the price of risk in insurance presented by michel m. dacorogna ?, moscow, russia, april 23-24,2008

The Price of Risk in Insurance

Presented by Michel M. Dacorogna

?, Moscow, Russia, April 23-24,2008

2Economy of Risk in InsuranceMichel M. DacorognaApril 23-24, 2008

Important disclaimerAlthough all reasonable care has been taken to ensure the facts stated herein are accurate and that the opinions contained herein are fair and reasonable, this document is selective in nature and is intended to provide an introduction to, and overview of, the business of Converium. Where any information and statistics are quoted from any external source, such information or statistics should not be interpreted as having been adopted or endorsed by Converium as being accurate. Neither Converium nor any of its directors, officers, employees and advisors nor any other person shall have any liability whatsoever for loss howsoever arising, directly or indirectly, from any use of this presentation.

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Outline of the Talk

A simple example

Risk and risk measures

Risk-based capital and economic capital

Valuation methods

Conclusions


A simple example for pricing risk

Assume an insurance customer approaches a company to insure the following risk:

He must pay 10 USD if he gets a six on a die, and nothing otherwise.

He must throw the die 6 times.

We will answer two questions:

1. What is the price for such a risk, independently of any other liability the insurer has?

2. How would the price change if the insurer assume many of the same risk?


Outcome of Throwing the Die

Out. Prob.Claim

6 1/6 10

Out. Prob.Claim

<6 5/6 0 Out. Prob.Claim

<6 5/6 0

Out. Prob.Claim

6 1/6 10

Out. Prob.Claim

<6 5/6 0

Out. Prob.Claim

6 1/6 10

Total

Probability

Claims

20

5/36 10

1 1 1

6 6 36

5/36 10

25/36 0

First Throw

Second Throw

And so on …


The Cumulative Distribution of Our Example

Value-at-Risk(1%) = 30 USDExpected Loss = 10 USD

Amount of Loss

Probability of Loss

Cumulative Probability

0 33.48% 33.48%10 40.17% 73.65%20 20.09% 93.75%30 5.36% 99.11%40 0.80% 99.91%50 0.06% 99.98%60 0.02% 100.00%


What is the Correct Price ?

Pricing the risk at the expected loss plus costs means running the risk of losing more (here there is 26% chances to pay more than the expected 10 USD).

The risk is to have a claim that far exceeds the expected loss.

We define the risk as the unexpected loss.

An insurer guarantees that he will pay the loss even if it is above expectation.

Thus the need to put up capital for covering this risk (Risk-Based Capital, RBC).


Determining the Capital to Cover the Risk

In order to quantify the risk, we need to define up to which probability the inusrer is willing to guarantee his payment

This is the confidence threshold at which the company wants to operate (let us choose here the “1 over 100 year event”)

Such a confidence threshold corresponds to a claim of 30 USD

In our case, the capital would be 20 USD (the 1% claim minus expected claim)

Providing capital has a cost – investors want a return on investment.

Let us assume in this case a cost of 15% before tax.


Computing the Premium

10

0.5

3.0

Expenses

Risk loading0.15 * (30-10)

Expected loss

13.5

CompanyStructure and Capital

Loss model

Premium


The Concept of Risk in Insurance

Risk describes the uncertainty of the future outcome of a current decision or situation.

The premium should reflect the risk assumed and the diversification of the insurer’s portfolio.

Insurance is the transfer of risk from an individual to a company (group).

We all have expectations about results – but the actual outcome is uncertain.

In a model, the possible outcomes can be adequately described by a probability distribution.


Risk and Risk Measures

This can be measured in terms of probability distributions but it is better to use one number to express it, called risk measure.

We want a measure that can give us a risk in form of a capital amount.

The risk measure should have the following properties (coherence):

1. Scalable (twice the risk should give a twice bigger measure),

2. Ranks risks correctly (bigger risks get bigger measure),

3. Allows for diversification (aggregated risks should have a lower measure),


Loss Model and Risk Measures

Standard DeviationMeasures typical size of fluctuations

Mean

Value-at-Risk (VaR)Measures position of 99th percentile, „happens once in a hundred years“

VaR

Expected Shortfall (ES) is the weighted average VaR beyond the 1% threshold.

