Chapter 14 Claims Reserving

14.1 Introduction

A crucial and central role of non-life actuaries is the calculation of provisions, primarily for outstanding claims. In particular, there are various technical methods of provisioning that we will explore in this chapter. These technical provisions are intended to facilitate the full settlement of commitments made to policyholders and beneficiaries of insurance contracts. They are closely tied to the very nature of insurance and are mandated by regulations to account for the “reverse production cycle,” as explained earlier. Technical provisions measure the remaining obligations to be fulfilled by the insurer and typically represent, on average in non-life insurance, 75% of the total balance sheet. However, this accounting concept requires an underlying probabilistic model since it involves forecasting a final liability, taking into account unreported or poorly known claims. For a long time, the role of actuaries was to find the best possible estimate of the required provisioning amount (the concept of the “best estimate”), but this approach quickly reveals its limitations (from a risk management and portfolio understanding perspective). Estimating the provisioning amount thus raises several issues for practitioners: should a safety margin be included (as claims handlers prefer to avoid re-provisioning)? How should atypical events (fraud, exceptionally severe claims, etc.) be taken into account? How can a regulator trust that a provision amount is not sufficient? What happens if we provision less? Does that pose a certain risk of bankruptcy?

The Chain-Ladder method (discussed in detail in Section 14.3 is probably the simplest and most widely used method by insurance companies in the last century. However, while it provides an estimator for the provisioning amount, it is challenging to extract more information from it. This provisioning calculation method (also referred to as the “claims development method”) is relatively old, as it can be traced back to legal books from the 1930s (see, for example, (Astesan 1938): “ […] ”). To do this, it relies on the fact that “,” noting that “.”

As noted by (Astesan 1938), describing the Chain-Ladder method, “. […] .” As (Henry 1937) pointed out, “.” Other methods will also be proposed later, introducing a approach, allowing for an error margin on the provisioning amount. We will particularly emphasize the various methods that provide a predictive distribution of this amount.

14.2 Notation and Motivation

Technical provisions are subject to regulatory constraints. They must, in particular, undergo two assessments. The first, known as the assessment, involves summing the outstanding amounts for all known claims. An estimated amount for late claims must be added to this. The second assessment is based on , i.e., the payment of claims to determine the amounts due for each year of occurrence.

Remark. The methods we will present here are in constant euros, meaning they do not take into account a provision discounting. This discounting has long been prohibited by legislation but is beginning to be introduced by European regulation for certain payment streams. In particular, in construction insurance, if we assume that payments are evenly distributed over 15 years (with an annual amount of \(P\)) and that money can be invested at a (fixed) rate \(r\) of \(3\%\), you only need to set aside \(12.3\) times the amount \(P\), instead of \(15\) without discounting, which is a 18% lower amount. The effect of this discounting factor is not negligible for very long-tail lines of business.

14.2.1 The Dynamics of Claim Life

For a type of risk (such as motor liability, health, marine, etc.), claims are reported (with varying delays) and then paid (also with varying time lags). The benefits to be paid by an insurance company cover several years of development (depending on the specific characteristics of the branch) and its seniority. The various aspects of the life of claims can be visualized in Figure ??, which corresponds to the analogue of Lexis diagrams used in life insurance.

The process strongly depends on the type of risk considered. Thus, the following table provides an idea of the payment rates for different lines of business:

Table 14.1: Payment developpement
n n+1 n+2 n+3 n+4
Homeowners 55% 90% 94% 95% 96%
Automobile Total 55% 79% 84% 99% 90%
Automobile Bodily injury 13% 38% 50% 65% 72%
Liability 10% 25% 35% 40% 45%

It is observed that for liability, automobile (bodily injury part), and general lines of business, less than 15% of claims are settled within the first year, and it takes 2 to 5 years for half of the claims to be settled. During this entire period, the balance sheet must reflect the likely cost of these claims.

The allocation of claims incurred (to assess the profitability of a line of business) will depend on the nature of the contract:

  • Incurred basis: For the vast majority of contracts (especially those covering mass risks such as automobile, homeowners, health, etc.). Earned premiums for a fiscal year are matched against claims incurred during the same period.
  • In some cases, claims may be analyzed based on the underwriting year or attached to the date of notification (e.g., in transportation or construction).

14.2.2 The Dynamics of Claims

For certain types of risks, especially in liability insurance, defining the exact “year of occurrence” can be relatively difficult, as demonstrated by the following example.

Example 14.1 In medical liability insurance, a large number of dates are involved in the life of a claim: the triggering event (a medical intervention), then the claim occurs (a side effect of the intervention is detected after a few months), then the claim is reported to the insurer, assessed, appraised, compensated, etc. The is therefore difficult to define precisely: should we consider the date of the intervention or the date when the claim is reported? Similarly, for water damage: should we consider the date when a pipe started leaking or the date when the damage was noticed on the lower floor?

14.2.3 Time Lags Before Reporting

For most claims, there is a more or less extended time lag before the policyholder reports the claim to the insurer. In Chapter 7 of Volume 1, we assumed that the of claims followed a homogeneous Poisson process. It may be legitimate to wonder if the result remains true for the reporting of these claims.

For a claim \(i\), let \(T_{i}\) denote the occurrence date, and \(D_{i}\) the time lag before reporting, such that \(\left(T_{i}+D_{i}\right)\) corresponds to the reporting date of the \(i\)th claim. If \(N_{t}\) denotes the number of claims occurring on the interval \((0,t)\), the number of claims not yet reported at time \(t\) is \[ M_{t}=\sum_{i=1}^{N_{t}}\mathbb{I}\left[ T_{i}+D_{i}>t\right]. \]

Proposition 14.1 If the time between occurrence and reporting follows a distribution \(F\), and if \(\left\{N_{t}, \hspace{2mm}t\geq 0\right\}\) is a homogeneous Poisson process with parameter \(\lambda\), then \(\left\{M_{t}, \hspace{2mm}t\geq 0\right\}\) is a non-homogeneous Poisson process with intensity \[ \Lambda _{M}\left( t\right) =\lambda \int_{0}^{t}F\left( x\right) dx. \]

In particular, note that \[ \mathbb{E}\left[ N_{t}\right] =\lambda t \text{ and } \mathbb{E}\left[ M_{t}\right] =\lambda \int_{0}^{t}F\left( x\right) dx=\mathbb{E}\left[ N_{t}\right] -\int_{0}^{t}\overline{F}\left( x\right) dx. \]

14.2.4 Run-off Triangles

Provisioning methods are all based on triangles, reflecting the dynamics of claims and allowing an aggregated view of them (for a history of methods used to provision non-life insurance claims, see (Reid 1986)). The notations used here are as follows:

  • \(i\) corresponds to the index of occurrence years \(i=1,...,n\).
  • \(j\) corresponds to the index of development years \(j=1,...,n\).
  • \(Y_{i,j}\) corresponds to the amount of claims occurring in year \(i\) and paid in year \(i+j\) (or after \(j\) years of development), also referred to as increments.
  • \(C_{i,j}\) corresponds to the aggregated payments of claims occurring in year \(i\), over \(j\) years of development, i.e., \(C_{i,j}=Y_{i,1}+Y_{i,2}+...+Y_{i,j}\), also referred to as cumulative amount.

