Best-Estimate User Guide
Contents
Contents ................................................................................................................................ 1
1 Introduction .................................................................................................................... 2
1.1
System Requirements ............................................................................................ 2
1.2
Disclaimer .............................................................................................................. 2
2 Run-Off Triangle ............................................................................................................. 3
3 Usage............................................................................................................................. 4
3.1
Basics .................................................................................................................... 4
3.2
Further Options ...................................................................................................... 5
3.3
Notes on Segmentation.......................................................................................... 9
4 Output of the Results .....................................................................................................10
4.1
Prediction ..............................................................................................................10
4.2
Residual Plots .......................................................................................................11
4.3
Prediction Error (Long-Term View) ........................................................................11
4.4
Prediction Error of the Claims Development Result (Short-Term View) .................14
4.5
Premium Risk........................................................................................................15
4.6
Tail Estimation ......................................................................................................15
4.7
Prediction Error Including Tail ...............................................................................17
4.8
Run-Off Time ........................................................................................................19
4.9
Discounting ...........................................................................................................19
4.10 p%-Quantile ..........................................................................................................21
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Best-Estimate User Guide
1 Introduction
1. Solvency II envisages a principle-based economic valuation of technical provisions,
where the best estimate, defined as the expected present value of future cash flows,
constitutes the main building block: It is about the estimation of the present value of
future cash flows where this relates to claims provisions in the field of non-life insurance.
The Excel macro presented is intended to assist participants in the valuation of the best
estimate. This document describes the usage of this macro.
2. This Excel macro has been designed as a simple tool with a view to assist those insurers
participating in the QIS which, up to now, have not used actuarial techniques or software
as their primary means for setting provisions in non-life insurance. Still, it is essential that
the user is capable of assessing the adequacy and quality of the given input data.
Furthermore, the results need to be checked for accuracy. For this purpose the paper
“Methoden zur Schätzung von Schaden- und Prämienrückstellungen“ (GDV circular
1215/2008 of 03.07.2008) provides assistance.
3. To make the access to this tool as easy as possible the Excel workbook contains a
spreadsheet called Übersicht. On the one hand there is a survey (see section 2 and
section 3.1) of the basic usage of this tool. There is an overview of all spreadsheets
contained in this workbook as well as their purpose. On the other hand the user can
change the language here. From the list in cell B3 the user can choose his preferred
language. Right now German and English are available. After choosing from the list the
translation is activated by clicking on the orange speech balloon.
Figure 1: Choosing a Language
1.1 System Requirements
4. The valuation tool has been developed using Excel 2003. In case other versions of Excel
(higher or lower, or with another language installed) are used, some menu items
mentioned in this documentation or shown in screenshots may have changed.
1.2 Disclaimer
5. This is a cost-free product provided by CEIOPS which is based on an Excel tool
developed by the German insurance industry association (GDV) for the use in Solvency II
quantitative impact studies. It has been validated by the competent staff members and
bodies to the best of their knowledge and belief. Nevertheless, the tool may include
technical or other mistakes, inaccuracies or typographical errors. Any feedback to this
effect would be welcomed by CEIOPS. CEIOPS assumes no responsibility for errors or
omissions in the tool, which is provided 'as is' without warranty of any kind, either express
or implied.
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Best-Estimate User Guide
2 Run-Off Triangle
Figure 2: Example of a run-off triangle with cumulative data
6. It is necessary to provide a run-off triangle with information on the paid claims – either
cumulative or incremental - in individual accident and development years. A run-off
triangle (see Figure 2) is a two-dimensional data structure consisting of columns and
rows determined to store information on the process of claims development. At this the
rows represent the accident years and the columns represent the development years.
Accident years declare the year in which a claim is incurred, whereas the development
years declare the elapsed time since the occurrence of the claim. A run-off triangle is
filled with data by determining the appropriate accident year and development year for
every claim payment. This claim payment is then added to the value in the corresponding
cell of the run-off triangle at the intersection of this particular accident year and
development year. In this manner a run-off triangle with incremental data is derived. In
order to get a run-off triangle with cumulative data one has to add up the contents of
every previous cell in the same row.
Note: This Excel tool requires (paid) claim amounts as input data. If incurred claims are
entered, the tool will produce incorrect results.
7. Instead of cumulative data as shown in Figure 2 incremental data can also be used.
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Best-Estimate User Guide
3 Usage
3.1 Basics
8. In the following the functionality of the tool will be explained. As shown in Figure 2, the
input data are entered in the range B2:AE31 of the worksheet Eingabe in the form of a
run-off triangle (or a trapezium). Note that the lower left hand corner of the run-off triangle
must be placed in the cell B31. A run-off triangle from another Excel workbook may be
entered by transferring it using the copy and paste function of Excel. It is recommended
to paste the values only. Otherwise the format of the input range will be changed
unintentionally.
Note: The lower left hand corner of the run-off triangle must be entered in cell B31.
9. All commands to run the macro and all the options to influence the results of the macro
can be found exclusively on the worksheet Eingabe. The other worksheets in the
spreadsheet of the tool are just designed to show the results of the calculations. In order
to run the macro one has to enter 2 accident years and 2 development years at least.
Moreover, no more than 30 accident years and development years may be entered. It is
possible to evaluate a run-off trapezium, that is a data structure that contains more
accident years than development years. However, it is not possible to evaluate a data
structure that contains more development years than accident years. Alongside with
evaluable input data the user must input whether the data is incremental or cumulative
(cell M34). If the user provides incorrect information about the input data, the results will
inevitably be wrong, too. In addition, the unit of the input data can be declared as well as
a unit for the output (cells I34 and I35). For each option there are three possibilities: euro,
thousands of euros and millions of euros. The units “thousands of euros” or “millions of
euros” can be used in case of large claim payments.
10. After entering these three essential pieces of information (run-off triangle, form of data
and unit) the macro can be started by pushing the button “Start!“ (see Figure 3). Having
calculated all results these are output on the three worksheets Ergebnisse,
kum.Schadenzahlung and Barwerte.
Figure 3: Start Button, Units of Input and Output, Form of Data
11. Once some input has been entered in the worksheet Eingabe it can be saved and
managed for later utilisation (see Figure 4). Thus, several run-off triangles can be
evaluated using different options within one spreadsheet. To save input data the user
enters a number into cell D90 identifying a column as labelled in row 94 and pushes the
button “save”. Thereafter, all input data is stored in the column underneath the given
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Best-Estimate User Guide
label. To retrieve data thus saved the user enters the corresponding digit from row 94 into
cell D91 and pushes the button “load“. Additionally, stored input data can be conveniently
deleted from its column by entering its column label from row 94 into cell D92 and then
pushing the button “delete“. Lastly, all data entered in the worksheet Eingabe can be
deleted by pushing the button “delete“ displayed over cell G90. Pushing that button
restores the default values of the options, too (as described in section 3.2). Note that
loading a saved data set only restores the input data and not the corresponding results
on the other worksheets. Results need to be calculated again by pushing the button
“Start!”.
Figure 4: Saving and Managing Input Data
3.2 Further Options
12. Beyond that the user has several options. These range from humble adjustments of
layout of the output to settings regarding the estimation of a tail that have major impact
on the calculated results. All options will be explained consecutively. Note that some
settings of options supersede other options. In that case these redundant options will be
marked gray shaded. Redundant input information is recognised and ignored by the
macro automatically, so there is no need to delete redundant input. It should be pointed
out that all options address features. To run the macro only the three essential pieces of
information (triangle, form and unit) are necessary as described in paragraphs 9 and 10.
3.2.1 General
13. The options of the category General affect the given run-off triangle in general. These
options can be found in the first row of the worksheet Eingabe:



