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Archives of Business Review – Vol. 8, No.12
Publication Date: December 25, 2020
DOI: 10.14738/abr.812.9460.
Berketis, N. G. (2020). A Chain-Ladder Analysis of P&I Claims. Archives of Business Research, 8(12). 50-62.
A Chain-Ladder Analysis of P&I Claims
Nicholas G. Berketis
Lecturer of Marine Insurance,
Frederick University, Cyprus
INTRODUCTION
According to a recent article on Lloyd’s List:
Ratings agency predicts rate hikes at most clubs, as mutuals feel double whammy squeeze from
string of big casualties and coronavirus pandemic. International Group pool claims have hit an all- time high for the halfway point in a claims year, and may already have reached threequarters of
the previous full-year outcome, according to senior club sources.
One well-placed industry source cited an unofficial estimate of $300m for the year to date,
compared with $400m for all of 2019-20. The development comes following publication of a new
note from ratings agency S&P Global, predicting that almost every IG affiliate will seek higher
pricing at the next renewal round, as the sector feels the combined pain of coronavirus and a string
of major casualties. The insurance industry, unlike other industries, does not sell products as such
but promises. An insurance policy is a promise by the insurer to the policyholder to pay for future
claims for an upfront received premium.
As a result, Insurers don’t know the upfront cost for their service, but rely on historical data
analysis and judgement to predict a sustainable price for their offering. In General Insurance (or
Non-Life Insurance, e.g. motor, property and casualty insurance) most Policies run for a period of
12 months. However, the claims payment process can take years or even decades. Therefore, often
not even the delivery date of their product is known to Insurers.
In particular, losses arising from casualty insurance can take a long time to settle and even when
the claims are acknowledged it may take time to establish the extent of the claims’ settlement cost.
Claims can take years to materialize. A complex and costly example are the claims from asbestos
liabilities, particularly those in connection with mesothelioma and lung damage arising from
prolonged exposure to asbestos. A research report by a working party of the Institute and Faculty
of Actuaries estimated that the un-discounted cost of UK mesothelioma-related claims to the UK
Insurance Market for the period 2009 to 2050 could be around £10bn. The cost for asbestos related
claims in the US for the worldwide insurance industry was estimated to be around $120bn in 2002.
Thus, it should come as no surprise that the biggest item on the liabilities side of an Insurer’s
balance sheet is often the provision or reserves for future claims payments. Those reserves can be
broken down in case reserves (or outstanding claims), which are losses already reported to the
insurance company and losses that are incurred but not reported (IBNR) yet.
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The analysis is based on R (Version 4.0.3 – 10th October, 2020), an integrated language and
environment for statistical computing and graphics. R provides a wide variety of statistical and
graphical techniques.
The estimated cost of notified pool claims (in USD 000,000) is as follows:
YEAR No. OF
CLAIMS 12M 24M 36M 48M 60M 72M 84M 96M 108M 120M
2010/11 22 1791 2411 2669 2525 2506 1590 2599 2540 2663 2662
2011/12 22 2310 2779 2808 2896 2893 2887 2844 2812 2802 NA
2012/13 14 3688 4539 4670 4651 4463 4186 4036 3927 NA NA
2013/14 22 2798 3270 3640 3649 4116 4086 4231 NA NA NA
2014/15 20 1796 1936 2045 2158 2213 2129 NA NA NA NA
2015/16 17 1984 2766 2840 2827 2914 NA NA NA NA NA
2016/17 14 840 1259 1450 1365 NA NA NA NA NA NA
2017/18 20 2272 2696 2897 NA NA NA NA NA NA NA
2018/19 28 3061 4558 NA NA NA NA NA NA NA NA
2019/20 24 2592 NA NA NA NA NA NA NA NA NA
This triangle shows the known values of loss from each origin year and of annual evaluations
thereafter. For example, the known values of loss originating from the 2015/16 exposure period
are 1984, 2766, and 2840 as of year ends 2015, 2014, and 2013, respectively. The latest diagonal –
i.e., the vector 2662, 2802, . . . 2592 from the upper right to the lower left – shows the most recent
evaluation available.
The column headings – 1, 2,. . . , 10 – hold the ages (in years) of the observations in the column
relative to the beginning of the exposure period. For example, for the 2016/17 origin year, the age
of the 1450 value, evaluated as of 20/02/2019, is three years.