Using these notations, the loss experience of a branch is represented by cumulative or non-cumulative triangles (referred to as run-off triangles) presented below: \[ \small{ \begin{array}{|ccccc} \hline C_{1,1} & C_{1,2} & \cdots & C_{1,n-1} & \multicolumn{1}{c|}{C_{1,n}} \\ \cline{5-5} C_{2,1} & C_{2,2} & \cdots & C_{2,n-1} & \multicolumn{1}{|c}{} \\ \cline{4-4} \vdots & \vdots & & & \\ C_{n-1,1} & \multicolumn{1}{c|}{C_{n-1,2}} & & & \\ \cline{2-2} \multicolumn{1}{|c|}{C_{n,1}} & & & & \\ \cline{1-1} \end{array}% } \] or \[ \small{ \begin{array}{|ccccc} \hline Y_{1,1} & Y_{1,2} & \cdots & Y_{1,n-1} & \multicolumn{1}{c|}{Y_{1,n}} \\ \cline{5-5} Y_{2,1} & Y_{2,2} & \cdots & Y_{2,n-1} & \multicolumn{1}{|c}{} \\ \cline{4-4} \vdots & \vdots & & & \\ Y_{n-1,1} & \multicolumn{1}{c|}{Y_{n-1,2}} & & & \\ \cline{2-2} \multicolumn{1}{|c|}{Y_{n,1}} & & & & \\ \cline{1-1} \end{array}% } \] The reading can be done by row, by column, or by diagonal:

\[\begin{equation*} \begin{tabular}{ccc} Development Year $j$ & Occurrence Year $i$ & Calendar Year $i+j$ \\ $% \begin{array}{|ccccc} \hline & \mathbf{\star } & & & \multicolumn{1}{c|}{} \\ \cline{5-5} & \mathbf{\star } & & & \multicolumn{1}{|c}{} \\ \cline{4-4} & \mathbf{\star } & & & \\ & \multicolumn{1}{c|}{\mathbf{\star }} & & & \\ \cline{2-2} \multicolumn{1}{|c|}{} & & & & \\ \cline{1-1} \end{array}% $ & $% \begin{array}{|ccccc} \hline & & & & \multicolumn{1}{c|}{} \\ \cline{5-5} \mathbf{\star } & \mathbf{\star } & \mathbf{\star } & \mathbf{\star } & \multicolumn{1}{|c}{} \\ \cline{4-4} & & & & \\ & \multicolumn{1}{c|}{} & & & \\ \cline{2-2} \multicolumn{1}{|c|}{} & & & & \\ \cline{1-1} \end{array}% $ & $% \begin{array}{|ccccc} \hline & & & \mathbf{\star } & \multicolumn{1}{c|}{} \\ \cline{5-5} & & \mathbf{\star } & & \multicolumn{1}{|c}{} \\ \cline{4-4} & \mathbf{\star } & & & \\ \mathbf{\star } & \multicolumn{1}{c|}{} & & & \\ \cline{2-2} \multicolumn{1}{|c|}{} & & & & \\ \cline{1-1} \end{array}% $% \end{tabular}% \end{equation*}\]

PAID = matrix(c(3209 ,4372 ,4411 ,4428 ,4435, 4456,
 3367, 4659, 4696, 4720, 4730  , NA,
 3871, 5345, 5338, 5420  , NA ,  NA,
 4239, 5917, 6020,   NA  , NA ,  NA,
 4929, 6794,   NA,   NA  , NA ,  NA,
 5217,   NA,   NA,   NA  , NA ,  NA),6,6,byrow=TRUE)
PAID
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 3209 4372 4411 4428 4435 4456
## [2,] 3367 4659 4696 4720 4730   NA
## [3,] 3871 5345 5338 5420   NA   NA
## [4,] 4239 5917 6020   NA   NA   NA
## [5,] 4929 6794   NA   NA   NA   NA
## [6,] 5217   NA   NA   NA   NA   NA

The rows (occurrence years) should, in particular, reflect changes related to underwriting and portfolio size effects. The columns (development years) allow for considering the fact that risks have varying durations. The diagonals reflect inflation, sometimes legal precedents. Note that every year, the triangle is completed by adding a new diagonal, reflecting payments made in year \(n+1\) for claims occurring in year \(n+1\) (bottom left), in year \(n\) (on the previous row), and so on.

Example 14.2 Consider the following payment triangle. \[\begin{equation*} \tiny{ \begin{array}{|cccccccc} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \cline{2-8} 1995 & \multicolumn{1}{|r}{23,758} & \multicolumn{1}{r}{25,356} & \multicolumn{1}{r}{19,468} & \multicolumn{1}{r}{11,258} & \multicolumn{1}{r}{ 6,458} & \multicolumn{1}{r}{4,268} & \multicolumn{1}{r}{2,312} \\ \cline{8-8} 1996 & \multicolumn{1}{|r}{31,245} & \multicolumn{1}{r}{32,496} & \multicolumn{1}{r}{27,034} & \multicolumn{1}{r}{15,664} & \multicolumn{1}{r}{ 8,615} & \multicolumn{1}{c|}{5,156} & \multicolumn{1}{r}{} \\ \cline{7-7} 1997 & \multicolumn{1}{|r}{26,312} & \multicolumn{1}{r}{31,467} & \multicolumn{1}{r}{24,672} & \multicolumn{1}{r}{13,055} & \multicolumn{1}{c|}{6,158} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} \\ \cline{6-6} 1998 & \multicolumn{1}{|r}{30,470} & \multicolumn{1}{r}{35,012} & \multicolumn{1}{r}{25,491} & \multicolumn{1}{c|}{12,589} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} \\ \cline{5-5} 1999 & \multicolumn{1}{|r}{49,756} & \multicolumn{1}{r}{51,831} & \multicolumn{1}{c|}{35,267} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} \\ \cline{4-4} 2000 & \multicolumn{1}{|r}{50,420} & \multicolumn{1}{c|}{52,315} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} \\ \cline{3-3} 2001 & \multicolumn{1}{|c|}{56,762} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} \\ \hline \end{array}% \vline } \end{equation*}\] in non-cumulative amounts, or, in cumulative amounts \[\begin{equation*} \tiny{ \begin{array}{|cccccccc} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \cline{2-8} 1995 & \multicolumn{1}{|r}{23,758} & \multicolumn{1}{r}{49,114} & \multicolumn{1}{r}{68,582} & \multicolumn{1}{r}{79,840} & \multicolumn{1}{r}{ 86,298} & \multicolumn{1}{r}{90,566} & \multicolumn{1}{r}{92,878} \\ \cline{8-8} 1996 & \multicolumn{1}{|r}{31,245} & \multicolumn{1}{r}{63,741} & \multicolumn{1}{r}{90,775} & \multicolumn{1}{r}{106,439} & \multicolumn{1}{r}{115,054} & \multicolumn{1}{c|}{120,210} & \multicolumn{1}{r}{} \\ \cline{7-7} 1997 & \multicolumn{1}{|r}{26,312} & \multicolumn{1}{r}{57,779} & \multicolumn{1}{r}{82,451} & \multicolumn{1}{r}{95,506} & \multicolumn{1}{c|}{101,604} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} \\ \cline{6-6} 1998 & \multicolumn{1}{|r}{30,470} & \multicolumn{1}{r}{65,482} & \multicolumn{1}{r}{90,973} & \multicolumn{1}{c|}{103,562} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} \\ \cline{5-5} 1999 & \multicolumn{1}{|r}{49,756} & \multicolumn{1}{r}{101,587} & \multicolumn{1}{c|}{136,854} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} \\ \cline{4-4} 2000 & \multicolumn{1}{|r}{50,420} & \multicolumn{1}{c|}{102,735} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} \\ \cline{3-3} 2001 & \multicolumn{1}{|c|}{56,762} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{} \\ \hline \end{array}% \vline } \end{equation*}\]

Also, note, for example, that in \(1999\), \(49,756\) Euros were paid for claims occurring in \(1999\), and \(6,458\) Euros for claims occurring in \(1995\). Thus, the amount of payments made for claims occurring in \(1998\) was, at the end of \(2000\), \(90,973\) Euros.