0.00%

0.40%

0.80%

1.20%

1.60%

450 498 547 595 643 691 739

Gross Losses Incurred ($ M)

Pro

babi

lity

/ B

in o

f 25

0'00

0


Appropriate Risk Measures

We want to measure the extreme risks so VaR and ES are more appropriate.

We want to ensure that diversification is appropriately accounted for: if two risks are added together the total risk should be at maximum equal the sum of both (sub-additivity):

Among the measures presented, only the Expected Shortfall or t-VaR has this property for the type of insurance risks we are facing. It is a coherent measure of risk.

In general ES is more conservative than VaR but one can choose the threshold (1% or 0.4%).

1 2 1 2( ) ( ) ( )R x x R x R x


Examples of Risk Evaluation

Measure Hurricane Earthquake Total

Expected 62 16 78

Std. Dev. 84 60 104

VaR(1%) 418 332 544

VaR(0.4%) 596 478 690

ES(1%) 575 500 678

ES(0.4%) 700 598 770

> <

Typical gross natural catastrophe exposures VaR and ES (in MUSD).


Diversification:Insuring Many Independent Risks together

Assume that the insurer takes on not only the risk of one policyholder but many

Each policyholder insures the risk that he has to pay EUR 10 in each case a 6 appears on a die at 6 throws

Many risks will constitute now a portfolio of risks

How will the premium change due to diversification?

Remember: The expected loss per policy was EUR 10, expense EUR 0.5 and the risk loading EUR 3 for one policy seen in isolation


Influence of Diversification on the Premium

As can be seen, if the risks diversify, the risk loading per policy reduces the more, the more policies are in the insurer’s portfolio

Number of Policies

Cost of Capital

1 3.00005 1.5000

10 1.050050 0.4638

100 0.32571000 0.1009

10000 0.0257


Limits to Diversification

Assume now that not all risk are diversifiable.

Assume that the policyholders play the die all in the same casino and that with a given probability p, they will have a crooked croupier. In that case, they will all lose EUR 60, i.e. each throw of a die will always show a 6.

No FraudNumber of Policies

Risk Premium 10% 5% 1% 0.10%

1 3.0000 6.7517 7.1249 6.7103 3.11635 1.5000 6.7529 7.1249 4.9527 1.4925

10 1.0500 6.7442 7.1253 4.5834 1.044150 0.4638 6.7623 7.1202 3.9828 0.4647

100 0.3257 6.7818 7.1286 3.6925 0.32391000 0.1009 6.7564 7.1192 3.5212 0.1803

10000 0.0257 6.5269 7.1266 1.3321 0.1251

Expected Loss 10.0000 15 12.5 10.5 10.05

Probability of Fraud

Diversification is significantly reduced if there are underlying risk factors affecting all policies simultaneously (e.g. a crooked croupier)


From Risk Loading to Cost of Capital

The traditional approach to pricing in insurance was to load the expected loss by the uncertainty of the outcome through a factor times the standard deviation or more generally:

Where L are the losses, is a risk measure (like , 2 or Value-at-Risk), k the risk loading factor and m the costs.

This approach is not compatible with premiums that depend on the losses, which is very common in reinsurance (reinstatements).

It also completely neglects portfolio effect, the cost of capital or target profitability and the payout patterns of the losses.

P E L k L m


From Pricing the Losses to Pricing for Profit

Moreover, the traditional approach with certain risk measures (standard deviation, VaR) is not always additive.

Introducing some basic finance idea we should price the profit to be expected rather than the loss.

Let be a reinsurance treaty with a profit P for the reinsurer:

We can now simply introduce the time value of money by computing the Net Present Value (NPV) of P discounting it to today.

P = Premium-Losses-Expenses


Some Alternative Pricing PrinciplesWe now use X=NPV(P) as the variable to compute the

premium.

Distorted Probability: Denneberg in 1988 and Wang independently in 1995 proposed to find a distortion function G:[0,1][0,1] increasing, surjective and concave such that:

They then define the technical premium as the one for which X satisfies the above equation.

Such a principle is additive.

Applying this methodology one can derive the risk neutral probability that is used in finance for pricing derivatives.

!

: ( )( ) 0XE X x d G F x


Coming up with a Quotation

Loss Model

Losses Expecte

d

Loss

Conditions

Expenses

Pure Losses

Expenses

Risk

Loading

Profitability

RoRBC

Performance

Excess

RBCTreaty Features &

Profit Distribution

NPV


Introducing Diversification and Discounting

The simple example before does not elaborate on two facts:

1. The insurer should price against his portfolio,

2. The payout patterns of the losses count: when does the insurer pay the loss?

We need to introduce here more complicated notions of capital allocation and discounting.

