Boot Camp on Reinsurance
Pricing Techniques –
Loss Sensitive Treaty Provisions
July 2005
Introduction to Loss Sensitive
Provision
 Definition: A reinsurance contract provision that varies the
ceded premium, loss, or commission based upon the loss
experience of the contract
 Purpose: Client shares in ceded experience & could be
incented to care more about the reinsurer’s results
 Typical Loss Sharing Provisions
• Profit Commission
• Sliding Scale Commission
• Loss Ratio Corridors
• Annual Aggregate Deductibles
• Swing Rated Premiums
• Reinstatements
2
Simple Profit Commission Example
 A property pro-rata contract has the following profit
commission terms
• 50% Profit Commission after a reinsurer’s margin of
10%.
• Key Point: Reinsurer returns 50% of the contractually
defined “profit” to the cedant
• Profit Commission Paid to Cedant =
50% x (Premium - Loss - Commission - Reinsurers
Margin)
• If profit is negative, reinsurers do not get any
additional money from the cedant.
3
Simple Profit Commission Example
 Profit Commission: 50% after 10% reinsurer’s Margin
 Ceding Commission = 30%
 Loss ratio must be less than 60% for us to pay a profit



commission
Contract Expected Loss Ratio = 70%
$1 Premium - $0.7 Loss - $0.3 Comm - $0.10 Reins Margin =
minus $0.10
Is the expected cost of profit commission zero?
4
Simple Profit Commission Example
 Answer: The expected cost of profit commission is not
zero
 Why: Because 70% is the expected loss ratio.
• There is a probability distribution of potential
outcomes around that 70% expected loss ratio.
• It is possible (and may even be likely) that the loss
ratio in any year could be less than 60%.
5
Cost of Profit Commission: Simple
Quantification
 Earthquake exposed California property pro-rata treaty
 LR = 40% in all years with no EQ
 Profit Comm when there is no EQ = 50% x ($1 of Premium $0.4 Loss - $0.30 Commission - $0.1 Reinsurers Margin)
= 10% of premium
 Cat Loss Ratio = 30%.
• 10% chance of an EQ costing 300% of premium, 90%
chance no EQ loss
Expected Cost of Profit Comm =
Profit Comm Costs 10% of Premium x 90% Probability of No EQ
+ 0% Cost of PC x 10% Probability of EQ Occurring = 9% of
Premium
6
Basic Mechanics of Analyzing Loss
Sensitive Provisions
 Build aggregate loss distribution
 Apply loss sensitive terms to each point on the
loss distribution or to each simulated year
 Calculate a probability weighted average cost
(or saving) of the loss sensitive arrangement
7
Example of Basic Mechanics: PC: 50% after 10%, 30%
Commission, 65% Expected LR
Cost of PC
Loss Ratio Band
at avg LR
Low
High
Avg in Band Probability in Band
20%
30%
25%
2.8%
17.5%
30%
40%
35%
9.4%
12.5%
40%
50%
45%
15.2%
7.5%
50%
60%
55%
20.9%
2.5%
60%
70%
65%
17.4%
0.0%
70%
80%
75%
15.1%
0.0%
80%
90%
85%
10.1%
0.0%
90%
100%
95%
5.8%
0.0%
100% 150%
125%
1.4%
0.0%
150% 200%
175%
1.1%
0.0%
200% 300%
250%
0.5%
0.0%
300% 400%
350%
0.3%
0.0%
Average:
65.0%
100.0%
3.3%
CR
at avg LR
in Band
72.5%
77.5%
82.5%
87.5%
95.0%
105.0%
115.0%
125.0%
155.0%
205.0%
280.0%
380.0%
98.3%
Cost of Profit Comm & CR at expected LR doesn't equal
expected Cost of Profit Comm and expected CR
8
Determining an Aggregate
Distribution - Two Methods
 Fit statistical distribution to on level loss ratios
• Reasonable for pro-rata treaties.
 Determine an aggregate distribution by modeling
frequency and severity
• Typically used for excess of loss treaties.
9
Fitting a Distribution to On Level
Loss Ratios
Most actuaries use the lognormal
distribution
• Reflects skewed distribution of loss ratios
• Easy to use
Lognormal distribution assumes that the
natural logs of the loss ratios are
distributed normally.
10
Incremental Probability
Skewness of Lognormal
Distribution
25.00%
20.00%
15.00%
10.00%
5.00%
0.00%
110-120%
100-110%
90-100%
80-90%
11
70-80%
60-70%
50-60%
40-50%
30-40%
20-30%
10-20%
0-10%
Loss Ratios
Fitting a Lognormal Distribution to
Projected Loss Ratios
 Fitting the lognormal

s^2 = LN(CV^2 + 1)
m = LN(mean) - s^2/2
Mean = Selected Expected Loss Ratio
CV = Standard Deviation over the Mean of the loss ratio
(LR) distribution.