Remark. While it is possible to assume that the row and column effects are independent, calendar effects will have impacts on both development and occurrence year claim amounts.

The calculation of reserves involves forecasting the ultimate claim amount to provision for payments that have not yet been made. In practice, under the assumption that \(n\) is the maximum time needed to close a year, this means completing the lower part of the triangle. We then seek the \(\widehat{Y}_{i,j}\) and \(\widehat{C}_{i,j}\) for \(i+j\geq n+2\), which means those below the diagonal:

\[\begin{equation*} \begin{array}{|ccccc|} \hline C_{1,1} & C_{1,2} & \cdots & C_{1,n-1} & \multicolumn{1}{c|}{C_{1,n}} \\ \cline{5-5} C_{2,1} & C_{2,2} & \cdots & C_{2,n-1} & \multicolumn{1}{|c|}{\widehat{C}% _{2,n}} \\ \cline{4-4} \vdots & \vdots & & & \\ C_{n-1,1} & \multicolumn{1}{c|}{C_{n-1,2}} & & \widehat{C}_{n-1,n-1} & \widehat{C}_{n-1,n} \\ \cline{2-2} \multicolumn{1}{|c|}{C_{n,1}} & \widehat{C}_{n,2} & & \widehat{C}_{n,n-1} & \widehat{C}_{n,n} \\ \hline \end{array}% \end{equation*}\]

or

\[\begin{equation*} \begin{array}{|ccccc|} \hline Y_{1,1} & Y_{1,2} & \cdots & Y_{1,n-1} & \multicolumn{1}{c|}{Y_{1,n}} \\ \cline{5-5} Y_{2,1} & Y_{2,2} & \cdots & Y_{2,n-1} & \multicolumn{1}{|c|}{\widehat{Y}% _{2,n}} \\ \cline{4-4} \vdots & \vdots & & & \\ Y_{n-1,1} & \multicolumn{1}{c|}{Y_{n-1,2}} & & \widehat{Y}_{n-1,n-1} & \widehat{Y}_{n-1,n} \\ \cline{2-2} \multicolumn{1}{|c|}{Y_{n,1}} & \widehat{Y}_{n,2} & & \widehat{Y}_{n,n-1} & \widehat{Y}_{n,n} \\ \hline \end{array}% \end{equation*}\]

Having estimated these future payments, the amount of reserves for the underwriting year \(i\) is given by:

\[\begin{equation*} \widehat{R}_{i}=\widehat{C}_{i,n}-C_{i,n+1-i}=\widehat{Y}_{i,n+2-i}+\widehat{% Y}_{i,n+3-i}+...+\widehat{Y}_{i,n} \end{equation*}\]

Thus, the total reserve amount required is:

\[\begin{equation*} R=\sum_{i=1}^{n}\widehat{R}_{i}=\sum_{i=1}^{n}\widehat{C}_{i,n}-C_{i,n+1-i}=% \sum_{\left( i,j\right) \in \Delta _{n}}\widehat{Y}_{i,j} \end{equation*}\]

where \(\Delta _{n}\) denotes the set ${ ( i,j) i+jn+2inj n} $.

Remark. This mathematical definition aligns with the accounting definition of reserves: it represents the amount that will allow the full settlement of claims (the sum of payments already made and the provision for the total liability). If \(C_i\) represents the ultimate liability for claims occurring in year \(i\), \(C_i=\lim_{n\rightarrow \infty }C_{i,n}\), then \(R_{i}\), the amount of reserves needed for claims occurring in year \(i\), is \(R_{i}=C_{i}-C_{i,n-i+1}\), and finally, the total provision amount, \(R=R_{1}+...+R_{n}\).

In the following, we will denote \(E_{i}\) as the exposure for year \(i\), \(\Pi _{i}\) as the earned premiums for year \(i\), and \(N_{i}\) as the number of policies for year \(i\).

14.3 Deterministic Methods

Deterministic methods are based on the assumption of stability in the time between the occurrence of a claim and its settlement(s), regardless of the occurrence year, in the absence of inflation, changes in portfolio structure, contract guarantees, deductibles, and more generally, claims management. If all these assumptions hold over a sufficiently long period (at least 5 years), deterministic methods can be an initial useful tool for forecasting the ultimate liability by using observed payment patterns from the past.

14.3.1 The Chain Ladder Method

This method is among the most popular because it is easy to implement and understand. The idea is that the development of payments is governed by development factors (denoted as \(\lambda_j\)) that depend only on the development year. The underlying model is then of the form \(C_{i,j} =\lambda_jC_{i,j-1}\). The parameters used in the Chain Ladder method have the advantage of having a clear interpretation and are easily estimable. The drawback is that this estimation is relatively less robust. This method does not make any assumptions about the distribution of claim costs or their frequency.

Given a triangle over \(n\) years (meaning there are \(1+2+...+n=n(n+1)/2\) observations), the goal is to consider models that involve a minimum number of parameters to best predict future payment amounts.

14.3.3 Bonuses and Penalties, or Updating Estimates

In the examples considered so far, we were at the end of the year \(2001\), and we sought to estimate the total reserve required at the close of the accounts. In practice, this exercise will be repeated at the end of the year \(2002\), which will generally result in an update of the estimates. Consider the payment triangle from Example ??, and suppose that the reserve amount was obtained using the Chain Ladder method (i.e., \(323,371\) at the end of \(2001\)). Suppose that at the end of 2002, we add the following diagonal to the cumulative payment triangle: {92,878} for 1995, {124,156} for 1996, {107,236} for 1997, {113,455} for 1998, {160,233} for 1999, {138,653} for 2000, {123,156} for 2001, and {61,262} for 2002. Also, the payment amounts made in \(2002\) are as follows for each of the previous years:

A slight difference can be noted compared to the predictions made by the Chain Ladder method (\(+1,970\) out of a total of \(138,563\), which is a higher amount of \(+1.5\%\)). However, these differences will have a significant impact on the total reserve amount. If we compare the estimates of the ultimate loss for the years prior to 2002, we obtain the following table:

Thus, for the previous years, by paying \(1,970\) more than what was predicted by the Chain Ladder method in 2002, the total amount of reserves required for the prior years will require an additional contribution of \(10,712\). This is referred to as a penalty of \(10,712\) on prior years.

