Aggregate Loss Generator
By Dom Yarnell
Purpose
This tool creates aggregate loss distributions for use in pricing primary insurance policies. While features
like occurrence limits are priced easily enough with increased limits factors, aggregate loss limits require
additional care, as their impact is a function of both severity and frequency. The aggregate loss
distribution enables the actuary to reflect the impact of expected frequency more accurately than
factors that don’t vary with the size of the risk. In addition to taking into account occurrence limits, the
VBA code underlying the model can be edited to include other features, like retentions, to price primary
insurance policies on-the-fly.
How it works
The algorithm underlying the Aggregate Loss Generator (ALG) relies on a frequency distribution and a
severity distribution, as does traditional Monte Carlo simulation. However, unlike most simulation tools,
the ALG must use a discrete severity distribution and evaluates all possible loss scenarios, so there is no
sampling error.
Let’s look at a very simple example: a frequency distribution that considers up to two losses, and a
discrete severity distribution with five points.
Claim
Count
0
1
2
PDF
70%
20%
10%
Loss
Amount
10,000
25,000
50,000
75,000
100,000
PDF
60%
20%
10%
7%
3%
Since a single loss implies five possible outcomes, and there are as many as two losses possible, there
are 31 = 50 + 51 + 52 loss scenarios. Such a small number of calculations lends itself to direct calculation,
but the distributions needed in the pricing process will likely include more points, and therefore require
many more calculations. For instance, a frequency distribution with 10 points and a severity distribution
with 12 points would imply over 67 billion loss scenarios to calculate. Clearly, a direct calculation of all
loss scenarios is not feasible for computers commonly found in the workplace.
In order to make the calculations more manageable, the ALG compresses the distribution of each claim
scenario to a certain number of points, and then uses this distribution to calculate the subsequent claim
scenario. In the simple example above, let’s assume we set the maximum number of points to four. If
there is a single loss, the five points on the severity distribution would be reduced to at most four
points. If the first two points were to be compressed, the Loss Amounts would be combined into a
weighted average and the probabilities would be totaled, so that the new, single point would have a
Loss Amount of 13,750 = [(10,000 x 60%) + (25,000 x 20%)] / (60% + 20%)], and a probability of 80% =
60% + 20%. This four-point, 1-claim scenario would then be used to calculate the 2-claim scenario, which
now requires 20 = 4 x 5 calculations instead of the original 25 = 5 x 5. In total, 26 = 50 + 51 + (4 x 5)
scenarios would be calculated instead of the original 31. If we look at the scenario with a 10-point
frequency distribution and a 12-point severity distribution, and limit the number of points to 100, the
number of loss scenarios to calculate drops from over 67 billion to 9,7571. In addition to compressing
the distribution of each claim scenario, the ALG compresses the resulting aggregate loss distribution
before recording it in Excel.
Loss scenario calculations
The calculation of loss and probability for a claim scenario is fairly straightforward.
1-Claim
Loss
Amount
13,750
13,750
13,750
13,750
13,750
50,000
50,000
50,000
50,000
50,000
75,000
75,000
75,000
75,000
75,000
100,000
100,000
100,000
100,000
100,000
1-Claim
Loss
Probability
80%
80%
80%
80%
80%
10%
10%
10%
10%
10%
7%
7%
7%
7%
7%
3%
3%
3%
3%
3%
Second
Loss
Amount
10,000
25,000
50,000
75,000
100,000
10,000
25,000
50,000
75,000
100,000
10,000
25,000
50,000
75,000
100,000
10,000
25,000
50,000
75,000
100,000
Second
Loss
Probability
60%
20%
10%
7%
3%
60%
20%
10%
7%
3%
60%
20%
10%
7%
3%
60%
20%
10%
7%
3%
2-Claim
Loss
Amount
23,750
38,750
63,750
88,750
113,750
60,000
75,000
100,000
125,000
150,000
85,000
100,000
125,000
150,000
175,000
110,000
125,000
150,000
175,000
200,000
2-Claim
Loss
Probability
48.0%
16.0%
8.0%
5.6%
2.4%
6.0%
2.0%
1.0%
0.7%
0.3%
4.2%
1.4%
0.7%
0.5%
0.2%
1.8%
0.6%
0.3%
0.2%
0.1%
Since we have four (after compression) scenarios in the event of one claim, and there are five possible
values for the next loss, calculating the distribution of two claims results in 20 loss scenarios. For each
scenario, the loss amounts are added together and the probability are multiplied together (since we
assume claims are independent of each other), and the result would be sorted and compressed.
