1. INTRODUCTION
This paper measures the risk from the volatility in a triangle of estimated ultimate losses rather than deriving metrics of risk from the volatility in the underlying paid or incurred triangles. Take a hypothetical omniscient actuary using the perfect crystal ball method. The resulting triangle of estimated ultimate losses would be constant along each accident year row. The development factors on the ultimate losses at each age would be 1. However, in the real world, the resulting estimates will change at each evaluation and the ultimate development factors will be different from 1. Intuitively, the degree of variation in the ultimate losses can be used to measure reserve volatility.
The correlation among the agetoage factors could have a significant impact on reserve risk. Although the correlations among the agetoage paid or incurred development factors tend to be positive, correlations among the agetoage ultimate development factors can be negative. In the context of reserve risk, this means that when using paid or incurred loss triangles, reserve risk with correlation included is often higher than reserve risk calculated without correlation terms. On the other hand, when based on ultimate loss triangles, reserve risk with correlation terms in the formula can sometimes be larger, and at other times lower, than when the correlation terms are omitted.
This paper will also highlight and diagnose irregularities that can arise from use of the variancecovariance matrix when it is derived from columns of development factors. Typically, each entry in an empirical variancecovariance matrix is based on taking the variance and covariance of vectors of the same size. Such a matrix will be wellbehaved and have desirable mathematical properties. However, all bets are off when the vectors are of different sizes. In a loss triangle, there are fewer observations in the more mature development age columns. As will be shown, a nominal variancecovariance matrix based on such triangular structure can exhibit surprising pathologies. Another aspect of the “triangle structure problem” is that the later age development factors are more leveraged, yet more impactful. Hence the resulting risk estimates are inherently more unstable. Tentative solutions will be proposed to these problems based on an approach of filling in the triangle of factors and recalibrating the covariance matrix to offset any bias introduced.
1.1. Research Context
There are several ways to estimate reserve risk using historical loss data. Mack (1993) provided the first example, estimating the total runoff reserve risk using paid or incurred loss triangles. Merz and Wuthrich (2008) extended this methodology to evaluate the oneyear reserve risk. Robbin (2012) provided a refinement to the Standard Formula used in Solvency II. Rehman and Klugman (2010) proposed using the triangle of estimated ultimate losses to estimate ultimate reserve risk. They assumed that the development factors, rather than losses, follow a lognormal distribution. Siegenthaler (2019) also used a triangle of estimated ultimate losses to derive both a oneyear and a total runoff formula, but he used different assumptions and obtained different resulting formulas than Rehman and Klugman.
1.2. Objectives
This paper will introduce the FengRobbin method, which makes several specific algorithmic enhancements in estimating reserve risk.^{[1]}
The approach starts with the triangle model promulgated by Rehman and Klugman. In this model, agetoage factors of ultimate losses are assumed to be lognormally distributed. The Rehman and Klugman model will be enhanced and extended in several ways:

Develop a oneyear reserve risk estimate.

Examine the covariance terms and derive a slightly different formula for the total runoff reserve risk.

Propose solutions to address the “triangle structure problem” that will eliminate potentially anomalous behaviors when raw variancecovariance matrices are used.

Propose an estimate of the parameter risk component of reserve risk.
1.3. Outline
The remainder of the paper proceeds as follows. Section 2 starts with a general discussion of reserve risk. Section 3 is devoted to presenting the formulas for oneyear and total runoff reserve risk estimates. Section 4 simplifies the formulas using matrix notation. Section 5 demonstrates how the raw variancecovariance matrix can go awry and yield a negative overall variance. It then presents several alternatives for addressing the “triangle structure problem.” Section 6 proposes an estimate for parameter error. Section 7 provides a comparison of the various formulabased methods as well as insights gained from examining the application of the methods to numerical data.
This paper will include simple examples to demonstrate the methodology described in each section. In the more substantial examples in Section 7 and Appendix C, different sets of ultimate triangles will be analyzed using a variety of reserving methods, and different methods will be applied to these ultimate loss triangles to calculate reserve risk standard errors.
2. Reserve Risk
What is reserve risk? In the context of this paper, it is the potential for adverse and favorable deviations between the eventual ultimate losses and the latest estimates of ultimate. The very mature accident years should have very little deviation, while the newer accident years could have sizeable volatility.
There are two main drivers of reserve risk. First is the volatility in the underlying loss development data. Such volatility of the underlying development process in turn translates into volatility in the difference between the eventual ultimate losses and the latest estimates on the diagonal. Second is the volatility that stems from the method used to project the ultimate estimates. While some methods are mathematically more stable than others, this does not necessarily imply they produce more accurate estimates of ultimate more quickly.
This paper will develop formulae for ultimate and for oneyear reserve risk. The oneyear reserve risk measures the volatility of the estimate of ultimate losses over the next year, whereas the ultimate reserve risk, also referred to as total runoff reserve risk, measures the variability of the estimated ultimate losses until all outstanding contract obligations are fulfilled. In the context of Solvency II, the expected unpaid loss is called the undiscounted best estimate and it is assumed to have no builtin prudential margin.