14.3.4 Critiques of the Chain Ladder Method

Despite being relatively simple, this method poses several problems:

  • The development pattern is the same for all accident years, meaning that the cost of a claim after \(j\) years of development is proportional to the cost in the previous year or any year \(i<j\). This proportionality coefficient does not change (and corresponds to \(\lambda_{j-1}\lambda_{j-2}...\lambda_i\), as mentioned in Equation (??). However, in practice, this is generally not the case in various situations:
    • Changes in jurisprudence: In such cases, there can be a jump in payments, which will be much higher if payments are made today compared to if they were made last year.
    • Management changes (in claims handling or underwriting): For various reasons, a company that thought it might be beneficial to delay claim settlements (and go to court rather than pay quickly) may later decide that it’s more advantageous to settle claims very quickly (especially for internal cost reasons).
  • For recent years, uncertainty is very high: the multiplicative coefficient for the last year is the product of \(n-1\) estimates of proportionality coefficients. This uncertainty is even greater for long-tail risks, where the first payments begin after a few years. If payments in the first year represent around 1% of the total amount, paying 0.8% or 1.2% will result in a 50% variation in the total provisions for that year. The use of payment triangles in such cases can be risky.
  • This deterministic method does not provide a measure of precision (or rather imprecision) for the estimates.

14.3.5 Variations on the Chain-Ladder Method

14.3.5.1 Weighting

It is possible to introduce weighting when estimating the \(\lambda_k\) values to give more or less importance to past years. Among the types of weighting used, you can consider:

  • Weighting that gives more weight to recent years and less to distant years.
  • Weighting that takes into account the actual exposure to risk for each year, meaning that the weights are related to the number of policies or the earned premium associated with policies in force in year \(i\).

In both cases, you consider link ratios of the form:

\[\begin{equation*} \widehat{\lambda }_{k}=\frac{1}{\sum_{i=1}^{n-k}\omega _{i,k}}\sum_{i=1}^{n-k}\omega _{i,k} \lambda_{i,k} \text{ for }\lambda_{i,k}=\frac{C_{i,k+1}}{C_{i,k}}\text{ for }k=1,...,n-1. \end{equation*}\]

It’s worth noting that if \(\omega _{i,k}=C_{i,k}\), you get the standard method.

Example 14.4 (Continuation of Example ??) The transition coefficients \(\lambda_{i,k}\) for each accident year are as follows: {\[\begin{equation*} \vline% \begin{tabular}{|c|cccccc|}\hline & $0\rightarrow 1$ & $1\rightarrow 2$ & $2\rightarrow 3$ & $3\rightarrow 4$ & $4\rightarrow 5$ & $5\rightarrow 6$ \\ \hline 1995 & $2.067$ & $1.396$ & $1.164$ & $1.080$ & $1.049$ & $1.025$ \\ 1996 & $2.040$ & $1.424$ & $1.172$ & $1.080$ & $1.044$ & \\ 1997 & $2.195$ & $1.427$ & $1.158$ & $1.064$ & & \\ 1998 & $2.149$ & $1.389$ & $1.138$ & & & \\ 1999 & $2.041$ & $1.347$ & & & & \\ 2000 & $2.037$ & & & & & \\ \hline \end{tabular} \end{equation*}\]}

14.3.5.1.1 London Chain Method

In the London Chain method, it is assumed that the dynamics of \(\left( C_{ij}\right) _{j=1,..,n}\) are given by a model of the form:

\[\begin{equation*} C_{i,k+1}=\lambda _{k}C_{i,k}+\alpha _{k}\text{ for all }i,k=1,..,n. \end{equation*}\]

In practical terms, it can be noted that the standard Chain Ladder method, based on a model of the form \(C_{i,k+1}=\lambda _{k}C_{i,k}\), could only be applied when the points \((C_{i,k},C_{i,k+1})\) are approximately aligned (at a fixed \(k\)) on a line passing through the origin. The London Chain method also assumes that the points are aligned on the same line, but it no longer assumes that the line passes through the origin.

Example 14.5 (Continuation of Example ??) The best-fit line passing through the point cloud and through the origin is obtained in Figure ??.

In this model, there are \(2n\) parameters to identify: \(\lambda _{k}\) and \(\alpha _{k}\) for \(k=1,...,n\). The most natural method is to estimate these parameters using the least squares method. That is, for each \(k\), we seek:

\[\begin{equation*} \left( \widehat{\lambda }_{k},\widehat{\alpha }_{k}\right) =\arg \min \left\{ \sum_{i=1}^{n-k}\left( C_{i,k+1}-\alpha _{k}-\lambda _{k}C_{i,k}\right) ^{2}\right\} \end{equation*}\]

Using the results from Chapter 9 on linear models, this gives:

\[\begin{equation*} \widehat{\lambda }_{k}=\frac{\frac{1}{n-k}\sum_{i=1}^{n-k}C_{i,k}C_{i,k+1}-\overline{C}_{k}^{\left( k\right) }\overline{C}_{k+1}^{\left( k\right) }}{\frac{1}{n-k}\sum_{i=1}^{n-k}C_{i,k}^{2}-\overline{C}_{k}^{\left( k\right) 2}}, \end{equation*}\]

where

\[\begin{equation*} \overline{C}_{k}^{\left( k\right) }=\frac{1}{n-k}\sum_{i=1}^{n-k}C_{i,k} \end{equation*}\]

and

\[\begin{equation*} \overline{C}_{k+1}^{\left( k\right) }=\frac{1}{n-k}\sum_{i=1}^{n-k}C_{i,k+1} \end{equation*}\]

The constant is given by:

\[\begin{equation*} \widehat{\alpha }_{k}=\overline{C}_{k+1}^{\left( k\right) }-\widehat{\lambda }_{k}\overline{C}_{k}^{\left( k\right) } \end{equation*}\]

Example 14.6 (Continuation of Example ??) The obtained coefficients are as follows: {\[\begin{equation*} \vline% \begin{array}{|r|cccccc|} \hline k& 1 & 2 & 3 & 4 & 5 & 6 \\\hline \widehat{\lambda }_{k}& 1.951 & 1.277 & 1.128 & 1.074 & 1.031 & 1.026\\\ \widehat{\alpha }_{k}& 4.468 & 7.709 & 2.515 & 0.111 & 1.603 & - \\ \hline \end{array}% \vline \end{equation*}\]\end{equation*}}%

It can be noted that the coefficients \(\widehat{\lambda }_{k}\) are relatively close to those obtained by the standard Chain Ladder method, which seems to confirm the impression provided by Figure ??, showing that the slopes of the lines are approximately the same. For estimating reserves, the obtained triangle is then given by:

\[\begin{equation*} \tiny{ \begin{tabular}{|l|rrrrrrr|} \hline & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$\\ \hline $1995$ & $23,758$ & $49,114$ & $68,582$ & $79,840$ & $86,298$ & $90,566$ & $92,878$ \\ $1996$ & $31,245$ & $63,741$ & $90,775$& $106,439$ & $115,054$ & $120,210$ & $\mathbf{123,278}$ \\ $1997$ & $26,312$ & $57,779$ & $82,451$ & $95,506$ & $101,664$ & $\mathbf{109,314}$ & $\mathbf{112,105}$ \\ $1998$ & $30,470$ & $65,482$ & $90,973$ & $103,562$ & $\mathbf{111,353}$ & $\mathbf{119,722}$ & $\mathbf{122,779}$ \\ $1999$ & $49,756$ & $101,587$ & $136,854$ & $\mathbf{156,849}$ & $\mathbf{168,592}$ & $\mathbf{181,206}$ & $\mathbf{185,832}$ \\ $2000$ & $50,420$ & $102,735$ & $\mathbf{138,854}$ & $\mathbf{159,104}$ & $\mathbf{171,015}$ & $\mathbf{183,809}$ & $\mathbf{188,501}$ \\ $2001$ & $56,762$ & $\mathbf{115,235}$ & $\mathbf{154,811}$ & $\mathbf{177,099}$ & $\mathbf{190,345}$ & $\mathbf{204,572}$ & $\mathbf{209,795}$ \\ \hline \end{tabular} } \end{equation*}\]

14.3.6 Projected Case Estimate Method

14.3.6.1 Principle

Note that provisioning methods, presented here as projections of future payments, can generally be adapted to model the total charge (payments and reserves, also known as case reserves, established by claims adjusters). These reserves are generally inadequate for two main reasons:

  1. Firstly, they are only established for claims reported to the insurance company (and do not take into account claims that have occurred but have not yet been reported, which should still be reserved for).
  2. Additionally, they may be established when a claim is reported, using fixed amounts (based on initial information reported to the insurer, such as the type of claim, number of people involved, etc.).

