1. Introduction
After the famous chain-ladder method, the Bornhuetter-Ferguson method (hereinafter “BF method”—see Bornhuetter and Ferguson 1972) is one of the methods practicing actuaries most use for the projection of non-life paid or incurred triangles. The BF method relies on two sets of parameters:
-
the claims development patterns derived from the incurred or paid triangles; and
-
the a priori estimates for the ultimate claims amount—such estimates can come from pricing models or from any other sources.
Using those sets of parameters, the BF method estimates the best estimate of the claims liability.
Following the development of the prediction error estimate for the chain-ladder method (Mack 1993, 1999), Mack (2008) proposed the estimate of the prediction error for BF. In addition, different skewness estimates for the chain-ladder method were developed (Salzmann, Wuthrich, and Merz 2012; Dal Moro 2013). All such estimates are done in a distribution-free framework.
To complete our knowledge of the BF method in a distribution-free framework, the missing piece is the skewness of the BF method. With knowledge of the first three moments, we can then estimate the different quantiles of the reserving distributions (see Dal Moro and Krvavych 2017). Such quantiles will certainly be useful in the context of International Financial Reporting Standard 17 (IFRS 17), which directs (re)insurance companies to publish additional quantile disclosures.
In this paper, we propose an approach to estimating this skewness relying mainly on the stochastic BF framework proposed by Mack (2008). The paper is divided into the following sections:
-
Section 2 describes the stochastic BF model of Mack (2008) and extends it to cover the assumptions necessary to estimate the third moment.
-
Section 3 provides an estimate of the skewness of the BF method per accident year.
-
Section 4 provides the estimate of the skewness over all accident years.
-
A final section provides a few numerical examples.
Remark: An Excel sheet developed to estimate the presented formulae is available at https://drive.google.com/open?id=1iRPEnD8eVOECPyR4oZl_Ezd89agVHYAa.
2. The stochastic model underlying the BF method
Let \(C_{i,k}\) denote the cumulative claims amount (either paid or incurred) of accident year i after k years of development, \(1 \leq i,\ k \leq n,\) and \(\nu_{i}\) be the premium volume of accident year i where n denotes the most recent accident year. Then \(C_{i,n + 1 - i}\) denotes the currently known claims amount of accident year i. Let further \(S_{i,k} = C_{i,k} - C_{i,k - 1}\) denote the incremental claims amount (with \(C_{i,0} = 0\)) and Ui the (unknown) ultimate claims amount of accident year i. Then \(R_{i} = U_{i} - C_{i,n + 1 - i}\) is the (unknown true) claims reserve for accident year i. Finally, let \(S_{i,n + 1} = U_{i} - C_{i,n}\) be the incremental claims amount after development year n (tail development).
Bornhuetter and Ferguson (1972) introduced their method to estimate Ri as follows:
\[{\widehat{R}}_{i}^{\text{BF}} = {\widehat{U}}_{i}\left( 1 - {\widehat{z}}_{n + 1 - i} \right),\]
where \({\widehat{U}}_{i} = \nu_{\text{i}}{\widehat{q}}_{i}\) with a prior estimate \({\widehat{q}}_{i}\) for the ultimate claims ratio \(q_{i} = \frac{U_{i}}{\nu_{i}}\) of accident year \(i\), \({\widehat{z}}_{k} \in \lbrack 0;1\rbrack\) is the estimated percentage of the ultimate claims amount that is expected to be known after development year k.
The BF stochastic model developed in Mack (2008) relies on the following assumptions related to the increments \(S_{i,k}\):
-
BF1: All increments \(S_{i,k}\) are independent.
-
BF2: There are unknown parameters xi, yk such that
-
BF3: There are unknown proportionality constants \(s_{k}^{2}\) with \(\text{Var}\left( S_{i,k} \right) = x_{i}s_{k}^{2}\).
On the basis of these three assumptions, one can estimate the prediction error of BF (see Mack 2008). The prediction error, usually denoted as the MSEP (mean squared error of prediction), consists of two components: the process error and the estimation error. Whereas one can basically always calculate the estimation error via the laws of error propagation, for the process error Mack (2008) developed a stochastic model of the claims process.
To estimate the skewness of the BF method, we need a fourth assumption, which is derived by analogy with the skewness estimation of the chain-ladder model (see Dal Moro 2013):
- BF4: There are unknown proportionality constants \(t_{k}^{3}\) with \(K\left( S_{i,k} \right) = x_{i}^{\frac{3}{2}}t_{k}^{3}\) where \(K\left( S_{i,k} \right) = E\left\lbrack \left( S_{i,k} - E\left( S_{i,k} \right) \right)^{3} \right\rbrack\).
Following Mack (2008), we have the following with x1,… xn known:
\[\ {\widehat{y}}_{k} = \frac{\sum_{i = 1}^{n + 1 - k}S_{i,k}}{\sum_{i = 1}^{n + 1 - k}x_{i}}\tag{1}\]
is a best linear unbiased estimate of yk, and
\[{\widehat{s}}_{k}^{2} = \frac{1}{n - k}\sum_{i = 1}^{n + 1 - k}\frac{\left( S_{i,k}\ - \ x_{i}{\widehat{y}}_{k} \right)^{2}}{x_{i}} \tag{2}\]
is an unbiased estimate of s2k.
And, following BF4:
\[{\widehat{t}}_{k}^{3} = \frac{1}{n - k}\sum_{i = 1}^{n + 1 - k}\frac{\left( S_{i,k}\ - \ x_{i}{\widehat{y}}_{k} \right)^{3}}{x_{i}^{\frac{3}{2}}} \tag{3}\]
is an unbiased estimate of t3k.
Having estimated the parameters \({\widehat{y}}_{k},\ {\widehat{s}}_{k}^{2},\ and\ {\widehat{t}}_{k}^{3}\) (see Mack (2008) for details on the estimation process), we can calculate the BF claims reserve by
\({\widehat{R}}_{i}^{\text{BF}} = {\widehat{U}}_{i}\left( {\widehat{y}}_{n + 2 - i} + ... + {\widehat{y}}_{n + 1} \right) = {\widehat{U}}_{i}\left( 1 - {\widehat{z}}_{n + 1 - i} \right)\) with \({\widehat{z}}_{k} = {\widehat{y}}_{1} + ... + {\widehat{y}}_{k}.\)
The properties of these estimators are given below:
-
\({\widehat{y}}_{1},...,{\widehat{y}}_{n + 1}\) are pairwise (slightly) negatively correlated as they have to add up to unity.
-
\({\widehat{y}}_{1},...,{\widehat{y}}_{n + 1},\) and \({\widehat{z}}_{1},...,{\widehat{z}}_{n + 1}\) are independent from \({\widehat{U}}_{1},...,\ {\widehat{U}}_{n}\).
