4.2.1. First Year Capital Comparison
A first value of interest is the required capital for the first year
The Merz–Wüthrich method only provides
, while the COT method was originally designed for
(
A4). An assumption concerning the link between
and
is therefore to be made. Since
follows a mixture of binomial distributions for a generally large number of rvs, its distribution can be approximated relatively well by a normal distribution (this approximation may lead up to 20% underestimation of the risk depending on the number of exposures
n). Note that the number of rvs from the binomial is not a uniform variable but a sum of uniform variables, which diminishes the probability of very large or very low values, thus making the normal approximation better than for a simple uniform number of rvs. Normal distribution fixes the relation between
and
:
In our case,
and
Combining Equations (
34) and (
35), we can obtain an approximate analytical value for
,
The COT method, such as explained in
Appendix D, was designed for real insurance data. In particular, parameter
b models the dependence between relative loss increments. In the case of our model, the relative loss increments are uncorrelated, which points to the choice of parameter
instead of the value chosen with mean time to payment. The choice of coefficient
is also arbitrary. Indeed,
determines the proportion of the risk that is due to the jump part of the process. For our process, there is no “special” type of behaviour that the model could have and that would increase the risk. Therefore, we choose
. In general, the type of data is not known, in particular the dependency between loss increments is not known. Thus, we are also interested in the results given by the COT method applied the standard way. We therefore also compute the COT estimator with
chosen according to Formula (
A5). In our case,
b cannot be chosen like in the formula, as the pattern used in the mean time to payment computation is a paid pattern that we do not have for our model. For the jump case, we choose
as for a long-tail process. Indeed, the (incurred) pattern of our n-step process corresponds generally to the type of patterns that one can find in long (or possibly medium) tail lines of business. We will refer to the two variations of the method as “COT method with jump part” for the version with standard
and
and “COT method without jump part” for the version with
and
.
In the case of an
n-step Bernoulli model triangle, we notice that the accident-year (incremental) patterns are given by
The first factor is simply the total number of exposures remaining to be realized, the first row not being counted because it is finished. Since, at each step but the last, half of the current exposures of the process are expected to be realized, the number of exposures remaining for each unfinished is expected to be the number of exposures remaining in the original triangle divided by 2 for each past step that is not the last step. Hence, the expected remaining exposures are
for the lines that are not finished after
i steps, plus
for the line that finishes precisely after
i steps. The pattern designates the results of the binomial random variable and not the number of exposures. However, since the rvs (random variable
X) are independent and have the same expectation, the numerator and denominator are both multiplied by
p leaving the result unchanged.
In particular, with the help of Equation (
37) and after few manipulations, we have that
This result uses the property of martingales that the variance of the sum of martingale increments is equal to the sum of variances. An analogous property is false for the
. However, we get for the COT model without jump part, an approximation (see
Ferriero 2016)
Moreover, if we assume that the normal approximation is not an approximation but indeed an exact distribution, it can be shown through straightforward calculations that this expression becomes an exact result for the required capital for the first year (see
Ferriero 2016).
The Merz–Wüthrich method is the one posing the most problems. Indeed, the triangles generated with our process are very noisy in the sense that the simulated triangles can quite often have many zeros. The Mack hypotheses, on which the Merz–Wüthrich method is based, are multiplicative in nature and, therefore, very sensitive to zeros. If there is a zero in the first column of a triangle, Mack’s estimation fails to compute the parameters
. There are more robust ways, such as the one developed in
Busse et al. (
2010), to calculate these estimators. However, all of them (except removing the line) fail if an entire row of the triangle is 0. This happens quite often for
n small. If
n is large, the problem becomes, as we explain in
Section 3.5, that the process terminates on average in time
, which means that the largest part of the triangle shows no variation at all and gives
. In order to eliminate all these problems, we simulate our test triangles with
n = 100,000 and work on the truncated top side of the triangle of size
, where 5 is a safety margin to insure that the run-off of the process is finished or at least almost finished.