Balance sheet year – Here, the user can specify the date of the current accident year. In
the output the accident years will be labelled according to this input instead of numbering
them from 0,1, , n . The default value is 2008.
Line of Business – Here, the user can specify the line of business relating to the current
run-off triangle. The line of business is important for the presetting of further information
regarding predefined chain-ladder factors and earned premiums. If no line of business is
given, the macro uses “other line of business“ as a default. Six lines of business are
available (see section 3.3).
Commentary – The user can comment on the run-off triangle and the settings here. This
memo can be useful especially if the input data is to be stored.
Figure 5: Options from Category “General”
3.2.2 Units and Data
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Best-Estimate User Guide
14. The options from the category Units and Data influence how the input data is dealt with.
Both options are essential for running the macro and they need to be set correctly before
pushing the start button. They have been addressed already in paragraph 9 but they are
also listed here for the sake of completeness (see Figure 3).



Unit: Conversion – Here, the user specifies the input unit of the given data (cell I34).
Moreover, the user can specify the output unit used for printing the results (cell I35).
There are three possibilities: euro, thousands of euros and millions of euros. The default
value is euro.
Data: Form – Here, the user must specify whether the input data of the run-off triangle
consists of cumulative or incremental claims amounts (M34). The default value is
cumulative.
Data: Years of Loss Occurrence Used – Here, the user can specify how many years of
loss occurrence are used to calculate the chain-ladder factors (cell P34). Thereby,
changed claims settlement can be taken into account. The subset of “years of loss
occurrence” is only used for the calculation of the chain-ladder factors (see paragraph
23). If the user does not specify this option, the whole run-off triangle will be used.
3.2.3 Tail
15. The options from the category Tail can be found in the rows from 33 to 44. With these
options different aspects of tail estimation can be controlled: fundamental questions,
techniques of tail estimation and calculation of the standard error including tail.






Tail: General
Consideration of Tail? – Yes/ No: Should a tail be included? If the user chooses No, all
other options from this category become redundant. The default value of this option is
Yes.
Tail Factor – If the user already has an estimate of a tail factor for the given run-off
triangle, he may use this factor directly instead of the estimation by means of the tail
function. If the user supplies a tail factor, all options from the category “Tail: Estimation”
become redundant. The use of a supplied tail factor has further impact on the
calculations, see paragraph 44.
Overall Length of Claims Settlement – How long does it take practically for all claims to
be settled completely (see section 4.5, paragraph 42)? Note that the overall length of
claims settlement including the tail is demanded. Please enter the number of
development years and not the label of the last development year. The default value is 50
years. If more than 50 years are entered, consider to predefine more interest rates for the
yield curve.
Tail: Estimation
Start Regression from Development Year – For the internal estimation of the tail of the
claims development, a development function is fitted to the chain-ladder factors. The user
can specify the development year from which the regression starts. All chain-ladder
factors of previous development years are ignored during the regression. Note that at
least two chain-ladder factors being larger than one are necessary to carry out the
regression (see section 4.5, paragraph 37). The default value is 5.
Use Predefined Chain-Ladder Factors for Regression – Yes/ No. If the user chooses
No, only the chain-ladder factors based on the observations in the run-off triangle are
used for the regression, except for those being explicitly excluded. Otherwise, the chainladder factors of the run-off triangle are supplemented consecutively by adding the
predefined chain-ladder factors (see section 4.5, paragraph 37). The default value is No.
Type of Curve for Development Function – Exponential/ Weibull/ Power/ Sherman.
Here, the type of curve for the development function can be chosen (see section 4.5,
Table 1). The default value is Exponential.
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Best-Estimate User Guide

