Why Larger Risks
Have
Smaller Insurance Charges
Ira Robbin, PhD
Partner RE
CAS Spring Meeting, May
8-10, 2006
2
CAS Spring Meeting,
May 8-10, 2006
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
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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
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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
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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?
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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


1
T ( r )   dFT ( t )( t  r )   dFR ( s )( s  r )
 r
r
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Why Larger Risks Have Smaller Insurance Charges
E[Max] and E[Min] Definitions
Expectation of Max and Min
E[max( 0, T  r )]
( r ) 
 E[max( 0, R  r )]

E[min( T, r )]
E[ T; r ]
( r )  1 
 1
 1  E[ R ; r ]


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Why Larger Risks Have Smaller Insurance Charges
Why Use the E[Min] Definition?
Two Useful Min Inequalities
i ) min( a  b, c  d )  min( a , c )  min( b, d )
ii ) min( a , c  d )  min( b, c  d )  min( a  b, c  d )
E[R;r] is a LEV. It may be easier to prove statements
about LEVs and then translate to results about charges.
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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
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Why Larger Risks Have Smaller Insurance Charges
Charges by Size Graph
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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
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Why Larger Risks Have Smaller Insurance Charges
CV Counterexample
index
i
1
2
3
4
5
Mean
index
i
1
2
3
4
5
Mean
point
ti
0.00
2.00
4.00
6.00
8.00
4.00
point
ti
0.00
2.00
4.00
6.00
8.00
4.00
density
f(ti)
20.0%
20.0%
20.0%
20.0%
20.0%
Random Variable T 1
square
ratio
CDF
2
ti
ri
F(ri)
0.00
0.00
20.0%
4.00
0.50
40.0%
16.00
1.00
60.0%
36.00
1.50
80.0%
64.00
2.00
100.0%
24.00
1.00
density
f(ti)
0.0%
60.0%
10.0%
0.0%
30.0%
Random Variable T 2
square
ratio
CDF
2
ti
ri
F(ri)
0.00
0.00
0.0%
4.00
0.50
60.0%
16.00
1.00
70.0%
36.00
1.50
70.0%
64.00
2.00
100.0%
23.20
1.00
Savings
ri
0.0%
10.0%
30.0%
60.0%
100.0%
Savings
ri
0.0%
0.0%
30.0%
65.0%
100.0%
Tail
G(ri)
80.0%
60.0%
40.0%
20.0%
0.0%
Charge
Tail
G(ri)
100.0%
40.0%
30.0%
30.0%
0.0%
Charge
ri
100.0%
60.0%
30.0%
10.0%
0.0%
ri
100.0%
50.0%
30.0%
15.0%
0.0%
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Why Larger Risks Have Smaller Insurance Charges
CV Counterexample Graph
Figure 2
RV with Smaller CV Has Larger Charge
at some Entry Ratios
1.00
0.75
T1
T2
0.50
0.25
0.00
0.00
0.50
1.00
1.50
2.00
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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
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Why Larger Risks Have Smaller Insurance Charges
Charge Inequality for
Sum of Two RVs
1
2
 T1  T2 ( r ) 
T 1 ( r ) 
 T2 ( r )
1   2
1   2
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]
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Why Larger Risks Have Smaller Insurance Charges
Charge Inequality for
Sum of Two ID RVs
T1  T2 ( r )  T ( r )
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
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Why Larger Risks Have Smaller Insurance Charges
Charge Inequality for
Sum of “n” IID RVs
Sn1 ( r )  Sn ( r )
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
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Proof Step 1:
Combinatoric Trick
n 1
n  S n 1   S(~ k / n  1)
k 1
2S3 = T1 + T2 + T3 + T1 +T2+ T3
2S3 = (T1 + T2 )+ ( T1 +T3) + (T2 + T3 )
n 1
E[ n  Sn 1 ; n( n  1)r ]  E[  S(~ k / n  1); n( n  1)r ]
k 1
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Proof: Step 2
Apply Min Inequality
n 1
E[  S(~ k / n  1); n( n  1)r ]
k 1
n 1
  E[ S(~ k / n  1); nr ]
k 1
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Use Sample Independence
and E[min] Formula to Finish
Sample Independen ce implies
E[ S(~ k / n  1); nr ]  E[ Sn ; nr ]
Also use : E[ Sn ; nr ]  n(1  n ( r ))
 n( n  1)(1  n 1( r ))  nE[ Sn 1 ; ( n  1)r ]
n 1
  E[ S(~ k / n  1); nr ]  ( n  1)n(1  n ( r ))
k 1
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Proof with n=3
6(1  3 ( r ))  2E[ S 3 ;3r ]
 E[( T1  T2  T3 )  ( T1  T2  T3 );6r ]
 E[( T1  T2 )  ( T1  T3 )  ( T2  T3 );6r ]
 3E[ T1  T2 ;2 r ]  6(1  2 ( r ))
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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.
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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
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Differentiable Decomposable Size Models
M is Differentiable if
FT ( t ) is differentiable with respect to 
Some Differential Decomposable Models
Poissons
Negative binomials common failure rate
Gammas with common scale
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Differentiable Size Model Inequalities on Partials
If M is Differentiable and Decomposable,
i)
∂
FTμ
∂
μ
≤0 ,
ii) 1 ≥
iii )
∂
E [ Tμ ;t ]
∂
μ
∂2 E [ Tμ ;t ]
∂
μ2
≥0
≤0
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Proof of One Inequality
Consider E[ T   ; t ]  E[ T ; t ]
 E[ T  T ; t ]  E[ T ; t ]
 E[ T ; t ]  E[ T ; t ]  E[ T ; t ]
 E[ T ; t ]  E[ T ]  
therefore 1st partial  1.
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LEV as Function of Risk SizeGraph of Poisson Example
Figure 3
Poisson Limited Expected Values E[T; 3]
3.000
2.500
2.000
1.500
1.000
0.500
0.000
0.000
1.000
2.000
3.000
Mean
4.000
5.000
6.000
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Result for Decomposable Models
Larger Risks have Smaller Charges
1   2  T1  T2
Proof: Let m1=1 and m2=2.
T1  T 1  T 2  ...  T m 1
T 2  T 1  T 2  ...  T m 1  ...  T m 2
Result follows from charge inequality for independent sum
of IID RVs.
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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
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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.
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Charge for Infinitely Large Risk in Decomposable Model
As   , T  0
where 0 ( r )  max( 0,1  r )
CV argument works to prove this.
Uses result:

