Why Larger Risks Have Smaller Insurance Charges

1. Why Larger Risks Have Smaller Insurance Charges Ira Robbin, PhD Partner RE

2. Why Larger Risks Have Smaller Insurance Charges Outline • Intro • Retros • Charge Definitions • Intuitions about Charge by Size • Charges for Sums • Sum of Two Risks • Finite Independent Sum • Result for Decomposable Risk Models • Bayesian Priors on Risk Model • Severity • General Result and Conclusion

3. Why Larger Risks Have Smaller Insurance Charges Retro Rating • Retro Premium • Loss sensitive • Subject to Max and Min Premiums • For our discussion, neglect loss limits • RP=SP*TM*(B+LCF*LR) • B=Basic • Expense in Basic + Net Insurance Charge

4. Why Larger Risks Have Smaller Insurance Charges Net Insurance Charge in Basic • NIC(% of SP) = LCF*ELR*NIC(% of EL) • NIC(% of EL) = Charge at Max – Savings at Min • NCCI Table of Insurance Charges • Enter Table with Entry Ratio • Max Entry Ratio = Max Loss/EL • Table Columns indexed by LG =Loss Group • LG 25 at 1.0 entry ratio, charge is 0.250 • Charges expressed as % of EL

5. Why Larger Risks Have Smaller Insurance Charges Retro Insurance Charge by Size • LG# determined by EL • LG Range Table • Adjustment for S/HG Severity • Increase in EL reduces LG # • As LG# declines, so do charges • Conclusion: Under Retro procedure: …Larger Risks get Smaller Charges… • Just a coincidence?

6. Why Larger Risks Have Smaller Insurance Charges Insurance Charge Definition • Insurance Charge Function (r) for T • T0 and =E[T]>0. • R=T/ = Normalized RV. • (r) =expected loss excess of r as ratio to  • Standard “integral” definition

7. Why Larger Risks Have Smaller Insurance Charges E[Max] and E[Min] Definitions Expectation of Max and Min

8. Why Larger Risks Have Smaller Insurance Charges Why Use the E[Min] Definition? Two Useful Min Inequalities E[R;r] is a LEV. It may be easier to prove statements about LEVs and then translate to results about charges.

9. Why Larger Risks Have Smaller Insurance Charges Intuition About Charges by Size • Larger risks ought to have smaller charges • Smaller XS Ratio function at every entry ratio • True in all actuarial literature • True for NCCI Table of Insurance Charges • Loss group look-up depends on E[L] • Law of Large Numbers • Take independent sum of “n” iid risks • CV decreases with sample size • Less likely to get extreme results

10. Why Larger Risks Have Smaller Insurance Charges Charges by Size Graph

11. Why Larger Risks Have Smaller Insurance Charges What is the Problem? • Intuition makes sense, but is no proof • Smaller CV does not imply smaller charges • Not at every entry ratio • Logic is not sufficiently general • Large Risk  Independent Sum of Small Risks • How to define risk size? • EL alone is insufficient to lead to desired result • E[Claim Count] should be key

12. Why Larger Risks Have Smaller Insurance Charges CV Counterexample

13. Why Larger Risks Have Smaller Insurance Charges CV Counterexample Graph

14. Why Larger Risks Have Smaller Insurance Charges Goals and Strategy • Prove what we can for sums • Charge for sum of two RVs • Charge for sum of “n” iid RVs • Generalize to decomposable models • Extend to handle parameter risk • Use Count results to prove results hold for CRM Loss

15. Why Larger Risks Have Smaller Insurance Charges Charge Inequality forSum of Two RVs • Proof: use E[T;r] =  (1-(r)) • Recall: min (a+c, b+d)  min(a,c)+min(b,d) • min(4+5, 3+6)=9 min(3,4)+min(5,6)=8 • E[T1+T2; r(1+2)]E[T1; r1]+ E[T2; r2]

16. Why Larger Risks Have Smaller Insurance Charges Charge Inequality forSum of Two ID RVs • T1, T2 and T are identically distributed • T1 and T2 are not necessarily independent • See proof for sum of two RVs • This is the first “Charge by Size” result! • Does not readily extend to “n+1” vs “n” result

17. Why Larger Risks Have Smaller Insurance Charges Charge Inequality forSum of “n” IID RVs • Sn= S(1,2,…,n) = T1 + T2 + …+ Tn • Assume sample selection independence • S(1,2,…,n) distributed same as S( i1, i2 ,…in ) • S(~k / n+1) = T1 + …+ Tk-1 + Tk+1 +…+ Tn+1 • S(~2/ 3) = T1 + T3