Constant c for Sherman Curve– Linear Regression is used to determine the
parameters of the curves fitting the chain-ladder factors. Since the Sherman curve
depends on three parameters, the user must specify the third parameter c by himself so
that the other two parameters can be estimated using linear regression (see section 4.5,
paragraph 38). If another type of curve has been selected, this option becomes
redundant.
Tail: Standard Error




  and ˆ
Preset by User? – Yes/ No. Does the user provide the two parameters se fˆult
ult
necessary for estimating the standard error including tail? If the answer is No, the macro
tries to estimate these two parameters (see section 4.7). If the answer is Yes, it is
redundant to specify a reference year. The default value is No.
se(f_ult) – Here, the user can specify the estimation error (standard deviation) of the tail
factors. Appropriate only if option “Preset by User” has been set to Yes.
sigma_ult – Here, the user can specify the variance parameter ˆ ult for the tail.
Appropriate only if option “Preset by User” has been set to Yes. Needs to be filled in the
same unit as the output data (see paragraph 9).
Reference Year – The user can mark an accident year (using numbers from 0 to n ) that
is representative of the whole run-off triangle for reasons of typical claims settlement.
This reference year is used as an “anchor point” to derive ˆ ult (see section 4.7,
paragraph 46). The macro is capable of marking a reference year if the user does not
supply a reference year.
Figure 6: Options from Category “Tail”
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Best-Estimate User Guide
3.2.4 Further Settings
16. To this category “Further Settings” all other options and settings belong. These can be
found starting from row 45.







p%-Quantile of the Overall Reserve Without Tail – The user can specify a number
between 1and 100. Then the corresponding p%-quantile of the overall reserve without tail
is calculated based on a log-normal distribution (see section 4.10 and Figure 7). The
default value is 75.
p%-Quantile of the Overall Reserve Including Tail – The user can specify a number
between 1 and 100. Then the corresponding p%-quantile of the overall reserve including
tail is calculated based on a log-normal distribution. Of course this entry will only be
evaluated if a tail is considered (see section 4.10 and Figure 7). The default value is 75.
Yield Curve – In these three rows yield curves can be provided which will be used for
calculating the present value of the future cash flows and duration. Three variants can be
given. The first row (row 50) shows the variant for interest rate reduction. The second row
(row 51) shows the standard yield curve and the third row (row 52) shows the variant for
interest rate increase (see section 4.9, paragraph 49 and Figure 8). If nothing is entered
here, every interest rate is automatically interpreted as 0 %.
Average Interest Rate in % - In addition to the yield curves the user can provide an
average interest rate for the calculation of the present values and duration. In doing so
only one interest rate is used during this calculation instead of many (see section 4.9,
paragraph 50 and Figure 8). If nothing is entered here, the interest rate is automatically
interpreted as 0 %.
Chain-Ladder Factors – The valuation tool allows the usage of pre-defined chain-ladder
factors, e.g. from market sources. The tool uses these pre-defined factors to supplement
the factors derived on the basis of the insurer-specific input triangle. This may be helpful
in cases where the available input triangle itself would be too short to determine an
appropriate tail function. For up to six lines of business the user can provide pre-defined
chain-ladder factors. To use one set of these the user must select the corresponding line
of business for the input triangle and the user must mark that pre-defined chain-ladder
factors should be used for estimating the tail function (see section 4.5, paragraph 35 and
Figure 9).
Earned Premiums – The user may supply earned premiums for the six lines of business
to calculate ultimate loss ratios (see section 4.5). To use one set of these the user must
select the corresponding line of business for the input triangle (see Figure 9). If nothing is
entered here, there will be no results regarding the ultimate loss ratios.
Overwriting Observed Chain-Ladder Factors – For every development year the user
can supply a chain-ladder factor that will be used instead of the corresponding chainladder factor calculated on the basis of the original run-off triangle. Particularly, one could
use factors from market sources or smoothed factors (see paragraph 39).
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Best-Estimate User Guide
Figure 7: Options from Category “Further Settings”: Quantiles
Figure 8: Options from Category “Further Settings”: Present Values
Figure 9: Options from Category “Further Settings”: Chain-Ladder Factors and Earned
Premiums
Figure 10: Options from Category “Further Settings”: Overwriting Observed Chain-Ladder
Factors
3.3 Notes on Segmentation
17. On the worksheet Eingabe one can supply parameters for six different lines of business.
While most are self-explanatory, there are two that need some explanation: third party
liability industry (m) and third party liability industry (l). In order to determine chain-ladder
factors being representative of the market the insurance association surveyed run-off
triangles from its members. Members were asked to provide two run-off triangles for third
party liability for each segment “private“ and “industry“. The segment “industry” seemed
to be inhomogeneous. Therefore, the association refined this segment even further. The
average run-off time of each run-off triangle in this segment was used as a criterion to
group the triangles. Those triangles yielding a medium run-off time were assembled in
the new segment “third party liability industry” (m) whereas, the other triangles yielding a
long run-off time were assembled in the new segment “third party liability industry” (l).
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Best-Estimate User Guide
4 Output of the Results
18. After the calculation, that takes some time, all results are output on the worksheets
designed especially for presenting the results: Ergebnisse, kum.Schadenzahlung,
Nachlauf and Barwerte. On the worksheet Ergebnisse a summary of the results is shown,
whereas on the worksheet kum.Schadenzahlung all results regarding the topics of
reserve risk and premium risk are displayed including standard error of prediction in the
short-term and long-term view (including tail or based on observations only) as well as
ultimate loss ratios and their temporal variance. On the worksheet Barwerte results
regarding the cash flows, present values and duration are shown. The next paragraphs
list in detail where every single result can be found.
19. The main results – the quantity of claims reserves – are summarized on the worksheet
Ergebnisse. This comprises the undiscounted overall reserve as well as the discounted
overall reserve based on a yield curve and its variants for interest rate increase and
reduction. Furthermore, the following results are shown:
 Prediction of the undiscounted cash flow for the next ten years to come,
 Estimation of the modified duration,
 Standard error of prediction of the reserve estimate and estimation of several quantiles
based on the results according to the model by Mack and a log-normal distribution,
 Standard error of prediction of the claims development result,
 Residual plots for analyzing the run-off triangle.
20. As stated in paragraph 9, it is possible to handle run-off trapeziums (accident years
0, , n , development years 0, , k0  n ). In the following formulary only the case of a
run-off triangle is described for the sake of simplicity. But it is always possible to limit the
formulae to the first k0  1 development years. As an example, the consequences of this
limiting are shown in paragraph 31 for the results described in the sections 4.1 and 4.2.
4.1 Prediction
21. We consider cumulative data which are arranged in the form of a run-off triangle Sik the
rows and columns of which are numbered from 0, , n in each case. The triangle is
output in the worksheet kum.Schadenzahlung!B2:AE31. Using this run-off triangle and
the formula
fˆk 
n  k 1
n  k 1