Var( R )  2  dr ( r )  1
0
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Risk Size Model
with Parameter Uncertainty
Separate 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).
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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).
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Bayesian Formula for Charge
Let  =RV with distribution H( |  )
Let h(| ) be the associated density
1
r
T(  ) ( r )   d h( | )  T(  )   

This represents the charge for risks with prior size equal to .
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Result for Scaled Priors
Larger Unconditional Risks have Smaller Charges if
Priors are a family of scaled distributions
Let 2= (1+c) 1, then
T( 2 ) ( r )  T( 1 ) ( r )
Note the Unconditional Risks do not in general form a
decomposable family
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Scaling Result Proof
Drop much of the conditional notation to simplify expressions and
write the Bayesian integral for the charge as:

1
 r 2 
T( 2 ) ( r )   d h 2 ( )    T(  ) 

2 0
  
Use scaling to relate the densities
   1
h 2 ( )  h1 

 1 c  1 c
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Scaling Result Proof-Next steps
Plug in to get
 T(  2 ) ( r )

1
   1
 r1(1  c ) 

d h 1 
   T(  ) 



1 (1  c ) 0

 1 c  1 c


Change variables to get

 r1 
1

T( 2 ) ( r )   d h1     ( T( ( 1c )) 
1 0
  
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Scaling Result Proof-Finale
T((1+c)) is a larger risk than T() in M. So it has a smaller charge:
 r1 
 r1 
( T( ( 1c ))    ( T(  )  
  
  
Arrive at conclusion
 T(  2 ) ( r )

 r1 
1
   T( 1 ) ( r )
  d h 1     ( T(  ) 
1 0
  
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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
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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.
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General Result in Math Symbols
If 2 ( r )  1 ( r ) when E[  2 ]  E[ 1 ]
then T( 2 ) ( r )  T( 1 ) ( r )
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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
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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.
CV( T(  ))  CV(  ) as   
Nothing we have assumed forces the CV of the prior to
approach 0.
Contagion model CV approaches -1/2
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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.
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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.
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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.
b is a positive continuous RV with E[1/b] =1
and Var(1/b) =b =mixing parameter in CRM.
b is independent of  and 
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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
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Aggregate Loss Model
–Key Result
Larger Risks have Smaller Charges
MT( N|Q, Y| bX) is the Risk Size Model
Risks of a priori mean =  E[X] have size  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.
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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.