(Taylor 2012) proposes the following method, called the “projected case” method. In this method, two development factors are used, one for payments and one for reserves. Note that using the Chain Ladder method on the total amounts (payments + reserves) of claims assumes that payments and reserves develop in the same way. The method studied in this section uses two types of development factors: one for reserves and one for payments. This method allows for the use of all available information while linking the developments of payments and reserves.

The use of the reserves triangle can be relevant in some slow-developing lines of business where very few claims are settled in the first year. The Chain Ladder method provides results that are not robust due to its high sensitivity to the first value for the last year. The use of established reserves is then a particularly interesting alternative.

14.3.6.1.1 Model for Reserves

Let \(Q_{ij}\) be the reserve for claims that occurred in year \(i\) and is recorded on the balance sheet liability at the end of year \(i+j-1\). The model chosen for the evolution of reserves is as follows:

\[\begin{equation} Q_{ij+1}=k_{j+1}\cdot Q_{ij}-Y_{ij+1}, \tag{14.2} \end{equation}\]

where \(k_{j+1}\) measures the variation that occurs between years \(j\) and \(j+1\) in the forecast for the total cost of claims occurring in accident year \(i\).

Indeed, \(Q_{ij+1}\) represents the estimate of what remains to be paid at the end of development year \(j+1\), and \(Q_{ij}\) represents the same estimate at the end of development year \(j\). If the estimate has not changed (i.e., \(k_{j+1}=1\)), then \(Q_{ij+1}\) is equal to the difference between \(Q_{ij}\) and what is paid in year \(j+1\), which is \(Y_{ij+1}\).

14.3.6.1.2 Estimation of \(k_{j+1}\)

The estimator for \(k_{j+1}\) chosen is a weighted average by individual coefficients \(Q_{ij}\):

\[\begin{equation} \hat{k}_{j+1}=\frac{\displaystyle\sum_{i=1}^{n-j}\left( Y_{ij+1}+Q_{ij+1} \right) } {\displaystyle\sum_{i=1}^{n-j}Q_{ij}}. \tag{14.3} \end{equation}\]

14.3.6.1.3 Model for Payments

The amount \(Y_{ij+1}\) paid during development year \(j+1\) is a fraction of \(Q_{ij}\), which was reserved at the end of the previous year:

\[\begin{equation} Y_{ij+1}=h_{j+1}\cdot Q_{ij}. \tag{14.4} \end{equation}\]

Once again, these development coefficients are estimated using a weighted average by individual coefficients \(Q_{ij}\):

\[\begin{equation} \hat{h}_{j+1}=\frac{\displaystyle\sum_{i=1}^{n-j}Y_{ij+1}} {\displaystyle\sum_{i=1}^{n-j}Q_{ij}}. \tag{14.5} \end{equation}\]

14.3.6.1.4 Extrapolation of Triangles

The triangle of payments and the triangle of reserves are simultaneously completed, diagonal by diagonal, using the two models (14.2) and ((14.4), used one after the other.

We start with the first unknown diagonal of the payment triangle:

\[ \hat{Y}_{i,n-i+2}=\hat{h}_{n-i+2}\cdot Q_{i,n-i+1} \text{ for } i=2,\ldots, n. \]

Then we complete the first unknown diagonal of the reserves triangle:

\[ \hat{Q}_{i,n-i+2}=\hat{k}_{n-i+2}\cdot Q_{i,n-i+1}-\hat{Y}_{i,n-i+2} \text{ for } i=2,\ldots, n. \]

We continue with the next diagonal of the payment matrix:

\[ \hat{Y}_{i,n-i+3}=\hat{h}_{n-i+3}\cdot \hat{Q}_{i,n-i+2} \text{ for } i=3,\ldots, n. \]

And so on…

14.3.6.1.5 Connection with the Chain Ladder Method

If we combine the two models (14.2) and (14.4) into one, we obtain:

\[ Q_{i,j+1}=\big(k_{j+1}-h_{j+1}\big)\cdot Q_{ij}. \]

By replacing the different terms of the factor \(k_{j+1}-h_{j+1}\) with the estimators (14.3) and (14.5), we obtain:

\[ \hat{Q}_{i,j+1}=\frac{\displaystyle\sum_{i=1}^{n-j}Q_{i,j+1}} {\displaystyle\sum_{i=1}^{n-j}Q_{ij}}\cdot Q_{ij}, \]

which is equivalent to applying the standard Chain Ladder method to the reserves triangle.

Instead of proceeding as explained above, we can simply complete the reserves triangle using the Chain Ladder method, calculate the coefficients \(\hat{h}_{j+1}\) using formula (14.5), and then complete the payment triangle.

Example 14.7 Let’s consider the annual payments triangle and the reserves triangle below:

Payments Triangle (\(\boldsymbol{Y}\)): \[ \boldsymbol{Y}= \left(\begin{array}{ccccc} 15.40 & 4.90 & 7.77 & 7.19 & 4.30 \\ 16.61 & 2.60 & 11.03 & 9.12 & \\ 21.35 & 7.29 & 5.59 & & \\ 24.52 & 8.49 & & & \\ 30.47 & & & & \\ \end{array} \right) \]

Reserves Triangle (\(\Qvec\)): \[ \Qvec= \left(\begin{array}{ccccc} 20.0 & 17.39 & 11.06 & 4.50 & 0.60 \\ 22.0 & 22.40 & 13.13 & 5.20 & \\ 22.5 & 18.66 & 15.22 & & \\ 25.0 & 20.32 & & & \\ 25.0 & & & & \\ \end{array} \right) \]

We can apply the Chain Ladder method to the cumulative charges triangle (obtained by adding cumulative payments and reserves), which is:

Cumulative Charges Triangle (\(\boldsymbol{C}\)): \[ \boldsymbol{C}=\left(\begin{array}{ccccc} 35.40 & 37.69 & 39.13 & 39.76 & 40.16 \\ 38.61 & 41.61 & 43.37 & 44.56 & \\ 43.85 & 47.30 & 49.45 & & \\ 49.52 & 53.33 & & & \\ 55.47 & & & & \\ \end{array} \right) \]

Let’s calculate the development coefficients \(\hat{\lambda}_k\), which are:

\(\hat{\lambda}_1=1.0750\), \(\hat{\lambda}_2=1.0423\), \(\hat{\lambda}_3=1.0221\), and \(\hat{\lambda}_4=1.0101\).