Allocating capital against the portfolio requires to know the dependence between treaties and to use a risk measure that accounts for diversification (sub-additive).


Some Conventions

For the sake of simplicity, we always assume sufficient differentiability, e.g.

Each random variable is assumed to have a density.

Empirical distributions can be approximated by smooth distributions (for our purpose, as exactly as we wish).

For a random variable S, we denote by FS the cumulative distribution function of S.

We use as our basic variable the NPV of the profit of a treaty :

X represents the random variable while is the full treaty.

X = NPV(Premium-Losses-Expenses)


What Is an Appropriate Amount of Profit?

Clearly the expectation of X, E(X), should be positive.

It should also cover the cost of capital to be paid back to the investors.

It should cover the expenses of the operation.

It should include a safety loading as seen before:

The higher the risk, the higher the loading, the higher the dependence with the portfolio the higher the

loading, And the longer it takes to develop to ultimate the more capital is

needed.


The Portfolio Viewpoint

Let us consider the following portfolio :

where i are the different risks (=treaties).

The portfolio is supported by a Risk Based Capital K.

An allocation of capital Ki to i requires a technical premium such that

where i is the duration of the risk i and is the profit target.

1, ,:i i m

X Z

[ ]i i iE X Kht= ×


Capital Allocation: Euler Principle

We allocate capital to a sub-portfolio (e.g., treaty, Line of Business) in according to the Euler principle:

Assume all i=1. Then, roughly speaking, this is the only allocation principle satisfying the following property (D. Tasche, 1999):

If the premium is higher (lower) than the technical one, then a small increase (decrease) of the participation in X will improve (lower) the return on RBC of the entire portfolio.

0

( )St

dK t

dt

Z S

Steering the portfolio through pricing.


Tasche’s result

Theorem (D. Tasche, 1999). Under the above assumptions and some mild differentiability assumptions we have:

1( | ( ))SK E F ZS Z

Thus we allocate capital to a line of business according to its contribution to the bad performance of the whole portfolio.

In order to use this principle in practise, we need a sound portfolio model!

To this end, we need a sound model to describe dependencies.

We use here copulae, CX,Y .


Allocation of Capital to a Treaty

C. Hummel (2002) showed that if we are given a treaty of and the copula CS,Z between S and Z, then:

with

We call H the Diversification Function of in .

The distorted probability depends on the diversification of within.

Note that we do not need to know F to calculate K..

1| ( ) ( )( )S SK E F s d H F s

S ZS Z

,

1( ) ( , )H u C u

S S Z


Interpretation

Consequently, the technical premium should insure a profit X for treaty X that satisfies

This is equivalent to (C. Hummel 2002):

Compare this to Denneberg and Wang’s premium principles: In our setup, the distorted probabilities differ from treaty to treaty and are determined from the diversification effect of the treaty In the

reinsurer’s portfolio.

( )( )( ) 0 ( )

1X

X X X

p H px d G F x G p

with

( ) ( )( )X X Xx dF x x d H F x


Hierarchical Dependences

11X

12X13X

14X15X

21X22X

23X

31X

32X33X

34X

Dependence between Line of Business (LoB)’s

1Y 3Y

Z

2Y

Dependence between contracts


A Central Assumption

Given this structure, the model is completely defined if we also require that:

for all LoB and all risks in .

In other words, given that the result of influences the information about the result in , the latter is not influenced by the distribution of in .

( | , ) ( | )P X x Y y Z z P X x Y y


Copula between the Risk X and the portfolio Z

From the model for the LoB we get CX,Y.

From the distribution of the LoB and its Copula to the portfolio , we get CY,Z.

It is then possible, with relatively mild assumptions, to compute the copula between and :

1, ,

,

0

( , ) ( , ) ( , )X Y Y ZX Z

C CC u w u v v w dv

v v


The Diversification Function

Given the copula, CX,Y, between the risk and the LoB , it is possible to define a diversification function, HX(u), as follows:

Assuming that the grid is fine enough.

