Comparison of Statistical Methods for Claims Reserve Estimation using
R Language
ENDRI RAÇO, KLEIDA HAXHI, ETLEVA LLAGAMI, ORIANA ZAÇAJ
Department of Mathematical Engineering,
Polytechnic University of Tirana,
ALBANIA
Abstract: - Stochastic methods of reserves estimation serve to assess the technical provisions of outstanding
claims and forecast cash payments of claims in the coming years. The chain ladder model developed by Mack
is the more prevalent model. The main deficiency in the chain-ladder model is that the chain-ladder model
depends on the last observation on the diagonal. If this last observation is an outlier, this outlier will be
projected to the ultimate claim. One of the possibilities to smooth outliers on the last observed diagonal is to
robustify such observations, making use of the maximum likelihood estimation along with the common Loss
Development Factor (LDF) curve fitting and Cape Cod (CC) techniques. This paper aims to highlight the
advantages of using these methods for the best estimate of claims reserves in the Domestic Motor Third Party
Liability portfolio. The maximum–likelihood parameter estimation and Chi-square test, are used to specify the
probability distribution that best fits the data. Using the Standard Chain Ladder method, LDF, and CC method
the claims reserve is calculated based on the run-off triangles of paid claims or the run-off triangles of the
incurred claims. Many times, the projections based on the paid claims are different than the projections based
on the incurred claims. The solution for this problem is the Munich Chain Ladder method.
Key-Words: - Chain ladder method, Maximum likelihood estimation, Loss Development Factor, Cape Cod,
Munich Chain ladder, Run off triangles
Received: September 23, 2021. Revised: May 19, 2022. Accepted: June 22, 2022. Published: July 14, 2022.
1 Introduction
The technical claims reserves, as all technical
reserves directly affect the profit loss statement, as
well as the technical balance of the insurance
company, it is required as a fair evaluation of them.
The correctness and dependability of data have a
significant impact on the outcomes of stochastic
techniques applications [6]. The reliability of the
data has a direct impact on the assessment of the
claims reserve. The absence of this reliability can
alter the results, either underestimating or
overestimating the final estimation. The actuary
knowing the progress and history of claims in a
portfolio, the market where are developed claims
payments over the years, the values of outstanding
claims, and claims in process court, decides which
values estimates are more appropriate to establish
technical reserves [1]. In addition, the insurance
company must hold sufficient assets to cover
technical reserves. The value of assets covering
technical provisions must at all times be not less
than the gross amount of technical reserves. For this
purpose, the estimation of claims reserves is a very
important issue. Since the Domestic Motor Third
Party Liability (DMTPL) is the most important
portfolio of general insurance in Albania, we used
the data of this product to apply Loss Development
Factor (LDF) curve fitting and Cape Cod (CC)
techniques. R programing languages facilitate the
comparison of methods.
2 Data and Fitting Distributions
We take into consideration DMTPL claims paid and
incurred from 2015 to 2021. The claims amounts are
in Albanian currency, Lek.
2.1 Paid Data Claims Distribution
We studied 9,262 Domestic Motor Third Party
Liability claims paid.
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Fig 1: The empirical density of paid claims 2015-2021
Based on the Chi - square tests and information
criteria the best distribution that fits the paid claims
data is the Weibull distribution [8].
Table 1. Chi-square test
Table 2. Information Criteria
Fig. 2: The empirical and theoretical cumulative
density function of paid claims data
2.2 Incurred Data Claims Distribution
We analyzed 8,413 incurred claims from 2015 to
2021.
Fig. 3: The empirical density of incurred claims
2015-2021
For the incurred claims too, if we go through the
distribution parameters for each theoretical
distribution using the fitteR library, it results that the
Weibull distribution fits better [8].
Table 3. Chi-square test
Table 4. Information Criteria
Fig. 4: The empirical and theoretical cumulative density
function of incurred claims data
3 Methods
3.1 Clark LDF and Cape Cod Methods
According to LDF e CC techniques to create a
proper model of claims reserving we have
considered the basic objectives as follows: [2]
Goodness of fit statistics Weibull Gamma Lognormal
Kolmogorov-Smirnov 0.19160 0.20402 0.27387
Cramer-von Mises 0.15910 0.19672 0.66494
Anderson-Darling 1.26022 1.33719 3.54576
Goodness of fit criteria Weibull Gamma Lognormal
Akaike Criterion AIC -21.5501 -28.2695 10.6406
Bayesian Criterion BIC -18.8857 -25.6051 13.3050
Goodness of fit statistics Weibull Gamma Lognormal
Kolmogorov-Smirnov 0.12030 0.14716 0.24891
Cramer-von Mises 0.04982 0.06765 0.35897
Anderson-Darling 0.55078 0.58125 2.35675
Goodness of fit criteria Weibull Gamma Lognormal
Akaike Criterion AIC -81.4963 -83.9391 -61.0574
Bayesian Criterion BIC -79.0585 -81.5014 -58.6195
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• Describing the loss emergence in easier
mathematical terms as a track for the selection of
the amounts related to carried reserves
• Providing estimating strategies for the predicted
reserve's range of probable outcomes.