Si ,k 1

Si ,k 1
i 0
S
i 0
i,k
, 0  k  n 1
or, equivalently,
fˆk 
n  k 1

i 0
Sik
n  k 1
S
j 0
Sik

n  k 1

i 0
S
 Fik , 0  k  n  1
ik
n  k 1
S
ik
j 0
ik
it is possible to calculate the n chain-ladder factors fˆk , 0  k  n  1 . By means of these
chain-ladder factors the run-off triangle is subsequently completed so as to estimate the
ultimate loss and the reserves:
Sˆin  Si ,n i  fˆni 

Rˆi  Sˆin  Si ,ni  Si ,ni fˆni 
 fˆn 1

 fˆn1  1 , with i  1,
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,n .
Best-Estimate User Guide
The chain-ladder factors are output in row 35 of the worksheet kum.Schadenzahlung
under the designation BE (“best estimate”). The reserves are displayed row by row,
starting in cell kum.Schadenzahlung!B44.
22. The chain-ladder factors are also used to estimate future payment flows Ẑ , which
appear for the discounting in the worksheet Barwerte:
Sˆi ,k 1  Sˆik  fˆk , to the diagonal element the following applies: Si ,n i  Sˆi ,n i ,
Zˆi ,k 1  Sˆi ,k 1  Sˆik ,
with k  n  i,
, n  1.
23. The chain-ladder factors can be calculated on a subset of the original run-off triangle, too.
Thus, only the most recent years of loss occurrence (balance sheet years) are used in
order to determine the current claims settlement. Cumulative claims Sik belong to a
balance sheet year J if they are located on the corresponding diagonal of the run-off
triangle: i  k  J . If only the most recent j balance sheet years are taken into account
while calculating the chain-ladder factors, the method of calculation changes as follows:
fˆk 
n  k 1

i 0
Si ,k 1   i ,k 1  j 
n  k 1
S
i 0
i ,k
1, if i  k  n  j  1
.
  i ,k  j  , with  i ,k  j   
0, else

To highlight the payments involved in calculating chain-ladder factors using a restricted
data set, these are displayed in a bold font on the worksheet kum.Schadenzahlung in the
range B2:AF31
4.2 Residual Plots
24. For a fixed development year k the chain-ladder method resembles a linear regression:
Si ,k 1  f k  Sik   i ,
with 0  i  n . Therefore, it is reasonable to examine the standardized residuals rik of
this regression in order to detect systematic influences in the input data:
rik 
Si ,k 1  Sik  fˆk
ˆ k Sik
, 0  i  n, 0  k  n  i  1 .
25. There are different ways to group and plot the residuals of the run-off triangle. The macro
is capable of plotting two residual plots on the worksheet Ergebnisse. The upper residual
plot groups the residuals by the accident years. That means the diagram shows the
points  i, rik  . Systematic influences would indicate a portfolio change or a change in the
development behaviour.
26. The lower residual plot groups the residuals by balance sheet years. That means the
diagram shows the points  const, rik i  k  const  . Systematic influences would show a
calendar year effect or inflation.
4.3 Prediction Error (Long-Term View)
27. The chain-ladder factors yield point estimators for the reserves and the ultimate claim.
Furthermore, information about the variability of these point estimators can be given.
Therefore, it is required to make an assumption on the variance of the claim amounts Sik :

Var Si ,k 1 Si 0 ,

, Si ,k  Si ,k k2
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Best-Estimate User Guide
The parameters  k2 are unknown and must be estimated in such a way that the term
Sik ˆ k2 produces an estimator of the variance of the term Si ,k 1 :
2
n  k 1
S

1
ˆ 
Sik  i ,k 1  fˆk 

n  k  1 i 0
 Sik

2
4
2
and ˆ n 1  min(ˆ n  2 ˆ n 3 , min(ˆ n23 , ˆ n2 2 )).
2
k
The parameters  k2 are displayed in the row 38 of the worksheet kum.Schadenzahlung.
  for the
28. Now, it is possible to calculate the mean square error of prediction mse Rˆi
accident year i . It is defined as follows:

 
mse Rˆi : E Rˆi  Ri
   Var  Rˆ   Var  R  .
2
i
Estimation Error
i
Process Error
The following formulas apply:
Var  Ri   Sˆin2
 
Var Rˆi  Sˆin2
n 1

k  n i
n 1

k  n i
ˆ k2 fˆk2
Sˆik
,
ˆ k2 fˆk2

n  k 1
j 0
, with i  1,
, n.
S jk
Adding up the estimation error and the process error the mean square error of prediction
is derived as:

ˆ  1
1
2
ˆ
ˆ
mse Ri  Sin 
 n  k 1
ˆ
ˆ
k  n i f  Sik
S jk


j 0

 
 
holds: se  Rˆ  
n 1
2
k
2
k


.



