Math 479 / 568
Casualty Actuarial Mathematics
Fall 2014
University of Illinois at Urbana-Champaign
Professor Rick Gorvett
Session 13: Reinsurance II –
Exposure Rating and Increased Limits Factors
October 14, 2014
1
Reinsurance Pricing
• Pro rata reinsurance
– Straightforward pricing, since a percentage
– Consider adequacy of primary pricing
• Excess reinsurance
– Price is generally specified as a percentage of
“subject premium”
– Two traditional approaches to pricing:
• Experience rating
• Exposure rating
2
Exposure Rating
• Allocation of primary premium to different
“layers” of coverage
• Consider adequacy of primary premium
• Depends upon:
– Attachment point and limit of reinsurance layer
– Primary premiums at different policy limits
– Increased limits factors (ILFs)
3
ILFs
• Consider a loss severity (size per claim)
distribution
• For example: exponential distribution:
f (x) 
1

x
e
F (x)  1  e

x

E[ X ]  
4
ILFs (cont.)
• Specifically, consider an exponential
distribution with a mean of 20:
f (x) 
1
x
e
20
 0 . 05 e
 0 . 05 x
20
F (x)  1  e
 0 . 05 x
5
ILFs (cont.)
• Suppose that the primary insurer writes a
“basic limits” policy, with a limit of 25 per
occurrence
• Notation (Y = insurer’s payment):
Y  X  25  min( X , 25 )
6
ILFs (cont.)
• One item of interest:
P [ X  25 ]  1  e
 25
20
 1 . 25
 1 e
 0 . 713
• Another item: what is the expected size of a
claim paid by the insurer?
E [Y ]  E [ X  25 ] 

25
xf ( x )dx 
0
  (1  e
 25

)  20 (1  e


25 f ( x )dx
25
 1 . 25
)  14 . 27
7
ILFs (cont.)
• Consider different policy limits written by the
primary insurance company:
Policy Limit
E[Y]
“ILF”
25
14.27
1.000
50
18.36
1.287
100
19.87
1.392
20.00
1.402
Unlimited
8
Exposure Rating
• Suppose basic limits (BL) premium = $100.
Policy Limit
Premium
25 (BL)
100.00
50
128.70
100
139.20
Unlimited
140.20
9
Exposure Rating (cont.)
• Consider 50 x/s 50 layer:
– Worth 139.20 - 128.70 = 10.50
– But this value can also be determined via ILFs:
100 (1.392-1.287) = 10.50
• Exposure rating involves using quantitative
relationships between limits to allocate
premium to different layers
10
Examples
• What proportion of the primary premium on a
100 limit policy should go to the 50 x/s 50
reinsurance layer?
• Suppose you determined that primary
premiums are, in general, 10% inadequate.
What proportion of the 50 limit policy
premium should go to the 25 x/s 25 layer?
11
Caveat
• A big issue here is the nature of the ILFs.
– How are expenses handled?
– How is risk load handled?
• A reinsurer might want to use loss-only ILFs,
without risk load, to determine the loss
portion of the layer, then determine the final
reinsurance premium by incorporating its
own expenses and risk load.
• Thus, adjustments may need to be made to
results associated with situations like the
12
examples.