As a main aspect of the LDF technique, the ultimate
loss amount in each accident year is independent of
claims in prior years. Instead, the Cape Cod
technique assumes that there is a connection
between the amounts of final loss projected in each
of the years in the prior documented period and that
an exposure base accurately detects that
relationship. Earned level premium is frequently
used as the exposure base. Although both of the
above methods can be used for the estimation of the
reserves, Cape Cod will be preferable. Because we
are dealing with data aggregated into annual blocks
as a development triangle, there will be relatively
few data points in the model,one data point for each
"cell" in the triangle.When the LDF method is used,
there is a real issue with over-parameterization. [2]
Fig. 5: The incremental triangle of paid claims
The statistical claims reserving model has two
primary elements: the emerging of the expected
value of the losses in specific periods and the
distribution of actual emerging regarding the
expected value. The projected amount of loss will
emerge based on the estimate of the ultimate loss by
year and the estimate of the sample of loss
emergence, according to this model. Based on an
estimate of the ultimate claim by year and an
assessment of the pattern of loss emergence, the
model calculates the predicted amounts of claims to
emerge. [2]
G(x) is the cumulative percentage of claims paid in
time x. The time index "x" represents the time from
the "average" accident date to the paid date.
(1)
The model will include a Weibull curve,
parameterized with a scale θ and a shape ω.
(2)
The next step in estimating the amount of loss
emergence by period is to apply the emergence
pattern G(x), to an estimate of the ultimate claim by
accident year. [2]
The final step is the estimation of the variances. It
comes because of process variance (random
amount) and parameter variance, which is the
uncertainty in the estimation. The assumption is that
the claim in each period has the same variance/mean
ratio and the incremental claims are independent
and identically distributed [2]
where p is the number of parameters
cAY,t is the actual incremental loss emergence, and
μAY,t is the expected incremental loss emergence.
This corresponds to the chi-square error term. [2].
Usually, the CC method is preferred since the LDF
method requires an estimation of parameters, one
for each accident year (AY) loss, as well as ω and θ.
Due to the additional information given by the
exposure base and the fewer parameters, the Cape
Cod method has a smaller parameter variance. The
process variance can be higher or lower than the
LDF method. Generally, the Cape Cod method
produces a lower total variance than the LDF
method. [2]
Results of the methods θ=0.50000 ω=1.459283
Table 5. LDF technique
Fig. 6: LDF standardized residuals
1 2 3 4 5 6 7
2015 92,415,152 78,075,705 16,871,685 2,825,878 1,470,000 39,200 102,436
2016 109,733,734 78,493,045 10,240,515 2,615,660 18,050,000 1,064,382
2017 116,159,869 71,964,169 8,203,749 3,800,000 7,497,779
2018 112,281,029 67,384,960 20,844,118 4,990,223
2019 137,313,347 87,328,648 9,487,241
2020 113,195,095 35,929,511
2021 135,245,016
Year Current value LDF Utimate value Future value Standard Error CV %
2015 191,800,056 1.002 192,152,611 352,555 1,703,131 483.1
2016 220,197,336 1.004 221,093,168 895,832 2,797,751 312.3
2017 207,625,566 1.009 209,555,566 1,930,000 4,209,341 218.1
2018 205,500,330 1.022 210,062,885 4,562,555 6,672,370 146.2
2019 234,129,236 1.057 247,455,742 13,326,506 11,948,301 89.7
2020 149,124,606 1.168 174,235,699 25,111,093 16,273,130 64.8
2021 135,245,016 1.815 245,498,753 110,253,737 42,637,449 38.7
Total 1,343,622,146 1,500,054,424 156,432,278 52,873,148 33.80
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Table 6. Cape Cod method
Fig. 7: Cape Cod standardized residuals
3.2 Munich Chain Ladder Method
We usually detect a considerable correlation
between the paid-to-incurred ratios and the
corresponding paid and incurred individual
development factors [3]. Consider a fixed
development year of the data triangle; in accident
years with a previous paid-to-incurred ratio that is
below average, we always see above-average paid
development factors and/or below-average incurred
development factors. With below-average paid and
above-average incurred development variables,
accident years with an above-average paid-to-
incurred ratio indicate the reverse pattern. This is to
be expected, and residual charts may be used to
confirm it [3].
