9  Calculations with Aggregate

Example calculations using Python and R.

This chapter reproduces many of the exhibits from the main text using Python and the aggregate package. It also describes how to extend to more realistic examples. Throughout, we have included only terse subsets of code so the reader can focus on the results and the output. Instructions for obtaining the full code are provided below.

9.1 Aggregate and Python

aggregate is a Python package that can be used to build approximations to compound (aggregate) probability distributions quickly and accurately, and to solve insurance, risk management, and actuarial problems using realistic models that reflect underlying frequency and severity. It delivers the speed and accuracy of parametric distributions to situations that usually require simulation, making it as easy to work with an aggregate (compound) probability distribution as the lognormal. aggregate includes an expressive language called DecL to describe aggregate distributions.

The aggregate package is available on PyPi, the source code is on GitHub at https://github.com/mynl/aggregate, and there is extensive documentation. The Aggregate class and DecL lanaguage are described in Mildenhall (2024). There is also an extensive series of videos introducing various capabilities available on YouTube.

9.2 Aggregate and R

Python packages can be used in R via the reticulate library. Python and R code can also be mixed in RMarkdown (now Quarto) files. This short video explains how to use aggregate from R.

9.3 Reproducing the code examples

To reproduce the Python code examples, you must set up your environment and then install aggregate using

pip install aggregate

The formatting of the examples relies on greater_tables, which can be installed in the same way. The installation process will also install several standard packages, such as numpy and pandas, that it depends on. If you get an error message about other missing packages, you can install them with pip. All the code in the monograph runs on aggregate version 0.24.0 and should run on any later version. If you have an older version installed, you can updated it with pip install -U aggregate.

Once aggregate is installed, start by importing basic libraries. The next code block shows the relevant aggregate function. The class GT from greater_tables is used to format DataFrames consistently.

from aggregate import (
    Portfolio,                  # creates multi-unit portfolios
    make_ceder_netter,          # models reinsurance
    Distortion                  # creates a Distortion function
)
from greater_tables import GT   # consistent table format

Online resources. Complete code samples can be extracted from the online version of the monograph, available at https://cmm.mynl.com/. Each code block has a small copy icon in the upper right-hand corner. The code blocks on each page can all be shown or hidden using the </>Code control at the top of the page. To download the original qmd file, use the View Source option under the same control. Alternatively, the entire Quarto (previously RMarkdown) source can be downloaded from GitHub at https://github.com/mynl/CapitalModeling (CAS TO INSERT OWN LINK).

9.4 Reproducing the discrete example

9.4.1 Input data and setup

The InsCo example running through this monograph is based on the 10 events shown in Table 9.1, which reproduces Table 2.1, and adds the mean and CV. For reference, here is the code to create the underlying dataframe.

Code 
loss_sample = pd.DataFrame(
    {
        'A': [ 5,  7, 15, 15, 13,  5, 15, 26, 17, 16],
        'B': [20, 33, 13,  7, 20, 27, 16, 19,  8, 20],
        'C': [11,  0,  0,  0,  7,  8,  9, 10, 40, 64],
    }
)
loss_sample['total'] = loss_sample.sum(axis=1)
loss_sample['p_total'] = 1/10
print(loss_sample)
    A   B   C  total  p_total
0   5  20  11     36      0.1
1   7  33   0     40      0.1
2  15  13   0     28      0.1
3  15   7   0     22      0.1
4  13  20   7     40      0.1
5   5  27   8     40      0.1
6  15  16   9     40      0.1
7  26  19  10     55      0.1
8  17   8  40     65      0.1
9  16  20  64    100      0.1
Table 9.1: The 10 equally likely simulations underlying the basic examples.
index A B C Total
0 5 20 11 72
1 7 33 0 80
2 15 13 0 56
3 15 7 0 44
4 13 20 7 80
5 5 27 8 80
6 15 16 9 80
7 26 19 10 110
8 17 8 40 130
9 16 20 64 200
EX 13.4 18.3 14.9 93.2
CV 0.453 0.412 1.324 0.455

Now we create an aggregate Portfolio class instance based on the loss_sample simulation output. The next block shows how to do this, using the Portfolio.create_from_sample method. TMA1 is a label, and loss_sample is the dataframe created previously. The other arguments specify working with unit-sized buckets bs=1 appropriate for our integer losses and to use \(256=2^8\) buckets, log2=8.