3. One Year and Ultimate Runoff Standard Errors
The main results will be presented in this section, following a summary of the notation shown in Table 1. The notation is largely consistent with that used by Mack, Merz and Wuthrich, and Siegenthaler,^{[2]} but differs from that used by Rehman and Klugman.^{[3]}
In addition, Table 2 has the notation for various lognormal and normal distribution parameters:
The estimators for these parameters will be denoted by an accent character
on top of the variable. The formula for the estimators will be given in Sections 3.3 and 3.4. Additionally, let denote the covariance between and If then is the variance of Even though the covariance terms could be negative, we used instead of the more common to simplify the collection of terms within the variancecovariance matrix.Using the above notation, the main result for the oneyear standard deviation for the entire triangle is as follows.
\[\begin{align} {\text{StDev}}\left( \sum_{y = 1}^{Y}U_{y,D  y + 2} \right) &= \sum_{y = 1}^{Y}U_{y,D  y + 1\ }\ \\ &\quad \times \ \sqrt{\left( e^{2\widehat{\alpha} + 2{\widehat{\theta}}^{2}}  e^{2\widehat{\alpha} + {\widehat{\theta}}^{2}} \right)} \end{align} \tag{3.01}\]
where
andThe main result for the total runoff standard deviation for the entire triangle is as follows.
\[\begin{align} {\text{StDev}}\left( \sum_{y = 1}^{Y}U_{y,D} \right) &= \sum_{y = 1}^{Y}U_{y,D  y + 1}\ \\ &\quad \times \ \sqrt{\left( e^{2\widehat{\omega} + 2{\widehat{\lambda}}^{2}}  e^{2\widehat{\omega} + {\widehat{\lambda}}^{2}} \right)} \end{align} \tag{3.02}\]
where
and3.1. Development Model Setup
The setup of this paper is largely consistent with that used by Rehman and Klugman.
For a given line of business with Table 3.
accident years and development periods, historical estimated ultimate losses are arrayed in the usual triangle format as shown inA frequently used quantity is the sum over the latest diagonal
which represents the sum of the best available estimate of ultimate over all accident years at the time of evaluation.^{[4]} Estimates in all future development periods are uncertain.Throughout this paper, the illustrative triangle of ultimate losses in Table 4 will be used to demonstrate the methodology developed.
3.1.1. Ultimate Loss Development Factors
Consistent with Rehman and Klugman’s, and Siegenthaler’s, definitions, the agetoage ultimate development factor, or UDF, is defined as
\[\begin{align} g_{y,d} ≔ \ \frac{U_{y,\ d + 1}}{U_{y,\ d}}\tag{3.03} \end{align}\]
Similar to the agetoage development factors for paid or incurred losses, a triangle of such ultimate development factors shown in Table 5 could be computed using the observed ultimate estimates, as illustrated using the same structure as Table 3.
Define the random variable
as the agetoage ultimate development factor applicable for ultimate losses from age to +1. This makes each an observed value for for As later sections of this paper would demonstrate, if the distribution of is known, variance of the ultimate losses in future development periods would follow.The observed agetoage ultimate development factors for the illustrative triangle are shown in Table 6. For example, 1.0502 in accident year 1 age 23 is calculated as 921÷877.
A related concept is the agetoultimate ultimate^{[5]} development factor,
defined as the product of all agetoage development from age up to maturity age That is,\[\begin{matrix} G_{d} = \prod_{j = d}^{D}g_{j}\tag{3.04} \end{matrix}\]
The estimation of agetoultimate UDF is computationally similar to the agetoultimate paid or incurred loss development factors and could be used similarly.
3.1.2. Accident Year Weight Factors
To address the different volume of exposure in each accident year, a weight factor
is calculated for each accident year as a proportion of the latest estimate of ultimate losses:\[\begin{matrix} r_{y} = \ \frac{U_{y,D  y + 1}}{\sum_{j = 1}^{Y}U_{j,\ D  j + 1\ }}\tag{3.05} \end{matrix}\]
It is easy to check that
For total runoff risks, a cumulative weight is defined for each age as follows:
\[\begin{matrix} R_{d} ≔ \sum_{j = Y  d + 1}^{Y}{r_{j}\ }\tag{3.06} \end{matrix}\]
For the illustrative triangle, the weight factors are shown in Table 7.
3.2. Model Assumptions
Model assumptions are entirely consistent with those used in Rehman and Klugman (2010). The main assumption is that agetoage UDFs for each development age follow a lognormal distribution for all origin periods. It follows that the logarithm of UDFs follow a normal distribution. No assumption is made regarding the independence of the agetoage ultimate development factors, and no assumption is required regarding the method of determining the ultimate loss estimates at each calendar year. Mathematically,
\[g_{d}\ \sim\ LogNormal\ (\mu_{d},\ {\sigma_{d}}^{2})\]
The mean, variance and covariance of
can be initially estimated by taking the logarithm of the triangle of UDFs.^{[6]}Following the methodology developed in Rehman and Klugman,
and could be estimated using the following sample mean and variance formulae.\[\begin{matrix} {\widehat{\mu}}_{d} ≔ \ \frac{\sum_{y = 1}^{Y  d}{{\text{ln}(g}_{y,d})\ }}{Y  d}\ \tag{3.07} \end{matrix}\]
\[\begin{matrix} {{\widehat{\sigma}}_{d}}^{2} ≔ \ \frac{\sum_{y = 1}^{Y  d}{{{(\text{ln}(g}_{y,d})  {\widehat{\mu}}_{d})}^{2}\ }}{Y  d  1}\ \tag{3.08} \end{matrix}\]
The observed logarithms of the agetoage ultimate development factors for the illustrative triangle are presented in Table 8.