1,

11 , ,

0

1,

0

, 1 ,1

1 1

( , )

( , ) ( , )

( , ) ( )

( , ) ( , )( ) ( ) .

X Z

X Y Y Z

X YY

MX Y m X Y m

Y m Y mm m m

C u

C Cu v v dv

v v

Cu v dH v

v

C u v C u vH v H v

v v

( )XH u

0 10 1Mv v v


Only the Diversification Function within the LoB is Relevant

We just saw that:

From this expression it follows:

To be able to price a risk within a line of business, we do not need to compute the copulae between the different LoB’s.

We only need to implement the diversification function, HY, with:

, 1 ,1

1 1

( , ) ( , )( ) ( ) ( )

MX Y m X Y m

X Y m Y mm m m

C u v C u vH u H v H v

v v

0 10 1Mv v v


Coming up with a Quotation

Loss Model

Losses Expecte

d

Loss

Conditions

Expenses

Pure Losses

Expenses

Risk

Loading

Profitability

RoRBC

Performance

Excess

RBCTreaty Features &

Profit Distribution

NPV


Using the Traditional Method for Pricing

Using the standard deviation loading makes all these programs lie on a straight line since they present very similar risk characteristics.

Risk Rate on Line

ExpectedLossRRoL

Granted Limit

Risk Loading for Various CAT Programs

0.0

0.1

0.2

0.3

0 1 2 3-Log(RRoL)

Lo

ad

ing

/ S

tDe

v

ABCDE


Active Portfolio-Management: An Example

The capital allocation taking into account the diversification effects within the portfolio results in different loading for similar risks.

Risk Loading for Various CAT-Programs

0.0

0.1

0.2

0.3

0.4

0 1 2 3

-Log(RRoL)

Lo

ad

ing

/ S

tDe

v

ABCDE

2nd Layer Prg. D

2nd Layer Prg. A

Example:

The distribution of the second layer A and D are almost identical.

A presents a stronger, D a weaker dependence to the rest of the Portfolio.


Active Portfolio-Management: An Example (II)

Diversification or risk accumulation are favored respectively penalized in the price.

As a result, the pricing mechanism implicitly optimizes the portfolio.

Risk Loading for Various CAT-Programs

0.0

0.1

0.2

0.3

0.4

0 1 2 3

-Log(RRoL)

Lo

ad

ing

/ S

tDe

v

ABCDE


The Die Example Priced in our Portfolio

The price for the example, we presented at the beginning is of course depending on the portfolio of the insurer.

We ran this example through our pricing tool MARS and got:11.5 for this example taken in our credit & surety book (dependence to the portfolio).

Let us modify the example by increasing the risk with the same expected loss: we pay 60 USD for one draw of a six.

The price standalone in this case would be: 10 + 0.5 + 7.5 = 18 and MARS would give 12.5.


Conclusion

The concept of Risk-Based Capital is central for understanding the value creation of an insurance company.

The definition of RBC depends on the risk measure used and the risk appetite.

Even if the measure and the threshold are defined: there are different ways of defining the RBC, and each of them is valid in a

certain context.

A sound capital allocation methodology allows to price the risk of an insurance contract to provide the appropriate return on equity.

Modeling the dependencies in a hierarchical way and using expected shortfall as a risk measure allow to price deals against the portfolio.


References

H. Bühlmann, An Economic Premium Principle, Astin Bulletin 11 (1980), 52-60.

M. Denault, Coherent Allocation of Risk Capital, Ecole des H.E.C Montreal, Sept. 1999, revised Jan. 2001, www.risklab.ch/Papers.html#Denault1999 .

D. Denneberg, Verzerrte Wahrscheinlichkeiten in der Versicherungsmathematik, quantilsabhängige Prämienprinzipien, Universität Bremen, 1989.

C. Hummel, Capital Allocation in the Presence of Tail Dependencies, May 2002, Presentation at the Eurandom Workshop on Reinsurance Eindhoven, The Netherlands.

D. Tasche, Risk contributions and performance measurement, Zentrum Mathematik (SCA), TU München, Jun. 1999, revised Feb. 2000, www-m4.mathematik.tu-muenchen.de/m4/pers/tasche/

D. Tasche, Conditional Expectation as Quantile Derivative, Nov. 2000, --- “ ---.

S. Wang, Premium Calculation by Transforming the Layer Premium Density, Astin Bulletin 26 (1996), 71-92.

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