Prob (LR  X) = Normal Dist(( LN(x) - m )/ s) i.e.. look up
(LN(x) - m )/ s) on a standard normal distribution table
 Producing a distribution of loss ratios
• For a given point i on the CDF, the following Excel
command will produce a loss ratio at that CDFi:
Exp (m + Normsinv(CDFi) x s)
12
Sample Lognormal Loss Ratio
Distribution
On Level
Year
LR
1998
65.5%
1999
70.0%
2000
55.0%
2001
48.0%
2002
72.0%
2003
65.0%
2004
55.0%
Mean LR:
61.5%
standard deviation:
8.92%
Calculated CV:
0.15
Selected CV:
0.17
Lognormal Mu:
(0.500)
Lognormal Sigma:
0.169
CDF
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
90.0%
95.0%
98.0%
99.0%
Modeled
LR
48.8%
52.6%
55.5%
58.1%
60.6%
63.3%
66.2%
69.9%
75.3%
80.0%
85.8%
89.8%
Modeled LR = Exp(MU+Normsinv(CDFi)*Sigma)
13
Is the resulting LR distribution reasonable?
 Compare resulting distribution to historical results




• Focus on level LR’s, but don’t completely ignore
untrended ultimate LR’s.
Potential for cat or shock losses not captured within
historical experience
Degree to which trended past experience is predictive of
future results for a book
Actuary and underwriter should discuss the above issues
If the distribution is not reasonable, adjust the CV
selection.
14
Process and Parameter
Uncertainty
 Process Uncertainty: Random fluctuation of results around the
expected value.
 Parameter Uncertainty: Do you really know the true mean of the
loss ratio distribution for the upcoming year?
• Are your trend, loss development & rate change
assumptions correct?
• For this book, are past results a good indication of future
results?
• Changes in mix and type of business
• Changes in management or philosophy
• Is the book growing, shrinking or stable
 Selected CV should usually be above indicated
• 5 to 10 years of data does not reflect full range of
possibilities
15
Modeling Parameter Uncertainty:
One Suggestion
 Select 3 equally likely expected loss ratios
 Assign weight to each loss ratio so that the weighted
average ties to your selected expected loss ratio
• Example: Expected LR is 65%, assume 1/3
probability that true mean LR is 60%, 1/3 probability
that it is 65%, and 1/3 probability that it is 70%.
• Simulate the “true” expected loss ratio (reflects
Parameter Uncertainty)
 Simulate the loss ratio for the year modeled using the
lognormal based on simulated expected loss ratio above
& your selected CV (reflects Process Variance)
16
Example of Modeling Parameter
Uncertainty
Simulated random variable from 0.33 to 0.67: Choose 65%
Simulated random variable from 0.67 to 1,00: Choose 70%
Simulated Random Variable:
0.8
Simulated Expected Loss Ratio:
70.0%
2) Calculate New Lognormal Parameters
Sigma (same as original selection):
Simulated Lognormal Mu:
Mu = LN(Expected LR) - Sigma^2/2
0.17
(0.37)
3) Simulate Loss Ratio for Year Based on New Lognormal Mu
Simulated Random Variable (CDFi):
0.842
# of St. Deviations Away from Mean [Normsinv(CDFi)]: 1.00
Simulated Loss Ratio:
81.7%
Exp (mu + Normsinv(CDFi) x sigma)
17
Common Loss Sharing Provisions
for Pro-rata Treaties
 Profit Commissions
• Already covered
 Sliding Scale Commission
 Loss Ratio Corridor
 Loss Ratio Cap
18
Sliding Scale Comm
 Commission initially set at Provisional amount
 Ceding commission increases if loss ratios are
lower than expected
 Ceding commission decreases if losses are
higher than expected
19
Sliding Scale Commission Example
 Provisional Commission: 30%
 If the loss ratio is less than 65%, then the commission


increases by 1 point for each point decrease in loss ratio up to a
maximum commission of 35% at a 60% loss ratio
If the loss ratio is greater than 65%, the commission decreases
by 0.5 for each 1 point increase in LR down to a minimum
comm. of 25% at a 75% loss ratio
If the expected loss ratio is 65% is the expected commission
30%?
20
Sliding Scale Commission Solution
Loss Ratio Band
Low
High
Ceding
Comm @ CR @ avg
Avg LR
avg LR in
LR in
in Band Probability Band
Band
Lognormal Parameters
0.0%
52.5%
45.0%
11.91%
35.0%
80.0%
Mean LR:
65.0%
52.5%
57.5%
55.0%
14.18%
35.0%
90.0%
Selected CV:
17.0%
57.5%
62.5%
60.0%
18.08%
35.0%
95.0%
Lognormal Mu:
(0.45)
62.5%
67.5%
65.0%
17.98%
30.0%
95.0%
Lognormal Sigma:
67.5% 72.5% 70.0%
72.5% 77.5% 75.0%
77.5% 87.5% 82.5%
87.5% 100.0% 93.8%
100.0% 200.0% 135.0%
200.0% 300.0% 228.0%
14.67%
10.22%
9.73%
2.82%
0.42%
0.00%
27.5%
25.0%
25.0%
25.0%
25.0%
25.0%
97.5%
100.0%
107.5%
118.8%
160.0%
253.0%
30.7%
95.5%
Prob Wtd Avg
64.9%
Conclusion: Expected cost of commission is not 30%.