The estimated unadjusted mean and variance of each log UDF factor is shown in Table 9.
The variancecovariance matrix presented in Table 10 is adjusted according to the fulladjustment procedure described in Section 5.2.3. This guarantees a positive calculated variance.
It follows from the definition of
that\[\begin{matrix} {\widehat{U}}_{y,\ d + 1} = \ U_{y,\ d}\ \times \ g_{d}\ \tag{3.09} \end{matrix}\]
The above equation implicitly assumes that future ultimate losses can be predicted using past ultimate losses. This relationship would fail to hold if there is extraneous information indicating future changes in reserving methodology, and such future changes are not reflected in the past data triangle of
s. In such situations, computed risk amounts must be manually adjusted. Retroactive adjustments of historical ultimate losses could also be made.3.2.1. Simplifying Assumptions
This paper follows Rehman and Klugman’s use of a firstdegree Taylor series approximation for the exponential and logarithmic functions. It is assumed that if
if For a triangle of ultimate losses, agetoage factors are approximately 1, thus is approximately 0. The exponential approximation is used where while the logarithmic approximation is used where Consequently, these approximations are appropriate. These approximations are utilized in the proof in Appendix A.3.3. Standard Error of Ultimate Runoff Reserve Risk
In this section, a slightly modified version of Rehman and Klugman’s formula is presented due to a more detailed analysis of covariance terms. The result is similar to Rehman (2016).
3.3.1. Total Runoff Reserve Risk – Single Accident Year
The ultimate estimate at maturity for a single accident year is
It has variance\[ \begin{align} \text{Var}\left\lbrack U_{y,D} \right\rbrack &= {U_{y,D  y + 1}}^{2} \times \ \text{Var}\left\lbrack \frac{U_{y,D}}{U_{y,D  y + 1}} \right\rbrack \\&= {U_{y,D  y + 1}}^{2}\ \times \text{Var}\ \left\lbrack G_{D  y + 1} \right\rbrack \end{align} \]
Here
is the latest known ultimate estimate for accident year Since is distributed lognormally, is distributed normally with associated mean and variance given by\[\begin{matrix} {\widehat{\omega}}_{y} ≔ E\left\lbrack \sum_{d = D  y + 1}^{D}{ln(}g_{d}) \right\rbrack = \sum_{d = D  y + 1}^{D}{\widehat{\mu}}_{d}\ \tag{3.10} \end{matrix}\]
\[\begin{align} {{\widehat{\lambda}}_{y}}^{2} &≔ Var\left\lbrack \sum_{d = D  y + 1}^{D}{ln(}g_{d}) \right\rbrack \\ &= \ \sum_{i = D  y + 1}^{D}{\sum_{j = D  y + 1}^{D}{{\widehat{\sigma}}_{i,j}}^{2}} \end{align} \tag{3.11}\]
In the above equations,
is the mean of the and is the covariance between and If is the sample variance ofThus, the standard error of total runoff reserve risk for a single accident year is given by
\[\begin{align} \text{StDev}\left( U_{y,D} \right) &= U_{y,D  y + 1}\ \\ &\quad \times \ \sqrt{\left( e^{2{\widehat{\omega}}_{y} + 2{{\widehat{\lambda}}_{y}}^{2}}  e^{2{\widehat{\omega}}_{y} + {{\widehat{\lambda}}_{y}}^{2}} \right)} \tag{3.12} \end{align}\]
Equations 3.10 through 3.12 produce identical results as the RehmanKlugman method for a single accident year.
Using the illustrative triangle, the results for each accident year are given in Table 11.