-
\({\widehat{R}}_{i}^{\text{BF}}\) and Ri are independent (due to BF1).
-
\(E\left( {\widehat{U}}_{i} \right) = E\left( U_{i} \right) = x_{i}\text{ for }1 \leq i \leq n\).
-
\(E\left( {\widehat{y}}_{k} \right) = y_{k}\text{ for }1 \leq k \leq n + 1\) and therefore
\(E\left( {\widehat{z}}_{k} \right) = z_{k}\text{ for }1 \leq k \leq n + 1\).
-
\(E\left( {\widehat{s}}_{k}^{2} \right) = s_{k}^{2}\text{ for }1 \leq k \leq n + 1\).
-
\(E\left( {\widehat{t}}_{k}^{3} \right) = t_{k}^{3}\text{ for }1 \leq k \leq n\).
For the last four bullet points, we simply assume that the actuary’s selections are unbiased.
Having described the stochastic BF model and its assumptions, we devote the next section to estimate the skewness of the BF method for one accident year.
3. Skewness of the BF method per accident year
The mean skewness of any reserve estimate \({\widehat{R}}_{i}\) is defined as
\[\text{SKEW}\left( {\widehat{R}}_{i} \right) = E\left\lbrack \left( {\widehat{R}}_{i} - R_{i} \right)^{3}│S_{i,1},...,S_{i,n + 1 - i} \right\rbrack.\]
Due to BF1, \({\widehat{R}}_{i}^{\text{BF}}\) and Ri are taken to be commonly independent from \(S_{i,1},...,S_{i,n + 1 - i}\). Hence,
\[\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) = E\left\lbrack \left( {\widehat{R}}_{i}^{\text{BF}} - R_{i} \right)^{3} \right\rbrack.\]
In Appendix A, it is proved that
\[\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) = K\left( {\widehat{R}}_{i}^{\text{BF}} \right) - K\left( R_{i} \right)\tag{4}.\]
Following BF4, we have
\[K\left( R_{i} \right) = K\left( S_{i,n + 2 - i} \right) + ... + K\left( S_{i,n + 1} \right) = x_{i}^{\frac{3}{2}}\left( t_{n + 2 - i}^{3} + ... + t_{n + 1}^{3} \right),\]
which is estimated by
\[\widehat{K}\left( R_{i} \right) = {\widehat{U}}_{i}^{\frac{3}{2}}\left( {\widehat{t}}_{n + 2 - i}^{3} + ... + {\widehat{t}}_{n + 1}^{3} \right).\]
As for \(K\left( {\widehat{R}}_{i}^{\text{BF}} \right) = K\left\lbrack {\widehat{U}}_{i}\left( 1 - {\widehat{z}}_{n + 1 - i} \right) \right\rbrack\), we use the formula below, which holds when X and Y are independent:
\(\begin{align}
K\left(XY\right) &= K(X)K(Y) + K(X)E(Y) \lbrack{3\text{Var}(Y) + E(Y)^{2}}\rbrack \\&\ \ \ \ + K(Y)E(X)\lbrack {3\text{Var}(X) + E(X)^{2}} \rbrack \\&\ \ \ \ + 6E(X)E(Y)\text{Var}(Y)\text{Var}(X).
\end{align}\)
Using the above formula, we have
\(\begin{align}
K\left( {\widehat{R}}_{i}^{\text{BF}} \right) =& - K\left\lbrack {\widehat{U}}_{i} \right\rbrack K\left( {\widehat{z}}_{n + 1 - i} \right) \\&+ K\left\lbrack {\widehat{U}}_{i} \right\rbrack E\left( 1 - {\widehat{z}}_{n + 1 - i} \right)\left\lbrack 3Var\left( {\widehat{z}}_{n + 1 - i} \right) + E\left( 1 - {\widehat{z}}_{n + 1 - i} \right)^{2} \right\rbrack\ \\& - K\left\lbrack {\widehat{z}}_{n + 1 - i} \right\rbrack E\left( {\widehat{U}}_{i} \right)\left\lbrack 3Var\left( {\widehat{U}}_{i} \right) + E\left( {\widehat{U}}_{i} \right)^{2} \right\rbrack \\& + 6\ E\left( {\widehat{U}}_{i} \right)E\left( 1 - {\widehat{z}}_{n + 1 - i} \right)\text{Var}\left( {\widehat{U}}_{i} \right)\text{Var}\left( {\widehat{z}}_{n + 1 - i} \right).
\end{align}\)
Estimators of \(E\left( {\widehat{U}}_{i} \right),\ Var\left( {\widehat{U}}_{i} \right), E\left( 1 - {\widehat{z}}_{n + 1 - i} \right),\) and \(Var\left( {\widehat{z}}_{n + 1 - i} \right)\) are provided in Mack (2008) (see Appendix B for details on these estimators). There remains to estimate \(K\left\lbrack {\widehat{U}}_{i} \right\rbrack\text{ and K}\left( {\widehat{z}}_{n + 1 - i} \right)\).
As for \(K\left\lbrack {\widehat{U}}_{i} \right\rbrack\), we have to make an assumption on the underlying distribution of \({\widehat{U}}_{i}\). As a generally accepted distribution used in reserving contexts, we will therefore assume a lognormal distribution with parameters μ and σ derived from \(E\left( {\widehat{U}}_{i} \right)\text{ and Var}\left( {\widehat{U}}_{i} \right)\). The parameters and resulting skewness are given by
\(\sigma = \sqrt{\ln\left( 1 + \frac{\text{Var}\left( {\widehat{U}}_{i} \right)}{{E\left( {\widehat{U}}_{i} \right)}^{2}} \right)}.\)
\(K\left\lbrack {\widehat{U}}_{i} \right\rbrack = \left( 2 +\text{exp}\left( \sigma^{2} \right) \right)\frac{{\text{Var}\left( {\widehat{U}}_{i} \right)}^{2}}{E\left( {\widehat{U}}_{i} \right)}\).