21
Max Comm
Prov Comm
Min Comm
0.17
LR
Comm
60%
35%
65%
30%
75%
25%
Loss Ratio Corridors
 A loss ratio corridor is a provision that forces the ceding

company to retain losses that would be otherwise ceded to
the reinsurance treaty
Loss ratio corridor of 100% of the losses between a 75%
and 85% LR
•
•
•
•
If gross LR equals 75%, then ceded LR is 75%
If gross LR equals 80%, then ceded LR is 75%
If gross LR equals 85%, then ceded LR is 75%
If gross LR equals 100%, then ceded LR is ???
22
Loss Ratio Cap
 This is the maximum loss ratio that could be ceded to the
treaty.
 Example: 200% Loss Ratio Cap
• If LR before cap is 150%, then ceded LR is
150%
• If LR before cap is 250%, then ceded LR is
200%
23
Loss Ratio Corridor Example
 Reinsurance treaty has
Loss Ratio Band
a loss ratio corridor of
50% of the losses
between a loss ratio of
70% and 80%.
 Use the aggregate
distribution to your
right to estimate the
expected ceded LR net
of the corridor
Low
0.0%
50.0%
60.0%
65.0%
70.0%
75.0%
80.0%
85.0%
100.0%
200.0%
24
Avg LR
in Band
High
Probability
50.0% 45.0% 14.23%
60.0% 55.0% 33.82%
65.0% 62.5% 17.47%
70.0% 67.5% 13.71%
75.0% 72.5%
9.28%
80.0% 77.5%
5.58%
85.0% 82.5%
3.05%
100.0% 92.5%
2.61%
200.0% 135.0%
0.25%
300.0% 228.0%
0.00%
Loss Ratio Corridor Example –
Solution
Loss Ratio Corridor
50.0% between 70.0%
&
80.0%
Loss Ratio Band
Low
High
0.0%
50.0%
50.0%
60.0%
60.0%
65.0%
65.0%
70.0%
70.0%
75.0%
75.0%
80.0%
80.0%
85.0%
85.0% 100.0%
100.0% 200.0%
200.0% 300.0%
Prob Wtd Avg:
Avg LR
in Band
45.0%
55.0%
62.5%
67.5%
72.5%
77.5%
82.5%
92.5%
135.0%
228.0%
61.5%
Savings
from
Probability Corridor
14.23%
0.0%
33.82%
0.0%
17.47%
0.0%
13.71%
0.0%
9.28%
1.3%
5.58%
3.8%
3.05%
5.0%
2.61%
5.0%
0.25%
5.0%
0.00%
5.0%
0.6%
25
LR Net
of
Corridor
45.0%
55.0%
62.5%
67.5%
71.3%
73.8%
77.5%
87.5%
130.0%
223.0%
60.9%
Modeling Property Treaties with
Significant Cat Exposure
 Model non-cat & cat LR’s separately
• Non Cat LR’s fit to a lognormal curve
• Cat LR distribution produced by commercial
catastrophe model
 Combine (convolute) the non-cat & cat loss ratio
distributions
26
Convoluting Non-cat & Cat LR’s Example
Non cat
LR
Prob
40%
10%
55%
25%
65%
35%
77%
25%
100%
5%
These probabilities
correspond to
these total LR's
Disretized Cat LR's
0%
30% 60%
60%
20% 15%
6.0%
2.0% 1.5%
15.0%
5.0% 3.8%
21.0%
7.0% 5.3%
15.0%
5.0% 3.8%
3.0%
1.0% 0.8%
100%
5%
0.5%
1.3%
1.8%
1.3%
0.3%
Total Loss Ratios
40%
70% 100%
55%
85% 115%
65%
95% 125%
77%
107% 137%
100%
130% 160%
140%
155%
165%
177%
200%
27
Truncated Loss Ratio Distributions
 Problem: To reasonably model the possibility of high LR
requires a high lognormal CV
 High lognormal CV often leads to unrealistically high
probabilities of low LR’s, which overstates cost of PC
 Solution: Don’t allow LR to go below selected minimum,
e.g.. 0% probability of LR<30%
• Adjust the mean loss ratio used to calculate the
lognormal parameters to cause the aggregate
distribution to probability weight back to initial
expected LR
28
Summary of Loss Ratio Distribution
Method
 Advantage:
• Easier and quicker than separately modeling
frequency and severity
• Reasonable for most pro-rata treaties
 Usually inappropriate for excess of loss contracts
• Does not reflect the hit or miss nature of many excess
of loss contracts
• Understates probability of zero loss
• May understate the potential of losses much greater
than the expected loss
29
Excess of Loss Contracts: Separate Modeling
of Frequency and Severity
 Used mainly for modeling excess of loss contracts
 Most aggregate distribution approaches assume that
frequency and severity are independent
 Different Approaches
•
•
Simulation (Focus of this presentation)
Numerical Methods
• Heckman Meyers – Fast calculating approximation to
aggregate distribution
• Panjer Method –
• Select discrete number of possible severities (i.e. create 5
possible severities with a probability assigned to each)
• Convolutes discrete frequency and severity distributions.