3.3.2. Total Runoff Reserve Risk – All Accident Years Combined
The ultimate estimate at maturity for the sum of all accident year is
This quantity has variance\[ \begin{align} \operatorname{Var}\left[\sum_{y=1}^Y U_{y, D}\right]&=\left(\sum_{y=1}^Y U_{y, Dy+1}\right)^2 \\ &\quad \times \operatorname{Var}\left[\frac{\sum_{y=1}^Y U_{y, D}}{\sum_{y=1}^Y U_{y, Dy+1}}\right] \end{align} \]
Lemma A.1 in the Appendix demonstrates that
\[\ln\left( \frac{\sum_{y = 1}^{Y}U_{y,\ D}}{\sum_{y = 1}^{Y}U_{y,D  y + 1\ }} \right) \approx \ \sum_{d = 1}^{D}{R_{d}\text{ln}(g_{d})}\]
Since
has a lognormal distribution, is normally distributed with mean and variance The linear combination of normally distributed random variables with coefficients is a normally distributed random variable with mean and variance given by:\[\begin{matrix} \widehat{\omega} ≔ E\left\lbrack \ \sum_{d = 1}^{D}R_{d}ln\left( g_{d} \right) \right\rbrack = \sum_{d = 1}^{D}R_{d}{\widehat{\mu}}_{d} \tag{3.13} \end{matrix}\]
\[\begin{matrix} {\widehat{\lambda}}^{2} ≔ Var\left\lbrack \sum_{d = 1}^{D}R_{d}ln\left( g_{d} \right) \right\rbrack = \ \sum_{i = 1}^{D}{\sum_{j = 1}^{D}R_{i}R_{j}}{{\widehat{\sigma}}_{i,j}}^{2} \tag{3.14} \end{matrix}\]
Note that
and are total runoff estimators for the entire triangle, whereas and are estimators for specific ages. It is also important to distinguish from and fromIn some cases, the formula for
above yields an anomalous negative variance if the raw variancecovariance matrix is used. This issue is further explored in Section 5. Starting from Equation 3.14, the FengRobbin method diverges from the RehmanKlugman method.Recovering the variance of the lognormal distribution yields
\[\text{Var}\left( \frac{\sum_{y = 1}^{Y}U_{y,\ D}}{\sum_{y = 1}^{Y}U_{y,D  y + 1\ }} \right) = e^{2\widehat{\omega} + 2{\widehat{\lambda}}^{2}}  e^{2\widehat{\omega} + {\widehat{\lambda}}^{2}}\]
The completed formula for the standard error of total runoff reserve risk is
\[\begin{align} \text{StDev}\left( \sum_{y = 1}^{Y}U_{y,D} \right) &= \sum_{y = 1}^{Y}U_{y,D  y + 1}\ \\ &\quad \times \ \sqrt{\left( e^{2\widehat{\omega} + 2{\widehat{\lambda}}^{2}}  e^{2\widehat{\omega} + {\widehat{\lambda}}^{2}} \right)}\tag{3.15} \end{align}\]
Using the illustrative triangle, the total runoff results for the sum of all accident years is given in Table 12.
3.4. Standard Error of OneYear Reserve Risk
This is a new result not covered by either Rehman and Klugman (2010) or Rehman (2016).
3.4.1. Oneyear Reserve Risk – Single Accident Year
The ultimate estimate for a single accident year evaluated 12 months in the future is
The variance of this quantity is\[ \begin{align} \text{Var}\left\lbrack U_{y,D  y + 2} \right\rbrack &= {U_{y,D  y + 1}}^{2} \times \ \text{Var}\left\lbrack \frac{U_{y,D  y + 2}}{U_{y,D  y + 1}} \right\rbrack \\ &= {U_{y,D  y + 1}}^{2}\ \times \text{Var}\ \left\lbrack g_{D  y + 1} \right\rbrack \end{align} \]
Here
is the latest known ultimate estimate for accident year Random variable has a lognormal distribution with associated mean and variance parameters given by\[\begin{matrix} {\widehat{\alpha}}_{y} ≔ \text{E}\left\lbrack \ln{(g_{D  y + 1})} \right\rbrack = {\widehat{\mu}}_{D  y + 1}\tag{3.16} \end{matrix}\]
\[\begin{matrix} {{\widehat{\theta}}_{y}}^{2} ≔ \text{Var}\left\lbrack \ln{(g_{D  y + 1})} \right\rbrack = \ {{\widehat{\sigma}}_{D  y + 1}}^{2}\tag{3.17} \end{matrix}\]
Thus, the standard error of oneyear reserve risk for a single accident year is given by
\[\begin{align} \text{StDev}\left( U_{y,D  y + 2} \right) &= U_{y,D  y + 1}\ \\ &\quad \times \ \sqrt{\left( e^{2{\widehat{\alpha}}_{y} + 2{{\widehat{\theta}}_{y}}^{2}}  e^{2{\widehat{\alpha}}_{y} + {{\widehat{\theta}}_{y}}^{2}} \right)}\tag{3.18} \end{align}\]
Using the illustrative triangle, the oneyear results for each accident year are given in Table 13.