As for \(K\left( {\widehat{z}}_{n + 1 - i} \right)\), we note that due to the slightly negative correlations between \({\widehat{y}}_{1},...,{\widehat{y}}_{n + 1}\), we have to make the following approximations:
\(\ K\left( {\widehat{z}}_{k} \right) \approx min\left\lbrack K\left( {\widehat{y}}_{1} \right) + ... + K\left( {\widehat{y}}_{k} \right);K\left( {\widehat{y}}_{k + 1} \right) + ... + K\left( {\widehat{y}}_{n + 1} \right) \right\rbrack.\)
As we have
\[K\left( {\widehat{y}}_{k} \right)\ = K\left( \frac{\sum_{i = 1}^{n + 1 - k}S_{i,k}}{\sum_{i = 1}^{n + 1 - k}{\widehat{U}}_{i}} \right) = \frac{\sum_{i = 1}^{n + 1 - k}{K\left( S_{i,k} \right)}}{\left( \sum_{i = 1}^{n + 1 - k}{\widehat{U}}_{i} \right)^{3}}\]
and
\[\widehat{K}\left( S_{i,k} \right) = {\widehat{U}}_{i}^{\frac{3}{2}}{\widehat{t}}_{k}^{3},\]
we finally get
\[K\left( {\widehat{y}}_{k} \right)\ = \frac{{\widehat{t}}_{k}^{3}\sum_{i = 1}^{n + 1 - k}{\widehat{U}}_{i}^{\frac{3}{2}}}{\left( \sum_{i = 1}^{n + 1 - k}{\widehat{U}}_{i} \right)^{3}}\]
This completes the estimation of the different elements of \(\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right)\).
4. Skewness of the BF method over all accident years
Having estimated the skewness of the BF method by accident year, we now aggregate these elements over all accident years. To do so, we use the Fleishman polynomials (see Fleishman 1978). First, we assume that the centralized and normalized copy of the risk value Xi of the ith class, \(\widehat{X_{i}} = \frac{X_{i} - E\left( X_{i} \right)}{E\left( X_{i} \right)\ \text{CoV}_{X_{i}}}\) (where CoV denotes the coefficient of variation), is estimated by the Fleishman polynomial structure of a standard normal random variable. In particular, we consider the following two different cases:
-
\({\widehat{X}}_{i} = P_{2}\left( Z_{i} \right) = a_{i}Z_{i} + b_{i}\left( Z_{i}^{2} - 1 \right)\), where \(Z_{i}\) denotes the standard normal distribution. Such a case is suitable for estimating the skewness of a risk portfolio profile when the confidence level is approximated using skewness only.
-
\({\widehat{X}}_{i} = P_{3}\left( Z_{i} \right) = a_{i}Z_{i} + b_{i}\left( Z_{i}^{2} - 1 \right) + c_{i}Z_{i}^{3}\). This is suitable for estimating skewness and kurtosis of the risk portfolio risk profile when the confidence level is approximated using both skewness and kurtosis.
The coefficients of the polynomials P2 and P3 are calibrated using the method of moments by matching the second and third moments of P2(Zi) and the second, third, and fourth moments of P3(Zi) to 1 (standard deviation of \(\widehat{X_{i}}\)), γi (skewness of Xi), and ιi + 3 (noncentralized or absolute kurtosis of Xi), respectively.
The coefficients of P2 can be analytically expressed by solving the following system of equations:
\(\begin{align}\left\{ \begin{matrix} 1 = a_{i}^{2} + 2b_{i}^{2}. \\ \gamma_{i} = 6a_{i}^{2}b_{i} + 8b_{i}^{3}. \\ \end{matrix} \right.\end{align} \tag{5}\)
The system (5) is reduced to
\(\left\{ \begin{matrix} a_{i} = \sqrt{1 - 2b_{i}^{2}.} \\ \gamma_{i} = 6b_{i} - 4b_{i}^{3}. \\ \end{matrix} \right.\tag{6}\)
Such a system is easily solved. The roots of the cubic equation (6) can be found using Cardano’s formula (see, e.g., Abramowitz and Stegun 1972). If we denote
\[\varphi = arccos\left( - \frac{\gamma_{i}^{3}}{\sqrt{8}} \right).\]
Then, the only real root of equation (6) is
\[b_{i} = \sqrt{2}\text{cos}\left( \frac{\varphi}{3} + 4\frac{\pi}{3} \right).\]
Having estimated the above parameters of the Fleishman polynomial, we define the total reserve value across the portfolio of m risks as
\[X_{\Sigma} = \sum_{i = 1}^{m}X_{i}\]
where each ith risk value is approximated by the Fleishman polynomial of a standard normal random variable:
\[X_{i} \approx \text{CE}_{i}\ \left( 1 + \text{CoV}_{i}\ P_{3}\left( Z_{i} \right) \right),\]
where CE denotes the central best estimate of X.
It is clear that \(\text{CE}_{\Sigma} = \sum_{i = 1}^{m}\text{CE}_{i}\). It should also be noted that setting \(c_i = 0\) reduces the problem to simply approximating:
\[X_{i} \approx \text{CE}_{i}\ \left( 1 + \text{CoV}_{i}\ P_{2}\left( Z_{i} \right) \right).\]
As discussed earlier, all the stand-alone risks interact between each other according to a Gaussian dependence structure, of which the linear correlations ρij (coefficients of a Gaussian copula) are given as in Mack (2008):
\[\rho_{\text{ij}} = \frac{{\widehat{z}}_{n + 1 - j}\left( 1 - {\widehat{z}}_{n + 1 - i} \right)}{{\widehat{z}}_{n + 1 - i}\left( 1 - {\widehat{z}}_{n + 1 - j} \right)}.\]
Skewness
We compute the third central moment of XΣ:
\[E\left\lbrack \left( X_{\Sigma} - \text{CE}_{\Sigma} \right)^{3} \right\rbrack = E\left\lbrack \left( \sum_{i = 1}^{m}{\sigma_{i}\ P_{3}\left( Z_{i} \right)} \right)^{3} \right\rbrack\]
\[\begin{align}E&\left\lbrack \left( X_{\Sigma} - \text{CE}_{\Sigma} \right)^{3} \right\rbrack \\&= \sum_{i = 1}^{m}{\sigma_{i}^{3}\ \gamma_{i}} + 3\sum_{\text{ij}}^{}{\sigma_{i}^{2}\sigma_{j}E\left\lbrack {P_{3}\left( Z_{i} \right)}^{2}P_{3}\left( Z_{j} \right) \right\rbrack} \\&\ \ \ \ + 6\sum_{\text{ijk}}^{}{\sigma_{i}\sigma_{j}\sigma_{\text{k}}E\left\lbrack P_{3}\left( Z_{i} \right)P_{3}\left( Z_{j} \right)P_{3}\left( Z_{k} \right) \right\rbrack}\end{align}\]
where \(E\left\lbrack {P_{3}\left( Z_{i} \right)}^{3} \right\rbrack = \gamma_{i}\), as the Fleishman polynomial coefficients are calibrated so that the polynomial has skewness γi for the ith stand-alone risk profile. In formula (7), the summation term with multiple 3 has \(\begin{pmatrix} m \\ 2 \\ \end{pmatrix}\) different subterms, and the summation term with multiple 6 is relevant if \(m \geq 3\) and has \(\begin{pmatrix} m \\ 3 \\ \end{pmatrix}\) different subterms.