• A detailed mathematical explanation of these methods is
beyond the scope of this session.
 Software that can be used for simulations
• @Risk
• Excel
30
Common Frequency Distributions
 Poisson
f(x|l) = exp(-l) l^x / x!
where l = mean of the claim count distribution
and x = claim count = 0,1,2,...
f(x|l) is the probability of x losses, given a
mean claim count of l
x! = x factorial, i.e. 3! = 3 x 2 x 1 = 6
Poisson distribution assumes the mean and
variance of the claim count distribution are
equal.
31
Fitting a Poisson Claim Count
Distribution
 Trend claims from ground up, then slot to reinsurance




layer.
Estimate ultimate claim counts by year by developing
trended claims to layer.
Multiply trended claim counts by frequency trend factor to
bring them to the frequency level of the upcoming treaty
year.
Adjust for change in exposure levels, i.e..
Adjusted Claim Count year i =
Trended Ultimate Claim Count i x
(SPI for upcoming treaty year / On Level SPI year i)
Poisson parameter l equals the mean of the ultimate,
trended, adjusted claim counts from above
32
Example of Simulated Claim Count
(Note)
Exposure
Adj
Factor
1.60
1.52
1.45
1.38
1.32
1.25
1.19
1.14
1.08
1.03
SPI at Trended Count Est Ult Annual Freq
Trended
2006 Rate Counts Devel Trended Freq Trend to Ult Claim
Year
Level
to Layer Factor Count Trend
2006
Count
1996
10,000
2.0
1.0
2.0
0.0% 1.104
2.21
1997
10,500
1.0
1.0
1.0
0.0% 1.104
1.10
1998
11,025
1.0
1.0
1.0
0.0% 1.104
1.10
1999
11,576
1.0
1.1
1.1
0.0% 1.104
1.16
2000
12,155
3.0
1.1
3.3
0.0% 1.104
3.64
2001
12,763
1.2
0.0% 1.104
2002
13,401
1.3
2.0% 1.082
2003
14,071
1.5
2.0% 1.061
2004
14,775
1.0
2.0
2.0
2.0% 1.040
2.08
2005
15,513
1.0
3.5
3.5
2.0% 1.020
3.57
2006
16,000
2.0%
Average:
Variance:
Note: Exposure Adj Factor Yr i = 2006 SPI / SPI year i
Selected Variance:
33
2006
Level
Claim
Count
3.53
1.68
1.60
1.60
4.80
2.25
3.68
1.92
2.82
3.11
Modeling Frequency- Negative
Binomial
 Negative Binomial: Same form as the Poisson distribution,
except that it assumes that l is not fixed, but rather has a
gamma distribution around the selected l
• Claim count distribution is Negative Binomial if the variance
of the count distribution is greater than the mean
• The Gamma distribution around l has a mean of 1
 Negative Binomial simulation
• Simulate l (Poisson expected count)
• Using simulated expected claim count, simulate claim count
for the year.
 Negative Binomial is the preferred distribution
• Reflects some parameter uncertainty regarding the true
mean claim count
• The extra variability of the Negative Binomial is more in line
with historical experience
34
Algorithm for Simulating Claim
Counts Using a Poisson Distribution
 Poisson
• Manually create a Poisson cumulative
distribution table
• Simulate the CDF (a number between 0 and 1)
and lookup the number of claims corresponding
to that CDF (pick the claim count with the CDF
just below the simulated CDF) This is your
simulated claim count for year 1
• Repeat the above two steps for however many
years that you want to simulate
35
Negative Binomial Contagion Parameter
 Determine contagion parameter, c, of claim count distribution:
(s^2 / m) = 1 + c m
If the claim count distribution is Poisson, then c=0
If it is negative binomial, then c>0, i.e. variance is
greater than the mean
 Solve for the contagion parameter:
c = [(s^2 / m) - 1] / m
36
Additional Steps for Simulating Claim
Counts using Negative Binomial
 Simulate gamma random variable with a mean of 1
• Gamma distribution has two parameters: a and b
a = 1/c; b = c; c = contagion parameter
• Using Excel, simulate gamma random variable as
follows: Gammainv(Simulated CDF, a, b)
 Simulated Poisson parameter =
=l x Simulated Gamma Random Variable Above
 Use the Poisson distribution algorithm using the above
simulated Poisson parameter, l, to simulate the claim
count for the year
37
Year 1 Simulated Negative
Binomial Claim Count
(A)
(B)
(C)
(D)
(E)
(F)
(G)
(H)
Selected Mean Claim Count (Poisson Gamma)
Selected Variance of Claim Count Distribution
