
Cash in a sub-portfolio reduces allocation accordingly
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Independent “first in”
•
•
•
Focus on “1 in 100” deviation
from mean for ease (128.4 for
whole portfolio for all measures)
Allocate to each sub-portfolio on
standalone basis
Scale to tie in with overall
portfolio measure?
Under-rewards diversification
Aggregating portfolios not
penalised
Not coherent
1st percentile
Actual
%
Scaled
A
60.1
33.3%
42.8
B
60.1
33.3%
42.8
C
60.1
33.3%
42.8
Total
180.3
100.0%
128.4
2.87 * Std Dev
Actual
%
Scaled
A
57.4
33.3%
42.8
B
57.4
33.3%
42.8
C
57.4
33.3%
42.8
Total
172.3
100.0%
128.4
TCE2.728%ile
Actual
%
Scaled
A
60.0
33.3%
42.8
B
60.0
33.3%
42.8
C
60.0
33.3%
42.8
Total
179.9
100.0%
128.4
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Marginal “last in”
• The additional ‘risk’
for incorporating
each sub-portfolio
• Scale to tie in with
overall portfolio
measure?
Over-rewards
diversification
1st
percentile
Marginal
impact
%
Scaled
A
79.6
(BC)
48.8
(ABC-BC)
46.1%
59.2
B
79.7
(AC)
48.7
(ABC-AC)
46.1%
59.2
C
120.2
(AB)
8.2
(ABC-AB)
7.8%
10.0
100.0%
128.4
Total
2.87 * Std
Deviation
105.8
Excluding
portfolio
Marginal
impact
%
Scaled
A
81.2
(BC)
47.2
(ABC-BC)
43.7%
56.1
B
81.2
(AC)
47.2
(ABC-AC)
43.7%
56.1
C
114.8
(AB)
13.6
(ABC-AB)
12.6%
16.2
100.0%
128.4
Total
TCE2.728%ile
Not coherent
Excluding
portfolio
107.9
Excluding
portfolio
Marginal
impact
%
Scaled
A
79.2
(BC)
49.2
(ABC-BC)
46.0%
59.1
B
79.2
(AC)
49.2
(ABC-AC)
46.0%
59.1
C
119.9
(AB)
8.5
(ABC-AB)
7.9%
10.2
100.0%
128.4
Total
106.8
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Shapley values
• The average of the “first in”, “second in”… and “last in”
TCE2.728%il
"1st in"
Average
"2nd in"
"Last in"
Average
A
60.0
39.6
49.2
49.6
59.9
(AB-B)
19.3
(AC-C)
B
60.0
39.6
49.2
49.6
60.0
(AB-A)
19.3
(BC-C)
C
60.0
19.2
8.5
29.2
19.2
(AC-A)
19.2
(BC-B)
Total
179.9
98.5
106.8
128.4
e
"2nd in" calculations
• Coherent (with coherent risk measure)
- although doesn’t deal with fractions of portfolios consistently
• A computational nightmare!
- Elegant algebraic simplification if variance used as risk measure
(not coherent!)
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Aumann-Shapley
•
Represents the rate of increase in the risk/capital allocation
- i.e. how much additional overall risk comes from a sub-portfolio for a tiny
increase in size
•
Easy to calculate from a scenario based simulation
- What weight was applied to each scenario in the calculation of the overall
measure?
- What contribution was made by each sub-portfolio in each scenario?
1st percentile
(0.57%-1.5%)
•
TCE2.728%ile
PHT5.1375
Concentration
Charge
A
54.7
54.7
58.9
63.3
B
54.8
54.7
58.3
63.3
C
19.0
19.0
11.2
1.8
Total
128.4
128.4
128.4
128.4
Coherent (with a coherent risk measure)
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Risk measurement in practice
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Risk measurement in practice
• Avoid all non-coherent risk measures?
• Not necessarily but…
- Be aware of the limitations
- Consider more than one measure and/or tolerance level?
• Beyond risk measurement theory…
- How relevant is a risk measure to the decision being made?
- Do the decision-makers understand the risk measurement
information?
- What if risk behaviours are poorly captured or not included in the
model in the first place?!
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Advice from Albert Einstein
• "Things should be made as simple as possible, but not any
simpler.”
• "Not everything that counts can be counted, and not everything
that can be counted counts."
• "Anyone who has never made a mistake has never tried
anything new.”
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