The following components of formula (7) above are provided here only for the partial case where P2 is used—i.e., ci = 0:
\[E\left\lbrack {P_{3}\left( Z_{i} \right)}^{2}P_{3}\left( Z_{j} \right) \right\rbrack = 2\rho_{\text{ij}}\left( 2a_{i}a_{j}b_{i} + \left( a_{i}^{2} + 4b_{i}^{2} \right)b_{j}\rho_{\text{ij}} \right).\tag{8}\]
\[\begin{align}E&\left\lbrack P_{3}\left( Z_{i} \right)P_{3}\left( Z_{j} \right)P_{3}\left( Z_{k} \right) \right\rbrack \\&= 2\left( a_{j}a_{k}b_{i}\rho_{\text{ij}}\rho_{\text{ik}} + a_{j}a_{i}b_{k}\rho_{\text{jk}}\rho_{\text{ik}} + a_{i}a_{k}b_{j}\rho_{\text{ij}}\rho_{\text{jk}} \right) \\&\ \ \ \ + 8b_{i}b_{k}b_{j}\rho_{\text{ij}}\rho_{\text{ik}}\rho_{\text{jk}}.\end{align}\tag{9}\]
The skewness γΣ is then calculated as follows:
\[\gamma_{\Sigma} = \frac{E\left\lbrack \left( X_{\Sigma} - \text{CE}_{\Sigma} \right)^{3} \right\rbrack}{\left( \text{CE}_{\Sigma}\ \text{CoV}_{\Sigma} \right)^{3}}.\]
In the numerical examples that follow, the parameters ai, bi are provided, the correlation matrices are provided in Appendix C, and the values for σi correspond to the msep(Ri), which are estimated using Mack (2008) formulae. Details of calculations can be found on the Excel sheet available at:
https://drive.google.com/open?id=1iRPEnD8eVOECPyR4oZl_Ezd89agVHYAa.
5. Numerical examples
The formulae above were tested on three triangles (see Appendix C). In all the cases, the coefficient of variation
of \({\widehat{U}}_{i}\) is kept constant at 10%. The correlation matrices used for applying the Fleishman
polynomials are also provided in Appendix C.
For the three triangles, it should be noted that one must add a tail factor as the cumulative patterns do not reach 100% (triangle 1 tail is 33.81%, triangle 2 tail is 32.45%, triangle 3 tail is 36.29%). As for these examples, the tail factors are the simple difference between the cumulated patterns coming from the application of the Mack (2008) method and 100%. Other methods could have been chosen to review the patterns such as increasing proportionally all incremental patterns to reach 100%.
As expected, the overall resulting skewness is positive in all cases. The skewness amounts are also reasonable as the corresponding distributions would not belong to extreme distributions (e.g., Pareto) but to smoother distributions (e.g., lognormal or gamma). Therefore, the proposed model seems to provide robust results on these three triangles.
Triangle 1
Triangle 1 |
Fleishmann polyn. |
|
i |
Ri BF |
msep(Ri) |
Skewness (Ri) |
bi |
ai |
k |
t3k |
s2k |
yk |
se(yk) |
zk |
se(zk) |
2017 |
193 654 347 |
12,0% |
0,172 |
0,0288 |
0,9992 |
1 |
2 386 009 |
22 883 |
4,47% |
0,35% |
4,47% |
0,35% |
2016 |
172 363 095 |
12,2% |
0,141 |
0,0235 |
0,9994 |
2 |
18 165 418 |
96 444 |
9,82% |
0,77% |
14,29% |
0,85% |
2015 |
132 336 647 |
12,9% |
0,168 |
0,0280 |
0,9992 |
3 |
-4 756 396 |
74 961 |
9,83% |
0,73% |
24,12% |
1,12% |
2014 |
117 146 970 |
12,9% |
0,151 |
0,0251 |
0,9994 |
4 |
5 727 410 |
166 684 |
9,21% |
1,16% |
33,33% |
1,61% |
2013 |
90 794 602 |
13,3% |
0,073 |
0,0121 |
0,9999 |
5 |
18 245 877 |
129 123 |
8,32% |
1,10% |
41,65% |
1,95% |
2012 |
96 347 577 |
12,0% |
-0,355 |
-0,0593 |
0,9965 |
6 |
127 640 479 |
225 383 |
6,48% |
1,57% |
48,13% |
2,23% |
2011 |
66 978 218 |
12,3% |
0,271 |
0,0452 |
0,9980 |
7 |
-902 325 |
23 032 |
3,28% |
0,56% |
51,41% |
2,16% |
2010 |
52 717"174 |
11,8% |
0,208 |
0,0346 |
0,9988 |
8 |
4 199 317 |
51 181 |
3,97% |
0,93% |
55,38% |
1,95% |
2009 |
50 921 257 |
11,5% |
0,316 |
0,0528 |
0,9972 |
9 |
-271 596 |
19 225 |
2,39% |
0,64% |
57,77% |
1,84% |
2008 |
37 067 585 |
11,7% |
0,303 |
0,0507 |
0,9974 |
10 |
298 |
4 798 |
2,03% |
0,37% |
59,80% |
1,80% |
2007 |
43 021 846 |
10,7% |
0,369 |
0,0617 |
0,9962 |
11 |
-695 855 |
25 322 |
1,01% |
0,98% |
60,81% |
1,51% |
2006 |
27 022 245 |
10,8% |
0,365 |
0,0610 |
0,9963 |
12 |
168 |
392 |
2,71% |
0,16% |
63,52% |
1,50% |
2005 |
26 090 865 |
10,9% |
- |
0,0000 |
1,0000 |
13 |
- |
- |
2,67% |
0,00% |
66,19% |
1,50% |
Total |
1 106 462 428 |
5,53% |
0,835
|
|
Ri BF: ultimate reserves resulting from the application of the BF method
msep(Ri): mean squared error of prediction according to Mack (2008)
t3k : proportionality constant between the third moment and the ultimate a priori BF reserves
s2k : proportionality constant between the second moment and the ultimate a priori BF reserves
yk : incremental incurred pattern
se(yk): standard error of yk according to Mack (2008)