The triangles used are the cumulative triangle of the
paid claims (Fig. 8) and the cumulative triangle of
incurred claims from 2015 to 2021 (Fig. 9)
Fig. 8: Triangle of cumulative claims paid
Fig. 9: Triangle of cumulative claims incurred
Fig. 10: Paid and incurred residual plots
The Munich chain ladder (MCL) approach
integrates paid claims P and incurred claims I data
by projecting P/I ratios. The proportion of paid and
incurred claims, as well as the fraction of incurred
claims that have been paid at the time of calculation,
is referred to as the P/I ratio. [3]
Fig. 11: Paid and incurred development period
The procedure of the Munich Chain Ladder is as
follows [3]:
• Plotting the residual plots of paid development and
incurred development factors for all development
years
• Drawing a regression line from the origin
• We calculate the residual and read the
accompanying development factor determined from
the average development factors for a specific P/I
ratio.
The P/I ratio of the year of the accident to the year
of development is [3]:
The average ratio for all years of accidents in year t
is [3]:
which is the weighted average of the reports (P/I) in
the year of development t with the value of claims
incurred. The development factors for paid claims
and incurred claims are respectively:
For the projected amounts, we will have [3]:
Year Current value ELR Earned Premiums Utimate value Future value Standard Error CV %
2015 191,800,056 0.233 891,779,071 192,235,619 435,563 1,843,393 423.2
2016 220,197,336 0.233 855,071,845 221,103,668 906,332 2,712,980 299.3
2017 207,625,566 0.233 892,394,125 209,737,359 2,111,793 426,579 202.0
2018 205,500,330 0.233 821,252,582 209,999,188 4,498,858 6,323,195 140.6
2019 234,129,236 0.233 888,137,830 245,960,473 11,831,237 10,444,722 88.3
2020 149,124,606 0.233 1,060,131,446 186,144,645 37,020,039 18,464,852 49.9
2021 135,245,016 0.233 1,094,152,645 251,586,915 116,341,899 31,766,420 27.3
Total 1,343,622,146 6,502,919,544 1,516,767,867 173,145,721 45,984,943 26.50
1 2 3 4 5 6 7
2015 92,415,152 170,490,856 187,362,541 190,188,420 191,658,420 191,697,620 191,800,056
2016 109,733,734 188,226,779 198,467,294 201,082,954 219,132,954 220,197,336
2017 116,159,869 188,124,038 196,327,787 200,127,787 207,625,566
2018 112,281,029 179,665,989 200,510,107 205,500,330
2019 137,313,347 224,641,995 234,129,236
2020 113,195,095 149,124,606
2021 135,245,016
1 2 3 4 5 6 7
2015 161,556,506 200,198,640 214,379,971 218,999,971 219,907,171 220,344,171 220,783,371
2016 96,662,685 106,874,242 124,512,808 128,312,808 128,612,808 128,634,408
2017 96,992,904 105,695,304 106,178,304 106,351,104 106,552,304
2018 103,104,902 114,068,621 117,428,621 117,548,621
2019 51,274,330 52,954,330 52,965,488
2020 128,792,520 133,817,587
2021 211,380,853
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The MCL technique takes advantage of the
historical connection between paid and incurred
claims to determine the extent to which they have
happened. It generates a paid and incurred prognosis
based on the information available. The MCL
approach gives the same result as the Standard
Chain Ladder when the correlations between paid
and incurred claims are not substantial [3]. The
results of the projections are the paid and the
incurred quadrangle.
Fig. 12: Projection of claims paid
Fig. 13: Projection of claims incurred
Estimation of the MCL parameters
Table 7. MCL method latest and ultimate claims and
ratios
4 Conclusion
The scope of this paper is the analysis of
distribution for the claims data and the best
estimates method for the claims reserves. The data
analyzed are the claims incurred and paid by the
DMTPL Albanian market. We noted that the best
distribution that fits incurred and paid claims are the
Weibull distribution. Usually, the Cape Cod method
has a smaller variance than the LDF method.
Table 8: Summary of Clark’s techniques
The Cape Cod method requires the estimation of
three parameters. The LDF method requires the
estimation of n+2 parameters. As a result of this, CC
method is easier even sometimes may have a higher
process variance estimated, but it will produce a
smaller estimation error.
The Munich Chain Ladder method considers the
correlation between paid claims and incurred
claims. The Munich chain ladder seeks to resolve
the differences that arise between the standard paid
claims and the incurred chain ladder indications.
MCL provides separate estimations for paid and
incurred, but they are closer to one another. In the
cases where the correlations are not significant, the
MCL method provides the same results as the
Standard Chain Ladder method.
Table 9. Summary of MCL method
References:
[1] T. Mack, [1997] Measuring the variability of
chain–ladder reserve estimates. In: Claims
Reserving Manual, vol. 2. London: Institute of
Actuaries.