The objective of a reserving exercise is to forecast the future claims development in the bottom
right corner of the triangle and potential further developments beyond development age 10.
Eventually all claims for a given origin period will be settled, but it is not always obvious to judge
how many years or even decades it will take. We speak of long and short tail business depending
on the time it takes to pay all claims. In order proceed with our analysis, we first plotted the data
to get an overview. Figure 1 that follows shows the claims development chart for the past 10 years.
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Berketis, N. G. (2020). A Chain-Ladder Analysis of P&I Claims. Archives of Business Research, 8(12). 50-62.
Figure 1
CHAIN-LADDER METHODS
The classical chain-ladder is a deterministic algorithm to forecast claims based on historical data.
It assumes that the proportional developments of claims from one development period to the next
are the same for all origin years.
Basic idea
Most commonly as a first step, the age-to-age link ratios are calculated as the volume weighted
average development ratios of a cumulative loss development triangle from one development
period to the next !!", i, k = 1, . . . , n.
"" = ∑∑&!(&!’(%’#%#$$!,#!,$#%
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[1] 1.2762415 1.0629387 0.9974655 1.0213301 0.9189056 1.0753785 0.9789007 1.0211136
0.9996245 Often it is not suitable to assume that the oldest origin year is fully developed. A typical
approach is to extrapolate the development ratios, e.g. assuming a log-linear model.
[1] 1.059312
Figure 2 below shows the Log-linear extrapolation of age-to-age factors.
Figure 2
The age-to-age factors allow us to plot the expected claims development patterns. This is shown
on Figure 3 that follows:
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Berketis, N. G. (2020). A Chain-Ladder Analysis of P&I Claims. Archives of Business Research, 8(12). 50-62.
Figure 3: Expected claims development pattern
Dev. Period
The link ratios are then applied to the latest known cumulative claims amount to forecast the next
development period. The squaring of the triangle is calculated below, where an ultimate column is
appended to the right to accommodate the expected development beyond the oldest age (10) of
the triangle due to the tail factor (1.059312) being greater than unity.
X12M X24M X36M X48M X60M X72M X84M X96M X108M X120M Ult
1 1791 2411 2669 2525 2506 1590 2599 2540 2663 2662 2820
2 2310 2779 2808 2896 2893 2887 2844 2812 2802 2801 2967
3 3688 4539 4670 4651 4463 4186 4036 3927 4010 4008 4246
4 2798 3270 3640 3649 4116 4086 4231 4142 4229 4228 4478
5 1796 1936 2045 2158 2213 2129 2289 2241 2288 2288 2423
6 1984 2766 2840 2827 2914 2678 2880 2819 2878 2877 3048
7 840 1259 1450 1365 1394 1281 1378 1349 1377 1377 1458
8 2272 2696 2897 2890 2951 2712 2916 2855 2915 2914 3087
9 3061 4558 4845 4833 4936 4535 4877 4774 4875 4873 5162
10 2592 3308 3516 3507 3582 3292 3540 3465 3538 3537 3747
The total estimated outstanding loss under this method is about 33,000. In particular, it was
calculated as 33,436.
This approach is also called Loss Development Factor (LDF) method. More generally, the factors
used to square the triangle need not always be drawn from the dollar weighted averages of the
0 5 10 15 20 25 30
70 75 80 85 90 95 100
Development % of ultimate loss
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triangle. Other sources of factors from which the actuary may select link ratios include simple
averages from the triangle, averages weighted toward more recent observations or adjusted for
outliers, and benchmark patterns based on related, more credible loss experience. Also, since the
ultimate value of claims is simply the product of the most current diagonal and the cumulative
product of the link ratios, the completion of interior of the triangle is usually not displayed in favor
of that multiplicative calculation.
MACK CHAIN-LADDER
Thomas Mack published in 1993 a method which estimates the standard errors of the chainladder
forecast without assuming a distribution under three conditions.
Following the notation of Mack let !!" denote the cumulative loss amounts of origin period (e.g.
accident year) i = 1, . . . ,m, with losses known for development period (e.g. development year) k ≤
n + 1 − i.
In order to forecast the amounts !!" for k > n+1−i the Mack chain-ladder-model assumes:
CL1: E[#!"I!!%, !!&,..., !!"] = "" with #!"= $!