zk : cumulative incurred pattern
se(zk): standard error of zk according to Mack (2008)
ai, bi: coefficients of the Fleishmann polynomials
Triangle 2
Triangle 2 |
Fleishmann polyn. |
|
i |
Ri BF |
msep(Ri) |
Skewness (Ri) |
bi |
ai |
k |
t3k |
s2k |
yk |
se(yk) |
zk |
se(zk) |
2017 |
258 640 859 |
11,0% |
0,224 |
0,0374 |
0,9986 |
1 |
23 900 |
853 |
0,31% |
0,06% |
0,31% |
0,06% |
2016 |
242 359 574 |
11,1% |
0,224 |
0,0374 |
0,9986 |
2 |
-786 889 |
34 579 |
5,85% |
0,42% |
6,17% |
0,42% |
2015 |
147 775 888 |
11,8% |
0,178 |
0,0297 |
0,9991 |
3 |
2 606 836 |
31 479 |
9,06% |
0,42% |
15,23% |
0,60% |
2014 |
91 588 543 |
13,1% |
0,155 |
0,0259 |
0,9993 |
4 |
-1 191 271 |
20 057 |
10,04% |
0,36% |
25,27% |
0,70% |
2013 |
71 936 537 |
14,2% |
0,130 |
0,0217 |
0,9995 |
5 |
-1 052 364 |
8 457 |
9,98% |
0,24% |
35,25% |
0,74% |
2012 |
122 712 262 |
12,9% |
0,154 |
0,0257 |
0,9993 |
6 |
338 178 |
17 065 |
8,20% |
0,36% |
43,44% |
0,82% |
2011 |
83 958 001 |
14,0% |
-0,055 |
-0,0091 |
0,9999 |
7 |
25 778 748 |
73 577 |
7,07% |
0,81% |
50,51% |
1,15% |
2010 |
72 027 374 |
13,5% |
-0,242 |
-0,0404 |
0,9984 |
8 |
40 375 942 |
173 275 |
5,93% |
1,35% |
56,44% |
1,77% |
2009 |
62 342 452 |
11,7% |
-0,545 |
-0,0913 |
0,9916 |
9 |
41 201 466 |
163 732 |
2,48% |
1,44% |
58,92% |
1,85% |
2008 |
58 602 275 |
11,7% |
0,307 |
0,0512 |
0,9974 |
10 |
613 285 |
8 138 |
1,84% |
0,36% |
60,76% |
1,81% |
2007 |
65 360 944 |
11,6% |
0,331 |
0,0552 |
0,9969 |
11 |
-77 515 |
3 469 |
1,35% |
0,27% |
62,11% |
1,79% |
2006 |
65 660 028 |
10,8% |
0,365 |
0,0610 |
0,9963 |
12 |
925 613 |
30 140 |
1,90% |
0,98% |
64,01% |
1,50% |
2005 |
42 271 185 |
11,0% |
- |
0,0000 |
1,0000 |
13 |
- |
- |
3,53% |
0,00% |
67,55% |
1,50% |
Total |
1 385 235 923 |
5,25% |
0,788
|
|
Triangle 3
Triangle 3 |
Fleishmann polyn. |
|
i |
Ri BF |
msep(Ri) |
Skewness (Ri) |
bi |
ai |
k |
t3k |
s2k |
yk |
se(yk) |
zk |
se(zk) |
2017 |
333 726 |
13,2% |
0,134 |
0,0224 |
0,9995 |
1 |
4 |
107 |
4,55% |
0,49% |
4,55% |
0,49% |
2016 |
269 832 |
13,9% |
-0,042 |
-0,0070 |
1,0000 |
2 |
3 703 |
282 |
12,08% |
0,82% |
16,63% |
0,96% |
2015 |
247 348 |
13,4% |
-0,197 |
-0,0328 |
0,9989 |
3 |
6 023 |
695 |
10,26% |
1,34% |
26,89% |
1,65% |
2014 |
219 403 |
13,6% |
-0,007 |
-0,0012 |
1,0000 |
4 |
2 294 |
292 |
9,41% |
0,91% |
36,30% |
1,88% |
2013 |
170 900 |
14,6% |
0,097 |
0,0162 |
0,9997 |
5 |
345 |
84 |
6,73% |
0,52% |
43,03% |
1,95% |
2012 |
164 880 |
14,7% |
0,117 |
0,0195 |
0,9996 |
6 |
244 |
118 |
4,29% |
0,64% |
47,32% |
2,06% |
2011 |
166 041 |
14,2% |
-0,155 |
-0,0259 |
0,9993 |
7 |
3 129 |
193 |
2,48% |
0,87% |
49,80% |
2,23% |
2010 |
163 283 |
14,3% |
0,181 |
0,0302 |
0,9991 |
8 |
-18 |
50 |
2,65% |
0,47% |
52,45% |
2,28% |
2009 |
169 810 |
13,0% |
0,008 |
0,0013 |
1,0000 |
9 |
2 801 |
310 |
3,64% |
1,29% |
56,09% |
2,25% |
2008 |
175 869 |
12,7% |
0,265 |
0,0442 |
0,9980 |
10 |
104 |
68 |
1,96% |
0,67% |
58,05% |
2,14% |
2007 |
188 390 |
10,7% |
0,254 |
0,0423 |
0,9982 |
11 |
2 418 |
246 |
2,21% |
1,51% |
60,26% |
1,52% |
2006 |
123 232 |
10,8% |
0,353 |
0,0589 |
0,9965 |
12 |
-1 |
3 |
1,71% |
0,23% |
61,97% |
1,50% |
2005 |
100 075 |
10,8% |
- |
0,0000 |
1,0000 |
13 |
- |
- |
1,74% |
0,00% |
63,71% |
1,50% |
Total |
2 492 791 |
5,82% |
0,27
|
|
6. Conclusion
This paper is a first attempt to estimate the skewness of the BF method. It relies on a few assumptions that have to be derived from external knowledge of the modeled reserving risk:
-
we assume the value of \({\widehat{U}}_{i}\);
-
we assume the coefficient of variation of \({\widehat{U}}_{i}\);
-
a lognormal distribution is assumed to estimate \(K\left\lbrack {\widehat{U}}_{i} \right\rbrack\); and
-
the slightly negative correlations between \({\widehat{y}}_{1},...,{\widehat{y}}_{n + 1}\) result also in some assumptions being made.
As for the first two points, the application of the proposed formulae could be refined if these parameters can be estimated externally, e.g., with the knowledge of the pricing distributions. However, such parameters are not always easily available. For the value of \({\widehat{U}}_{i}\), however, the application of the Cape Cod method can provide a good starting point.
Overall, refining the above assumptions may provide slightly better results, but we should bear in mind that in the context of IFRS 17 and of the quarterly disclosure requirements, simple and easily implementable models should be favored. Therefore, the proposed formulae will certainly fit with IFRS 17 requirements.
