Pragmatic Insurance Option Pricing by Jon Holtan If P&C Insurance Company Norway/Sweden/Denmark/Finland

1. Pragmatic Insurance Option Pricingby Jon HoltanIf P&C Insurance CompanyNorway/Sweden/Denmark/Finland CAS Spring Meeting, Puerto Rico, May 8th 2006

2. From only fair risk valuation to also include market-oriented pricing Market based insurance pricing Formulated within a market based context Financial option pricing Insurance risk/ cost fair valuation • Main objectives of the paper: • First of all: Point out a pragmatic market based approach of insurance pricing. • Why do different insurance players offer different prices for unique risks in the same market ? • Important bi-effect: From a practical point of view describe the common headlines of the theories of insurance and option pricing. • Gives valuable pragmatic insight to (stochastic) value versus (market) price relationships !

3. Well-known: Insurance contract as a call option Call option => C is a Euro call option on stock price S(t) and strike price K Insurance => Z is excess-of-loss contract on claims process N and Y => Z* is stop loss contract on claims process • Insurance call contracts => "Insurance option pricing" • Z is a stochastic sum of N single European call options • Z* is an ordinary European call option • But: Crucial difference between options and insurances: • Options => based on geometric Brownian stock processes • Insurances => based on compound Poisson claims processes

4. Hard insight: Dynamic hedging of insurance contracts • General definitions: • Dynamic hedging => continuously buying and selling • Hedging strategy => remove risk by replicating a risk free payoff • Main difference between financial options and insurance: • Options => hedging to remove risk => do not rely on the law of large numbers • Insurance => diversification to manage risk => rely on the law of large numbers • However: insurance hedging may be seen as an approach to establish a relevant and efficient insurance market place and hence better understand the market information dynamics. • Example: a company reduces its insurance risk through reassurance. Martingale !

5. Key setup: Risk-neutral martingale valuation • Help: The arbitrage-free (Black-Scholes) hedging approach helps us to better understand market based insurance pricing. • But: Arbitrage-free dynamic hedging of insurance contracts is only possible when buying and selling martingale values. • Hence: A risk-adjusted probability measure transformation from P to Q makes a risk-neutral martingale valuation of the insurance contracts Fair value excess of loss: Fair value stop loss: = Fair value call option: What "market" information should we put into Q?

6. Key question: What information should we put into the probability measure Q? 1) Traditional actuarial information approach I: • Risk and risk loading Delbaen & Haezendonck (1989) approach: Give more weight to unfavourable events • Expected value premium principle => E(X) + a E(X) • Variance premium principle => E(X) + b Var(X) • Esscher premium principle => E(X exp(aX)) / E(exp(aX)) 2) Traditional actuarial approach II • Add expenses, reassurance costs, investment returns => cost allocation 3)Untraditional actuarial information approach => add supply and demand information like purchasing preferences and insurer's price position in the market • Very relevant information in incomplete markets ! • Actuaries should not let the real market information be irrelevant to them !!

7. Market based insurance pricing Formulated within a market based context Insurance risk valuation Financial option pricing OK

8. Finn & Lane (1997): "There are no right price of insurance.... ...there is simply the transacted market price which is high enough to bring forth sellers and low enough to induce buyers" Key question: Complete or incomplete market? • 1) Complete markets => Most financial markets • Unique price per unique risk => Unique market prices => The law of one price • Optimal price per risk is pure risk and cost price based • 2) Incomplete markets => Most insurance markets • Optimal price per risk is also market adjusted => Different market prices per risk • Because: Each combination of buyer and seller generates valuable uniqueness • ...ref Harrison & Pliska (1983): • "A market is complete if and only if there is only one equivalent martingale measure of the underlying stochastic process (stocks or claims) describing the market" • Hence: Purchasing preferences should be part of the price models!

9. Key pricing challenge: How to optimize bottom line in markets with profitable and non-profitable segments? • Driving elements: • True risk/cost prices versus market prices of different markets/customer segments • Price sensitivity of different markets/customer segments Risk segments where the market makes losses High price Best knowledge of true risk/cost price level Average price level in market Risk segments where the market earns money Low price Low risk High risk

10. Market based insurance price models A general set up: Net sales price per risk = pure risk price + internal cost allocation price adjustments + external market price adjustments A more specific set up => Should be expressed with respect to price p: Net present value V(T) of the insurance portfolio over the time period (0,T):

11. Parameter estimation = real insurance pricing! • Risk, expenses, capital & return pricing estimation => Well-known! • Risk selection multi-factor: GLM; Bayesian; Credibility; etc • Price level: RR and CR targets based on capital & return allocation (DFA) per LoB • Claims level trend: ARIMA time series forecasting, Claims reserving, ++ • Expense allocation: Internal "ABC"++ methods (from simple to complex) • Market pricing estimation => Not well-known! • Customer purchasing rate (multi-factor based): LogReg on hitrates/renewals • Soft/hard opinions of the price and product positions in the market • Dynamic, flexible, short time-to-market, data driven IT price calculation systems • No final answers and no fixed rules

12. Market based insurance pricing OK Formulated within a market based context Insurance risk valuation Financial option pricing OK

13. Summary insurance versus option pricing