risk measurement in insurance paul kaye cas 2005 spring meeting phoenix, arizona 17 may 2005

Risk Measurement in Insurance

Paul Kaye

CAS 2005 Spring Meeting

Phoenix, Arizona

17 May 2005

The information contained in this document is strictly proprietary and confidential. © Benfield 2005

Company ABC

• Made up of 3 similar risk portfolios A, B and C

Premium 100 Expenses 25 Losses 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


ABC Financial Result - frequency distribution

-200 -150 -100 -50 0 50 100 150

Distribution Mean Median

Company ABC – how risky?


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


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


Coherence

A risk should measure…err…risk!


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


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


Properties of Coherent Allocation Methods

Later…


Category 1

Point measures



-200 -150 -100 -50 0 50 100 150

Distribution Mean 1 in 100

Point measures

• The value of a distribution of outcomes at a single point


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


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


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


Category 2

Standard deviation and higher moments



-200 -150 -100 -50 0 50 100 150

Distribution Mean 1 * Std Dev 2.87 * Std Dev

Standard deviation and higher moments

• Probability weighted deviation from mean


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


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!


Category 3

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



-200 -150 -100 -50 0 50 100 150

Distribution Mean Threshold (1 in 37) Expected Exceedence


• TCE2.728% / TVAR2.728%


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


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


Category 4

Transform measures


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



-200 -150 -100 -50 0 50 100 150

Distribution Mean PHT Transformed mean

Proportional Hazard Transform

• PHT5.1375


ABC Financial Result - cumulative distribution

0%

20%

40%

60%

80%

100%

-200 -150 -100 -50 0 50 100 150

Distribution Mean PHT Transformed mean

Proportional Hazard Transform

• PHT5.1375


ABC Financial Result - cumulative distribution

0%

20%

40%

60%

80%

100%

-200 -150 -100 -50 0 50 100 150

Distribution Mean Conc Charge

CC mean Scaled CC mean

Concentration Charge

• Possible weights: *8 if <0 and *4 if >100, otherwise *1


Transform measures - theory

• Strong technical properties - coherent

• Transforming percentiles is equivalent to transforming results

In both cases giving different weights to different outcomes


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?


Category 5

Performance ratios


Performance ratios

• Performance rather than merely risk focused

• Risk Coverage Ratio (aka R2R – Reward to Risk)†

† Downside: PV < 0

• Omega function‡

‡ For any threshold

Expected Result

Probability of downside * Expected result given downside

Probability of upside * Expected result given upside

Probability of downside * Expected result given downside



0 .0 %

2 .0 %

4 .0 %

6 .0 %

8 .0 %

1 0 .0 %

-200 -150 -100 -50 0 50 100 150

0.001

0.01

0.1

1

10

100

1000

Om

eg

a v

alu

e (

log

sc

ale

)

Distribution Mean 1 in 100 Omega

Omega function

• Omega value based on threshold of the mean (always 1)



0 .0 %

2 .0 %

4 .0 %

6 .0 %

8 .0 %

1 0 .0 %

-200 -150 -100 -50 0 50 100 150

0.001

0.01

0.1

1

10

100

1000

Om

ega

valu

e (l

og

sca

le)

Distribution Mean 1 in 100 Omega

Omega function

• Omega values based on full range of thresholds



0 .0 %

2 .0 %

4 .0 %

6 .0 %

8 .0 %

1 0 .0 %

-200 -150 -100 -50 0 50 100 150

Distribution(1) Mean(1) 1 in 100(1)


Omega function

• Introduce a new distribution – more volatile but higher mean



0 .0 %

2 .0 %

4 .0 %

6 .0 %

8 .0 %

1 0 .0 %

-200 -150 -100 -50 0 50 100 150

0.001

0.01

0.1

1

10

100

1000

Om

ega

valu

e (l

og

sca

le)



Omega(1) Omega(2)

Omega function

• Introduce a new distribution – more volatile but higher mean


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


Risk / Capital allocation between sub-portfolios


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


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

1st percentile Actual

A 60.1

B 60.1

C 60.1

Total 180.3


Marginal “last in”

• The additional ‘risk’ for incorporating each sub-portfolio

• Scale to tie in with overall portfolio measure?

Over-rewards diversification

Not coherent

1st percentile

Excluding portfolio

Marginalimpact

% 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

Total 105.8 100.0% 128.4

2.87 * Std Deviation

Excluding portfolio

Marginalimpact

% 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

Total 107.9 100.0% 128.4

TCE2.728%ileExcluding portfolio

Marginalimpact

% 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

Total 106.8 100.0% 128.4

1st percentile

Excluding portfolio

Marginalimpact

A 79.6 (BC) 48.8 (ABC-BC)

B 79.7 (AC) 48.7 (ABC-AC)

C 120.2 (AB) 8.2 (ABC-AB)

Total 105.8


Shapley values

• The average of the “first in”, “second in”… and “last in”

• 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!)

TCE2.728%ile "1st in"Average "2nd in"

"Last in" Average "2nd in" calculations

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


"Last in" Average "2nd in" calculations

A 60.0 39.6 49.2 59.9 (AB-B) 19.3 (AC-C)

B 60.0 39.6 49.2 60.0 (AB-A) 19.3 (BC-C)

C 60.0 19.2 8.5 19.2 (AC-A) 19.2 (BC-B)

Total 179.9 98.5 106.8


"Last in" Average

A 60.0 49.2

B 60.0 49.2

C 60.0 8.5

Total 179.9 106.8


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?

• Coherent (with a coherent risk measure)

1st percentile (0.57%-1.5%)

TCE2.728%ile PHT5.1375Concentration

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


Risk measurement in practice


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?!


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.”

risk measurement in insurance paul kaye cas 2005 spring meeting phoenix, arizona 17 may 2005

Documents

risk slide

risk monotonicity

coherent slide

chosen risk measure

risk measure chosen

risk free portfolio

similar risk portfolios

robust slide