Further to refinements, it should be noted that the proposed skewness estimator is sensitive to the choice of the \({\widehat{U}}_{i}\) and to the tail factor retained for the pattern estimation. The sensitivity to the choice of \({\widehat{U}}_{i}\) is the result of the formulae given in section 3. As for the tail factor, it comes from the application of the parametrization of the BF model by Mack (2008). In general, the cumulative payment or incurred pattern does not reach 100% and, as a result, it is necessary to put a tail factor. This tail factor (which is automatically calculated in this paper) may be different from the one that an actuary would have applied based on his or her judgment. Such differences in the tail factor and in the overall pattern could affect the overall skewness. When applying the proposed formulae, it is therefore important to test the sensitivity of the results to both the choice of \({\widehat{U}}_{i}\) and to changes in patterns.
As a next step, we must recognize also that the BF method is usually used together with the chain-ladder method. Therefore, as for the hybrid chain-ladder method (see Arbenz and Salzmann 2014), a skewness formula for the mixed BF/chain-ladder method should be developed in a further paper.
Appendices
Appendix A. Proof of Equation (4)
\(\begin{align}\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) &= E\left\lbrack \left( {\widehat{R}}_{i}^{\text{BF}} - R_{i} \right)^{3} \right\rbrack \\& = E\left( {\widehat{R}}_{i}^{BF\ \ 3} \right) - 3E\left( {\widehat{R}}_{i}^{BF\ \ 2}R_{i} \right) \\&\ \ \ \ + 3E\left( {\widehat{R}}_{i}^{\text{BF}}\ R_{i}^{2} \right) - E\left( R_{i}^{3} \right)\end{align}.\)
As \({\widehat{R}}_{i}^{\text{BF}}\) and \(R_{i}\) are independent, we have
\(\begin{align}\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) &= E\left( {\widehat{R}}_{i}^{BF\ \ 3} \right) \\&\ \ \ \ - 3E\left( {\widehat{R}}_{i}^{BF\ \ 2} \right)E\left( R_{i} \right) \\&\ \ \ \ + 3E\left( {\widehat{R}}_{i}^{\text{BF}} \right)E\left( \ R_{i}^{2} \right) - E\left( R_{i}^{3} \right).\end{align}\)
As we have defined
\(\begin{align}K\left( R_{i} \right) &= E\left\lbrack \left( R_{i} - E\left( R_{i} \right) \right)^{3} \right\rbrack \\&= E\left( R_{i}^{3} \right) - 3E\left( R_{i}^{2} \right)E\left( R_{i} \right) + 2E\left( R_{i} \right)^{3},\end{align}\)
we have
\(\begin{align}\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) &= K\left( {\widehat{R}}_{i}^{\text{BF}} \right) - K\left( R_{i} \right) +3E ( \widehat{R}_{i}^{BF\text{ }2})E(R_i^{BF})\\&\ \ \ \ -2E(R_i)^3 -3E(\widehat{R}_i^2)E(R_i)+2E(R_i)^3 \\&\ \ \ \ -3E\left( {\widehat{R}}_{i}^{BF\text{ }2} \right)E\left( R_{i} \right) + 3E\left( {\widehat{R}}_{i}^{\text{BF}} \right)E\left( \ R_{i}^{2} \right).\end{align}\)
After rearranging, we find
\[\begin{align}\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) =\ & K\left( {\widehat{R}}_{i}^{\text{BF}} \right) - K\left( R_{i} \right) + E\left( {\widehat{R}}_{i}^{\text{BF}} - R_{i} \right)\\
&\times \Bigl\lbrack 3E\left( {\widehat{R}}_{i}^{BF\ \ 2} \right) + 3E\left( \ R_{i}^{2} \right) - 2{E\left( {\widehat{R}}_{i}^{{BF}} \right)}^{2}\\
&- 2{E\left( R_{i} \right)}^{2} - 2E\left( {\widehat{R}}_{i}^{\text{BF}} \right)E\left( R_{i} \right) \Bigr\rbrack.\end{align}\]
As \(E\left( {\widehat{R}}_{i}^{\text{BF}} - R_{i} \right) = 0\), we finally have
\(\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) = K\left( {\widehat{R}}_{i}^{\text{BF}} \right) - K\left( R_{i} \right).\)
Appendix B. Estimators of \(E\left( {\widehat{U}}_{i} \right)\), \(Var\left({\widehat{U}}_{i} \right)\), \(E\left(1 -\widehat{z}_{n + 1 - i} \right)\) and \(Var\left( {\widehat{z}}_{n + 1 - i} \right)\)
\(E\left( {\widehat{U}}_{i} \right)\) is estimated by \(\nu_{\text{i}}{\widehat{q}}_{i},\) where \(\nu_{i}\) is the premium volume of accident year i and \({\widehat{q}}_{i}\) is the ultimate claims ratio.
As mentioned in Mack (2008), \(\text{Var}\left( {\widehat{U}}_{i} \right)\) is best obtained from the pricing distribution (when available).
As for \(\text{E}\left( 1 - {\widehat{z}}_{n + 1 - i} \right)\), we use: \({\widehat{z}}_{k} = {\widehat{y}}_{1} + ... + {\widehat{y}}_{k}\).
Finally, \(\text{Var}\left( {\widehat{z}}_{k} \right) \approx min\left\lbrack \text{Var}\left( {\widehat{y}}_{1} \right) + ... + Var\left( {\widehat{y}}_{k} \right);Var\left( {\widehat{y}}_{k + 1} \right) + ... + Var\left( {\widehat{y}}_{n + 1} \right) \right\rbrack,\) where
\(\text{Var}\left( {\widehat{y}}_{k} \right) = \frac{{\widehat{s}}_{k}^{2}}{\sum_{i = 1}^{n + 1 - k}{\widehat{U}}_{i}}\).
Appendix C.