Contagion Parameter [(Variance / Mean -1) / Mean]
Gamma Distribution Alpha
Gamma Distribution Beta
Simulated Gamma CDF
Simulated Gamma Random Variable
Simulated Poisson Parameter (A) X (G)
38
1.92
3.11
0.32
3.08
0.32
0.412
0.78
1.50
Year 1 Simulated Negative
Binomial Claim Count
Simulated Poisson Gamma
Simulated Poisson CDF:
Year 1 Simulated Claim Count:
Prob
Claim Poisson
Count ClaimPoisson
Count Probability <= X CountProbability
0
22.39% 22.39% 5
1.40%
1
33.51% 55.90% 6
0.35%
2
25.07% 80.97% 7
0.07%
3
12.51% 93.48% 8
0.01%
4
4.68% 98.16% 9
0.00%
39
1.50
0.808
2
Prob
Count
<= X
99.56%
99.91%
99.98%
100.00%
100.00%
Year 2 Simulated Negative
Binomial Claim Count
Selected Mean Claim Count (Poisson Gamma)
Simulated Gamma CDF
Simulated Gamma Random Variable
Simulated Poisson Gamma (A) X (G)
40
1.92
0.668
1.15
2.20
Year 2 Simulated Negative
Binomial Claim Count
Simulated Poisson Gamma
Simulated Poisson CDF:
Year 2 Simulated Claim Count:
2.20
0.645
3
Prob
Prob
Claim Poisson
Count Claim Poisson
Count
Count Probability <= X Count Probability <= X
0
11.13% 11.13% 5
4.73%
97.53%
1
24.44% 35.57% 6
1.73%
99.26%
2
26.83% 62.40% 7
0.54%
99.80%
3
19.63% 82.03% 8
0.15%
99.95%
4
10.77% 92.80% 9
0.04%
99.99%
41
Modeling Severity –
Common Severity Distributions





Lognormal
Mixed Exponential (currently used by ISO)
Pareto
Truncated Pareto.
This curve was used by ISO before moving to the Mixed
Exponential and will be the focus of this presentation.
• The ISO Truncated Pareto focused on modeling the larger
claims. Typically those over $50,000
42
Truncated Pareto
 Truncated Pareto Parameters
t = truncation point.
s = average claim size of losses below truncation point
p = probability claims are smaller than truncation point
b = pareto scale parameter - larger b results in larger
unlimited average loss
q = pareto shape parameter - lower q results in thicker tailed
distribution
 Cumulative Distribution Function
F(x) = 1 - (1-p) ((t+ b)/(x+ b))^q
Where x>t
43
Algorithm for Simulating Severity to
the Layer
 For each loss to be simulated, choose a random number
between 0 and 1. This is the simulated CDF
 Transformed CDF for losses hitting layer (TCDF) =
Prob(Loss < Reins Att. Pt) +
Simulated CDF x Prob (Loss > Reins Att. Pt)
•
If there is a 95% chance that loss is below attachment point, then
the transformed CDF (TCDF) is between 0.95 and 1.00.
 Find simulated ground up loss, x, that corresponds to simulated
TCDF
Doing some algebra, find x using the following formula:
x = Exp{ln(t+b) - [ln(1-TCDF) - ln(1-p)]/Q} - b
 From simulated ground up loss calculate loss to the layer
44
Year 1 Loss # 1 Simulated Severity
to the Layer
Pareto Parameters
B
79,206
Q
1.39
P
0.858
Reinsurance Layer:
750,000
Pareto Probability of Loss < Reins Att Point:
Simulated CDF:
Transformed CDF for Losses Simulated to the Excess Layer:
Simulated Loss:
Simulated Loss to Layer:
45
S
6,090
T
50,000
xs
250,000
96.13%
0.4029
0.9769
397,876
147,876
Year 1 Loss # 2 Simulated Severity
to the Layer
Pareto Parameters
B
79,206
Q
1.39
P
0.858
Reinsurance Layer:
750,000
Pareto Probability of Loss < Reins Att Point:
Simulated CDF:
Transformed CDF for Losses Simulated to the Excess Layer:
Simulated Loss:
Simulated Loss to Layer:
46
S
6,090
xs
T
50,000
250,000
96.13%
0.8400
0.9938
1,151,131
750,000
Simulation Summary
Year 1 Simulation
Year 2 Simulation
Claim Losses
Count to Layer
2 147,876
750,000
Total: 897,876
3 576,745
281,323
54,726
Total: 912,794
Run about 1,000 more years and we have
our aggregate distribution to the excess of
loss layer
47
Common Loss Sharing Provisions
for Excess of Loss Treaties
 Profit Commissions
• Already covered
 Swing Rated Premium
 Annual Aggregate Deductibles
 Limited Reinstatements
48
Swing Rated Premium
 Ceded premium is dependent on loss experience
 Typical Swing Rating Terms
• Provisional Rate: 10%
• Minimum/Margin: 3%
• Maximum: 15%
• Ceded Rate = Minimum/Margin +
Ceded Loss as % of SPI x 1.1;
subject to a maximum rate of 15%.
 Why did 100/80 x burn subject to min and
max rate become extinct?
