CAS Spring Meeting, Puerto Rico, May 8th 2006 Pragmatic Insurance Option Pricing by Jon Holtan If P&C Insurance Company Norway/Sweden/Denmark/Finland

From only fair risk valuation to also include market-oriented pricing Insurance risk/ cost fair valuation Financial option pricing Market based insurance pricing 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 ! Formulated within a market based context

Well-known: Insurance contract as a call option Call option Insurance call contracts => "Insurance option pricing" Z is a stochastic sum of N single European call options Z* is an ordinary European call option => Z* is stop loss contract on claims process => 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 But: Crucial difference between options and insurances: Options => based on geometric Brownian stock processes Insurances => based on compound Poisson claims processes

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 !

Key setup: Risk-neutral martingale valuation = Fair value excess of loss: Fair value stop loss: Fair value call option: What "market" information should we put into Q? 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

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 !!

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

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! 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?

Key pricing challenge: How to optimize bottom line in markets with profitable and non-profitable segments? Low riskHigh risk High price Low price Average price level in market Best knowledge of true risk/cost price level 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 Risk segments where the market earns money

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):

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

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

Summary insurance versus option pricing