3.4.2. Oneyear Reserve Risk – All Accident Years Combined
The ultimate estimate for the sum of all accident years evaluated 12 months in the future is
The variance of this quantity is expressed as\[ \begin{align} \operatorname{Var}\left[\sum_{y=1}^Y U_{y, Dy+2}\right]&=\left(\sum_{y=1}^Y U_{y, Dy+1}\right)^2 \\ &\quad \times \operatorname{Var}\left[\frac{\sum_{y=1}^Y U_{y, Dy+2}}{\sum_{y=1}^Y U_{y, Dy+1}}\right] \end{align} \]
Lemma A.2 in the Appendix demonstrates that
\[\ln\left( \frac{\sum_{y = 1}^{Y}U_{y,D  y + 2}}{\sum_{y = 1}^{Y}U_{y,D  y + 1\ }} \right) \approx \ \sum_{y = 1}^{Y}r_{y}\ln\left( g_{D  y + 1} \right)\]
Since
has a lognormal distribution, has a normal distribution with mean and variance The linear combination of such normal distributions with coefficients is another normal distribution with mean and variance given as follows:\[\begin{matrix} \widehat{\alpha} ≔ \text{E}\left\lbrack \ \sum_{y = 1}^{Y}r_{y}\ln\left( g_{D  y + 1} \right) \right\rbrack = \sum_{y = 1}^{Y}r_{y}{\widehat{\mu}}_{D  y + 1}\tag{3.19} \end{matrix}\]
\[\begin{align} {\widehat{\theta}}^{2} &≔ \text{Var}\left\lbrack \sum_{y = 1}^{Y}r_{y}\ln\left( g_{D  y + 1} \right) \right\rbrack \\ &= \ \sum_{i = 1}^{Y}{\sum_{j = 1}^{Y}r_{i}r_{j}}{{\widehat{\sigma}}_{D  i + 1,D  j + 1}}^{2}\tag{3.20} \end{align}\]
Note that
and are estimators for the entire triangle, whereas and are estimators for specific ages. Additionally, is a covariance term between development age and when It is also important to distinguish and from and and from and used for the total runoff reserve risk.Recovering the variance of the lognormal distribution yields
\[\text{Var}\left( \frac{\sum_{y = 1}^{Y}U_{y,D  y + 2}}{\sum_{y = 1}^{Y}U_{y,D  y + 1\ }} \right) = e^{2\widehat{\alpha} + 2{\widehat{\theta}}^{2}}  e^{2\widehat{\alpha} + {\widehat{\theta}}^{2}}\]
The completed formula for the standard error of oneyear reserve risk is
\[\begin{align} \text{StDev}\left( \sum_{y = 1}^{Y}U_{y,D  y + 2} \right) &= \sum_{y = 1}^{Y}U_{y,D  y + 1\ }\ \\ &\quad \times \ \sqrt{\left( e^{2\widehat{\alpha} + 2{\widehat{\theta}}^{2}}  e^{2\widehat{\alpha} + {\widehat{\theta}}^{2}} \right)} \end{align} \tag{3.21}\]
Using the illustrative triangle, the oneyear results for the sum of all accident years is given in Table 14:
4. Matrix Implementation
This section simplifies the formula in the previous section using matrix notation and allows for an easier spreadsheet implementation to yield the same numerical results as previous sections. Matrix operations significantly simplify spreadsheet implementation through the use of SUMPRODUCT( ) and MMULT( ) function in Microsoft EXCEL. This section adopts the notation in Table 15.
Under the above notation, Equations 3.13, 3.14, 3.19 and 3.20 for the estimators for oneyear and ultimate runoff risk in Section 3.3.2 and 3.4.2 may be rewritten according to Table 16.
5. The Triangle Structure Problem
The variancecovariance matrix by development age calculated using a nominal variancecovariance matrix
could sometimes give rise to an overall value for the variance that is negative. Computationally, this happens because matrix is not necessarily positive semidefinite. In a triangle, has a different number of data points for each age, and such inconsistent data size among different ages does not naturally give rise to a positive semidefinite variancecovariance matrix. This is the “triangle structure problem.”An example of this occurrence is the triangle of
previously given in section 3.2. Section 5.1 illustrates that raw variancecovariance matrix computed using this triangle gives rise to negative computed variance. Solutions are proposed in Section 5.2, whereby the raw variancecovariance matrix is replaced by a modified variancecovariance matrix that is positive semidefinite. It follows from the properties of positive semidefinite matrices that the calculated variance must be nonnegative. Several options for matrix are presented in Section 5.2.5.1. Negative Calculated Variance
This section demonstrates the problems with the unadjusted variancecovariance matrix. The nominal variancecovariance matrix Table 17 is computed using the triangle of logarithmic agetoage ultimate development factors from Table 8 in Section 3.2.
inThis raw variancecovariance is problematic, as it would imply that the variance of the sum of age 3 plus age 4 is negative, computed by 0.0384732×0.057611+0.055301=0.021449. Additionally, the variance of the sum of age 1 through 4 is also negative, as the sum of the variancecovariance matrix is 0.033669. Using Table 17, the RehmanKlugman method would result in a negative variance.
The root cause is that this raw variancecovariance matrix is not positive semidefinite. This happens because each age has different number of observations. The solutions posed in the next section will address this problem.
5.2. Solutions for the Triangle Structure Problem
In this section, extensive references are made to matrix theory in Appendix B.
The basic approach is to first calculate the variancecovariance matrix
using a square of “fillin” factors. Any factor in the square that is not present in the original triangle is filled in using the average value of known factors of the same development age. The diagonal variance entries of matrix are smaller than the corresponding entries of matrix Furthermore, the offdiagonal covariance entries of matrix tend to be of lower magnitude than those in matrix To scale the variances closer to the sample variances, two procedures are proposed in Sections 5.2.2 and 5.2.3. These procedures are crafted to yield positive semidefinite matrices. Theorem B.2 in Appendix B shows that the variance of a linear combination of random variables calculated using a positive semidefinite variancecovariance matrix is always nonnegative.5.2.1. The “Fillin” Procedure
The fillin procedure relies on the fact that a variancecovariance matrix calculated using datasets of same size is always positive semidefinite. The fillin procedure starts from the triangle of logUDF and fills in any blank using the means of each development age. The result is a logUDF square. The variancecovariance matrix computed using this square is labelled
and is positive semidefinite according to Theorem B.2 in the appendix.Starting from Table 8, the completed square of logUDF is presented in Table 18. Note that the lowerright corner is filled in with averages of the respective development age.