29. Let se Rˆi denote the prediction error of the reserve of accident year i . The following
i
 
mse Rˆi . The prediction error is important because it has the same
currency unit as the cumulative payments of the run-off triangle. In addition, it can be
used to obtain a confidence interval for the reserves and the ultimate claims:
 
 
 Rˆi  2  se Rˆi , Rˆi  2  se Rˆi  . Finally, the estimation error can be used to construct a


confidence interval for the mean of the reserve to assess the difference to other reserve
ˆ
estimates:  Rˆi  2 Var  Ri  , Rˆi  2 Var  Ri   . Every other reserve estimate Ri obtained


by another method does not differ significantly form the reserve estimate Rˆi derived
ˆ
above, if Ri   Rˆi  2 Var  Ri  , Rˆi  2 Var  Ri  . The prediction error is displayed as a


column starting in cell kum.Schadenzahlung!C44. The adjacent column from B44 shows
the respective estimated reserve. The following column from D44 shows the ratio
between the prediction error and the reserve. The columns below E44:H44 show the
roots of the process error and of the estimation error as well as the ratios between the
random error and the prediction error and between the estimation error and the prediction
error.
30. Then, the variance parameters can be used to specify the standard deviation of the
chain-ladder factors:
- 12 -
Best-Estimate User Guide
 
Var fˆk  ˆ k2
n  k 1
S
j 0
jk
, with k  0, , n  1 ,
these results are shown in row 41 of worksheet kum.Schadenzahlung for each
development year. The variation coefficient of the chain-ladder factors is calculated as
follows:
 
Vkok  Var fˆk
fˆk , with k  0,
, n  1.
It is displayed in row 36 of the worksheet.
31. If a run-off trapezium with n  1 accident years and k0  n  1 development years is
evaluated, the following applies to the terms just discussed:
ˆ k 
0
 ˆ n 1  0,
 
Var fˆk0 
Vkok0 
 
 Var fˆn 1  0,
 Vkon 1  0,
 
 
mse Rˆ1 
 mse Rˆ k0  0,
Var  R1  
 Var Rk0  Var Rˆ1 
 
 
 
 Var Rˆ k0  0.
32. On the basis of the prediction error for individual accident years i the prediction error of
the overall reserve Rˆ 
n
 Rˆ
i 0
i
may be derived as follows:


mse( Rˆ )    se Rˆi
i 1 

n
  
2

2
ˆ


2 ˆ f k
.
 Sˆin   Sˆ jn  
 j i 1  k  n i  S 
mk

m0
n
n 1
2
k
n  k 1
The overall reserve is displayed in the cell kum.Schadenzahlung!J44. Adjacent to it in the cell
 
 
K44 the prediction error of the overall reserve se Rˆ  mse Rˆ is output and in cell L44 the
ratio between the prediction error of the overall reserve and the overall reserve is displayed.
- 13 -
Best-Estimate User Guide
4.4 Prediction Error of the Claims Development Result (Short-Term
View)
33. Considering the short-term view, the claims development result of one calendar year is
estimated. The claims development result is defined as the difference of reserve estimate
at the beginning of the calendar year and payments during this calendar year and the
reserve at the end of the calendar year.
Let Dn  Sik k  n  i denote the known claims at the beginning of the calendar year.
Then the set Dn 1  Sik k  n  i  1 denotes the data available one period later. An
estimator for the claims development result is defined by:


ˆ
ˆ
ˆ
CDR
i : Ri Dn  Zi , n i 1  Ri Dn 1 .
Here Rˆi Dn denotes the reserve estimate for accident year i at the beginning of the
calendar year and Rˆi Dn 1 denotes the reserve estimate of the same accident year
updated by the information gained during the calendar year.
Particularly, the short-term view reflects the changes of the reserve estimate due to new
information gathered within the next calendar year. Thus, the mean of the claims
development result is 0. The mean square error of prediction for the claims development
result can be calculated, too:



ˆ
ˆ
mse CDR
CDR
i : E
i 0

2
 

ˆ
Dn  se CDR
i
 .
2
Consider:
n  k 1
fˆ 
n
k
S
i 0
i , k 1
Tkn
, mit Tkn 
n  k 1
S
i 0
ik
nk
fˆ
n 1
k

S
i 0
i , k 1
Tkn 1
nk
, mit Tkn 1   Sik
i 0
Then, the mean square error of prediction for the claims development result is derived as:

  
ˆ
ˆ
mse CDR
i  Sin
2
 
 ˆ 2
fˆnni
 n i

Si , n i

2

ˆ n2i
 fˆ 
n
n i
n
n i
T
2
n 1


Sn k ,k
k  n i 1
ˆ k2
Tkn 1
 fˆ   .
n
k
Tkn
2


Besides, the mean square error of prediction for the overall claims development result
can be computed by means of the following formula:
 
