Why Larger Risks Have Smaller Insurance Charges

CAS Spring Meeting, May 8-10, 2006

Why Larger Risks Have Smaller Insurance Charges

Ira Robbin, PhD

Partner RE

Why Larger Risks Have Smaller Insurance Charges2 CAS Spring Meeting, May 8-10, 2006

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


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


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


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?


Insurance Charge Definition

Insurance Charge Function (r) for T T0 and =E[T]>0. R=T/ = Normalized RV. (r) =expected loss excess of r as ratio to

Standard “integral” definition

)rs()s(dF)rt()t(dF1

)r(r

Rr

TT


E[Max] and E[Min] Definitions

)]rR,0[max(E)]rT,0[max(E

)r(

Expectation of Max and Min

]r;R[E1]r;T[E

1)]r,T[min(E

1)r(


Why Use the E[Min] Definition?

)dc,bamin()dc,bmin()dc,amin()ii

)d,bmin()c,amin()dc,bamin()i

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.


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


Charges by Size Graph


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


CV Counterexample

index point density square ratio CDF Savings Tail Charge

i ti f(ti) ti 2 ri F(ri) ri

G(ri) ri

1 0.00 20.0% 0.00 0.00 20.0% 0.0% 80.0% 100.0%2 2.00 20.0% 4.00 0.50 40.0% 10.0% 60.0% 60.0%3 4.00 20.0% 16.00 1.00 60.0% 30.0% 40.0% 30.0%4 6.00 20.0% 36.00 1.50 80.0% 60.0% 20.0% 10.0%5 8.00 20.0% 64.00 2.00 100.0% 100.0% 0.0% 0.0%

Mean 4.00 24.00 1.00

index point density square ratio CDF Savings Tail Charge

i ti f(ti) ti 2 ri F(ri) ri

G(ri) ri

1 0.00 0.0% 0.00 0.00 0.0% 0.0% 100.0% 100.0%2 2.00 60.0% 4.00 0.50 60.0% 0.0% 40.0% 50.0%3 4.00 10.0% 16.00 1.00 70.0% 30.0% 30.0% 30.0%4 6.00 0.0% 36.00 1.50 70.0% 65.0% 30.0% 15.0%5 8.00 30.0% 64.00 2.00 100.0% 100.0% 0.0% 0.0%

Mean 4.00 23.20 1.00

Random Variable T2

Random Variable T1


CV Counterexample Graph

RV with Smaller CV Has Larger Charge at some Entry Ratios

0.00

0.25

0.50

0.75

1.00

0.00 0.50 1.00 1.50 2.00

T1T2

Figure 2


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


Charge Inequality forSum of Two RVs

)r()r()r(2121 T

21

2T

21

1TT

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; r1]+ E[T2; r2]


Charge Inequality forSum of Two ID RVs

)r()r( TTT 21

T1, T2 and T are identically distributedT1 and T2 are not necessarily independentSee proof for sum of two RVs

This is the first “Charge by Size” result!Does not readily extend to “n+1” vs “n” result


Charge Inequality forSum of “n” IID RVs

)r()r(n1n SS

Sn= S(1,2,…,n) = T1 + T2 + …+ Tn

Assume sample selection independenceS(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


Proof Step 1:Combinatoric Trick

)1n/k(~SSn1n

1k1n

2S3 = T1 + T2 + T3 + T1 +T2+ T3

2S3 = (T1 + T2 )+ ( T1 +T3) + (T2 + T3 )

]r)1n(n);1n/k(~S[E]r)1n(n;Sn[E1n

1k1n


Proof: Step 2Apply Min Inequality

]nr);1n/k(~S[E

]r)1n(n);1n/k(~S[E

1n

1k

1n

1k


Use Sample Independenceand E[min] Formula to Finish

] nr ;S [E] nr );1n/k(~S [E

impliesceIndependen Sample

n

))r(1(n] nr ;S [E :use Also nn

))r(1(n)1n(]nr);1n/k(~S[E

]r)1n(;S[nE))r(1)(1n(n

n

1n

1k

1n1n


Proof with n=3

))r(1(6]r2;TT[E3

]r6);TT()TT()TT[(E

]r6);TTT()TTT[(E

]r3;S[E2))r(1(6

221

323121

321321

33


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 T1M and T2M, then T1+ T2 M

Complete: T in M for every >0.