49
Swing Rated Premium - Example
 Burn (ceded loss / SPI) = 10%. Rate = 3% + 10% x 1.1 =
14%
 Burn = 2%. Rate = 3% + 2% x 1.1 = 5.2%.
 Burn = 14%. Calculated Rate = 3% + 14% x 1.1 = 18.4%.
Rate = 15% maximum rate
50
Swing Rated Premium Example
 Swing Rating Terms: Ceded
premium is adjusted to equal
to a 3% minimum rate +
ceded loss times 1.1 loading
factor, subject to a maximum
rate of 15%
 Use the aggregate distribution
to your right to calculate the
ceded loss ratio under the
treaty
51
Band of Burns
Low
High
Average Probability
0.0%
0.0%
0.0%
9.0%
0.0%
2.5%
1.3%
6.0%
2.5%
5.0%
3.8%
9.0%
5.0%
7.5%
6.3%
10.2%
7.5% 10.0%
8.8%
11.4%
10.0% 12.5% 11.3%
15.0%
12.5% 15.0% 13.8%
12.0%
15.0% 17.5% 16.3%
9.0%
17.5% 20.0% 18.8%
7.8%
20.0% 25.0% 21.9%
6.0%
25.0% 50.0% 30.3%
4.8%
Swing Rated Premium Example Solution
Loss
Load
Min/Margin Prov Rate Max Rate Factor
Swing Rated Terms
3.0%
10.0%
15.0% 110.0%
Band of Burns
Low
High
Average Probability
0.0%
0.0%
0.0%
9.0%
0.0%
2.5%
1.3%
6.0%
2.5%
5.0%
3.8%
9.0%
5.0%
7.5%
6.3%
10.2%
7.5%
10.0%
8.8%
11.4%
10.0%
12.5%
11.3%
15.0%
12.5%
15.0%
13.8%
12.0%
15.0%
17.5%
16.3%
9.0%
17.5%
20.0%
18.8%
7.8%
20.0%
25.0%
21.9%
6.0%
25.0%
50.0%
30.3%
4.8%
Prob Wtd Avg:
11.1%
Final
Rate
3.0%
4.4%
7.1%
9.9%
12.6%
15.0%
15.0%
15.0%
15.0%
15.0%
15.0%
11.8%
Proj LR = Expected Burn/Expected Final Rate
52
93.8%
Annual Aggregate Deductible
 The annual aggregate deductible (AAD) refers to a retention by
the cedant of losses that would be otherwise ceded to the treaty
 Example: Reinsurer provides a $500,000 xs $500,000 excess
of loss contract. Cedant retains an AAD of $750,000
• Total Loss to Layer = $500,000. Cedant retains all
$500,000. No loss ceded to reinsurers
• Total Loss to Layer = $1 mil. Cedant retains $750,000.
Reinsurer pays $250,000.
• Total Loss to Layer =$1.5 mil. Cedant retains? Reinsurer
pays?
53
Annual Aggregate Deductible
 Discussion Question: Reinsurer writes a $500,000 xs
$500,000 excess of loss treaty.
• Expected Loss to the Layer is $1 million (before AAD)
• Cedant retains a $500,000 annual aggregate
deductible.
• Cedant says, “I assume that you will decrease your
expected loss by $500,000.”
• How do you respond?
54
Annual Aggregate Deductible
Example
 Your expected burn to a
$500K xs $500K reinsurance
layer is 11.1%. Cedant adds
an AAD of 5% of subject
premium
 Using the aggregate
distribution of burns to your
right, calculate the burn net of
the AAD.
55
Band of Burns
Low
High
Average Probability
0.0%
0.0%
0.0%
9.0%
0.0%
2.5%
1.3%
6.0%
2.5%
5.0%
3.8%
9.0%
5.0%
7.5%
6.3%
10.2%
7.5%
10.0%
8.8%
11.4%
10.0%
12.5% 11.3%
15.0%
12.5%
15.0% 13.8%
12.0%
15.0%
17.5% 16.3%
9.0%
17.5%
20.0% 18.8%
7.8%
20.0%
25.0% 21.9%
6.0%
25.0%
50.0% 30.3%
4.8%
Prob Wtd Avg:
11.1%
Annual Aggregate Deductible
Example - Solution
Annual Aggregate Deductible as % of SPI:
5.0%
Band of Burns
Low
High Average Probability
0.0% 0.0%
0.0%
9.0%
0.0% 2.5%
1.3%
6.0%
2.5% 5.0%
3.8%
9.0%
5.0% 7.5%
6.3%
10.2%
7.5% 10.0%
8.8%
11.4%
10.0% 12.5% 11.3%
15.0%
12.5% 15.0% 13.8%
12.0%
15.0% 17.5% 16.3%
9.0%
17.5% 20.0% 18.8%
7.8%
20.0% 25.0% 21.9%
6.0%
25.0% 50.0% 30.3%
4.8%
Prob Wtd Avg:
11.1%
Savings
from
AAD
0.0%
1.3%
3.8%
5.0%
5.0%
5.0%
5.0%
5.0%
5.0%
5.0%
5.0%
4.2%
56
Burn
Net of
AAD
0.0%
0.0%
0.0%
1.3%
3.8%
6.3%
8.8%
11.3%
13.8%
16.9%
25.3%
6.8%
Limited Reinstatements
 Limited reinstatements refers to the number of times that the
risk limit of an excess can be reused.