The variancecovariance matrix Table 19 is calculated using the sample covariance of each development age in Table 18.
in5.2.2. The “Diagonal Calibrated” Procedure
The diagonal calibrated procedure is based on the fillin procedure. The modified variancecovariance matrix
replaces the diagonal in matrix with the diagonal in matrix thereby preserving the diagonal variance terms from the raw triangle while reducing the magnitude of covariance terms in larger ages. Lemma B.5 in Appendix B demonstrates that is positive semidefinite.The variancecovariance matrix Table 20.
is presented in5.2.3. The “Full Adjustment” Procedure
The full adjustment procedure modifies the entire variancecovariance matrix, in contrast to the diagonal calibrated procedure, which only replaces the diagonal.
Let
be a by diagonal matrix defined below. is not data dependent.\[\begin{matrix} \mathbf{X}_{ij} = \left\{ \begin{matrix} \sqrt{\frac{D  2}{D  i  1}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ i = j\ and\ i \leq D  2 \\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise \\ \end{matrix} \right. \end{matrix} \tag{5.1}\]
The “full adjustment” variancecovariance matrix is defined as
\[\begin{matrix} \mathbf{M}_{full} = \mathbf{X}^{T}\mathbf{M}_{fill  in}\mathbf{X\ } \end{matrix} \tag{5.2}\]
has the following desirable properties:
 It is positive semidefinite. For proof refer to Lemma B.8.
 The th element in the diagonal of is equal to the raw sample variance of the corresponding age in the triangle of This is demonstrated in Lemma B.9.
Matrix Table 21 and Table 22, respectively.
and the full adjustment variancecovariance matrix, are presented inNote that the diagonal elements of
are the same as those in Using matrix the variance of age 3 plus age 4 is computed as 0.0384732×0.040737+0.055301=0.012299, and the variance of the sum of age 1 through 4 is 0.017530, both positive.5.3. Assessment
Although the fillin procedure resolves potential negative calculated variance, it tends to understate the overall variance, therefore creating bias. However, this procedure is useful in stabilizing triangles with highly leveraged tail factors. The diagonal calibrated procedure resolves the negative variance and fixes the major potential source of bias present in the matrix
The full adjustment procedure theoretically eliminates all bias, but it is more prone to modeling noise than the other two procedures.Loss trianglebased methods are in general vulnerable to modeling noise arising from the use of tail variance and covariance terms. The upper right corners of the triangle are inherently based on small sample sizes leading to instability and lack of statistical significance. There is a tradeoff between the loss of information contained in the covariance terms, versus the instability in using them.
6. Optional Parameter Estimation Error
The formulae presented thus far assume that there is no parameter error. This section proposes a formula for parameter errors. Only the estimation error of
is considered, as this is the parameter that directly impacts the ultimate estimates. Conceptually, if reserves are perfectly set, all ultimate development factors should equal 1 and there would be no parameter error. The farther is from 1, the more difficult it is to make a true estimation of A natural estimation error for a given accident year with ultimate loss of for oneyear reserve risk would be\[\begin{matrix} U_{y,D  y + 1} \times \left g_{d}  1 \right \end{matrix} \tag{6.01}\]
Similarly, for total runoff reserve risk, an estimate for parameter risk would be
\[\begin{matrix} U_{y,D  y + 1} \times \left G_{d}  1 \right \end{matrix}\tag{6.02}\]
Siegenthaler derived these intuitive results using different sets of assumptions than those used in this paper.
7. Comparison of Methods
In this section, comparisons are made between the FengRobbin method and other closedform formulae quantifying reserve risk, including the RehmanKlugman Method, the Siegenthaler Method, the Mack Method, and the MerzWuthrich method. Section 7.1 compares the formulae themselves along with model assumptions. Section 7.2 compares model results when the aforementioned methods are applied to several sets of ultimate triangles. Section 7.3 provides a summary of comparisons.
7.1. Model Assumptions and Formula
All formulae discussed in this section are expressed in terms of notations defined in this paper using additional definitions where necessary.
7.1.1. The RehmanKlugman Method
Because the FengRobbin approach uses the basic setup of the RehmanKlugman model, the assumptions are identical. The FengRobbin approach also quantifies a oneyear reserve risk. Additionally, due to the more refined treatment of covariance terms, there are small differences in the formula for total runoff reserve risk between the FengRobbin approach and the RehmanKlugman approach.
Recall from Section 3 that the FengRobbin standard error of total runoff reserve risk is
\[ \begin{align} \text{StDev}\left( \sum_{y = 1}^{Y}U_{y,D} \right) &= \sum_{y = 1}^{Y}U_{y,D  y + 1}\ \\ &\quad \times \ \sqrt{(e^{2\widehat{\omega} + 2{\widehat{\lambda}}^{2}}  e^{2\widehat{\omega} + {\widehat{\lambda}}^{2}})} \end{align} \]
where
andIn RehmanKlugman’s approach, the formula for the standard error formula is the same; however,
is estimated differently.^{[7]} Using the notations in this paper, RehmanKlugman’s estimation of is expressed as the following:\[\begin{matrix} {\widehat{\lambda}}^{2} ≔ \sum_{y = 1}^{D}{{r_{y}}^{2}{{\widehat{\lambda}}_{y}}^{2}} \end{matrix} \tag{7.01}\]
In this paper,
= in accordance with the definition in Section 3. Computationally, Rehman and Klugman add fewer covariance terms than in this paper since the FengRobbin method also accounts for covariance terms among accident years. The standard error for total runoff risk computed using the RehmanKlugman (2010) method is usually lower than the FengRobbin result.Rehman (2016) recognized these additional covariances between accident year and proposed constructing an additional variancecovariance matrix at the accident year level. This is a different approach than the FengRobbin method, which expressed accident year random variables as sums of development age random variables.