Decomposable Risk Size Models

Let M be a Unique Risk Size model

Decomposable If =1+2 then T1 , T2 ,T M where

the independent sum, T1 + T2 = T

Closed and Complete Decomposable


Differentiable Decomposable Size Models

M is Differentiable if

to respect with abledifferenti is )t(FT

Some Differential Decomposable Models Poissons Negative binomials common failure rate Gammas with common scale


Differentiable Size Model Inequalities on Partials

0≤

0≥≥1

, 0≤

2μ

2

μ

μT

μ∂

]t;T[E∂

μ∂

]t;T[E∂

μ∂

F∂

)iii

ii)

)i

If M is Differentiable and Decomposable,


Proof of One Inequality

1.partial 1st therefore

]T[E]t;T[E

]t;T[E]t;T[E]t;T[E

]t;T[E]t;TT[E

]t;T[E]t;T[E Consider


LEV as Function of Risk Size-Graph of Poisson Example

Poisson Limited Expected Values E[T; 3]

0.000

0.500

1.000

1.500

2.000

2.500

3.000

0.000 1.000 2.000 3.000 4.000 5.000 6.000

Mean

Figure 3


Result for Decomposable Models

Larger Risks have Smaller Charges

21TT21

Proof: Let m1=1 and m2=2.

212

11

mm21

m21

T ...T ... T TT

T ... T TT

Result follows from charge inequality for independent sum of IID RVs.


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


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.


Charge for Infinitely Large Risk in Decomposable Model

)r1,0max()r( where

, As

0

0T

CV argument works to prove this. Uses result:

1 )r( dr2)R(Var0


Risk Size Model with Parameter UncertaintySeparate true mean from a priori mean

=true mean = a priori mean of a risk

M = all risks in model with prior mean

Prior distribution, H( | ). E[|]= where expectation uses H.

Covers risk heterogeneity and predictive uncertainty


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).


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).


Bayesian Formula for Charge

Let =RV with distribution H( | )

Let h(| ) be the associated density

r)(T)(T )|h( d

1)r(

This represents the charge for risks with prior size equal to .


Result for Scaled Priors

Larger Unconditional Risks have Smaller Charges if Priors are a family of scaled distributions

Let 2= (1+c) 1, then

)r()r( )(T)(T 12

Note the Unconditional Risks do not in general form a decomposable family


Scaling Result Proof

Drop much of the conditional notation to simplify expressions and write the Bayesian integral for the charge as:

2)(T2

02)(T

r)(h d

1)r(

2

c11

c1h)(h 12

Use scaling to relate the densities


Scaling Result Proof-Next steps

Plug in to get

)c1(r

c11

c1h d

)c1(1

)r(

1)(T1

01

)(T 2

1))c1((T(1

01)(T

rh d

1)r(

2

Change variables to get


Scaling Result Proof-Finale

T((1+c)) is a larger risk than T() in M. So it has a smaller charge:

)r(r

h d1

)r(

)(T1

)(T(101

)(T

1

2

Arrive at conclusion

1)(T(

1))c1((T(

rr


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


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.


General Result in Math Symbols

)r()r( then

]E[]E[ when )r()r( If

)(T)(T

12

12

12


Comments on ProofThe 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


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.

as )(CV))(T(CV

Nothing we have assumed forces the CV of the prior to approach 0.

Contagion model CV approaches -1/2


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.


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.


Independent Severity with Scale Parameter Uncertainty

Each risk has a particular and associated severity RV, Y=X/

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/] =1

and Var(1/) =b =mixing parameter in CRM.

is independent of and


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


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 = E[X] have size E[X] though each risk has true mean E[X/

The introduction of severity and the “”increase the charges, but do not change the relation between charges of different size risks.


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.

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Why Larger Risks Have Smaller Insurance Charges