$,#!#$% (2)
CL2: Var($!
$,#$% I!!%, !!&,..., !!") = (!’
##)$!*# (3)
CL3: {!!%,..., !!)}, {!*%,..., !*)}, are independent for origin period i1 j (4)
with $!" ∈ [0; 1], α ∈ {0, 1, 2}. If these assumptions hold, the Mack chain-ladder model gives an
unbiased estimator for IBNR (Incurred But Not Reported) claims.
The Mack chain-ladder model can be regarded as a weighted linear regression through the origin
for each development period: lm(y ~ x + 0, weights=w/x^(2- alpha)), where y is the vector of claims
at development period k + 1 and x is the vector of claims at development period k.
The Mack method is implemented in the ChainLadder package via the function MackChainLadder.
We therefore applied the MackChainLadder function to our triangle:
Latest Dev. To. Date Ultimate IBNR Mack. S.E. CV (IBNR)
1 2,662 1.000 2,662 0.00 0.00 NaN
2 2,802 1.000 2,801 -1.05 34.9 -33.16
3 3,927 0.980 4,008 81.41 163.3 2.01
4 4,231 1.001 4,228 -3.41 173.6 -50.88
5 2,129 0.931 2,288 158.63 701.2 4.42
6 2,914 1.013 2,877 -36.79 942.9 -25.63
7 1,365 0.992 1,377 11.51 636.7 55.30
8 2,897 0.994 2,914 17.03 977.6 57.40
9 4,558 0.935 4,873 315.36 1,346.7 4.27
10 2,592 0.733 3,537 944.89 1,172.0 1.24
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Berketis, N. G. (2020). A Chain-Ladder Analysis of P&I Claims. Archives of Business Research, 8(12). 50-62.
Totals
Latest: 30,077.00
Dev: 0.95
Ultimate: 31,564.58
IBNR: 1,487.58
Mack. S.E.: 3,282.85
CV(IBNR): 2.21
Executing Mack Chain Ladder will print the following columns of information per accident year
(origin period):
1. Latest: the claim amount for the last development period
2. Dev.To.Date: the development to date or the ratio of the latest over the predicted ultimate
3. Ultimate: predicted ultimate claim
4. IBNR: the predicted IBNR reserve
5. Mack.S.E.: the standard error, or the standard deviation of the bounds for the predicted
ultimate and IBNR since the estimate is unbiased (shown in Mack's 1999 paper). In other
words, since the S.E given is equal to one standard deviation, a confidence interval for the
true ultimate value can be found using the standard error and the predicted ultimate.
6. CV(IBNR): coefficient of variation, or the ratio of the standard error over the predicted IBNR
The bottom output gives a total or sum of the latest, ultimates, IBNRs. It also gives the standard
error of the total ultimate (this is not the total of the standard errors). The development to date
factor is the ratio of the total latest against the total ultimate, and the CV(IBNR) is the percentage
of the total standard error in the total IBNR.
If the CV (absolute value) is greater than 25%, then another model or a log linear regression should
be used.
We can access the loss development factors and the full triangle via:
[1] 1.2762415 1.0629387 0.9974655 1.0213301 0.9189056 1.0753785 0.9789007 1.0211136
0.9996245 1.0000000
No. X12M X24M X36M X48M X60M X72M X84M X96M X108M X120M
1 1791 2411.0 2669.000 2525.000 2506.000 1590.000 2599.000 2540.000 2663.000 2662.000
2 2310 2779.0 2808.000 2896.000 2893.000 2887.000 2844.000 2812.000 2802.000 2800.948
3 3688 4539.0 4670.000 4651.000 4463.000 4186.000 4036.000 3927.000 4009.913 4008.407
4 2798 3270.0 3640.000 3649.000 4116.000 4086.000 4231.000 4141.729 4229.176 4227.588
5 1796 1936.0 2045.000 2158.000 2213.000 2129.000 2289.481 2241.174 2288.494 2287.634
6 1984 2766.0 2840.000 2827.000 2914.000 2677.691 2879.531 2818.775 2878.290 2877.209
7 840 1259.0 1450.000 1365.000 1394.116 1281.061 1377.625 1348.558 1377.031 1376.514
8 2272 2696.0 2897.000 2889.657 2951.294 2711.960 2916.384 2854.850 2915.126 2914.032
9 3061 4558.0 4844.874 4832.595 4935.674 4535.419 4877.292 4774.384 4875.189 4873.358
10 2592 3308.0 3516.220 3507.308 3582.119 3291.629 3539.747 3465.061 3538.221 3536.892
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To plot that Mack’s assumption are valid review the residual plots, we see no trends in either of
them. Please refer to the Figure 4 that follows:
Figure 4
BOOTSTRAP CHAIN-LADDER
The BootChainLadder function uses a two-stage bootstrapping / simulation approach following
the paper by England and Verral. In the first stage an ordinary chain-ladder method is applied to
the cumulative claims’ triangle. From this we calculate the scaled Pearson residuals which we
bootstrap R times to forecast future incremental claims payments via the standard chain-ladder
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Berketis, N. G. (2020). A Chain-Ladder Analysis of P&I Claims. Archives of Business Research, 8(12). 50-62.