7.1.2. The Siegenthaler Method
Similar to the RehmanKlugman and FengRobbin methods, the Siegenthaler method uses a triangle of estimated ultimate losses. Instead of analyzing the mean and variance of
Siegenthaler develops reserve risk in a similar manner as the Mack method and the MerzWuthrich method, and uses the volume weighted mean defined as\[\begin{matrix} {\widehat{g}}_{d} ≔ \ \frac{\sum_{y = 1}^{Y  d}{U_{y,d + 1}\ }}{\sum_{y = 1}^{Y  d}{U_{y,d}\ }} \end{matrix}\tag{7.02}\]
A key assumption in the Siegenthaler method is
\[\begin{matrix} \text{Var}\left( U_{y,d + 1} \right) = {s_{d}}^{2}U_{y,d} \end{matrix}\tag{7.03}\]
Where
\[\begin{matrix} {{\widehat{s}}_{d}}^{2}: = \ \frac{\sum_{y = 1}^{Y  d}{{U_{y,d} \times \left( g_{y,d}  {\widehat{g}}_{d} \right)}^{2}\ }}{Y  d  1}\end{matrix}\tag{7.04}\]
Siegenthaler did not assume the lognormal distribution of the UDFs as Rehman and Klugman did. Siegenthaler demonstrates that under his assumptions, many covariance terms for ultimate losses reduce to zero. In contrast, this paper takes the approach that these covariances between the current estimate of the ultimate losses and the eventual fully developed ultimate losses are potentially statistically significant and should be measured. For example, covariances between accident years can stem from similar claims handling practices, retroactive legal impacts, inflation, etc.
Siegenthaler’s process variance for total runoff risk (using notation in this paper)^{[8]} is:
\[\begin{align} \sum_{y = 2}^{Y}&\biggl( \biggl\{ \sum_{d = Y  y + 1}^{D  1}{{G_{d + 1}}^{2}{{\widehat{s}}_{d}}^{2}\prod_{j = Y  y}^{d  1}{\widehat{g}}_{j}} \biggr\} \\ &\quad \quad U_{y,Y  y + 1}\ \biggr) \end{align}\tag{7.05}\]
Siegenthaler’s parameter variance for total runoff reserve risk is:
\[ \begin{align} &\sum_{y = 2}^{Y}\left( \{{G_{D  y + 1}  1\}}^{2}{U_{y,Y  y + 1}}^{2}\ \right) \\&+ 2\sum_{2 \leq i < j \leq Y}^{}{\left( G_{D  i + 1}  1 \right)\left( G_{D  j + 1}  1 \right)U_{i,Y  i + 1}U_{j,Y  j + 1}} \end{align} \]
Using the matrix implementation, the above could be rewritten as:
\[\begin{matrix} \left( \overrightarrow{G}  \overrightarrow{1} \right)^{T} \times \overrightarrow{U} \times {\overrightarrow{U}}^{T} \times \left( \overrightarrow{G}  \overrightarrow{1} \right) \end{matrix}\tag{7.06}\]
Siegenthaler’s process variance for oneyear reserve risk is:
\[\begin{matrix} \sum_{y = 2}^{Y}\left( U_{y,Y  y + 1}{{\widehat{s}}_{D  y + 1}}^{2}\ \right) = \ \overrightarrow{U} \cdot {\overrightarrow{s}}^{2} \end{matrix}\tag{7.07}\]
Siegenthaler’s parameter variance for oneyear reserve risk is:
\[ \begin{align} &\sum_{y = 2}^{Y}\left( \{{g_{D  y + 1}  1\}}^{2}{U_{y,Y  y + 1}}^{2}\ \right) \\&+ 2\sum_{2 \leq i < j \leq Y}^{}{(g_{D  i + 1}  1)(g_{D  j + 1}  1)U_{i,Y  i + 1}U_{j,Y  j + 1}} \end{align}\]
Using the matrix implementation, the above could be rewritten as:
\[\begin{matrix} {(\overrightarrow{g}  \overrightarrow{1})}^{T} \times \overrightarrow{U} \times {\overrightarrow{U}}^{T} \times (\overrightarrow{g}  \overrightarrow{1}) \end{matrix}\tag{7.08}\]
Computationally, the Siegenthaler method yields larger standard errors when UDFs are significantly different from 1, whereas the FengRobbin method without parameter error yields larger standard errors when UDFs are significantly different from the mean of each age. As a consequence, when there is consistent adverse development or consistent favorable development, the FengRobbin method yields lower standard error than the Siegenthaler method since a consistent reserve development can cluster around a mean and is therefore more “predictable.”
