why larger risks have smaller insurance charges
DESCRIPTION
Why Larger Risks Have Smaller Insurance Charges. Ira Robbin, PhD Partner RE. 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 - PowerPoint PPT PresentationTRANSCRIPT
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
Why Larger Risks Have Smaller Insurance Charges3 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges4 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges5 CAS Spring Meeting, May 8-10, 2006
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?
Why Larger Risks Have Smaller Insurance Charges6 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges7 CAS Spring Meeting, May 8-10, 2006
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 Larger Risks Have Smaller Insurance Charges8 CAS Spring Meeting, May 8-10, 2006
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.
Why Larger Risks Have Smaller Insurance Charges9 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges10 CAS Spring Meeting, May 8-10, 2006
Charges by Size Graph
Why Larger Risks Have Smaller Insurance Charges11 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges12 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges13 CAS Spring Meeting, May 8-10, 2006
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
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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
Why Larger Risks Have Smaller Insurance Charges15 CAS Spring Meeting, May 8-10, 2006
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]
Why Larger Risks Have Smaller Insurance Charges16 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges17 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges18 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges19 CAS Spring Meeting, May 8-10, 2006
Proof: Step 2Apply Min Inequality
]nr);1n/k(~S[E
]r)1n(n);1n/k(~S[E
1n
1k
1n
1k
Why Larger Risks Have Smaller Insurance Charges20 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges21 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges22 CAS Spring Meeting, May 8-10, 2006
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.
Why Larger Risks Have Smaller Insurance Charges23 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges24 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges25 CAS Spring Meeting, May 8-10, 2006
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,
Why Larger Risks Have Smaller Insurance Charges26 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges27 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges28 CAS Spring Meeting, May 8-10, 2006
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.
Why Larger Risks Have Smaller Insurance Charges29 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges30 CAS Spring Meeting, May 8-10, 2006
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.
Why Larger Risks Have Smaller Insurance Charges31 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges32 CAS Spring Meeting, May 8-10, 2006
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
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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).
Why Larger Risks Have Smaller Insurance Charges34 CAS Spring Meeting, May 8-10, 2006
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).
Why Larger Risks Have Smaller Insurance Charges35 CAS Spring Meeting, May 8-10, 2006
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 .
Why Larger Risks Have Smaller Insurance Charges36 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges37 CAS Spring Meeting, May 8-10, 2006
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
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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
Why Larger Risks Have Smaller Insurance Charges39 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges40 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges41 CAS Spring Meeting, May 8-10, 2006
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.
Why Larger Risks Have Smaller Insurance Charges42 CAS Spring Meeting, May 8-10, 2006
General Result in Math Symbols
)r()r( then
]E[]E[ when )r()r( If
)(T)(T
12
12
12
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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
Why Larger Risks Have Smaller Insurance Charges44 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges45 CAS Spring Meeting, May 8-10, 2006
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.
Why Larger Risks Have Smaller Insurance Charges46 CAS Spring Meeting, May 8-10, 2006
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.
Why Larger Risks Have Smaller Insurance Charges47 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges48 CAS Spring Meeting, May 8-10, 2006
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
Why Larger Risks Have Smaller Insurance Charges49 CAS Spring Meeting, May 8-10, 2006
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.
Why Larger Risks Have Smaller Insurance Charges50 CAS Spring Meeting, May 8-10, 2006
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.