1. INTERNATIONAL RATE-MAKING APPLICATIONS OF DFA Chris Daykin, UK Government Actuary CAS Rate-making Seminar, San Diego, 9-10 March 2000

2. CHARACTERISTICS OF DFA use of cash-flow projections inclusion of assets as well as liabilities may be deterministic or stochastic lends itself to sensitivity analysis regulatory requirements built in many possible types of model

3. ADVANTAGES OF DFA FOR RATE-MAKING requires explicit consideration of uncertainty takes account of all cash-flows (& incidence) allows for the impact of capital requirements uses realistic investment assumptions shows impact of investment returns on rating explores trade-off of risk and return permits explicit modelling of market and cycles

4. LEVELS OF DFA by individual contract by line of business by portfolio for whole company

5. SINGLE PRODUCT DFA “profit-testing” cohort analysis detailed cash-flow impact consider matching investments identify capital requirements and return uncertainty reflected in higher price

6. MODEL OFFICE DFA for line of business, portfolio or company add new business to existing model office consider impact on investment strategy model impact on financial strength consider price/volume sensitivity introduce impact of insurance cycles

7. MODELLING MARKET BEHAVIOUR multi-player game simplify to two-player game company in question and rest of market model impact of differential pricing elasticity of demand or gearing

8. PRICE ELASTICITY exponential price elasticity  B/B = (1 -  ) -  - 1   where  is an empirical elasticity parameter, also known as a gearing factor, and -100  is the % deviation from the level of market prices linear price elasticity - change in volume is proportional to difference in premiums input-output model:  B =  m ·E m where E m is an investment to generate extra business and  m is an empirically determined return coefficient

9. NATURE OF CYCLES capacity fluctuations new entrants to the market competitive strategies interest rates and asset fluctuations economic cycle influences on loss propensity unforeseen changes in inflation time-lag of premium changes behind losses

10. MODELLING CYCLES model cycle in underlying loss ratios model cyclical variation in profitability of premium rates charged company may adopt different stance * because of incorrect anticipation of cycle * in an attempt to buck the cycle * or in an attempt to recover past losses allow for different behaviour of company

11. ADEQUACY OF CAPITAL test capital requirement using DFA ensure target solvency is met after a period… ….at a satisfactory probability level DFA provides distribution of outcomes requirement will depend on position in cycle allow additional capital for new business risk

12. PRICING FOR RISK DFA at company level provides overview can incorporate loss uncertainty into model combine with other tools to price for risk need to allocate capital or at least profit target avoid artificial arbitrage situations

13. PROFIT OBJECTIVE set overall profit objective from new business allocate to line using pricing model Proportional Hazards Pricing Basis (Wang) set risk aversion level... ….and hence premium level

14. Proportional Hazards (P-H) Pricing The Risk Adjusted Premium  X  at Risk Aversion Level (RAL) , is given by:  (X) = - S(x)=Survival Function = Probability ( X > x ) - X is the pv of the cash flows at the risk-free rate At  Risk Adj. Prem = Expected Value As  Risk Adj. Prem  Max Value All moments incorporated in calculation  1/ (S(x)) dx

15. P-H Graphical Representation AREA UNDER CHART EQUALS MEAN VALUE S(x)^(1/  )

16. Impact of increasing Risk Aversion Level  VALUES IN THE TAIL GIVEN PROGRESSIVELY MORE WEIGHT

17. WHY USE PROPORTIONAL HAZARDS MODEL? uniquely satisfies set of desirable conditions provides sound basis for quantifying risk incorporates skewness as well as variability generalisation of standard deviation principle risk aversion level is equivalent to varying risk discount rate can use implicit RALs to compare price adequacy

18. Actuarial Models: Standard Premium = Pure Premium + Safety(Risk) Loading Expected Value:  E (P) = ( 1 +  ) * E(P) Standard Deviation:  S (P) = E(P) +  * SD(P) Variance Principle:  V (P) = E(P) +  * Var(P) Generalisation (see Daykin et al PRTFA) Risk Load = 1 * E(P) + 2 * SD(P) + 3 * Var(P) Important Observation: Standard Dev.  *Coefficient of Variation

19. SHAREHOLDER ADDED VALUE present value of future new cash flows test for sensitivity analyse by diversifiable and systematic risk offers clarity in setting objectives identifies contributions of new and old business overall return on capital

20. LLOYD’S AND LONDON MARKET DFA used as a tool to support underwriters permits scientific rate-making mechanisms market determines feasible prices does pricing allow enough for risk? use to measure impact of reinsurance program use in design of financial reinsurance

22. Economic Pricing Requirements Pricing is “at the margin” –supply and demand (changing risk aversion level?) Prices have no opportunity for arbitrage –market forces determine prices (& implicit risk aversion level) –prices for parts of a whole must add up to the price of the whole –often a problem with actuarial bases

23. Actuarial Pricing Requirements Positive Loading and No Rip-Off E(P)   (P)  Max (P) Preservation of Stochastic Order If S P (t)  S Q (t) then  (P)  (Q) Linearity and scale invariance  (aX+b) = a  (X)+b (Currency, QS RI,.. ) Sub-additivity  (X+Y)  (X) +  (Y) Additive for Comonotonic Risks  (X+Y) =  (X) +  (Y) Layer additivity for Excess of Loss etc