We set
, which is a more realistic value given the high number of policies, simulate 500 triangles and, for each of them, calculate the first year capital
using the theoretical value, the COT method with and without jumps and the Merz–Wüthrich method. We display, in
Table 1, the mean capital and the standard deviation of the capital around that mean over the 500 triangles. We also calculate the mean absolute deviation (MAD)
and mean relative absolute deviation (MRAD)
with respect to the theoretical value using standard and robust mean estimation.
Note that, in our example, the relative risk, i.e., the first year capital relative to the reserves volume, is about 18% (the reserves are approximately 100 and the first year capital approximately 18), which is a realistic value. The reserves in our model can be easily computed by the close formula
, as proved in
Appendix C. As a point of comparison, using the prescription of the Solvency Standard Formula, we find a stand-alone capital intensity (SCR/Reserves) between 14% to 26% for the P&C reserves. Given the type of risks we are considering here, it is logical that the capital intensity should be at the lower end of the range. By the way, we also see, as expected, that the average claim is much smaller than the maximum claim (100,000) given the fact that the chances that 100,000 independent policies claim at the same time with such a low probability of claims (
) is practically nil.
The results presented in
Table 1 are striking. While the COT method gives answers close to the theoretical value with a slight preference, as expected, for the COT without jumps, the Merz–Wüthrich method is way off (1356.6% off the true value), and the true result is not even within one standard deviation away. The coefficient of variation
for this method is more than 59% while in all the other cases it hovers around 21%. There are many explanations for this. Looking at triangles and analysing the properties of the methods allows us to understand those results. The true risk depends on the number of rvs remaining to be realized. In most cases, only the few last underwriting years are truly important in that matter because the others will be almost fully developed. For Merz–Wüthrich, as most of the volatility of the process will appear on the first step, the most crucial part of the triangle is the last line of the triangle, which is the only process representation at this stage of development. If the latter is large, it influences the Merz–Wüthrich capital in the same direction. Merz–Wüthrich interprets a large value as: “Something happened on that accident year, there is going to be more to pay than expected”. The logic behind our model is different, through the “fixed number of rvs” property, it is: “What has been paid already needs not to be paid anymore”. A large value on the last line of the triangle is therefore likely to indicate that few rvs remain to be realized, which implies smaller remaining risk. This explains negative correlation because the same cause has the exact opposite effect on the result.
How can we explain the difference of magnitude in the estimated capitals? This may be due to the fact that our model is additive while Merz–Wüthrich is multiplicative. If a small number appears in the first column and then the situation reestablishes on the second step by realizing a larger number of exposures, we know that this is irrelevant for future risk. However, Mack and subsequently Merz–Wüthrich don’t consider the increase but the ratio. If the first value is small, the ratio may be large. However, the estimated ratio
is the mean of the ratios weighted with the value of the first column, i.e., Equation (
A2) can be rewritten as
Therefore, cases with large ratios, such as described before, will not appear in but in . The Merz–Wüthrich (Mack) method considers that small and large values are as likely to be multiplied by a factor, which is not the case with our model for which small values are likely to be multiplied by large factors and large values are likely to be multiplied by small factors.
This is confirmed by plotting and comparing the distributions of the different capital measurements (
Figure 3), we can notice that, while the true capital and the two COT capitals seem to follow a normal distribution, the distribution of the Merz–Wüthrich capital seems to follow rather a log-normal distribution.
Another interesting statistic to understand how related these capital measurements are is the correlation between them. Computing the correlation matrix yields the results presented in
Table 2, we see that the correlation is almost 100% for the two COT estimates and the true value. Indeed, with or without jumps, the COT method is very close to the theoretical result. This is partially due to the fact that the ultimate distribution is known and that all these methods simply multiply the ultimate risk by a constant. The correlation is not exactly 100% due to the stochasticity induced by the simulations used to calculate ultimate risk in the COT methods. The Merz–Wüthrich capital however shows a negative correlation. The standard and robust estimators are very different, which suggests the presence of very large values of Merz–Wüthrich capital and departure from normality. This is confirmed by
Figure 3 where the distribution in the bottom right plot is very different from a Gaussian. It indicates in particular that the robust estimator is more representative of the data. A correlation of −46% is rather strong. It is not true though that a small true capital implies a large Merz–Wüthrich capital, nor the opposite, but there is a real tendency among large values of true capital to coincide with relatively small values of Merz–Wüthrich capital.
