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The Price of Risk in Insurance

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The Price of Risk in Insurance

Presented by Michel M. Dacorogna

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

Although 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.

The content of this document should not be seen in isolation but should be read and understood in the context of any other material or explanations given in conjunction with the subject matter.

This document contains forward-looking statements as defined in the US Private Securities Litigation Reform Act of 1995. It contains forward-looking statements and information relating to the Company's financial condition, results of operations, business, strategy and plans, based on currently available information. These statements are often, but not always, made through the use of words or phrases such as 'expects', 'should continue', 'believes', 'anticipates', 'estimated' and 'intends'. The specific forward-looking statements cover, among other matters, the reinsurance market, the outcome of insurance regulatory reviews, the Company's operating results, the rating environment and the prospect for improving results, the amount of capital required and impact of our capital improvement measures and our reserve position. Such statements are inherently subject to certain risks and uncertainties. Actual future results and trends could differ materially from those set forth in such statements due to various factors. Such factors include general economic conditions, including in particular economic conditions; the frequency, severity and development of insured loss events arising out of catastrophes; as well as man-made disasters; the outcome of our regular quarterly reserve reviews, our ability to raise capital and the success of our capital improvement measures, the ability to exclude and to reinsure the risk of loss from terrorism; fluctuations in interest rates; returns on and fluctuations in the value of fixed income investments, equity investments and properties; fluctuations in foreign currency exchange rates; rating agency actions; the effect on us and the insurance industry as a result of the investigations being carried out by US and international regulatory authorities including the US Securities and Exchange Commission and New York’s Attorney General; changes in laws and regulations and general competitive factors, and other risks and uncertainties, including those detailed in the Company's filings with the US Securities and Exchange Commission and the SWX Swiss Exchange. The Company does not assume any obligation to update any forward-looking statements, whether as a result of new information, future events or otherwise.

Please further note that the Company has made it a policy not to provide any quarterly or annual earnings guidance and it will not update any past outlook for full year earnings. It will however provide investors with perspective on its value drivers, its strategic initiatives and those factors critical to understanding its business and operating environment.

This document does not constitute, or form a part of, an offer, or solicitation of an offer, or invitation to subscribe for or purchase any securities of the Company. Any securities to be offered as part of a capital raising will not be registered under the US securities laws and may not be offered or sold in the United States absent registration or an applicable exemption from the registration requirements of the US securities laws.

- A simple example
- Risk and risk measures
- Risk-based capital and economic capital
- Valuation methods
- Conclusions

- 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.

- What is the price for such a risk, independently of any other liability the insurer has?
- How would the price change if the insurer assume many of the same risk?

Second Throw

First Throw

And so on …

Value-at-Risk(1%) = 30 USD

Expected Loss = 10 USD

- Pricing the risk at the expected lossplus 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).

- 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 capitalhas a cost – investors want a return on investment.
- Let us assume in this case a cost of 15% before tax.

Premium

Expenses

0.5

Company

Structure and Capital

3.0

Risk loading

0.15* (30-10)

10

Loss model

Expected loss

13.5

- 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.

- 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):
- Scalable (twice the risk should give a twice bigger measure),
- Ranks risks correctly (bigger risks get bigger measure),
- Allows for diversification (aggregated risks should have a lower measure),

VaR

Standard Deviation

Measures typical size of fluctuations

+s

Value-at-Risk (VaR)

Measures position of 99th percentile, „happens once in a hundred years“

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

Mean

- 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%).

>

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

- 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

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

- 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.

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

- 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 Lare the losses, ris a risk measure (like s, s2 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.

- 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 Xbe a reinsurance treaty with a profit Pfor the reinsurer:
- We can now simply introduce the time value of moneyby computing the Net Present Value (NPV) of Pdiscounting it to today.

P = Premium-Losses-Expenses

- We 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 concavesuch that:
- They then define the technical premium as the one for which Xsatisfies 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.

Profitability

RoRBC

Treaty Features &

Profit Distribution

RBC

PerformanceExcess

NPV

RiskLoading

Loss Model

Expenses

Expenses

Conditions

Losses

Expected

Loss

Pure Losses

- The simple example before does not elaborate on two facts:
- The insurer should price against his portfolio,
- The payout patterns of the losses count: when does the insurer pay the loss?

- 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).

X = NPV(Premium-Losses-Expenses)

- Clearly the expectation of X, E(X), should be positive.
- It should also cover the cost of capitalto 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.

- Let us consider the following portfolio Z:
- where Xiare the different risks (=treaties).
- The portfolio Z is supported by a Risk Based CapitalK.
- An allocation of capitalKito Xirequires a technical premium such that
- where ti is the durationof the riskXi and h is the profit target.

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

Steering the portfolio through pricing.

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

- Thus we allocate capital to a line of business according to its contributionto 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 .

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

- We call HS the Diversification Functionof S in Z.
- The distorted probability depends on the diversification of S withinZ.
- Note that we do not need to know FZ to calculate KS..

- Consequently, the technical premium should insure a profit Xfor 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.

Dependence between Line of Business (LoB)’s

Dependence between contracts

- Given this structure, the model is completely defined if we also require that:
for all LoB Y and all risks X in Y.

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

- From the model for the LoB Y we get CX,Y.
- From the distribution of the LoB Yand its Copula to the portfolio Z, we get CY,Z.
- It is then possible, with relatively mild assumptions, to compute the copula between X and Z :

- Given the copula, CX,Y, between the risk X and the LoB Y, it is possible to define a diversification function, HX(u), as follows:
- Assuming that the grid is fine enough.

- 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:

Profitability

RoRBC

RBC

Treaty Features &

Profit Distribution

PerformanceExcess

NPV

RiskLoading

Loss Model

Expenses

Expenses

Conditions

Losses

Expected

Loss

Pure Losses

- 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

2nd LayerPrg. A

- Example:
- The distribution of the second layer A and D are almost identical.
- A presents a stronger, D a weakerdependence to the rest of the Portfolio.

2nd Layer Prg. D

Active Portfolio-Management: An Example

- The capital allocation taking into account the diversification effects within the portfolioresults in different loadingfor similar risks.

- Diversification or risk accumulation are favored respectively penalized in the price.
- As a result, the pricing mechanism implicitly optimizes the 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.

- 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.

- 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.