Using these development coefficients, we can complete the cumulative charges triangle \(C\):

Cumulative Charges Triangle (\(C\)): \[ C=\left(\begin{array}{ccccc} 35.40 & 37.69 & 39.13 & 39.76 & 40.16 \\ 38.61 & 41.61 & 43.37 & 44.56 & \textbf{45.01} \\ 43.85 & 47.30 & 49.45 & \textbf{50.54} & \textbf{51.05} \\ 49.52 & 53.33 & \textbf{55.58} & \textbf{56.81} & \textbf{57.38} \\ 55.47 & \textbf{56.63} & \textbf{62.15} & \textbf{63.52} & \textbf{64.16} \\ \end{array} \right) \]

Finally, we can estimate the reserves \(R_i\) for \(i=2,\ldots,5\) using \(\hat{R}_i=\hat{C}_{i5}-C_{i,6-i}\), which gives the values in the following table:

{

}

This approach, however, imposes the same development on both payments and reserves. To overcome this constraint, we can apply the Projected Case Estimate method. The coefficients of the reserves model, calculated by (14.3), are as follows: \[ \hat{k}_2=1.1402 \quad \hat{k}_3=1.0915 \quad \hat{k}_4=1.0752 \quad \hat{k}_5=1.0889. \]

The coefficients of the payments model, calculated by (??), are as follows: \[ \hat{h}_2=0.2601 \quad \hat{h}_3=0.4173 \quad \hat{h}_4=0.6742 \quad \hat{h}_5=0.9556. \]

Both the payments and reserves triangles can then be completed diagonally to obtain: Payments Triangle (\(\boldsymbol{Y}\)): \[ \boldsymbol{Y} = \left(\begin{array}{ccccc} 15.40 & 4.90 & 7.77 & 7.19 & 4.30 \\ 16.61 & 2.60 & 11.03 & 9.12 & {\bf 4.97}\\ 21.35 & 7.29 & 5.59 & {\bf 10.26} & {\bf 5.83}\\ 24.52 & 8.49 & {\bf 8.48} & {\bf 9.24} & {\bf 5.25}\\ 30.47 & {\bf 6.50} & {\bf 9.18} & {\bf 10.00} & {\bf 5.68}\\ \end{array} \right) \]

Reserves Triangle (\(\Qvec\)): \[ \Qvec= \left(\begin{array}{ccccc} 20.0 & 17.39 & 11.06 & 4.50 & 0.60 \\ 22.0 & 22.40 & 13.13 & 5.20 & {\bf 0.69}\\ 22.5 & 18.66 & 15.22 & {\bf 6.10} & {\bf 0.81}\\ 25.0 & 20.32 & {\bf 13.70} & {\bf 5.49} & {\bf 0.73}\\ 25.0 & {\bf 22.00} & {\bf 14.84} & {\bf 5.95} & {\bf 0.79}\\ \end{array} \right). \]

If we recalculate the payments and add them to the reserves of claims, we obtain the following total results: \[ \left(\begin{array}{ccccc} 35.40 & 37.69 & 39.13 & 39.76 & 40.16 \\ 38.61 & 41.61 & 43.37 & 44.56 & {\bf 45.02}\\ 43.85 & 47.30 & 49.45 & {\bf 50.60} & {\bf 51.14}\\ 49.52 & 53.33 & {\bf 55.19} & {\bf 56.22} & {\bf 56.71}\\ 55.47 & {\bf 58.98} & {\bf 60.99} & {\bf 62.11} & {\bf 62.63}\\ \end{array} \right). \]

So, we obtain results of the same order of magnitude as with the Chain Ladder method, but with more noticeable differences for the later years.

Note that the development coefficients of the Chain Ladder method calculated for the reserves triangle above are: \[ \hat{\lambda}_1=0.8801 \quad \hat{\lambda}_2=0.6743 \quad \hat{\lambda}_3=0.4010 \quad \hat{\lambda}_4=0.1333. \]

14.3.6.1.6 Comment

The Projected Case Estimate method uses more information than the Chain Ladder method. However, the amounts reserved by the company may not have the same objectivity as the amounts paid by the company. In some cases, provisioning policies are guided by tax considerations, and the use of balance sheet reserves can be questionable.

14.3.7 De Vylder’s Least Squares Method

This methodology is based on modeling the increments (rather than aggregated payments as done so far) in the form: \[\begin{equation*} Y_{i,j}=r_{j}\cdot p_{i} \end{equation*}\] where \(p_{i}\) corresponds to the ultimate loss of claims occurring in year \(i\), and \(r_{j}\) is the proportion of the amount \(p_{i}\) paid in year \(j\). The payments triangle is then written as: \[\begin{equation*} \begin{array}{|ccccc} \hline r_{1}\cdot p_{1} & r_{2}\cdot p_{1} & \cdots & r_{n-1}\cdot p_{1} & \multicolumn{1}{c|}{ r_{n}\cdot p_{1}} \\ \cline{5-5} r_{1}\cdot p_{2} & r_{2}\cdot p_{2} & \cdots & r_{n-1}\cdot p_{2} & \multicolumn{1}{|c}{} \\ \cline{4-4} \vdots & \vdots & & & \\ r_{1}\cdot p_{n-1} & \multicolumn{1}{c|}{r_{2}\cdot p_{n-1}} & & & \\ \cline{2-2} \multicolumn{1}{|c|}{r_{1}\cdot p_{n}} & & & & \\ \cline{1-1} \end{array}% \end{equation*}\]

The coefficients \(r_{j}\) and \(p_{i}\) are obtained by minimizing the sum of squares of the differences between the observed values \(Y_{ij}\) and their theoretical form \(r_jp_i\), i.e., \[\begin{equation*} \sum_{i+j\leq n}\left( Y_{i,j}-r_{j}\cdot p_{i}\right) ^{2}, \end{equation*}\] with the constraint of identifiability \(r_{1}+...+r_{n}=1\). This leads to: \[\begin{equation*} \widehat{p}_{i}=\frac{\sum_{j}\widehat{r}_{j}Y_{i,j}}{\sum_{j}\widehat{r}% _{j}^{2}}\text{ and }\widehat{r}_{j}=\frac{\sum_{i}\widehat{p}_{j}Y_{i,j}}{% \sum_{i}\widehat{p}_{j}^{2}}\text{.} \end{equation*}\]

Remark. This method remains stable assuming that there is a constant (calendar) inflation of rate \(\nu\), i.e., \[ Y_{i,j}=r_{j}\cdot p_{i}\cdot \nu ^{i+j}. \] Inflation is then integrated into the coefficients \(r_j\) and \(p_i\).