1
The actual number of points may be less than the maximum number of points, which means the number of loss
scenarios calculated above represents the maximum the computer might have to calculate.
2-Claim
Loss
Amount
27,500
74,965
121,331
179,412
2-Claim
Loss
Probability
64.0%
28.2%
7.3%
0.5%
We’ve calculated each loss amount and the probabilities of each loss amount in the event of one-loss
and two-loss scenarios, so the next step is to incorporate the probabilities of each claim scenario by
looking to the frequency distribution.
Claim Frequency
Count Probability
0
70%
1
20%
1
20%
1
20%
1
20%
2
10%
2
10%
2
10%
2
10%
Loss
Severity
Loss
Amount Probability Probability
0
100.0%
70.00%
13,750
80.0%
16.00%
50,000
10.0%
2.00%
75,000
7.0%
1.40%
100,000
3.0%
0.60%
27,500
64.0%
6.40%
74,965
28.2%
2.82%
121,331
7.3%
0.73%
179,412
0.5%
0.05%
The Loss Probability is the product of the Frequency Probability and the Severity Probability, since we
assume frequency and severity are independent. The aggregate loss distribution is then sorted and
compressed.
Loss
Loss
Amount Probability
4,286
92.40%
66,945
6.22%
111,701
1.33%
179,412
0.05%
The compressed aggregate loss distribution has an expected value of 9,700, which equals the product of
the expected frequency = 0.4, and the expected severity = 24,250.
Generating the frequency distribution
Although it’s common practice to assume one severity distribution for pricing purposes, specific
frequency distributions need to be created for each risk when using the ALG, as exposure varies from
risk to risk (and year to year). Using a Poisson distribution, the ALG creates a frequency distribution
based on the Expected Frequency and Probability Limit.
Expected Probability
Frequency
Limit
5.00
0.10%
A Probability Limit of 0.10% will limit the number of claims in the frequency distribution to the number
associated with a cumulative distribution function (CDF) that exceeds 1 – 0.10%. For example, a Poisson
distribution with an expected value of 5.00 implies a CDF of 99.80% at 12, and 99.93% at 13. Since
99.93% is the first value of the CDF to exceed 1 – 0.10% = 99.9%, the ALG will create a frequency
distribution with a maximum of 13 claims. Lowering the Probability Limit will increase the likeliness of
higher claim count scenarios being calculated.
Since a Poisson distribution includes probabilities for numbers above the limit imposed by the ALG, the
probabilities are further adjusted so that 1) they add up to 100%, and 2) they yield a mean that matches
the Expected Frequency.
Occurrence limit and maximum scenario points
The ALG also takes into account an Occurrence Limit and an Aggregate Limit when calculating loss
scenarios.
Occurrence
Limit
80,000
Maximum
Scenario
Points
4
The Maximum Scenario Points (MSP) selected will limit the number of points in 1) the outcome of each
claim count scenario, and 2) the number of points in the final distribution. The higher the MSP, the more
refined your distribution, and the longer it will take ALG to produce the aggregate loss distribution.
In order to compress the distribution, the algorithm first determines the largest loss amount in the
scenario. For instance, in the one-claim scenario without an occurrence limit, the largest loss in the
scenario is going to be the largest loss amount in the severity distribution. Using our simple severity
distribution, we see that the maximum loss for a one-claim scenario is 100K. The algorithm then
calculates the size of the loss buckets by dividing 100K by the MSP = 4, meaning the loss buckets are
25K.
Loss
Amount
10,000
25,000
50,000
75,000
100,000
PDF
60%
20%
10%
7%
3%
Loss
Bucket
1
1
2
3
4
Since the first two Loss Amounts are less than or equal to 25K, they are assigned to the first bucket, and
each of the other points gets their own bucket. If the third point were 55K instead of 75K, it would be
assigned to the third bucket, and the second bucket would be empty, so that the resulting distribution
would have only three points instead of four.
Loss
Amount
10,000
25,000
55,000
75,000
100,000
Loss
Bucket
1
1
3
3
4
In other words, the more evenly distributed the severity distribution, the more refined the compressed
distributions.
