A Bayesian Approach to Pricing with worked example

Transcription

1 A Bayesian Approach to Pricing with worked example Win-Li Toh & Andrew Kwok Taylor Fry Pty Ltd The Institute will ensure that all reproductions of the paper acknowledge the Author/s as the author/s, and include the above copyright statement:

2 Aim of the paper To determine loss ratios for pricing purposes using a Bayesian* approach, combining prior knowledge and claims experience To apply this theory on real data

3 Data available A professional indemnity book Five years data for claims reserving Split of claims and premium data by occupational groups Historical year-to-year premium increases

4 Basic theory Use a combination of: the B-F method (some prior knowledge) chain ladder (data driven) in deriving the loss ratios Determine (using a mathematically derived credibility formula) how much reliance to place on results from each method

5 Assumed: A GLM framework Incremental claims C ij distributed as over-dispersed Poisson: E(Cij) = xiyj and Var (Cij) = φ xiyj xi ~ independent Gamma (αi, βi): E(xi) = αi/βi and Var(xi) = αi/βi 2

6 A GLM framework (cont d) Theoretical distribution for Cij has mean: [ZijDi,j-1 + (1-Zij)*αi/βi*(1/ λjλj+1 λn)] * (λj-1) Chain-ladder estimate, assumes no prior information of loss ratios BF estimate, assumes perfect prior information about loss ratios where Z ij is the credibility factor: Zij = (1/λjλj+1 λn) / (1/p*β i φ + (1/λjλj+1 λn))

7 A GLM framework (cont d) Z ij = (1/λjλj+1 λn) / (1/p*βiφ + (1/λjλj+1 λn)) Z effects a trade-off between the BF method and chain ladder results: λ j λ j+1 λ n - how far through development 1/p - how large is occupational group β i - how uncertain are initial loss ratios φ - how variable are incurred costs

8 Worked example selecting initial assumptions Assumptions* needed for BF method: initial projected ultimate loss ratio, expected pattern of time lags Assumptions* needed for chain-ladder: development factor for reported incurred costs to ultimate *We have selected these assumptions from the most recent outstanding claims valuation carried out for the total portfolio, for underwriting years

9 Worked example selecting initial assumptions (cont d) Initial loss ratio assumption for BF Reported incurred time lag patterns for BF Incurred development factors for CL In 31/12/2004 values Underwriting Gross Paid to Case Estimated Estimated Gross IBNR Development year premium date estimates outstanding ultimate loss factor factor on $m $m $m $m losses, $m ratio (g)= incurred (h)= (a) (b) (c) (d) (e)=(b)+(d) (f)=(e)/(a) [(d)-(c)]/(e) (e)/[(b)+(c)] % 8% 108% % 11% 112% % 39% 165% % 67% 301% % 88% 819%

10 Worked example variability between years, β Used variance of the normalised loss ratios over the last 5 underwriting years In 31/12/2004 values Underwriting Gross written Estimated Gross Increase Revised β i year, i premium ultimate loss in renewal gross $m losses, $m ratio premium rates loss ratio (a) (b) (c)=(b)/(a) (d) (e) (f)=α/(b) % 73% 8.88E % 5% 70% 8.11E % 7% 99% 6.94E % 12% 103% 7.82E % 15% 106% 7.16E-06 Arithmetic mean = 90% Corresponding variance = 3% α = mean 2 /variance = 27.3

11 Worked example results under 1) BF and 2) chain ladder Assume Accountants represent 64% of book in 2002 Basic details BF results Chain ladder results In 31/12/2004 values Underwriting Gross written Paid to Case Incurred year premium date estimates gross loss $ $ $ ratio (d)= (a) (b) (c) [(b)+(c)]/(a) ,812, , ,874 38% BF method Initial gross IBNR Ultimate Ultimate loss ratio factor losses gross loss $ $ $ ratio (e) (f) (g)=(b)+(c) (h)=(g)/(a) +(a)*(e)*(f) 89% 39% 2,055,554 73% Chain-ladder method Development Ultimate Ultimate factor losses gross loss $ $ ratio (i) (j)=(i)*[(b)+(c)] (k)=(j)/(a) 165% 1,769,485 63%

12 Worked example what ppn of each method to adopt? Determined by formula for Z where: Z = 79% (from Table 3) / (1/64% * 6.94E-06 (from Table 1) * (from Table 2) + 79% (from Table 3)) = 38%. Gross written Proportion Z Ultimate Ultimate premium of book losses gross loss $ $ (n)= ratio (a) (l) (m) (1-Z)*(g)+Z*(j) (o)=(n)/(a) 2,812,147 64% 38% 1,946,504 69%

13 Worked example normalisation of results Process above to determine loss ratios repeated for each underwriting year, and occupational group Resulting loss ratios from prior years then adjusted as if 2004 premium rates had applied throughout Underwriting Ultimate gross Year on year "Normalised" year loss ratio premium gross loss increase ratio % 5% 52% % 7% 52% % 12% 54% % 15% 59% % 64%

14 Some concluding comments Premium reductions possible from our example Intended to complement obvious pricing methods e.g. competitor rates, hardness of the market, management desire for growth in particular markets.