 Example: $1 million xs $1 million layer
• 1 reinstatement: It means that after the cedant uses up the
first limit, they also get a second limit
 Treaty Aggregate Limit =
= Risk Limit x (1 + number of Reinstatements)
57
Limited Reinstatements Example
$1 million xs $1 million layer
1 reinstatement
Simulated Year 1
Individual Ceded
Losses Loss
$000's
$000's
2,000
1000
2,000
1000
2,000
0
Simulated Year 2
Individual Ceded
Losses Loss
$000's
$000's
3,000 1000
1,500
500
1,500
500
58
Simulated Year 3
Individual Ceded
Losses Loss
$000's
$000's
3,000
?
1,500
?
1,500
?
2,000
?
Reinstatement Premium
 In many cases to “reinstate” the limit, the cedant is
required to pay an additional premium
 Choosing to reinstate the limit is almost always
mandatory
 Reinstatement premium should simply be viewed as
additional premium that reinsurers receive
depending on loss experience
59
Reinstatement Premium Example 1
$1 million xs $1 million layer
1 reinstatement at 100%
Upfront Ceded Premium = $250,000
Prorata as to amount 100% as to time
Simulated Year 1
Simulated Year 2
Simulated Year 3
Individual Ceded Reinst Individual Ceded Reinst Individual Ceded Reinst
Losses Loss
Prem Losses Loss Prem Losses Loss Prem
$000's $000's $000's $000's $000's $000's $000's $000's $000's
2,000 1,000
250
1,500
500
125
1,250
?
?
2,000 1,000
1,500
500
125
2,000
?
?
2,000
1,500
500
2,000
?
?
60
Reinstatement Premium Example 2
$1 million xs $1 million layer
2 reinstatements: 1st at 50%, 2nd at 100%.
Upfront Ceded Premium = $250,000
Simulated Year 1
Simulated Year 2
Simulated Year 3
Individual Ceded Reinst Individual Ceded Reinst Individual Ceded Reinst
Losses Loss
Prem Losses Loss Prem Losses Loss Prem
$000's
$000's $000's $000's $000's $000's $000's $000's $000's
3,000
1,000
125
1,500
500 62.5
1,250
?
?
2,000
1,000
250
1,500
500 62.5
2,000
?
?
2,000
1,000
1,500
500 125.0
2,000
?
?
2,000
-
61
Reinstatement Example 3
 Reinsurance Treaty:

Loss
$1 mil xs $1 mil
$000's
Probability
Upfront Premium = 400K
2 Reinstatements: 1st at 50%,
68.00%
2nd at 100%
Using the aggregate distribution
1,000
25.00%
to the right, calculate our
expected ultimate loss, premium,
2,000
4.00%
and loss ratio
3,000
2.00%
4,000
1.00%
62
Reinstatement Example 3 –
Solution
Upfront Premium = 400K
2 Reinstatements: 1st at 50%, 2nd at 100%
Total
Loss Net
Loss
of Reinst
Reinst Total
$000's
Probability Limitation Premium Premium
68.00%
400
1,000
25.00%
1,000
200
600
2,000
4.00%
2,000
600
1,000
3,000
2.00%
3,000
600
1,000
4,000
1.00%
3,000
600
1,000
Prob Wtd Avg:
420
92
492
Projected Loss Ratio:
85.4%
63
Reinstatement Example 4
 Note: Reinstatement provisions are typically found on high excess

layers, where loss tends to be either 0 or a full limit loss.
Assume: Layer = 10M xs 10M, Expected Loss = 1M, Poisson
Frequency with mean = .1
Upfront Premium = 1.2M
1 Reinstatement at 50%
# of
Expected
Loss Net of
Reinst
Total
Clms
Prob
Loss (000's) Reinst Limit Premium Premium
0 90.48%
0
0
0
1,200
1
9.05%
10,000
10,000
600
1,800
2
0.45%
20,000
20,000
600
1,800
3
0.02%
30,000
20,000
600
1,800
4
0.00%
40,000
20,000
600
1,800
5
0.00%
50,000
20,000
600
1,800
Prob Wtd Avg
100.0%
1,000
998
57
1,257
Projected Loss Ratio:
79.5%
64
Deficit Carry forward
 Treaty terms may include Deficit Carry forward Provisions, in
which some losses are carried forward to next year’s
contract in determining the commission paid.
 Example:
Min
Comm
25.0%
LR
75.0%
Slide
0.5 to 1
Prov
30.0%
65.0%
1 to 1
Max
35.0%
60.0%
Defecit Carry Forward: 5% of Premium
65
Deficit Carry forward Example
• Solution - Shift Sliding Scale Commission terms.