 ˆ 2
fˆnni
n i



ˆ
ˆ
ˆ ˆ
mse   CDR
i    mse CDRi  2  Sin S kn 
Tnni
k i  0
 i 0
 i 1

n
n


2

n 1

j  n i 1
2
Sn  j , j ˆ j
T jn 1
 fˆ   .
n
j
T jn


Again, the prediction error of the claims development result is defined as the square root
of the mean square error. It is output as a column starting at cell AC44 of the worksheet
kum.Schadenzahlung. In the adjacent column starting in cell AD44 the prediction error in
the long-term view is output again. In the column starting at cell AE44 the ratio between
the prediction error in the short-term view and the prediction error in the long-term view is
given. The overall prediction error of the claims development result is displayed in cell
kum.Schadenzahlung!AF44. In the adjacent cell AG44 the prediction error in the long- 14 -
Best-Estimate User Guide
term view is displayed one more time while the cell AH44 holds the ratio between these
two terms.
4.5 Premium Risk
34. Using earned gross premiums Pi provided by the user and the ultimate losses
determined by the chain-ladder method (without or with tail), ultimate loss ratios can be
calculated :
Sˆin
Sˆ  fˆ
and in ult resp. with i  0,
Pi
Pi
,n.
The arithmetic average of these ratios is calculated and output. In addition, the variation
coefficient of the ultimate loss ratios is determined in order to quantify temporal variability
of these ratios.
The ultimate loss ratios are displayed in a column starting at cell
kum.Schadenzahlung!AJ44. The average ultimate loss ratio is displayed in cell AK44
whereas the variation coefficient is displayed in cell AL44.
4.6 Tail Estimation
35. Possibly, the data contained in the run-off triangle does not characterize the complete
run-off of all claims, which means that the claims are not fully run-off after the last
observed development year. In order to examine the run-off of future payments beyond
the known development years a development function f  k  is fitted to the chain-ladder
factors using a log-linear regression. The available development functions can be found
in Table 1.
Name
Exponential
Definition
Regression
f  k   1  a  exp  b  k 

Weibull
f  k   1 1  exp  a  k b 
Power
f  k   ab^ k 
Sherman
f k   1 a k  c
b

ln  f k  1 vs. k
ln   ln 1 1 f k   vs. ln  k 
Result
ln  a  , b
ln  a  , b
ln ln  f  k   vs. k
ln  ln  a   ,ln  b 
ln  f k 1 vs. ln  k  c 
ln  a  , b


Table 1: Development Functions
The development functions Exponential, Weibull and Power depend on two parameters
a , b . Hence, the procedure to estimate these two parameters is to carry out a log-linear
regression.
36. For the regression the numbers of the development years or derived values are used as
independent variables x whereas the chain-ladder factors or derived values are used as
explanatory variables. The column Regression of Table 1 shows the variables used for
each development function (using the notation: explanatory variable against independent
variable).
37. If necessary, externally predefined chain-ladder factors can be used to supplement the
calculated chain-ladder factors on the basis of the given run-off triangle. Thus,
- 15 -
Best-Estimate User Guide
n : max  n, w chain-ladder factors are available for the regression, with w denoting the
sum of the number of original chain-ladder factors and the number of additional
predefined chain-ladder factors (if any). Moreover, the user can specify a development
year r from which the regression starts. Anyway, the regression needs at least two
chain-ladder factors the values of which must be larger than 1.
Example (independent and explanatory variables for the Exponential development
function)
x   r , r  1,

, n  ,
y  ln( fˆr  1), ln( fˆr 1  1),

, ln( fˆn  1) .
38. The Sherman development function depends on three parameters a, b, c . That is why a
linear regression cannot be used to determine all parameters. If the user specifies c , the
parameters a , b can be calculated using a log-linear regression according to the
description from Table 1.
39. In order to assist the user in choosing a proper development function the macro offers
two facilities. On the one hand, the coefficient of determination R 2 of the regression is
output. The coefficient of determination R 2 is a measure of how well the regression line
approximates the real data points. It is defined as follows:
  fˆ
n
R2  1 
k r
k

n
k r
 f k 
fˆk  f

2

2
,
In this formula f denotes the arithmetic average of the chain-ladder factors fˆk . For the
coefficient of determination the following holds: R 2  0,1 . If the coefficient of
determination equals 0, there is no linear relation between the independent and the
explanatory variables of the regression. However, if the coefficient of determination
equals 1, this indicates that the regression line perfectly fits the data. The coefficient of
determination of the regression for the chosen development function is output on the
worksheet kum.Schadenzahlung in the cell Z49.
40. On the other hand, all different development functions are shown on the worksheet
Nachlauf both as a chart using a logarithmic scale allowing for a graphical comparison
and compared with each other in table form. At the same time the development functions
are not only used to predict future chain-ladder factors but also to provide smoothed
chain-ladder factors for the past. If the user does not provide a parameter c for the
Sherman curve as development function, the macro chooses the value for that parameter
c 0.5,0,1,3,5 yielding the largest coefficient of determination. Because of its
definition it is not possible to specify a chain-ladder factor for the development year 0
when using the Weibull curve.
41. If the development function f  k  is evaluated at the point k   n , one gets the chainladder factor corresponding to the development year k  . The infinite product of the chainladder factors of the development years larger than or equal to n is called tail factor f ult .
The tail factor is the ratio between the expected ultimate loss and the accumulated claims
amount at development year n :
- 16 -
Best-Estimate User Guide

fˆult   f (k ) .
k n
42. In practice, the tail factor is calculated using a fixed development year e : E 1
(numbering starts at 0) provided by the user. At this year the calculation of the infinite
product is aborted. Note: The user specifies the whole run-off time of run-off triangle E
and not only the run-off time of the tail:
e 1
fˆult   f (k )
k n
43. Using the tail factor the accumulated total claims amount and the reserve including tail
can be calculated for each accident year i  0, , n :
Sˆi ,ult  Sˆin  fˆult
Rˆiult  Sˆi ,ult  Si ,n i
n
Rˆ ult   Rˆiult
i 0
The two parameters b, a are output in the cells kum.Schadenzahlung!W44 and X44
alternatively. The tail factor is displayed in cell Y44. The development year r is displayed
in cell Z45. The reserve including tail for each accident year Rˆiult is output in the column
starting at cell N44. The overall reserve including tail Rˆ ult is output in cell S44.
Note: Sometimes the choice of a development function for estimating the tail and the overall
length of claims settlement have a huge impact on the reserve estimate. It is therefore
recommended to be careful when choosing a development function and when fitting a tail.
44. Instead of estimating a tail factor by means of a development function it is possible to
provide a tail factor. Doing so affects the computations done by the macro. Since external
presetting of a tail factor yields no information about the temporal development of the
claims settlement in the tail, the discounting and the calculation of the duration is carried
out based on the original run-off triangle. Likewise, no payments for the tail are output on
the worksheet Barwerte so that the output is consistent with the method of calculation in
case no tail is considered.
4.7 Prediction Error Including Tail
45. In this section, unless otherwise stated, the following applies:
i  0, , n and k  0, , n  1 . The prediction error of the estimated ultimate losses Sˆin is
identical with the prediction error of the reserves Rˆi :