Triangle 1 |
Development Year |
Accident Year |
Premium |
Initial LR |
Ui |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
2005 |
110 940 316 |
69,6% |
77 176 365 |
4 626 711 |
7 833 956 |
9 237 575 |
4 075 137 |
10 828 853 |
2 000 000 |
1 916 455 |
1 000 000 |
2 136 161 |
873 469 |
-696 077 |
1 970 249 |
2 060 435 |
2006 |
136 881 755 |
54,1% |
74 081 078 |
2 497 127 |
11 277 229 |
7 424 316 |
5 612 201 |
3 410 710 |
930 785 |
952 588 |
3 311 966 |
2 527 888 |
2 177 772 |
655 961 |
2 129 708 |
|
2007 |
148 066 614 |
74,1% |
109 785 557 |
6 733 949 |
16 001 488 |
8 581 545 |
16 094 884 |
4 740 896 |
2 946 401 |
5 320 553 |
8 191 596 |
858 217 |
1 921 349 |
2 674 149 |
|
|
2008 |
142 419 083 |
64,8% |
92 216 629 |
5 495 872 |
10 634 243 |
9 357 874 |
3 687 937 |
5 459 783 |
7 244 685 |
3 514 985 |
1 278 885 |
1 130 235 |
2 201 251 |
|
|
|
2009 |
141 000 201 |
85,5% |
120 589 443 |
5 513 563 |
11 892 818 |
13 879 064 |
5 892 620 |
6 906 455 |
19 110 413 |
4 127 163 |
4 465 916 |
4 691 799 |
|
|
|
|
2010 |
148 748 619 |
79,4% |
118 144 218 |
4 642 178 |
9 611 735 |
16 674 760 |
7 170 953 |
13 301 042 |
7 576 809 |
1 561 553 |
5 247 638 |
|
|
|
|
|
2011 |
181 023 013 |
76,1% |
137 843 628 |
5 088 492 |
12 712 715 |
12 492 986 |
16 125 708 |
17 424 813 |
9 530 326 |
6 511 744 |
|
|
|
|
|
|
2012 |
195 545 471 |
95,0% |
185 764 666 |
5 911 892 |
14 485 648 |
15 199 655 |
20 501 488 |
15 352 506 |
10 000 000 |
|
|
|
|
|
|
|
2013 |
212 536 401 |
73,2% |
155 613 255 |
7 261 343 |
13 285 662 |
16 307 615 |
12 557 638 |
11 727 223 |
|
|
|
|
|
|
|
|
2014 |
210 377 519 |
83,5% |
175 714 686 |
5 210 360 |
23 258 993 |
10 144 201 |
23 137 200 |
|
|
|
|
|
|
|
|
|
2015 |
219 950 578 |
79,3% |
174 402 671 |
6 709 466 |
14 819 711 |
20 470 286 |
|
|
|
|
|
|
|
|
|
|
2016 |
230 357 704 |
87,3% |
201 091 730 |
13 556 673 |
13 519 627 |
|
|
|
|
|
|
|
|
|
|
|
2017 |
238 706 732 |
84,9% |
202 706 418 |
8 255 514 |
|
|
|
|
|
|
|
|
|
|
|
|
Triangle 2 |
Development Year |
Accident Year |
Premium |
Initial LR |
Ui |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
2005 |
227 650 968 |
57,2% |
130 246 487 |
89 281 |
5 826 729 |
10 642 655 |
13 443 290 |
12 861 714 |
11 016 529 |
5 941 853 |
15 777 922 |
-2 016 362 |
3 897 878 |
2 017 884 |
3 991 344 |
4 599 524 |
2006 |
248 800 723 |
73,3% |
182 459 114 |
253 461 |
10 141 425 |
15 502 109 |
14 653 424 |
19 038 614 |
13 526 518 |
11 920 567 |
11 637 234 |
3 152 799 |
2 832 256 |
1 596 518 |
1 957 742 |
|
2007 |
241 098 844 |
71,6% |
172 508 136 |
285 602 |
11 280 177 |
13 467 685 |
17 570 170 |
16 533 966 |
11 438 498 |
10 882 011 |
11 879 412 |
12 716 240 |
3 081 753 |
2 959 545 |
|
|
2008 |
183 864 522 |
81,2% |
149 329 997 |
056 |
11 224 085 |
14 034 547 |
17 843 506 |
16 022 003 |
11 874 150 |
11 319 798 |
4 775 666 |
1 954 680 |
1 837 360 |
|
|
|
2009 |
188 240 854 |
80,6% |
151 761 168 |
964 209 |
600 107 |
16 442 016 |
14 689 707 |
15 510 264 |
11 733 827 |
17 713 234 |
3 013 169 |
3 668 265 |
|
|
|
|
2010 |
196 095 |
84,3% |
165 366 633 |
671 876 |
10 913 835 |
16 157 793 |
14 372 512 |
17 147 487 |
14 286 587 |
11 287 031 |
9 345 582 |
|
|
|
|
|
2011 |
194 276 106 |
87,3% |
169 372 |
335 616 |
8 956 636 |
12 122 076 |
394 437 |
17 350 796 |
16 863 903 |
10 246 783 |
|
|
|
|
|
|
2012 |
245 584 590 |
88,3% |
216 964 899 |
762 265 |
11 836 973 |
17 712 645 |
23 477 706 |
18 519 857 |
18 934 294 |
|
|
|
|
|
|
|
2013 |
174 627 992 |
63,6% |
111 092 658 |
859 883 |
9 284 060 |
13 453 513 |
11 872 282 |
11 664 474 |
|
|
|
|
|
|
|
|
2014 |
201 255 277 |
60,9% |
122 553 304 |
1 002 904 |
8 720 759 |
13 041 075 |
12 514 625 |
|
|
|
|
|
|
|
|
|
2015 |
273 145 637 |
63,8% |
174 317 096 |
285 729 |
7 943 122 |
15 639 699 |
|
|
|
|
|
|
|
|
|
|
2016 |
406 620 096 |
63,5% |
258 284 240 |
536 767 |
9 569 288 |
|
|
|
|
|
|
|
|
|
|
|
2017 |
413 560 514 |
62,7% |
259 455 540 |
513 235 |
|
|
|
|
|
|
|
|
|
|
|
|
Triangle 3 |
Development Year |
Accident Year |
Premium |
Initial LR |
Ui |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
2005 |
306 442 |
90,0% |
275 798 |
13 487 |
50 621 |
30 204 |
33 570 |
24 290 |
12 173 |
5 826 |
10 832 |
21 869 |
10 000 |
1 048 |
4 023 |
4 801 |
2006 |
360 077 |
90,0% |
324 069 |
20 508 |
56 047 |
17 758 |
33 570 |
24 290 |
12 173 |
5 826 |
10 832 |
21 869 |
10 000 |
17 621 |
6 244 |
|
2007 |
526 754 |
90,0% |
474 078 |
14 156 |
50 362 |
53 811 |
43 285 |
28 710 |
31 867 |
31 430 |
18 001 |
12 657 |
6 330 |
5 052 |
|
|
2008 |
465 852 |
90,0% |
419 267 |