method. In the second stage we simulate the process error with the bootstrap value as the mean
and using the process distribution assumed. The set of reserves obtained in this way forms the
predictive distribution, from which summary statistics such as mean, prediction error or quantiles
can be derived.
BootChainLadder(Triangle = GGG21, R = 999, process.distr = "gamma")
Latest Mean Ultimate Mean IBNR IBNR S.E. IBNR 75% IBNR 95%
1 2,662 2,662 0.00 0 0.00e+00 0.0
2 2,802 2,795 -6.75 215 1.15e-24 17.1
3 3,927 3,996 69.44 706 9.63e+01 880.4
4 4,231 4,221 -10.09 749 9.36e+01 1,074.1
5 2,129 2,272 142.51 627 2.49e+02 1,144.6
6 2,914 2,872 -42.39 801 2.09e+02 1,178.5
7 1,365 1,382 16.56 537 1.31e+02 923.0
8 2,897 2,914 17.47 896 2.97e+02 1,494.2
9 4,558 4,847 289.19 1,419 8.77e+02 2,341.6
10 2,592 3,469 877.35 1,290 1.50e+03 3,074.8
Totals
Latest: 30,077
Mean Ultimate: 31,430
Mean IBNR: 1,353
IBNR S.E. 4,390
Total IBNR 75%: 3,422
Total IBNR 95%: 6,928
The BootChainLadder is a model that provides a predicted distribution for the IBNR values for a
claims’ triangle. However, this model predicts IBNR values by a different method than the previous
model. First, the development factors are calculated and then they are used in a backwards
recursion to predict values for the past loss triangle. Then the predicted values and the actual
values are used to calculate Pearson residuals.
Using the adjusted residuals and the predicted losses from before, the model solves for the actual
losses in the Pearson formula and forms a new loss triangle. The steps for predicting past losses
and residuals are then repeated for this new triangle. After that, the model uses chain ladder ratios
to predict the future losses then calculates the ultimate and IBNR values like in the previous Mack
model. This cycle is performed R times, depending on the argument values in the model (default is
999 times). The IBNR for each origin period is calculated from each triangle (the default 999) and
used to form a predictive distribution, from which summary statistics are obtained such as mean,
prediction error, and quantiles.
The output has some of the same values as the Munich and Mack models did. The Mean and SD
IBNR is the average and the standard deviation of the predictive distribution of the IBNRs for each
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origin year. The output also gives the 75% and 95% quantiles of the predictive distribution of
IBNRs, in other words 95% or 75% of the predicted IBNRs lie at or below the given values.
The above also appear on following Figure 5:
Figure 5:
The above Figure 5 shows four graphs, starting with a histogram of the total simulated IBNRs over
all origin periods, including a rug plot; a plot of the empirical cumulative distribution of the total
IBNRs over all origin periods; a box-whisker plot of simulated ultimate claims costs against origin
periods; and a box-whisker plot of simulated incremental claims cost for the latest available
calendar period against actual incremental claims of the same period. In the last plot the simulated
data should follow the same trend as the actual data, otherwise the original data might have some
intrinsic trends which are not reflected in the model.
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Berketis, N. G. (2020). A Chain-Ladder Analysis of P&I Claims. Archives of Business Research, 8(12). 50-62.
