Transcript Download Handout 2

Additional Topics
Chris Svendsgaard
Casualty Exposure Rating
CARe Boot Camp 2005
Page 1

Pricing Umbrella and Excess on Excess

The ISO Mixed Exponential
Pricing Umbrella and XS on XS

Issues
– Who wrote the primary contract?
- Over Own
- Over Other
– How does the reinsurance contract handle Over
Own/Over Other?
– How does the reinsurance contract handle ALAE
(esp. Over Own)?
– Shares
Chris Svendsgaard
Casualty Exposure Rating
CARe Boot Camp 2005
Page 2
IMPORTANT!
You
must understand the proposed reinsurance
contract wording
Chris Svendsgaard
Casualty Exposure Rating
CARe Boot Camp 2005
Page 3
Example
5
Umbrella
1
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 4
Primary
Reinsured Layer:
2x1
Example
5
Reinsured Layer:
2x1
2.5
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 5
Loss
1
fgu loss
is 2.5
Example: Over Other
5
Reinsured Layer:
2x1
2.5
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 6
Loss
1
Reinsurance
pays 0.5
Example: Over Own
5
Reinsured Layer:
2x1
2.5
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 7
Loss
1
Primary and Umbrella
losses are combined
to define loss
Reinsurance pays 1.5
ALAE: Some Considerations
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 8

Large loss ALAE tends to be lower (as a %) than fgu
ALAE

Umbrella policies are relatively passive on defense so
ALAE may be low.

When Over Own, how does the client/reinsurance
contract allocate ALAE between the primary policy
and the umbrella policy?
Shares: Some considerations

Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 9
Need to inflate reinsurance limits, etc. to 100% basis
in order to use the severity distribution correctly.
Further Considerations

For treaty, the Subject Premium depends on the
reinsurance contract wording.
– If the S.P. is the Umbrella premium, then the
reinsurance rate will be much higher (because
the denominator is smaller)

Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 10
Reinsurance contract wording has many variations.
You have to figure out each one on its own.
The ISO Mixed Exponential
Distribution
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 11

What it is

Why ISO uses it

Pitfalls
What it is
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 12

Distribution function

Exponential distribution

Mixing
Distribution function

If X is a random variable, then

F(x) = Probability (X <= x) is its distribution function
– Density function f(x) is derivative with respect to
x of F(x)
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 13
A distribution function
1
0.9
0.8
0.7
F(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 14
100
200
300
400
500
x
600
700
800
A density function
0.9
0.8
0.7
f(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 15
100
200
300
400
500
x
600
700
800
Exponential distribution

F(x) = 1 – exp(-X/μ)
“exp” is the exponential function
μ is the mean of the exponential distribution

Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 16
f(x) looks like a slide
Exponential density function
μ = 400
0.003
0.0025
f(x)
0.002
0.0015
0.001
0.0005
0
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 17
0
100
200
300
400
x
500
600
700
800
NO HUMP!
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 18
What it is: MIXING
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 19
1.
Randomly pick a type of claim by flipping a coin that
has a certain probabilty for each claim type.
2.
Each type of claim has its own severity distribution.

Overall severity distribution is the result of steps 1
and 2—it is a “mixture” of the the claim types
What it is: MIXING—the math

If F and G are distribution functions, then so is
wF+(1-w)G

for
0<w<1
More generally, so is

k
w
iFi
i 1
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 20
if all the w’s are between 0 and 1 and they sum to 1.
ISO-like mixed exponential
distribution
Distribution
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 21
Mean
Weight
1
1,500
38%
2
5,000
31%
3
25,000
18%
4
90,000
9%
5
400,000
3%
6
1,700,000
0.9%
7
10,000,000
0.1%
Weights and means are fake.
Why ISO uses the mixed
exponential distribution

Lots of parameters

Some theory
– Can approximate any (sufficiently nice, humpless)
distribution with a mixture of exponentials
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 22
Pitfalls

A single term by itself does not correspond to any
particular claim characteristic
– Vs. old “Pareto Soup”, where each term
corresponded to the settlement lag of the claim
– You can’t tell much about a distribution by looking
at a single term (μ and w).
- The term might be low-severity but the
overall distribution might be high-severity
– Can’t compare terms from year to year
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 23
- At least, not meaningfully
- Term 2 from this year does not
correspond to term 2 from last year
More pitfalls

Tail is problematic
– A finite mixture of exponentials will always be
less severe in the tail than any Pareto distribution
– ISO caps the largest μ at $10,000,000
- In practice, most ILF tables have a capped
largest μ
Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 24
Still more pitfalls

Curves do not reflect parameter risk

ISO data falls off dramatically at a relatively low level
(say, $2,000,000—depends on the line)
– Anti-selection picked up?

Chris Svendsgaard, Swiss Re
Casualty Exposure Rating
CARe Boot Camp 7-28-05
Page 25
ISO nudges the curves for the highest layers