18. Why Larger Risks Have Smaller Insurance Charges Proof Step 1:Combinatoric Trick 2S3 = T1 + T2 + T3 + T1 +T2+ T3 2S3 = (T1 + T2 )+ ( T1 +T3) + (T2 + T3 )

19. Why Larger Risks Have Smaller Insurance Charges Proof: Step 2Apply Min Inequality

20. Why Larger Risks Have Smaller Insurance Charges Use Sample Independenceand E[min] Formula to Finish

21. Why Larger Risks Have Smaller Insurance Charges Proof with n=3

22. Why Larger Risks Have Smaller Insurance Charges Risk Size Models • Initially identify risk size with the mean • E[T]=   T has size  • Risk Size Model, M is a set of RVs • Unique Risk Size Model, M • If T1 and T2 are the same size, T1 = T2 • Closed Under Independent Summation • If T1M and T2M, then T1+ T2M • Complete: T in M for every >0.

23. Why Larger Risks Have Smaller Insurance Charges Decomposable Risk Size Models • Let M be a Unique Risk Size model • Decomposable If =1+2 then  T1 , T2 ,TM where the independent sum, T1 + T2 = T • Closed and Complete  Decomposable

24. Why Larger Risks Have Smaller Insurance Charges Differentiable Decomposable Size Models • M is Differentiable if • Some Differential Decomposable Models • Poissons • Negative binomials common failure rate • Gammas with common scale

25. Why Larger Risks Have Smaller Insurance Charges Differentiable Size Model Inequalities on Partials • If M is Differentiable and Decomposable,

26. Why Larger Risks Have Smaller Insurance Charges Proof of One Inequality

27. Why Larger Risks Have Smaller Insurance Charges LEV as Function of Risk Size-Graph of Poisson Example

28. Why Larger Risks Have Smaller Insurance Charges Result for Decomposable Models • Larger Risks have Smaller Charges Proof: Let m1=1 and m2=2. Result follows from charge inequality for independent sum of IID RVs.

29. Why Larger Risks Have Smaller Insurance Charges Comments on Proof • Decomposability needed to write out decomposition of T’s as iid sums • Proof is technically valid only for rational values of risk size • Proof extends to all risk sizes due to continuity of charge as function of size • Converse of Proposition is not true: Examples easy to construct where larger risks have smaller charges and model is not decomposable

30. Why Larger Risks Have Smaller Insurance Charges Application of Result to Commonly Used Distributions • Larger Risks have Smaller Charges in the following models • M= {Poissons} • M= {T| T=NegBi(,q) with q fixed} • M={T| T=Gamma(,) with  fixed} • Proof: These are all decomposable.

31. Why Larger Risks Have Smaller Insurance Charges Charge for Infinitely Large Risk in Decomposable Model CV argument works to prove this. Uses result:

32. Why Larger Risks Have Smaller Insurance Charges Risk Size Model with Parameter Uncertainty • Separate true mean from a priori mean • =true mean • = a priori mean of a risk • Mm = all risks in model with prior mean  • Prior distribution, H( |  ). • E[|]=  where expectation uses H. • Covers risk heterogeneity and predictive uncertainty

33. Why Larger Risks Have Smaller Insurance Charges Parameter Uncertainty Setup • Q={ |  > 0 } •  is an RV with cdf H( |  ) • Assume Q is a unique risk size model not necessarily decomposable. • Suppose these priors act on a decomposable risk size model of conditionals, M={T()} • Let M(Q) = resulting set of unconditional RVs • Use properties of Q and M to get results on M(Q).

34. Why Larger Risks Have Smaller Insurance Charges Parameter Uncertainty Example • Exponentials on Poissons • Q={Exponential with mean  |  > 0 } • H( |  )= 1-exp(- /  ) • M={Poisson with mean  |  >0} • M(Q) = {Geometric RVs} • Unconditional Density f(n| )=(1-q)qn • Where q=  /( +1).

35. Why Larger Risks Have Smaller Insurance Charges Bayesian Formula for Charge • Let  =RV with distribution H( |  ) • Let h(| ) be the associated density This represents the charge for risks with prior size equal to .