[2] David R. Clark, [2003] LDF Curve-Fitting
and Stochastic Reserving: A Maximum
Likelihood Approach
[3] G. Quarg, T. Mack [2008] Munich Chain
Ladder: A Reserving Method that Reduces
the Gap between IBNR Projections Based on
Paid Losses and IBNR Projections Based on
Incurred Losses. In: CAS, Volume 2, Issue 2,
pp. 266 -299
[4] P. England, R.Verrall, [1999] Analytic and
Bootstrap estimates of prediction error in
claims reserving. Insurance: Mathematics and
Economics 25, 281-293
[5] M.V. Wüthrich, M. Merz [2013] Financial
Modeling, Actuarial Valuation and Solvency
in Insurance, Springer
[6] M.V. Wüthrich, M. Merz [2008] Stochastic
Claims Reserving Methods in Insurance,
Wiley
[7] Moro, M. G., ChainLadder: Statistical
Methods and Models for Claims Reserving in
General, Retrieved from https://CRAN.R-
project.org/package=ChainLadder,2022
[8] Boenn, M. fitteR: Fit Hundreds of Theoretical
Distributions to Empirical Data, 2017
[9] Team, R. C, R: A Language and Environment
for Statistical Computing. Vienna, Austria: R
Foundation for Statistical Computing.
Retrieved 01 24, 2022, from https://www.R-
project.org/
[10] “Handbook on Loss Reserving", Springer,
Science and Business Media LLC, 2016
1 2 3 4 5 6 7
2015 92,415,152 170,490,856 187,362,541 190,188,420 191,658,420 191,697,620 191,800,056
2016 109,733,734 188,226,779 198,467,294 201,082,954 219,132,954 220,197,336 220,315,008
2017 116,159,869 188,124,038 196,327,787 200,127,787 207,625,566 208,186,234 208,297,489
2018 112,281,029 179,665,989 200,510,107 205,500,330 214,916,453 215,496,337 215,611,498
2019 137,313,347 224,641,995 234,129,236 23,839,545 249,396,961 250,075,102 250,208,752
2020 113,195,095 149,124,606 159,391,590 162,285,116 169,670,161 170,124,557 170,215,465
2021 135,245,016 218,318,876 33,318,391 237,548,803 248,309,405 248,971,100 249,104,134
1 2 3 4 5 6 7
2015 161,556,506 200,198,640 214,379,971 218,999,971 219,907,171 220,344,171 220,783,371
2016 96,662,685 106,874,242 124,512,808 128,312,808 128,612,808 128,634,408 128,890,314
2017 96,992,904 105,695,304 106,178,304 106,351,104 106,552,304 106,614,464 106,826,378
2018 103,104,902 114,068,621 117,428,621 117,548,621 117,854,568 117,941,622 118,176,158
2019 51,274,330 52,954,330 52,965,488 53,069,706 53,115,742 53,183,960 53,494,330
2020 128,792,520 133,817,587 144,219,209 147,067,485 14,758,500 147,817,574 148,112,242
2021 211,380,853 241,683,162 262,293,591 267,915,617 268,940,932 269,440,368 269,977,927
Year Latest Paid Lates t incurred Latest P/I Ratio UltimatePaid Ultimate incurred Ultimate P/I Ratio
2015 191,800,056 220,783,371 0.8687 191,800,056 220,783,371 0.8687
2016 220,197,336 128,634,408 1.7118 220,315,008 128,890,314 1.7093
2017 207,625,566 106,552,304 1.9486 208,297,489 106,826,378 1.9499
2018 205,500,330 117,548,621 1.7482 215,611,498 118,176,158 1.8245
2019 234,129,236 53,494,330 4.3767 250,208,752 53,069,706 4.7147
2020 149,124,606 133,817,587 1.1144 170,215,465 148,112,242 1.1492
2021 135,245,016 211,380,853 0.6398 249,104,134 269,977,927 0.9227
Total 1,343,622,146 972,211,474 1.3820 1,505,552,402 1,045,836,096 1.4396
Clark's Current value Utimate value Future value Standard Error CV %
LDF 1,343,622,146 1,317,409,914 156,432,278 52,873,148 33.80
Cape Cod 1,343,622,146 1,516,767,867 173,145,721 45,984,943 26.50
MCL Paid Incurred P/I Ratio
Latest 1,343,622,146 972,211,474 1.38203
Ultimate 1,505,552,402 1,045,836,096 1.43957
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[11] Martinek. L.: Analysis of Stochastic
Reserving Models by Means of NAIC Claims
Data, Risks, 2019
[12] Gesmann, M.: Claims Reserving and IBNR.
Computational Actuarial Science with R
(2014) Chapman and Hall/CRC.
[13] www.casact.org
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Endri Raço carried out the simulations in R and
graph plotting.
-Kleida Haxhi has worked on paper structure,
algorithm choice, and conclusions.
-Etleva Llagami has organized and executed the
experiments of Section 3.1.
-Oriana Zaçaj has organized and executed the
experiments of Section 3.2.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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