30 721 |
41 275 |
19 637 |
30 725 |
28 823 |
21 106 |
4 683 |
8 735 |
5 000 |
2 938 |
|
|
|
2009 |
429 724 |
90,0% |
386 751 |
7 305 |
30 942 |
29 580 |
25 095 |
17 541 |
9 402 |
8 953 |
4 637 |
7 104 |
|
|
|
|
2010 |
381 545 |
90,0% |
343 390 |
13 601 |
40 569 |
22 200 |
20 380 |
21 140 |
10 123 |
-572 |
5 787 |
|
|
|
|
|
2011 |
367 539 |
90,0% |
330 785 |
9 657 |
38 809 |
47 837 |
31 987 |
22 077 |
6 375 |
7 262 |
|
|
|
|
|
|
2012 |
347 770 |
90,0% |
312 993 |
7 085 |
41 073 |
55 745 |
46 024 |
17 225 |
19 820 |
|
|
|
|
|
|
|
2013 |
333 314 |
90,0% |
299 982 |
18 237 |
38 891 |
51 624 |
39 890 |
29 183 |
|
|
|
|
|
|
|
|
2014 |
382 678 |
90,0% |
344 410 |
15 008 |
41 664 |
40 047 |
25 834 |
|
|
|
|
|
|
|
|
|
2015 |
375 905 |
90,0% |
338 314 |
21 592 |
40 788 |
26 595 |
|
|
|
|
|
|
|
|
|
|
2016 |
359 604 |
90,0% |
323 644 |
17 061 |
33 068 |
|
|
|
|
|
|
|
|
|
|
|
2017 |
388 474 |
90,0% |
349 627 |
17 291 |
|
|
|
|
|
|
|
|
|
|
|
|
Correlation matrix for Skewness - Triangle 1 |
ρ ij |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
1 |
100% |
94% |
89% |
87% |
84% |
80% |
74% |
69% |
60% |
51% |
40% |
29% |
15% |
2 |
94% |
100% |
94% |
92% |
89% |
84% |
78% |
73% |
64% |
54% |
43% |
31% |
16% |
3 |
89% |
94% |
100% |
98% |
94% |
89% |
83% |
77% |
68% |
57% |
45% |
33% |
17% |
4 |
87% |
92% |
98% |
100% |
96% |
91% |
84% |
79% |
69% |
58% |
46% |
33% |
18% |
5 |
84% |
89% |
94% |
96% |
100% |
95% |
88% |
82% |
72% |
60% |
48% |
35% |
18% |
6 |
80% |
84% |
89% |
91% |
95% |
100% |
92% |
86% |
76% |
63% |
51% |
37% |
19% |
7 |
74% |
78% |
83% |
84% |
88% |
92% |
100% |
94% |
82% |
69% |
55% |
40% |
21% |
8 |
69% |
73% |
77% |
79% |
82% |
86% |
94% |
100% |
88% |
73% |
59% |
42% |
22% |
9 |
60% |
64% |
68% |
69% |
72% |
76% |
82% |
88% |
100% |
84% |
67% |
48% |
26% |
10 |
51% |
54% |
57% |
58% |
60% |
63% |
69% |
73% |
84% |
100% |
80% |
58% |
31% |
11 |
40% |
43% |
45% |
46% |
48% |
51% |
55% |
59% |
67% |
80% |
100% |
72% |
38% |
12 |
29% |
31% |
33% |
33% |
35% |
37% |
40% |
42% |
48% |
58% |
72% |
100% |
53% |
13 |
15% |
16% |
17% |
18% |
18% |
19% |
21% |
22% |
26% |
31% |
38% |
53% |
100% |
Correlation matrix for Skewness - Triangle 2 |
ρ ij |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
1 |
100% |
92% |
89% |
86% |
83% |
79% |
70% |
61% |
51% |
40% |
29% |
18% |
4% |
2 |
92% |
100% |
96% |
93% |
90% |
85% |
76% |
66% |
55% |
44% |
32% |
19% |
4% |
3 |
89% |
96% |
100% |
97% |
94% |
89% |
79% |
68% |
58% |
45% |
33% |
20% |
4% |
4 |
86% |
93% |
97% |
100% |
96% |
91% |
81% |
70% |
59% |
47% |
34% |
21% |
5% |
5 |
83% |
90% |
94% |
96% |
100% |
95% |
84% |
73% |
62% |
49% |
35% |
21% |
5% |
6 |
79% |
85% |
89% |
91% |
95% |
100% |
89% |
77% |
65% |
51% |
37% |
23% |
5% |
7 |
70% |
76% |
79% |
81% |
84% |
89% |
100% |
87% |
73% |
58% |
42% |
25% |
6% |
8 |
61% |
66% |
68% |
70% |
73% |
77% |
87% |
100% |
84% |
66% |
48% |
29% |
6% |
9 |
51% |
55% |
58% |
59% |
62% |
65% |
73% |
84% |
100% |
79% |
57% |
35% |
8% |
10 |
40% |
44% |
45% |
47% |
49% |
51% |
58% |
66% |
79% |
100% |
73% |
44% |
10% |
11 |
29% |
32% |
33% |
34% |
35% |
37% |
42% |
48% |
57% |
73% |
100% |
60% |
13% |
12 |
18% |
19% |
20% |
21% |
21% |
23% |
25% |
29% |
35% |
44% |
60% |
100% |
22% |
13 |
4% |
4% |
4% |
5% |
5% |
5% |
6% |
6% |
8% |
10% |
13% |
22% |
100% |
Correlation matrix for Skewness - Triangle 3 |
ρ ij |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
1 |
100% |
96% |
93% |
89% |
85% |
79% |
75% |
72% |
66% |
57% |
46% |
34% |
16% |
2 |
96% |
100% |
96% |
92% |
89% |
82% |
78% |
74% |
68% |
59% |
48% |
35% |
17% |
3 |
93% |
96% |
100% |
96% |
92% |
85% |
81% |
77% |
71% |
61% |
49% |
36% |
18% |
4 |
89% |
92% |
96% |
100% |
96% |
89% |
85% |
81% |
74% |
64% |
52% |
38% |
19% |
5 |
85% |
89% |
92% |
96% |
100% |
93% |
88% |
84% |
77% |
67% |
54% |
40% |
19% |
6 |
79% |
82% |
85% |
89% |
93% |
100% |
95% |
90% |
83% |
72% |
58% |
43% |
21% |
7 |
75% |
78% |
81% |
85% |
88% |
95% |
100% |
95% |
87% |
76% |
61% |
45% |
22% |
8 |
72% |
74% |
77% |
81% |
84% |
90% |
95% |
100% |
92% |
80% |
64% |
47% |
23% |
9 |
66% |
68% |
71% |
74% |
77% |
83% |
87% |
92% |
100% |
87% |
70% |
51% |
25% |
10 |
57% |
59% |
61% |
64% |
67% |
72% |
76% |
80% |
87% |
100% |
80% |
59% |
29% |
11 |
46% |
48% |
49% |
52% |
54% |
58% |
61% |
64% |
70% |
80% |
100% |
74% |
36% |
12 |
34% |
35% |
36% |
38% |
40% |
43% |
45% |
47% |
51% |
59% |
74% |
100% |
49% |
13 |
16% |
17% |
18% |
19% |
19% |
21% |
22% |
23% |
25% |
29% |
36% |
49% |
100% |