36. Why Larger Risks Have Smaller Insurance Charges Result for Scaled Priors • Larger Unconditional Risks have Smaller Charges if Priors are a family of scaled distributions • Let 2= (1+c) 1, then Note the Unconditional Risks do not in general form a decomposable family

37. Why Larger Risks Have Smaller Insurance Charges Scaling Result Proof • Drop much of the conditional notation to simplify expressions and write the Bayesian integral for the charge as: Use scaling to relate the densities

38. Why Larger Risks Have Smaller Insurance Charges Scaling Result Proof-Next steps • Plug in to get Change variables to get

39. Why Larger Risks Have Smaller Insurance Charges Scaling Result Proof-Finale • T((1+c)) is a larger risk than T() in M. So it has a smaller charge: Arrive at conclusion

40. Why Larger Risks Have Smaller Insurance Charges Contagion Model of Counts • Scaled Gammas on Poissons • Q={Gamma(, /)|  fixed and  > 0 } • C = contagion = 1/ from CRM • M={Poisson with mean  |  >0} • M(Q) = {Negative Binomial RVs} • Parameters of unconditional density •  fixed and q = /(+) • Larger Risks have Smaller Charges for Claim Count RVs in CRM model

41. Why Larger Risks Have Smaller Insurance Charges General Result in Words • Suppose the priors form a Unique Risk Size Model, Q, in which Larger Risks have Smaller Charges • Assume the priors act on a Decomposable Differentiable Model, M • Then in the Unconditional Model, M(Q), it follows that Larger Risks have Smaller Charges • Note the Unconditional Model is not generally decomposable. Large risks are not the independent sum of small risks.

42. Why Larger Risks Have Smaller Insurance Charges General Result in Math Symbols

43. Why Larger Risks Have Smaller Insurance Charges Comments on Proof The proof makes use of integration by parts and the inequalities on the partials of the LEVs. We will not give it here. Simple argument: If it works for scaling, where all priors have the same charge, then it ought to be true when the charges on the priors decline with risk size

44. Why Larger Risks Have Smaller Insurance Charges Infinitely Large Risks The CV for infinitely large risks does not approach 0, but rather the CV of the prior for an infinitely large risk. • Nothing we have assumed forces the CV of the prior to approach 0. • Contagion model CV approaches -1/2

45. Why Larger Risks Have Smaller Insurance Charges Aggregate Loss Model-IID Severity • T(N,X) = X1 + X2 +…+ XN • Independent Severity • all risks share common severity RV, X • {X1, X2, … XN} is an independent set. • Xi’s of different risks are independent. • Xi is independent of N. • Xi is independent of , where  is the true mean of N for a risk.

46. Why Larger Risks Have Smaller Insurance Charges Loss Model with IID Severity Inherits Decomposability MT(N,X) = {T(N, X) N MN } Result: If the count model, MN,is decomposable, then so is the loss model MT(N,X) when severities are iid. Conclusion: Larger risks have smaller charges in any Loss Size model based on decomposable counts and iid severity.

47. Why Larger Risks Have Smaller Insurance Charges Independent Severity with Scale Parameter Uncertainty • Each risk has a particular b and associated severity RV, Y=X/b • The Xi satisfy the usual independence properties: • {X1, X2, … XN} is an independent set. • Xi’s of different risks are independent. • Xi is independent of N. • is a positive continuous RV with E[1/b] =1 and Var(1/b) =b =mixing parameter in CRM. • b is independent of  and m

48. Why Larger Risks Have Smaller Insurance Charges Aggregate Loss Model –Assumptions for Key Result • Assume decomposable count model MN • Let Q = {} be a unique risk size model of priors on the count distributions in MN • Assume larger prior risks have smaller(not necessarily strictly) charges. • Assume Independent Severity with Scale Parameter Uncertainty

49. Why Larger Risks Have Smaller Insurance Charges Aggregate Loss Model –Key Result • Larger Risks have Smaller Charges • MT( N|Q,Y| X) is the Risk Size Model • Risks of a priori mean = m E[X] have size m E[X] though each risk has true mean  E[X/b] • The introduction of severity and the “b”increase the charges, but do not change the relation between charges of different size risks.

50. Why Larger Risks Have Smaller Insurance Charges Conclusions • 1st result: Larger risks have smaller charges in decomposable model. • Adding in parameter risk with priors doesn’t change the relation - assuming the priors have smaller charges by size. • The resulting final model does not require large risks to be the independent sum of small risks. • Introducing Severity does not change charge by size relationships • Larger risks have smaller charges under CRM • It can be a lot harder than you think to prove what everyone knows is true.