 
mse Rˆi  E Rˆi  Ri
   E  Sˆ
in  Si , n i   Sin  Si , n i 
2

With the abbreviations mse  Fik   se  Fik 
    
mse fˆk  se fˆk
2


ˆ k2
n  k 1
j 0
2

ˆ k2
Sik
2
in
and
the formula for the mean square error of prediction can
S jk
be expressed in an equivalent way:
 

   mse  Sˆ  .
mse Sˆin  Sˆin2
  mse  F   mse  fˆ  
n 1
ik
k  n i
- 17 -
k
fˆk2 .
Best-Estimate User Guide
The two abbreviations indicated may be interpreted as follows. The term se  Fik 
estimates the extent to which the individual settlement factors
Si ,k 1
deviate on average
Sik
  describes the extent to which the estimated
from the chain-ladder factor f k while se fˆk
chain-ladder factor fˆk deviates from the actual chain-ladder factor f k .
From the formula a recursive definition of the standard errors of the estimated
accumulated total claims amounts may be derived:



   mse  Sˆ  fˆ
mse Sˆi ,k 1  Sˆi2,k mse  Fik   mse fˆk
ik
2
k
.
The initial value for the recursion for each accident year is the diagonal element for which


mse Sˆi ,ni  0 is assumed. By means of the recursive definition it is possible to calculate
the prediction error for the ultimate loss including tail as the recursive formula is
evaluated one step beyond the last development year n interpreting Sˆi ,n 1 as Sˆi ,ult and
fˆn as fˆult :

 
   mse  Sˆ  fˆ
mse Sˆi ,ult  Sˆin2 mse  Fi ,ult   mse fˆult
in
2
ult
.
46. Unfortunately, three unknown terms are necessary for this purpose: tail factor fˆult ,
 
estimation error of the tail factor se fˆult
and an estimation for the variance parameter of


the tail ˆ ult (in order to calculate se Fi ,ult ). Either these terms must be given by the user
or they must be estimated by the macro.
The method to estimate the tail factor has been described in the last section. If the tail
factor is known, the other two terms are determined as follows. First an index k  is
determined for which fˆk   fˆult  fˆk 1 holds. Then, by using a linear regression, a straight
line is drawn through the two points
 fˆ , se  fˆ  ,  fˆ
k
k
k  1
  and this line is
, se fˆk 1
 
evaluated at fˆult . The result is an estimate for se fˆult . The next thing needed is a
reference accident year that is representative of the run-off triangle as a whole. This
reference year is either given by the user or the macro determines such a year using the
following algorithm: The mean of all claims amounts of the whole run-off triangle is
calculated. This mean is compared to the mean claims amount of each accident year.
The accident year i that has the least deviation from the overall mean is taken as the
reference year. After that a straight line g is drawn trough the points
 fˆ , se  F  ,  fˆ
k
i, k 
k 1
 
, se  Fi,k 1 


 and this line is evaluated at the point
equation g fˆult  se Fi,ult  ˆult

estimators se Fi ,ult

fˆult . With the
Sˆi,n an estimator for ˆ ult is derived. Using ˆ ult
for each accident year i are derived.
47. The recursive formula for the calculation of the prediction error of the overall reserve
including tail is given by:
2
2
  
n
n
  n ˆ    n ˆ  ˆ2
ˆ
ˆ 2   se  F  2  
ˆ 
se
S

se
S

f

S
   i ,k 1      ik   k  ik
ik
  Sik   se f k
i nk
    i  n 1 k  
 ink 
  i nk
2
- 18 -
2
,
Best-Estimate User Guide
interpreting Sˆi ,ult  Sˆi ,n 1 again. The initial value for the recursion is k  n and the criterion



n
Sˆ


for aborting the recursion is k  0   se 

i  n 1
i0


   0  .




The prediction errors of the reserve including tail se Sˆi ,ult are displayed in a column
starting at cell kum.Schadenzahlung!O44. In the column starting at cell P44 the ratio
between the prediction error of the reserve including tail and the reserve including tail is
output. The output of the prediction error of the overall reserve including tail is placed in
cell T44. Adjacent to it in cell U44 the ratio of the prediction error of the reserve including
  and ˆ
tail and the overall reserve including tail is output. The terms se fˆult
ult
are output
in the cells kum.Schadenzahlung!W49 and X49 respectively. The reference year i is
displayed in cell Y49.
Note: Special accuracy is needed when determining fˆult and ˆ ult . Therefore, particular
caution is needed when interpreting the results, too.
4.8 Run-Off Time
48. The run-off time of payment flows for observed years k  0,..., n  i and forecasted years
k  n  i  1,..., n is calculated using the annual median points tk  k 
1
separately for
2
each accident year:
e
di : 
k 0
Z ik
tk 
e
Z
p 0
1
Sie
e
Z
k 0
t , i  0,...., n .
ik k
ip
To calculate the run-off time for subsequent years greater than n the development
function for estimating the future payments is required:
n2
 n 1