When we include an Occurrence Limit = 80K, the maximum loss amount for the one-claim scenario is
80K, and the bucket size becomes 20K = 80K / 4.
Limited
Loss
Amount
10,000
25,000
55,000
75,000
80,000
Limited
Loss
Bucket
1
2
3
4
4
Now the compressed distribution becomes a four-point distribution, as only the last two points are in
the same 20K bucket, and the remaining three points get their own bucket. So the lower the Occurrence
Limit, the more refined the compressed distribution.
Aggregate limit
In the initial example, the 70% chance of zero loss (the chance of having no claims) was compressed into
other scenarios such that the smallest Loss Amount in the aggregate loss distribution is 4,286 with a
probability of 92.4%. But the ALG actually preserves the probability of a loss amount of zero, excluding it
from the compression calculations.
Loss
Loss
Amount Probability
0
70.00%
22,597
89,480
159,375
25.80%
4.04%
0.16%
Since the MSP is four, the ALG preserves the probability of zero loss at each scenario and compresses
the remaining distribution in to three points.
Aggregate
Limit
150,000
Likewise, if an Aggregate Limit is included, the ALG preserves the probability of the Loss Amount equal
to the Aggregate Limit, so that that probability is not compressed with any other points.
Loss
Amount
0
23,295
90,921
150,000
PDF
70.00%
26.10%
3.80%
0.10%
The resulting expected value is 9,685, which reveals that the impact of the Aggregate Limit using the
parameters above would be a discount of 15, since the original expected value was 9,700.
How to use the Aggregate Loss Generator
The ALG is meant to be implemented behind the scenes, either directly in an Excel model or as a highly
functional spec for IT to use as a reference. The VBA code that drives the model is not passwordprotected, and users are highly encouraged to examine the code and customize it to their purposes.
Further potential enhancements include



Calculating the effects of deductibles and self-insured retentions. This could be achieved in VBA
or the severity distribution itself could be entered as net of limits and retentions.
Calculating the effect of retention caps. This calculation would likely take place in VBA, as it’s not
possible to calculate from the aggregate loss distribution.
Implementing an alternative compression algorithm. The ALG does a good job of preserving the
tails of aggregate loss distributions, but you may prefer distributions that are more “spread
out,” with more evenly distributed probabilities per bucket.
Discrete severity distributions
Loss modeling often makes use of continuous severity distributions (lognormal, Pareto, etc.) that are
fully defined using a couple parameters. Since these distributions imply an infinite number of loss
amounts, the ALG, which calculates all possible loss scenarios, cannot make use of these continuous
distributions. Some would view having to use discrete severity distributions as a weakness, but there are
some clear advantages.
Continuous distributions can be altered by changing two or three parameters, but these adjustments
may be in sufficient to get a distribution you want. On the other hand, you can always add another point
to a discrete distribution and edit the loss amounts and probabilities, so you’re more likely to arrive at a
distribution you find acceptable because you have an unlimited number of parameters with which to
work. To put it another way, if you’re happy with a lognormal distribution, it’s not too difficult to turn
that into a discrete distribution, and then you can edit the parameters to change the distribution in ways
you couldn’t change a lognormal distribution.
Discrete distributions are very easy to understand, even for non-actuaries. If audit or underwriting takes
an interest in the pricing assumptions of your model, they are more likely to understand a table of losses
and probabilities than parameters of a continuous distribution. And if aggregate loss generation were
included in a rate filing, I’m betting that a table of losses and probabilities would have a much easier
time getting approved than would values assigned to Greek letters.
Furthermore, actuaries might be tempted to calculate probabilities for very large losses using
continuous distributions that were fit to losses in lower layers. But there’s no reason to believe that the
tail of a continuous distribution is appropriate for pricing large losses simply because it works well for
smaller losses. Discrete distributions can be edited so that tail fits expectations, which are, by their
nature, explicitly stated.
More generally, it’s arguably unreasonable to believe that real life probability distributions conform to
continuous distributions. Actuaries just might be spending too much time learning and applying curvefitting techniques when they could instead simply bucket losses by size, use the claim counts as a proxy
for probability, and make some explicit assumptions about missing data that was censored by retentions
or truncated by limits.