4.2.2. Risk Margin Comparison
Another important quantity to study is the risk margin defined in Equation (
21). We compare here the results obtained with the COT method described in
Appendix D to those obtained from theory. Note that we cannot do this for the Merz–Wüthrich method as it is only giving the variation for the first year.
Our methodology is very close to the one for the first-year capital using normal approximation and Equations (
34) and (
35). Assume
known, we can then generalise Equation (
36), to get the following expression
We then use the normality assumption to write
thus obtaining a theoretical form for the tail value at risk, given the triangle developed up to calendar-year
. Our methodology, starting from a triangle of realized rvs, is to complete it
R times using the Bernoulli model and to calculate on each completed triangle
according to the formula of Equation (
39). By taking the mean over all
R triangles, we obtain the required capital for calendar-year
i that we sum up and multiply by the cost of capital (chosen here at 6%, as in the Solvency II directive) to obtain the risk margin.
In this case, we do not need to simulate truncated large triangles to make our comparison. Indeed, both the COT method and the theoretical simulation method work on small triangles. However, for the results to be similar and to avoid too frequent “zero risk left” situations, we still use a truncated large triangle like for the first-year capital comparison, i.e., triangles of size 19 and with
rvs. Like for the first-year capital, we simulate 500 triangles from the process and, for each of them, calculate the capital required at each consecutive year and the risk margin using for the theoretical simulation method
triangle completions and for the COT
and
(without jump part) and
(long tail) and
from Equation (
A5) (with “jump part”). In
Table 3 and
Figure 4, we can observe the results obtained on average and the measures of deviation over the 500 triangles.
As we just saw, assuming normality, the first year capital of the COT method without jump part is an exact result. However, for
(still assuming normality), the method gives
However, from Schwarz inequality, which also holds for conditional expectation, for any positive integrable random variable
Y and any
algebra
, the COT method without jumps is systematically overestimating the true capital, as we can see in
Figure 4. However, the overestimation is not very big (
Table 3) and the method replicates reasonably well the form of the actual yearly capital. The average relative absolute error of the risk margin is 10.57% (see results in
Table 3). We do not show the results for the capital at each year, but they lead to a similar message with the error increasing with the years as the capital itself decreases. The same method with jumps has less success with 26.52% of absolute error. This error is always underestimation, which is also true for each year. The error on the capital is always bigger with jumps. It is only at the end (calendar year 13 here) that the capital estimation is better with jumps and these values are almost 0, so they are not very relevant for the risk margin.
In general, one can see (
Table 1 and
Table 3) that, for our
n-steps model, the COT method without jumps is the one that performs the best. If we compute the autocorrelation of consecutive loss increments, we obtain 5% of mean correlation, which is close to independence. The independence situation corresponds to the calibration of the COT method with
, thus explaining why the COT method without jumps provides the best results. This raises the question of what value of
b would give the risk margin the closest to the benchmark. To answer this, we simulate another 100 triangles and calculate each time the risk margin with the benchmark method and with both COT methods with and without jumps, for all values of parameter
b between 0.3 and 1 by steps of 0.01. For the COT method without jumps, we find that the fitted values for
b are between 0.52 and 0.53, which is very close to the
that we have been using. For the COT method with jumps, the mean best
b is also close to 0.5. However, we find some best
b observations that are below 0.5, which stands for negative dependence between accident years and is not accepted by the COT method. In this case, the best
b is much further than the one that has been used (0.75) by SCOR for real data. This is not unexpected since the method yields rather poor results (
Table 3) for our
n-steps model. (In the
Appendix E we discuss the numerical stability of the above approximations. Furthermore, in the
Appendix F we discuss the one-year capital for the first period as proportion to the sum of all the one-year capitals over all the periods.)