Zˆin  Sˆin  Sˆi ,n 1  Sˆin    f  k    f  k   , with n  1  n  e.
k n
 k n

The overall run-off time is displayed depending on the accident year in one column from
the cell kum.Schadenzahlung!Q44, regardless of whether a tail was included or not. If a
tail is included, the exclusive run-off time of the tail is displayed in cell Z44.
4.9 Discounting
49. For a given term structure zk , k  0,1, 2,
value of the future payments Z
Z ikd : Z ik
d
for the balance sheet year J the present
is calculated as follows:
1
1  zk ni  
k  n  i  0.5
, i  0,..., n, k  n  i  1,......, e
1
The interest rate zk , k  0, applies to the period  J , J  k  while the interest rate z0
applies to a half year’s period.
- 19 -
Best-Estimate User Guide
50. In the special case of a constant interest rate the present value of a future payment flow
in the accident year i ( Z i ,k  n 1 , Z i ,k  n  2 , ) is calculated analogically by means of the
formula:
Bi  z  :
e

k  n i 1
1
1  z 
k  n i  0.5
 Zik für i  0,
,n .
The present value calculated by means of this formula is an approximative solution
because the summation is only made until the development year e .
51. Thus, the present value function Bi  z  determines the present value of a payment flow
at a fixed interest rate with an observation period of E years (development years 0 till e ).
The change of this function in the case of a change in the interest rate is described by a
parameter which is referred to as absolute duration Di abs :
Di abs  z  : 
e
dBi  z 
 k  n i  0,51
   k  n  i  0, 5  Zik  1  z 
 0 with i  0,
dz
k  n i 1
,n
To interpret this parameter the Taylor expansion of the present value function is
considered:
Bi  z  h   Bi  z  
dBi  z 
 h  Rest  h    Diabs  h
dz
0 for h 0
The above equation means that the present value Bi is reduced by Diabs 100 euro if the
interest rate rises by 1 percentage point. Vice versa, the present value increases by
Diabs 100 euro if the interest rate decreases by 1 percentage point.
52. If the absolute duration is divided by the present value Bt  z  , the modified duration
Di mod is derived:
e
Di mod  z  : 
Di  z 

Bi  z 
abs

  k  n  i  0, 5 Z 1  z 
 k  n  i  0,5  1
ik
k  n i 1
e

k  n i 1
Zik 1  z 
with i  0,
 k  n  i  0,5 
,n .
This parameter may be interpreted as follows: The present value is reduced by
Dimod  h % if the interest rate rises by h percentage points. However, the present value




increases by Dimod  h % if the interest rate drops by h percentage points. Moreover, in
structural terms, the modified duration may be compared to the average run-off time:
Dimod corresponds to an average run-off time exclusively weighted by the future
discounted payment flows.
Example: Dimod  4 : In the case of an interest rate increase of 0.5 percentage points (for


instance, from 3.5% to 4.0%) the present value is reduced by Dimod  0.5 %  2.0% .
In calculating this parameter depending on the accident year an interest rate should be
used which matches the average term. The guideline for the average term is the overall
run-off time. To obtain parameters which are independent of the accident year the
following procedure would be appropriate: For a fixed interest rate the modified durations
for each accident year, weighted by the sums of all future payments in an accident year,
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Best-Estimate User Guide
may be averaged. D mod is derived. By analogy, the formula for the modified duration
when using a non-constant term structure is derived:
e
Di mod  z1 ,
, zt  
  k  n  i  0, 5 Z 1  z
ik
k  n i 1
e

k  n i 1
Z ik 1  zk n i 1 
 k  n  i  0,5  1
k  n  i 1

 k  n  i  0,5 
From this, in turn, a modified duration which is independent of the accident year may be
calculated in the form just described.
All of the described results are output on the worksheet Barwerte. The output in this
worksheet shifts depending on the size of the evaluated run-off triangle. Hence, the
output cannot be described exactly but only relatively. All results are presented in the
range Barwerte!B6:AS35. First of all, the future cash flows as payments are shown in a
triangular form in this range. Next to it there are the payments for the tail. This column is
labelled “tail payments“. The overall sum of all future cash flows in an accident year are
output in the adjacent column labelled “sum“. This sum is equal to the overall reserve
output on the worksheet kum.Schadenzahlung – either including a tail or not depending
on the options set by the user. The future cash flows are discounted using up to three
different yield curves provided by the user as described above. The default order should
be: variant for interest rate reduction, standard yield curve and variant for interest rate
increase. In this order the output of the calculations is carried out. In each case the
discounted values are labelled “disc.(downw.)“, “disc.“, “disc.(upw.)“, respectively.
Underneath the sums of all the three columns are displayed as well as the difference to
the undiscounted reserves as percentage (including tail or not including a tail). In the
same way the discounted overall reserve using an average interest rate is output
together with the difference to the undiscounted overall reserve as percentage.
Afterwards the absolute duration is output. This column is labelled “abs. dur.“. After that
the modified duration is output labelled “mod. dur.“, followed by the modified overall
duration (“total“). These three columns are calculated using the standard yield curve. The
next three columns contain the absolute and modified duration and overall modified
duration calculated using the constant average interest rate. These columns are therefore
labelled “(z const.)“ in addition.
4.10 p%-Quantile
53. For the overall reserve without and with tail a p%-quantile q is calculated based on a log-

normal distribution using the parameters  ,  2 ln R


 . , 2 :
q  exp       1  p   , 0  p  1 .
The parameters are derived, for instance, according to the method of moments from the
expected value and the standard deviation of the log-normal distribution. For the
determination of the two parameters estimators from the chain-ladder method are used:
n
 Rˆ
i 0
i
 R for the mean and the mean square error of prediction mse  R  for the
variance. According to the formulas the following results are yielded:
 2  ln 1  mse  R  R 2 
  ln  R    2 2
The percentage p may be freely determined by the user both for the overall reserve
without tail and for the overall reserve with tail independently of each other. The p%-
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Best-Estimate User Guide
quantile for the overall reserve without tail is displayed in cell kum.Schadenzahlung!K46
whereas the p%-quantile of the overall reserve including tail can be found in cell T46.
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