Code 
wport = Portfolio.create_from_sample('TMA1', loss_sample, bs=1, log2=8)

The next line of code displays Table 9.2, which shows summary output from the object wport by accessing its describe attribute. The summary includes the mean, CV, and skewness. The model and estimated (Est) columns are identical because we specified the distribution directly; no simulation or Fast Fourier Transform-based numerical methods are employed.

Code 
fGT(wport.describe.drop(columns=['Err E[X]', 'Err CV(X)']))
Table 9.2: Summary statistics for the base example.
unit X E[X] Est E[X] CV(X) Est CV(X) Skew(X) Est Skew(X)
A Freq 1 0
Sev 13.4 13.4 0.453 0.453 0.281 0.281
Agg 13.4 13.4 0.453 0.453 0.281 0.281
B Freq 1 0
Sev 18.300 18.300 0.412 0.412 0.269 0.269
Agg 18.300 18.300 0.412 0.412 0.269 0.269
C Freq 1 0
Sev 14.9 14.900 1.324 1.324 1.603 1.603
Agg 14.9 14.900 1.324 1.324 1.603 1.603
total Freq 3 0
Sev 15.533 15.533 0.827 1.884
Agg 46.6 46.600 0.471 0.471 1.177 1.177

Next, we need to calibrate distortion functions to achieve the desired pricing. The example uses full capitalization throughout (assets equal the maximum loss), which is equivalent to a 100% percentile level (see Table 9.5) and assumes the cost of capital is 15%. The code runs the calibration by calling wport.calibrate_distortions with arguments for the capital percentile level Ps and cost of capital COCs. The arguments are passed as lists because the routine can be used to solve for several percentile levels and costs simultaneously. Table 9.3 shows the resulting parameters, which match those in Table 4.5.

Code 
# calibrate to 15% return at p=100% capital standard
wport.calibrate_distortions(Ps=[1], COCs=[.15]);
Table 9.3: Distortion functions calibrated to 15% return on full capital.
method P param error
ccoc 53.565 0.150 0
ph 53.565 0.720 0.000
wang 53.565 0.343 0.000
dual 53.565 1.595 -0.000
tvar 53.565 0.271 0.000

Figure 9.1 shows the resulting distortion \(g\) functions (top) and the probability adjustment \(Z=q/p\) (bottom). The red star indicates a probability mass at \(p=1\).

Figure 9.1: Distortion functions (top row) and distortion probability adjustment functions (bottom row).

For technical reasons (discussed in Section 4.3 as well as chapter 14.1.5 in Mildenhall and Major (2022), see especially Figure 14.2), we must summarize the 10 events by distinct totals. Routine code produces Table 9.4, reproducing Table 4.2.

Table 9.4: Assumptions for portfolio, summarized by distinct totals.
index A B C p Total
0 15 7 0 0.1 22
1 15 13 0 0.1 28
2 5 20 11 0.1 36
3 10 24.000 6.000 0.4 40
4 26 19 10 0.1 55
5 17 8 40 0.1 65
6 16 20 64 0.1 100
EX 13.4 18.3 14.9 1 46.6
Plan 13.9 18.7 19.6 1 52.2

The examples describe a VaR- or TVaR-based capital standard (see Table 3.1), but for this example it works out to the the same as a fully capitalized 100% standard, as shown by the values in Table 9.5. The wport methods q and tvar compute quantiles and TVaRs for each unit and the total.

Table 9.5: Quantiles (VaR) and TVaR at different probability levels.
p Quantile TVaR
0.8 55 82.500
0.850 65 88.333
0.9 65 100
1 100 100

9.4.2 Premiums by distortion

Table 4.11 shows premium assuming a CCoC distortion with a 15% return applied using a 100% capital standard. The premium P row shows what Mildenhall and Major (2022) calls the linear NA premium. Table 9.6 reproduces this exhibit, confirming an equal 15% returns across all allocated capital. The code illustrates the power of aggregate. The first line creates a CCoC distortion using the Distortion class; note the argument is risk discount \(\delta\). The second line uses the price method to compute the linear allocation, which it returns in a structure with various diagnostic information, including Table 9.6.