Example 14.8 (Continuation of Example ??) The estimated \(r_j\) and \(p_i\) values are given in the following table: \[\begin{equation*} \tiny{ \begin{tabular}{|l|lllllll|} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline $\widehat{r}_{j}$& $0.261$ & $0.278$ & $0.207$ & $0.118$ & $0.065$ & $0.043$ & $0.025$ \\ $\widehat{p}_{i}$& $92,056$ & $121,776$ & $109,585$ & $120,709$ & $183,958$ & $190,118$ & $217,383$ \\ \hline \end{tabular}% } \end{equation*}\]

These values are used to reconstitute the lower part of the annual payments table: \[\begin{equation*} {\tiny \begin{array}{|c|lllllll|}\hline & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ \hline 1995 & 24,037 & 49,720 & 68,796 & 79,670 & 85,713 & 89,744 & 92,056 \\ 1996 & 31,797 & 65,772 & 91,007 & 105,392 & 113,386 & 118,719 & \mathbf{121,776} \\ 1997 & 28,614 & 59,188 & 81,896 & 94,841 & 102,034 & \mathbf{106,833} & \mathbf{109,585} \\ 1998 & 31,519 & 65,196 & 90,209 & 104,469 & \mathbf{112,392} & \mathbf{117,679} & \mathbf{120,709} \\ 1999 & 48,034 & 99,358 & 137,477 & \mathbf{159,208} & \mathbf{171,283} & \mathbf{179,339} & \mathbf{183,958} \\ 2000 & 49,642 & 102,685 & \mathbf{142,080} & \mathbf{164,539} & \mathbf{177,019} & \mathbf{185,344} & \mathbf{190,118} \\ 2001 & 56,762 & \mathbf{117,411} & \mathbf{162,456} & \mathbf{188,136} & \mathbf{202,405} & \mathbf{211,924} & \mathbf{217,383} \\ \hline \end{array}} \end{equation*}\]

This method thus yields a total required provision of approximately \(321,383\text{Euros}\), which is very close to the value obtained using the Chain Ladder method. :::

14.3.8 Taylor’s Separation Method

This method was proposed by (Taylor 1977). The idea is to consider inflation as an endogenous factor in the loss development triangles. To do this, we work with the disaggregated triangles and assume that the amount of payments \(Y_{ij}\) related to claims occurring in year \(i\) and paid in year \(i+j-1\) has the following form: \[\begin{equation*} Y_{ij}=r_{j}\cdot \mu _{i+j-1}\text{ for all }i,j. \end{equation*}\] The payment triangle is then expressed as: \[\begin{equation*} \begin{array}{|ccccc} \hline r_{1}\cdot \mu _{1} & r_{2}\cdot \mu _{2} & \cdots & r_{n-1}\cdot \mu _{n-1} & \multicolumn{1}{c|}{r_{n}\cdot \mu _{n}} \\ \cline{5-5} r_{1}\cdot \mu _{2} & r_{2}\cdot \mu _{3} & \cdots & r_{n-1}\cdot \mu _{n} & \multicolumn{1}{|c}{} \\ \cline{4-4} \vdots & \vdots & & & \\ r_{1}\cdot \mu _{n-1} & \multicolumn{1}{c|}{r_{2}\cdot \mu _{n}} & & & \\ \cline{2-2} \multicolumn{1}{|c|}{r_{1}\cdot \mu _{n}} & & & & \\ \cline{1-1} \end{array}% \end{equation*}\]

In other words, payments are functions of a payment pattern \(r_{j}\), which depends on the development time \(j\) (found throughout the \(j\)-th column), and a calendar (or inflation) factor \(\mu _{i+j-1}\), which depends on the payment year \(i+j-1\) (found throughout the \(i+j\)-th diagonal).

The goal is to estimate the development coefficients \(r_{1}, r_{2}, \ldots, r_{n}\) and the inflation factors \(\mu _{1}, \mu _{2}, \ldots, \mu_{n}\). To estimate these \(2n\) coefficients, we make the normalization assumption for \(r_{j}\) as before, namely \(r_{1}+r_{2}+\ldots+r_{n}=1\) (some authors refer to this as arithmetic separation). Also, the sum along the diagonal is \[\begin{equation*} d_{n}=Y_{1,n}+Y_{2,n-1}+\ldots+Y_{n,1}=\mu_{n}\left(r_{1}+r_{2}+\ldots+r_{k}\right)=\mu_{n}. \end{equation*}\] For the first superdiagonal, we can note that \[\begin{equation*} d_{n-1}=Y_{1,n-1}+Y_{2,n-2}+\ldots+Y_{n-1,1}=\mu_{n-1}\left(1-r_{n}\right) \end{equation*}\] and, furthermore, by considering the \(n\)-th column, \(\gamma_{n}=Y_{1,n}=r_{n}\mu_{n}\), which implies \[\begin{equation*} r_{n}=\frac{\gamma_{n}}{\mu_{n}}\text{ and }\mu_{n-1}=\frac{d_{n-1}}{1-r_{n}}. \end{equation*}\] In general, for the \(i\)-th superdiagonal, we can note that \[\begin{eqnarray*} d_{n-i}&=&Y_{1,n-i}+Y_{2,n-i-1}+\ldots+Y_{n-i,1}\\ &=&\mu_{n-i}\left(r_{1}+r_{2}+\ldots+r_{n-i}\right)\\ &=&\mu_{n-i}\bigg(1-\left(r_{n}+r_{n-1}+\ldots+r_{n-i+1}\right)\bigg), \end{eqnarray*}\] and, furthermore, by considering the \(n-i+1\)-st column, \[\begin{eqnarray*} \gamma_{n-i+1}&=&Y_{1,n-i+1}+Y_{2,n-i+1}+\ldots+Y_{i-1,n-i+1}\\ &=&r_{n-i+1}\mu_{n-i+1}+\ldots+r_{n-i+1}\mu_{n-1}+r_{n-i+1}\mu_{n}, \end{eqnarray*}\] which implies \[\begin{equation*} r_{n-i+1}=\frac{\gamma_{n-i+1}}{\mu_{n}+\mu_{n-1}+\ldots+\mu_{n-i+1}}, \end{equation*}\] and \[\begin{equation*} \mu_{k-i}=\frac{d_{n-i}}{1-\left(r_{n}+r_{n-1}+\ldots+r_{n-i+1}\right)}. \end{equation*}\]

Example 14.9 (Continuation of Example ??) It is possible to extract the payment patterns and inflation factors. This results in \[\begin{equation*} \tiny{ \vline% \begin{array}{rccccccc} \hline k\text{ \ }\vline & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline d_{k}\text{ \ }\vline & \multicolumn{1}{r}{23,758} & \multicolumn{1}{r}{ 56,601} & \multicolumn{1}{r}{78,276} & \multicolumn{1}{r}{100,229} & \multicolumn{1}{r}{131,562} & \multicolumn{1}{r}{153,680} & \multicolumn{1}{r}{170,559} \\ \gamma _{k}\text{ \ }\vline & \multicolumn{1}{r}{268,723} & \multicolumn{1}{r}{228,477} & \multicolumn{1}{r}{131,932} & \multicolumn{1}{r}{52,566} & \multicolumn{1}{r}{21,231} & \multicolumn{1}{r}{ 9,424} & \multicolumn{1}{r}{2,312} \\ \mu _{k}\text{ \ }\vline & \multicolumn{1}{r}{73,705} & \multicolumn{1}{r}{ 90,855} & \multicolumn{1}{r}{95,440} & \multicolumn{1}{r}{109,926} & \multicolumn{1}{r}{137,391} & \multicolumn{1}{r}{155,791} & \multicolumn{1}{r}{170,559} \\ r_{k}\text{ \ }\vline & \multicolumn{1}{r}{0.322} & \multicolumn{1}{r}{0.300} & \multicolumn{1}{r}{0.197} & \multicolumn{1}{r}{0.091} & \multicolumn{1}{r}{ 0.045} & \multicolumn{1}{r}{0.028} & \multicolumn{1}{r}{0.013} \\ \hline \end{array}% \vline } \end{equation*}\]%