Quantiles of the bootstrap IBNR can be calculated via the quantile function:
$ByOrigin
IBNR 75% IBNR 95% IBNR 99% IBNR 99.5%
1 0.000000e+00 0.00000 0.0000 0.0000
2 1.151800e-24 17.07965 338.7176 604.7815
3 9.626126e+01 880.43892 1,961.0599 2,229.0305
4 9.356708e+01 1,074.14256 2,168.5127 2,615.0994
5 2.490389e+02 1,144.58741 2,029.4400 2,302.8436
6 2.089716e+02 1,178.45709 2,198.1897 2,602.8466
7 1.313045e+02 922.99808 1,717.0711 2,142.4096
8 2.974393e+02 1,494.23171 2,475.7708 2,770.7317
9 8.774595e+02 2,341.60747 3,938.6746 4,353.9703
10 1.503892e+03 3,074.76561 4,433.8774 4,972.3641
Totals
IBNR 75%: 3,421.616
IBNR 95%: 6,928.246
IBNR 99%: 11,252.789
IBNR 99.5%: 11,733.925
The distribution of the IBNR appears to follow a log-normal distribution, so let’s fit it:
meanlog sdlog
7.56248311 1.15389021
(0.04367539) (0.03088316)
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Figure 6
Figure 7
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Berketis, N. G. (2020). A Chain-Ladder Analysis of P&I Claims. Archives of Business Research, 8(12). 50-62.
CONCLUSIONS
1. The Loss Development Factor (LDF) is above unity, i.e. 1.059312, which shows an increasingly
positive trend for I.B.N.R.’s. The LDF compared to the previous period has shown a remarkable
increase as it was then estimated at 1.0127889;
2. The claim amount for the last development period is estimated by both Mack and Bootstrap
chain ladder methods at 30,077;
3. The predicted ultimate claim is estimated 33,436 under the chain ladder method, Mack chain
ladder estimated it at 31,564.58, while Bootstrap chain ladder method showed 31,430;
4. The predicted I.B.N.R. reserve was estimated at 1,487.58 under the Mack chain ladder method
and 1,353 under Bootstrap chain ladder method;
5. Since the coefficient of variation of I.B.N.R.’s was estimated in absolute value well above 25%,
i.e. 221%, we followed the Bootstrap chain ladder method, which also justified the increasingly
positive trend of I.B.N.R.’s.
6. Hence, the results do follow the trend for a general increase in the forthcoming 2021 renewal.
After all and to use the comments made by the Executive Director of Gallagher Marine Division:
“If we put COVID-19 aside for a moment, we would have expected to see the outcome of 2019-
20 result in a further upward adjustment in rates as part of an exercise that had begun at the
2020-21 renewal.”
References
1. Carrato, Alessandro, Concina, Fabio, Gesmann, Markus, Murphy, Dan, Wuthrich, Mario & Zhang, Wayne, (2018),
“Claims Reserving with R: ChainLadder-0.2.9 Package Vignette”, 6, December.
2. England, P. D., & Verrall, Richard J., (2002), “Stochastic Claims Reserving in General Insurance”, Presented to the
Institute of Actuaries, 28 January.
3. Gallagher, (2020), “Marine P&I Market Overview 2020”, London, U.K.
4. IUMI, (2020), “Global Marine Insurance Report”, Stockholm.
5. Mack, Thomas, (1993), “Distribution-free Calculation of the Standard Error of Chain Ladder Reserve Estimates”,
ASTIN Bulletin, Vol. 23(2):213– 225.
6. Mack, Thomas, (1999), “The Standard Error of Chain Ladder Reserve Estimates:
7. Recursive Calculation and Inclusion of a Tail Factor”. ASTIN Bulletin, Vol. 29(2): 361366.
8. Merz, Michael and Mario V. W ̈uthrich, (2014), “Claims Run-off Uncertainty: The Full Picture”, SSRN Manuscript,
2524352.
9. Osler, David, (2020), “P&I Pool Claims Hit Record $300m at Halfway Mark”, Lloyd’s List, 22 October.
10. http://opensourcesoftware.casact.org/chain-ladder
11. Weindorfer, Bjorn, (2012), “A Practical Guide to the Use of the Chain-Ladder Method for Determining Technical
Provisions for Outstanding Reporting Claims in Non-Life Insurance”, Working papers series by University of
Applied science BFI Vienna, October, No. 77/2012.