Code 
ccoc = Distortion.ccoc(.15 / 1.15)
pricing_info = wport.price(1, ccoc, allocation='linear')
Table 9.6: Industry standard approach pricing using the CCoC distortion.
index A B C Total
L 13.4 18.3 14.900 46.6
a 16.000 20.000 64.000 100
Q 2.261 1.478 42.696 46.435
P 13.739 18.522 21.304 53.565
M 0.339 0.222 6.404 6.965
COC 0.150 0.150 0.150 0.150

Table 4.3, Table 4.4 and Table 4.10 compute \(\rho(X)\) for \(\rho\) the CCoC SRM, using the \(\rho(X)=\int g(S(x))\,dx\) (second table) and \(\rho(X)=\int xg'(S(x))f(x)\,dx\) (third) representations. The numbers needed for these calculations are shown in Table 9.7, where \(q=g'(S(x))f(x)\). These are extracted from the dataframe wport.density_df, which contains the probability mass, density, and survival functions for the portfolio, among other facts. The last row carries out the calculations and confirms the two methods give the same result, the total under \(gS\) using the former representation and under \(q\) using the latter representation.

Table 9.7: Industry standard approach pricing: raw ingredients and computed means.
loss p_total F S gS q ΔX
22 0.1 0.1 0.9 0.913 0.087 6
28 0.1 0.2 0.8 0.826 0.087 8
36 0.1 0.3 0.7 0.739 0.087 4
40 0.4 0.7 0.3 0.391 0.348 15
55 0.1 0.8 0.2 0.304 0.087 10
65 0.1 0.9 0.1 0.217 0.087 35
100 0.1 1 0 0 0.217 0
\(\rho(X)\) 53.565 53.565

Table 9.8 shows the results for the other standard distortions. The calculations match those of the Wang in Table 9.8 (c). All of the calibrated distortions are carried in the dictionary wport.dists. These tables also include the NA of capital (a process we do not recommend, but that is described in Section 8.3). The Q row matches the calculation of allocated capital shown in Table 8.10.

Table 9.8: Pricing by distortion by unit.
(a) CCoC
statistic A B C Total
L 13.400 18.300 14.900 46.600
a 18.613 23.810 57.577 100.0
Q 4.874 5.288 36.273 46.435
P 13.739 18.522 21.304 53.565
M 0.339 0.222 6.404 6.965
COC 0.070 0.042 0.177 0.150
(b) PH
statistic A B C Total
L 13.400 18.300 14.900 46.600
a 21.133 24.914 53.952 100.000
Q 7.074 6.565 32.796 46.435
P 14.060 18.349 21.156 53.565
M 0.660 0.049 6.256 6.965
COC 0.093 0.008 0.191 0.150
(c) Wang
statistic A B C Total
L 13.400 18.300 14.900 46.600
a 22.520 27.328 50.152 100.000
Q 8.411 8.691 29.333 46.435
P 14.109 18.637 20.819 53.565
M 0.709 0.337 5.919 6.965
COC 0.084 0.039 0.202 0.150
(d) Dual
statistic A B C Total
L 13.400 18.300 14.900 46.600
a 22.999 28.260 48.741 100.0
Q 8.873 9.143 28.419 46.435
P 14.127 19.117 20.322 53.565
M 0.727 0.817 5.422 6.965
COC 0.082 0.089 0.191 0.150
(e) TVaR
statistic A B C Total
L 13.400 18.300 14.900 46.600
a 22.817 29.659 47.525 100.000
Q 9.034 9.247 28.154 46.435
P 13.783 20.412 19.371 53.565
M 0.383 2.112 4.471 6.965
COC 0.042 0.228 0.159 0.150

9.4.3 Reinsurance analysis

Chapter 5 analyzes a possible 35 xs 65 aggregate stop loss reinsurance contract. The setup to analyze this contract, using a separate Portfolio object, is shown next. The first two lines apply reinsurance; [1, 35, 65] specifies 100% of the 35 xs 65 layer, and c and n are functions mapping gross to ceded and net. The last line creates a new Portfolio object from the net and ceded aggregate losses.