7.1.3. The Mack Method and the MerzWuthrich Method
The Mack method is a wellknown closedform method used to evaluate reserve risk for an entire triangle. The MerzWuthrich method is a oneyear implementation of the Mack method. Both the Mack and MerzWuthrich methods are applied to paid or reported triangles of
unlike the other approaches mentioned in this section. The Mack and MerzWuthrich methods assume ultimate losses are developed using the chain ladder method. Paid and reported agetoage loss development factors are used in a manner familiar to most property and casualty actuaries in calculating reserve risk, as follows:\[\begin{matrix} {\widehat{f}}_{d} ≔ \ \frac{\sum_{y = 1}^{Y  d}{C_{y,d + 1}\ }}{\sum_{y = 1}^{Y  d}{C_{y,d}\ }} \end{matrix}\tag{7.09}\]
The variance parameter
is calculated as follows:\[\begin{matrix} {{\widehat{s}}_{d}}^{2} ≔ \ \frac{\sum_{y = 1}^{Y  d}{{C_{y,d} \times \left( f_{y,d}  {\widehat{f}}_{d} \right)}^{2}\ }}{Y  d  1} \end{matrix}\tag{7.10}\]
Computationally, these estimators are identical to those used by Siegenthaler, except that in Siegenthaler’s method,
is replaced with and is replaced with When the formulae are compared, Mack and MerzWuthrich’s formulae are significantly more complex than Siegenthaler’s. One of the reasons for such complexity is that while the ultimate losses used in the Siegenthaler method at development age might be decent estimates for ultimate losses at maturity, paid or reported claims at age are often poor estimates for ultimate losses.7.2. Numerical Results
This section presents a summary of numerical comparison of the results obtained using the FengRobbin method (without parameter error), the RehmanKlugman method, the Siegenthaler method, the Mack method, and the MerzWuthrich method.
There are seven studies in total. Each study is based upon a particular set of ultimate loss triangles and is evaluated using all of the aforementioned reserve risk methods. Furthermore, each ultimate loss triangle is generated using one of three sets of paid triangles and by applying one of four ultimate loss selection approaches. These approaches include chain ladder, BF, and two hybrid selection methods which we call actuarial best estimate and management booked estimate. The rules for the hybrid selection methods are shown in Appendix C.1, which also derives the ultimate loss triangle from the paid loss triangle and presents all the underlying assumptions. Appendix C.2 presents the worksheets that quantify the reserve risk based on the ultimate loss triangles derived in Appendix C.1.
The following are observations based on a review of the numerical results in Table 23.

Study 1 is performed using data from Siegenthaler (2019) and produced identical results for the Siegenthaler method, the Mack method and the MerzWuthrich method as published in Siegenthaler (2019).

The FengRobbin standard error for total runoff risk is often higher than the RehmanKlugman result due to the inclusion of additional covariance terms.

The FengRobbin result for oneyear risk is often comparable to the Siegenthaler result.

Compared with the total runoff risk using the Siegenthaler method, the total runoff risk using the FengRobbin method is sometimes larger and other times smaller.

The FengRobbin method and the Siegenthaler method usually indicate a lower ratio of oneyear CV to total runoff CV than the ratio of MerzWuthrich oneyear CV to Mack total runoff CV. Lower ratios of oneyear CV to total runoff CV are usually appropriate for longertailed lines of business, especially when reserves are set using an initial expected loss ratio or the BF method.

Studies 4 – 7 use the same triangles and demonstrate that reserve risk based on the actuarial central estimate (Study 6) or management booked estimates (Study 7) can be very different from reserve risk assuming reserves are set using a particular method such as chainladder (Study 4) or BF (Study 5).

Study 7 shows that when there is consistent adverse development or consistent favorable development, the FengRobbin method yields lower standard error than the Siegenthaler method.
7.3. Overall Evaluation
The FengRobbin method’s treatment of covariance terms is beneficial for evaluating reserve risk because it fully considers the impact of covariance. All of the procedures proposed in this paper eliminate the possibility of negative calculated variance that arises from the triangle structure problem.
In comparison with the Siegenthaler method, both the FengRobbin and the RehmanKlugman methods measure variances of
factors around their own means, whereas the Siegenthaler method measures variances of factors around 1. In the FengRobbin method, the deviation of the mean of from 1 is captured in the parameter risk.Both the FengRobbin and Siegenthaler methods tend to compute a lower ratio of oneyear risk as a portion of total runoff risk than the ratio indicated by the Mack method and the MerzWuthrich method. The Mack method and the MerzWuthrich method theoretically only apply when reserves are set using the chainladder method, whereas the FengRobbin method, the Siegenthaler method and the RehmanKlugman method will be applicable in more general settings.
8. CONCLUSION
This paper has been an exploration into fully accounting for the covariance terms in computing reserve risk based on triangles of estimated ultimate losses. It has expanded the RehmanKlugman method from the totalrun off standard error to a oneyear standard error. Procedures were proposed for modifying the covariance matrix to eliminate the possibility of negative calculated variance that arises from the triangle structure problem. A broader examination of the fillin procedure and its derivatives for analysis of triangle data deserves research of its own. This work points to the need for further investigation into the statistical significance of variance and covariance terms for mature ages. Overall, this paper has attempted to provide readers with additional insights into alternative ways to estimate risk from ultimate loss triangles.