Exp LR
40.0%
57.5%
62.5%
67.5%
72.5%
77.5%
85.0%
95.0%
150.0%
225.0%
71.5%
Prob
3.49%
8.23%
15.22%
19.77%
19.30%
14.94%
14.79%
3.60%
0.66%
0.00%
Ceding
Comm
35.0%
32.5%
28.8%
26.3%
25.0%
25.0%
25.0%
25.0%
25.0%
25.0%
26.8%
Last Year's Treaty LR:
Deficit Carryforward:
(80.0% - 75.0% = 5.0%)
80.0%
5.0%
Original
Min
Prov
Max
Comm
25.0%
30.0%
35.0%
LR
75.0%
65.0%
60.0%
Slide
50.0%
100.0%
Shifted
Min
Prov
Max
Comm
25.0%
30.0%
35.0%
LR
70.0%
60.0%
55.0%
Slide
50.0%
100.0%
66
DCF/Multi-Year Block
• Question: How much credit do you give an account for
Deficit Carry forwards, other than using the CF from the
previous year (e.g. unlimited CFs)?
• Can estimate using an average of simulated
“years”, but this method should be used with caution:
– Assumes independence (probably unrealistic)
– Accounts for both Deficit and Credit carry forwards
– Deficits are often forgiven, treaty terms may change, or
treaty may be terminated before the benefit of the deficit
carry forward is felt by the reinsurer.
67
DCF/Multi-Year Block - Example
Average LR
Std Dev
Avg Comm
Year 1
71.52%
9.98%
Year 2
71.39%
9.95%
Year 3
71.69%
10.08%
3-Year
Block
71.54%
5.84%
28.25%
28.28%
28.20%
27.39%
69.62%
67.96%
77.54%
73.85%
88.54%
55.43%
67.49%
71.83%
63.93%
75.92%
69.42%
63.91%
71.13%
58.66%
91.61%
79.21%
78.55%
78.42%
59.58%
70.11%
52.09%
68.91%
74.77%
46.96%
72.24%
65.86%
80.54%
73.05%
47.51%
72.82%
63.71%
66.93%
74.48%
59.82%
84.13%
66.83%
75.53%
74.43%
57.01%
72.95%
Simulation
1
2
3
4
5
6
7
8
9
10
68
Technical Summary
 Modeling loss sensitive provisions is easy.
 Selecting your expected loss and aggregate
distribution is hard
 Steps to analyzing loss sensitive provisions
• Build aggregate loss distribution
• Apply loss sensitive terms to each point on the
loss distribution or to each simulated year
• Calculate probability weighted average of treaty
results
69
Additional Issues & Uses of
Aggregate Distributions
 Correlation between lines of business
 Reserving for loss sensitive treaty terms
 Some companies Use aggregate distributions to measure risk &

allocate capital. One hypothetical example:
Capital = 99th percentile Discounted Loss x Correlation Factor
Fitting Severity Curves: Don’t Ignore Loss Development
• Increases average severity
• Increases variance – claims spread as they settle.
• See “Survey of Methods Used to Reflect Development in
Excess Ratemaking” by Stephen Philbrick, CAS 1996
Winter Forum
70
Risk transfer
 FASB 113: A reinsurance contract should be booked using deposit


accounting unless:
• “The reinsurer assumes significant insurance risk”
• Insurance risk not significant if “the probability of a significant
variation in either the amount or timing of payments by the
reinsurer is remote”
• “It is reasonably possible that the reinsurer may realize a significant
loss from the transaction.
• 10/10 Rule of Thumb: Is there a 10% chance that the reinsurer
will have a loss of at least 10% of premium on a discounted
basis
• Calculation excludes brokerage and reinsurer internal
expense.
SFAS 62 governs statutory accounting. Requirements are similar to
FASB 113.
Recent regulator concerns have centered on pro-rata reinsurance.
71
Risk Transfer Recent
Developments
 New York State Draft Bifurcation Proposal:
•
•
•
•
•
Bifurcation applies to any pro-rata treaty that contains one of the
following features: profit commissions, sliding scale commissions,
loss ratio corridors or caps, occurrence limits below an unspecificed
% of premium, etc.
Excess of loss and facultative contracts are excluded
If above conditions are met, premium must be split as follows:
• Premium covering exposure in excess of the 90th percentile of
the loss distribution counts as reinsurance.
• The remaining premium should be booked as a deposit.
Rule would be applied retroactively to business written 1/1/94 and
later.
Appears unlikely that NAIC will approve this proposal, but proposal
emphasizes regulators concerns.
72
Concluding Comment
 Aggregate distributions are a critical element in




evaluating the profitability of business.
They are frequently produced by (re)insurers as a risk
management tool.
They are being used on a broader spectrum of contracts
to review risk transfer.
Some accountants and regulators seem to treat these
aggregate distributions as if they were gospel.
Critical to effectively communicate the difficulties in
projecting aggregate distributions of future results.
• Need to make regulators and accountants understand
the degree of parameter uncertainty.
73