Code 
loss_sample_re = loss_sample.copy()
c, n = make_ceder_netter([(1, 35, 65)])
loss_sample_re['Net'] = n(loss_sample_re.total)
loss_sample_re['Ceded'] = c(loss_sample_re.total)
loss_sample_re = loss_sample_re[['Net', 'Ceded', 'p_total']]

# build Portfolio object with Net, Ceded units
wport_re = Portfolio.create_from_sample('WGS3',
            loss_sample_re[['Net', 'Ceded', 'p_total']], bs=1, log2=8)

Table 9.9 shows the standard summary statistics. It should be compared to Table 5.1. Table 9.10 shows the allocated pricing to the reinsurance and net across the five standard distortions, compare with the last row of Table 5.1 and discussion in Section 5.2.

Table 9.9: Summary statistics created by the reinsurance Portfolio object.
unit X E[X] Est E[X] CV(X) Est CV(X) Skew(X) Est Skew(X)
Net Freq 1 0
Sev 43.1 43.100 0.317 0.317 0.369 0.369
Agg 43.1 43.100 0.317 0.317 0.369 0.369
Ceded Freq 1 0
Sev 3.5 3.500 3 3.000 2.667 2.667
Agg 3.5 3.500 3 3.000 2.667 2.667
total Freq 2 0
Sev 23.3 23.300 0.998 0.340
Agg 46.6 46.600 0.370 0.370 0.788 0.788
Table 9.10: Pricing by distortion for ceded, net, and total (gross).
(a) CCoC
statistic Ceded Net Total
L 3.500 43.100 46.600
a 30.013 69.987 100.0
Q 22.405 24.030 46.435
P 7.609 45.957 53.565
M 4.109 2.857 6.965
COC 0.183 0.119 0.150
(b) PH
statistic Ceded Net Total
L 3.500 43.100 46.600
a 23.383 76.617 100.000
Q 16.721 29.714 46.435
P 6.662 46.903 53.565
M 3.162 3.803 6.965
COC 0.189 0.128 0.150
(c) Wang
statistic Ceded Net Total
L 3.500 43.100 46.600
a 20.237 79.763 100.000
Q 14.150 32.284 46.435
P 6.087 47.478 53.565
M 2.587 4.378 6.965
COC 0.183 0.136 0.150
(d) Dual
statistic Ceded Net Total
L 3.500 43.100 46.600
a 18.539 81.461 100.0
Q 13.125 33.310 46.435
P 5.415 48.151 53.565
M 1.915 5.051 6.965
COC 0.146 0.152 0.150
(e) TVaR
statistic Ceded Net Total
L 3.500 43.100 46.600
a 17.679 82.321 100.000
Q 12.876 33.559 46.435
P 4.803 48.762 53.565
M 1.303 5.662 6.965
COC 0.101 0.169 0.150

9.5 A more realistic example

In this section, we create a series of exhibits analogous to those in Section 9.4 for an example with more realistic assumptions. It is included to show how aggregate can be used to solve real-world problems, hopefully motivating you to explore it further. The analysis steps are:

  1. Create realistic by-unit frequency and severity distributions using Fast Fourier Transforms with independent units.
  2. Sample the unit distributions with correlation induced by Iman-Conover, Section 8.2.
  3. Build a Portfolio from the correlated sample.
  4. Calibrate distortions and compute unit pricing for each distortion.
  5. Apply by-unit, per-occurrence reinsurance and examine pricing impact.

The aggregate DecL programming language makes it easy to specify frequency-severity compound distributions. In the code chunk below, the four lines inside the build statement are the DecL program. The triple quotes are Python shorthand for entering a multiline string. The first program line, beginning agg Auto, creates a distribution with an expected loss of 5000 (think: losses in 000s), severity from the 5000 xs 0 layer of a lognormal variable with a mean of 50 and a CV of 2, and gamma mixed-Poisson frequency with a mixing parameter (CV) of 0.2. The other two lines are analogous. The build statement runs the DecL program and creates a Portfolio object with relevant compound distributions by unit. It also sums the unit distributions as though they were independent—we will introduce some correlation later.

