Transcript Download Handout 3

Introduction to Exposure Rating
Kathy H. Garrigan, FCAS, MAAA
March 8, 2007
Agenda
1.
2.
3.
4.
5.
6.
7.
What is exposure rating?
Uses for exposure rating
Data needed for exposure rating
How exposure rating works
Real live example!
Advantages / disadvantages of exposure rating
Questions?
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What is exposure rating and what is it
used for?
 Exposure rating uses a client’s limits profile, hazard
characteristics, a severity curve, and a gross loss and
ALAE ratio to estimate losses in excess layers.
 Exposure rating does NOT consider actual client loss
experience in those excess layers; nice complement to
experience rating
 Enables pricing for start-up opportunities purchasing
XOL protection
 Addresses “free cover” issues
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What is “free cover”?
 “Free cover” exists when the top of your reinsurance
layer exceeds the largest trended loss in your data
Free Cover
200,000
100,000
1
2
3
4
5
Trended Losses
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Data needed for exposure rating
 Current limit / attachment point profile with premium
 Hazard characteristics for subject business
• Important for severity curve selection
• Heavy Commercial Trucks? Products B?
 Prospective gross loss and ALAE ratio for subject
business
 Prospective Subject Premium
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How exposure rating works
 Calculate the percentage of total expected losses that
fall into the excess layer
 This equates to:
•
•
•
•
•
Expected losses limited to the top of the layer (or policy)
Minus
Expected losses limited to the bottom of the layer
Divided by
Expected losses limited to the policy itself
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100K x 100K Layer
100K x100K Layer
250K
200K
150K
100K
50K
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100K x 100K Layer – $100K policy
100K x100K Layer
$100K policy
250K
200K
150K
100K
50K
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100K x 100K Layer – $150K policy
100K x100K Layer
$150K policy
250K
200K
150K
100K
50K
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100K x 100K Layer – $200K policy
100K x100K Layer
$200K policy
250K
200K
150K
100K
50K
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Limited expected value
 This “losses limited to some L” idea is called a limited
expected value, also known as:
• LEV
• E[x;L]
• Limited average severity
• LAS
• That equation on part 4B that I skipped
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L
Ex; L   xfX x dx  L1  FX ( L)
0
 For any x, only one of two things can happen:
1. x is less than or equal to the limitation, L
2. x is greater than the limitation, L
•
•
First part of equation tackles (1) by calculating the
expected loss limited to L when x <= L.
Second part of equation tackles (2). For any x > L, if
L is the limitation, you just get L (multiplied by the
probability that x > L)
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Real live example!
 Exposure rate a 100K x100K casualty per occurrence
layer for Bellini Mutual, an Atlanta based insurer
 Assume:
• Gross loss and ALAE ratio = 50%
• 100% Prem / Ops 1, LEVs provided
• Current limits profile provided
• Prospective subject premium is $40M
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Current Limits Profile for Bellini Mutual
Policy Limit
25,000
50,000
100,000
150,000
250,000
500,000
Total
Policy
Premium
500,000
1,000,000
7,500,000
9,500,000
10,500,000
11,000,000
40,000,000
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Which policies expose my 100K x 100K?
Policy Limit
25,000
50,000
100,000
150,000
250,000
500,000
Total
Policy
Premium
500,000
1,000,000
7,500,000
9,500,000
10,500,000
11,000,000
Expose
100Kx100K?
No
No
No
Yes
Yes
Yes
40,000,000
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Limited expected values
LEV @
LEV @
LEV @
LEV @
LEV @
LEV @
LEV @
LEV @
LEV @
25,000
50,000
100,000
150,000
200,000
250,000
300,000
400,000
500,000
2,787
3,942
6,421
6,957
7,306
7,558
7,753
8,039
8,243
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100K x 100K Layer – $150K policy
100K x100K Layer
$150K policy
250K
200K
150K
100K
50K
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What fraction of a 150K policy’s total losses fall
into my 100K x 100K layer?
= E[x;150K] – E[x;100K]
------------------E[x;150K]
=
LEV@150K – LEV@100K
------------------------------LEV@150K
= (6,957-6,421)/(6,957)
= 7.69%
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Completing the table…
Policy Limit
25,000
50,000
100,000
150,000
250,000
500,000
Total
Policy
Premium
500,000
1,000,000
7,500,000
9,500,000
10,500,000
11,000,000
Expose
E[Layer Losses] /
100Kx100K? Total E[Losses]
No
No
No
Yes
Yes
Yes
0.00%
0.00%
0.00%
7.69%
11.70%
10.73%
40,000,000
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What does a “loss to loss” ratio do for me
anyway?
 Not as much as you might like
 We want our denominator to be premium, not losses,
aka:
• Burn
• Layer loss cost
• Burning cost
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Not done yet….
Have:
(1) Expected Layer Losses
-------------------------------Total Expected Losses
Need:
(2) Expected Layer Losses
-------------------------------Total Expected Premium
Solution?
Multiply (1) by your prospective gross loss and ALAE
ratio; total expected losses cancel out, yielding what
you want.
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Really completing the table…
Policy Limit
25,000
50,000
100,000
150,000
250,000
500,000
Total
Policy
Premium
500,000
1,000,000
7,500,000
9,500,000
10,500,000
11,000,000
40,000,000
Gross Loss
Expose E[Layer Losses] / + ALAE
100Kx100K? Total E[Losses]
Ratio
No
No
No
Yes
Yes
Yes
0.00%
0.00%
0.00%
7.69%
11.70%
10.73%
50.00%
50.00%
50.00%
50.00%
50.00%
50.00%
Burning
Cost
0.00%
0.00%
0.00%
3.85%
5.85%
5.36%
3.92%
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Now what?
 Compare the results from the experience and exposure
rating exercises for the desired layer to be priced
 Make a selection by considering the advantages and
disadvantages of each method
 Load up final loss cost for profit load and expenses to
arrive at a final rate
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What about umbrella?
 What if your client writes umbrella policies? How would
losses from these policies fall into my 100K x 100K
layer?
 Assume $200K umbrella policy attaching over a $100K
underlying policy
• If they write “over their own”, 100K x 100K may attach
“from the ground up”
• If they write “over someone else”, 100K x 100K will
attach on top of underlying
 Be careful to read contract wording or ask the broker or
underwriter for an example!
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“Over their own”
 $200K umbrella policy over $100K underlying policy
• Client writes the underlying policy as well
• 100K x 100K attaches “from ground up”
100x100 Layer attaching from the ground up
200Kx100K umbrella policy
400K
300K
200K
100K
 (LEV@200K – LEV@100K)/(LEV@300K – LEV@100K) =66.4%
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“Over someone else”
 $200K umbrella policy with $100K underlying
• Client does not write the underlying policy
• 100K x 100K attaches on top of underlying
• This is 100K x 100K x 100K = 100K x 200K
100x100 Layer attaching on top of underlying
200Kx100K umbrella policy
400K
300K
200K
100K
 (LEV@300K – LEV@200K)/(LEV@300K – LEV@100K) = 33.6%
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Advantages of exposure rating
 Relatively easy to do
 Any excess layer can be priced
 Use of current limits profile enables “up to date” view of
excess layer pricing. Shifts in limits and attachment
points over time, which may make experience rating
difficult, are irrelevant here.
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Disadvantages of exposure rating
 Selected severity curve may not properly reflect client’s
subject business
 Selected gross loss and ALAE ratio may not
appropriately reflect exposed risks
 Exposure rating does not consider client’s actual loss
experience in excess layers
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