Code 
from aggregate import build
port = build("""
port Units
    agg Auto 5000 loss  5000 xs 0 sev lognorm 50 cv  2 mixed gamma 0.2
    agg GL   3000 loss  2500 xs 0 sev lognorm 75 cv  3 mixed gamma 0.3
    agg WC   7000 loss 25000 xs 0 sev lognorm  5 cv 10 mixed gamma 0.25
""")

Table 9.11 shows the by-unit frequency and severity statistics and the total statistics assuming the units are independent. The deviation between the Fast Fourier Transform-generated compound distributions and the requested specifications are negligible. Figure 9.2 plots the unit and total densities on a nominal and log scale. The effect of WC driving the tail, via thicker severity and higher occurrence limit, is clear on the log plot.

Table 9.11: Unit frequency, severity and compound assumptions, and portfolio total, showing requested and model achieved and key statistics.
unit X E[X] Est E[X] CV(X) Est CV(X) Skew(X) Est Skew(X)
Auto Freq 100.0 0 0.224 0 0.402 0
Sev 49.982 49.982 1.973 1.973 10.678 10.678
Agg 5,000 5000.0 0.298 0.298 0.699 0.699
GL Freq 41.008 0 0.338 0 0.604 0
Sev 73.156 73.156 2.377 2.378 7.385 7.385
Agg 3,000 3000.0 0.502 0.502 1.024 1.024
WC Freq 1401.1 0 0.251 0 0.500 0
Sev 4.996 4.945 9.103 9.198 147.9 147.9
Agg 7,000 7000.0 0.350 0.350 1.776 1.775
total Freq 1542.1 0 0.229 0 0.496 0
Sev 9.727 9.681 6.123 0 68.886 0
Agg 15,000 15000.0 0.216 0.216 0.939 0.938
Figure 9.2: By-unit loss densities (left) and logdensities (right).

Next, we sample from the unit distributions and shuffle using Iman-Conover to achieve the desired correlation shown in Table 9.12. This matrix comes from a separate analysis. The revised statistics (note higher total CV), quantiles, and achieved correlation are shown in Table 9.13 and Table 9.14.

Table 9.12: Desired correlation matrix, as input to Iman-Conover.
index Auto GL WC
Auto 1 0.5 0.4
GL 0.5 1 0.1
WC 0.4 0.1 1
Table 9.13: Key statistics from sample with Iman-Conover induced correlation.
index Auto GL WC Total
count 1,000 1,000 1,000 1,000
mean 5010.4 2991.9 6996.2 14998.6
std 1469.6 1488.6 2603.5 4065.6
min 1,722 469 3,293 7,156
25% 3977.8 1933.8 5267.2 12098.8
50% 4897.5 2742.5 6,387 14412.5
75% 5896.8 3781.2 8043.2 17272.5
max 11,136 11,249 31,008 42,501
CV 0.293 0.498 0.372 0.271
Table 9.14: Achived between-unit correlation.
index Auto GL WC Total
Auto 1 0.491 0.349 0.765
GL 0.483 1 0.072 0.590
WC 0.391 0.093 1 0.793
total 0.788 0.591 0.758 1

Table 9.15 shows the output of using the correlated sample to build a Portfolio object. This uses the samples directly, proxying the aggregate loss as a compound with degenerate frequency distribution identically equal to one and severity equal to the desired distribution. Hence the frequency rows show expectation one with zero CV.

Table 9.15: Portfolio statistics reflecting the correlation achieved by the Iman-Conover sample.
unit X E[X] Est E[X] CV(X) Est CV(X) Skew(X) Est Skew(X)
Auto Freq 1 0
Sev 5010.4 5010.4 0.293 0.293 0.636 0.636
Agg 5010.4 5010.4 0.293 0.293 0.636 0.636
GL Freq 1 0
Sev 2991.9 2991.9 0.497 0.497 1.222 1.222
Agg 2991.9 2991.9 0.497 0.497 1.222 1.222
WC Freq 1 0
Sev 6996.2 6996.2 0.372 0.372 2.202 2.202
Agg 6996.2 6996.2 0.372 0.372 2.202 2.202
total Freq 3 0
Sev 4999.5 4999.5 0.505 1.487
Agg 14998.6 14998.6 0.223 0.223 1.206 1.206

Table 9.16 shows the parameters for distortions calibrated to achieve an overall 15% return at a 99.5% capital standard. Figure 9.3 plots the distortions and probability adjustment functions, compare this with Figure 9.1. These distortions are slightly more risk averse (higher parameters except PH, driving higher indicated prices).

Table 9.16: Distortion parameters for to target total return at the regulatory capital standard.
method P param error
ccoc 16777.2 0.150 0
ph 16777.2 0.657 0
wang 16777.2 0.435 0.000
dual 16777.2 1.759 -0.000
tvar 16777.2 0.301 0.000
Figure 9.3: Distortion functions and distortion probability adjustment functions.

Table 9.17 compares the indicated pricing by unit and distortion. It is computed using exactly the same method described for the simple discrete example. The calculations are easy enough that they can be performed in a spreadsheet, but that is not recommended unless you are very patient. Table 9.18 shows just the implied loss ratios. The total loss ratios are all the same because the distortions are all calibrated to the same overall total premium. Since WC has the heaviest tail, it gets the lowest target loss ratio under CCoC, which is the most tail-risk averse. In fact, CCoC is so tail-risk averse that GL (which offers some diversification because of the correlation structure) gets a target over 100%. This is typical CCoC behavior—cat is really expensive; everything else is cheaper. The reason is clear when you remember CCoC pricing is just a weighted average of the mean and the max. The other distortions are more reasonable. TVaR and dual are quite similar and focus more on volatility risk. Since all three lines are similarly volatile, it is no surprise their target loss ratios are also similar under this view.

This analysis—and indeed the whole monograph—has ignored several important elements: expenses, taxes, and the time value of money. Expenses are relatively easy to incorporate: acquisition expenses are typically amortized over the policy term, administration expenses span the policy term and claim payout period, and claim expenses can be bundled with losses. Taxes, however, are a Byzantine labyrinth requiring careful modeling at the level of legal entity and country. Their treatment is up to each insurer, and it’s hard to offer even vague general guidance. Finally, time value of money brings real conceptual and computational difficulty (as opposed to the man-made complexity of taxes). It is the quantum gravity of insurance pricing. For some early thoughts, see the forthcoming monograph Pricing Multi-Period Insurance Risk by your intrepid second author.

Table 9.17: Pricing by distortion.
(a) CCoC
statistic Auto GL WC Total
L 5007.2 2990.0 6984.5 14981.7
a 5780.1 3351.0 19615.9 28747.0
Q 783.1 380.0 10806.7 11969.8
P 4997.0 2971.0 8809.1 16777.2
M -10.134 -19.019 1824.6 1795.5
COC -0.013 -0.050 0.169 0.150
(b) PH
statistic Auto GL WC Total
L 5007.2 2990.0 6984.5 14981.7
a 8632.6 5552.1 14562.3 28747.0
Q 3189.4 2214.4 6566.0 11969.8
P 5443.2 3337.7 7996.3 16777.2
M 436.0 347.7 1011.8 1795.5
COC 0.137 0.157 0.154 0.150
(c) Wang
statistic Auto GL WC Total
L 5007.2 2990.0 6984.5 14981.7
a 8952.3 5686.4 14108.3 28747.0
Q 3451.3 2308.1 6210.4 11969.8
P 5501.0 3378.3 7897.9 16777.2
M 493.8 388.3 913.4 1795.5
COC 0.143 0.168 0.147 0.150
(d) Dual
statistic Auto GL WC Total
L 5007.2 2990.0 6984.5 14981.7
a 9081.1 5759.9 13906.0 28747.0
Q 3536.2 2354.4 6079.2 11969.8
P 5544.9 3405.5 7826.8 16777.2
M 537.7 415.5 842.3 1795.5
COC 0.152 0.176 0.139 0.150
(e) TVaR
statistic Auto GL WC Total
L 5007.2 2990.0 6984.5 14981.7
a 9129.4 5782.5 13835.1 28747.0
Q 3546.4 2355.2 6068.2 11969.8
P 5583.0 3427.3 7766.9 16777.2
M 575.8 437.3 782.4 1795.5
COC 0.162 0.186 0.129 0.150
Table 9.18: Implied loss ratio by unit by pricing by distortion (extract from previous tables).
Method Auto GL WC Total
Dist ccoc 100.2% 100.6% 79.3% 0.893
Dist ph 92.0% 89.6% 87.3% 0.893
Dist wang 91.0% 88.5% 88.4% 0.893
Dist dual 90.3% 87.8% 89.2% 0.893
Dist tvar 89.7% 87.2% 89.9% 0.893