In this form, the inflation factor may not appear explicit. By converting to a base of \(100\), it is possible to better visualize the impact of inflation. However, in this case, to be able to calculate the provisions, it is necessary to make forecasts of future inflation. We obtain \[\begin{equation*} \tiny{ \vline% \begin{array}{rccccccc} \hline k\text{ \ }\vline & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \mu _{k}\text{ \ }\vline & \multicolumn{1}{r}{73,705} & \multicolumn{1}{r}{ 90,855} & \multicolumn{1}{r}{95,440} & \multicolumn{1}{r}{109,926} & \multicolumn{1}{r}{137,391} & \multicolumn{1}{r}{155,791} & \multicolumn{1}{r}{170,559} \\ \text{(base 100) \ }\vline & \multicolumn{1}{r}{100} & \multicolumn{1}{r}{123 } & \multicolumn{1}{r}{129} & \multicolumn{1}{r}{149} & \multicolumn{1}{r}{ 186} & \multicolumn{1}{r}{211} & \multicolumn{1}{r}{231} \\ \hline \end{array}% \vline } \end{equation*}\] by forecasting an annual inflation rate of approximately \(+10\%\), we obtain the lower part of the following triangle. Indeed, the coefficient \(\mu _{k}\) then becomes% \[\begin{equation*} \tiny{\vline% \begin{array}{rcccccc} \hline k\text{ \ }\vline & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \mu _{k}\text{ \ }\vline & \multicolumn{1}{r}{170,559} & \multicolumn{1}{r}{ 187,615} & \multicolumn{1}{r}{206,375} & \multicolumn{1}{r}{227,014} & \multicolumn{1}{r}{249,715} & \multicolumn{1}{r}{274,687} \\ \text{(base 100) \ }\vline & \multicolumn{1}{r}{231} & \multicolumn{1}{r}{24} & \multicolumn{1}{r}{307} & \multicolumn{1}{r}{338} & \multicolumn{1}{r}{409} & \multicolumn{1}{r}{450} \\ \hline \end{array}% \vline} \end{equation*}\]

which makes it possible to reconstitute future payments

\[\begin{equation*} \tiny{ \vline% \begin{array}{rrrrrrrr} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline 1995\text{ \ }\vline & 23,758 & 25,356 & 19,468 & 11,258 & 6,458 & 4,268 & 2,312 \\ \cline{8-8} 1996\text{ \ }\vline & 31,245 & 32,496 & 27,034 & 15,664 & 8,615 & \multicolumn{1}{c|}{5,156} & \mathbf{2,543} \\ \cline{7-7} 1997\text{ \ }\vline & 26,312 & 31,467 & 24,672 & 13,055 & \multicolumn{1}{c|}{6,158} & \mathbf{5,417} & \mathbf{2,797} \\ \cline{6-6} 1998\text{ \ }\vline & 30,470 & 35,012 & 25,491 & \multicolumn{1}{c|}{12,589} & \mathbf{8,589} & \mathbf{5,959} & \mathbf{3,077} \\ \cline{5-5} 1999\text{ \ }\vline & 49,756 & 51,831 & \multicolumn{1}{c|}{35,267} & \mathbf{17,191} & \mathbf{9,448} & \mathbf{6,555} & \mathbf{3,384} \\ \cline{4-4} 2000\text{ \ }\vline & 50,420 & \multicolumn{1}{c|}{52,315} & \mathbf{36,993} & \mathbf{18,910} & \mathbf{10,393} & \mathbf{7,211} & \mathbf{3,723} \\ \cline{3-3} 2001\text{ \ }\vline & \multicolumn{1}{c|}{56,762} & \mathbf{56,404} & \mathbf{40,692} & \mathbf{20,801} & \mathbf{11,432} & \mathbf{7,932} & \mathbf{4,095} \\ \hline \end{array}% \vline} \end{equation*}\]%

As this table is not cumulative, we then add it together to obtain total amount of reserves required,

\[\begin{equation*} \tiny{ \vline% \begin{array}{rrrrrrrr|} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline 1995\text{ \ }\vline & 23,758 & 49,114 & 68,582 & 79,840 & 86,298 & 90,566 & 92,878 \\ \cline{8-8} 1996\text{ \ }\vline & 31,245 & 63,741 & 90,775 & 106,439 & 115,054 & \multicolumn{1}{r|}{120,210} & \mathbf{122,753} \\ \cline{7-7} 1997\text{ \ }\vline & 26,312 & 57,779 & 82,451 & 95,506 & \multicolumn{1}{r|}{101,664} & \mathbf{107,081} & \mathbf{109,879} \\ \cline{6-6} 1998\text{ \ }\vline & 30,470 & 65,482 & 90,973 & \multicolumn{1}{r|}{103,562 } & \mathbf{112,151} & \mathbf{118,110} & \mathbf{121,188} \\ \cline{5-5} 1999\text{ \ }\vline & 49,756 & 101,587 & \multicolumn{1}{r|}{136,854} & \mathbf{154,045} & \mathbf{163,493} & \mathbf{170,049} & \mathbf{173,434} \\ \cline{4-4} 2000\text{ \ }\vline & 50,420 & \multicolumn{1}{r|}{102,735} & \mathbf{% 139,728} & \mathbf{158,638} & \mathbf{169,031} & \mathbf{176,242} & \mathbf{% 179,966} \\ \cline{3-3} 2001\text{ \ }\vline & \multicolumn{1}{r|}{56,762} & \mathbf{113,166} & \mathbf{153,859} & \mathbf{174,660} & \mathbf{186,093} & \mathbf{194,025} & \mathbf{198,121}\\ \hline \end{array}% \vline } \end{equation*}\]%

This gives a reserve requirement of around €283.555. By way of information, with different inflation forecasts, we obtain we obtain the following reserve amounts

\[\begin{equation*} \tiny{ \vline% \begin{array}{rccccc} \hline \text{taux prévu \ }\vline & +5\% & +10\% & +15\% & +20\% & +25\% \\ \text{montant de réserves \ }\vline & 258,388 & 283,555 & 310,832 & 340,412 & 372,501 \\ \hline \end{array}% \vline } \end{equation*}\]%

It is worth noting that the average inflation rate was, per annum, \[ \left( 170.559/73.705\right) ^{1/6}-1\approx 15\% \] The estimate obtained in the case of the ordinary Chain Ladder corresponds to a forecasted inflation rate of approximately \(17\%\).

14.4 Stochastic Methods

Here, it is important to recall that the goal of calculating provisions is to a total charge for a given occurrence year. The problem is then identical to any forecasting problem.

14.4.1 The Mack Model

14.4.1.1 Connection with the De Vylder Model

The standard Chain Ladder method assumes that we have a model of the form \(C_{i,k+1}=\lambda _{k}C_{i,k}\) for all \(i,k=1,..,n\). The stochastic version of this relationship is written as follows: \[\begin{equation} \mathbb{E}\left[ C_{i,j+1}\right] =\lambda _{j}\mathbb{E}\left[ C_{i,j}\right] \text{ for all }i,j=1,..,n. \tag{14.6} \end{equation}\] The model (14.6) is closely related to the De Vylder model, as shown by the following result.

Proposition 14.2 The model @ref(eq:mack 93) is equivalent to \(\mathbb{E}\left[ Y_{ij}\right] =c_{i}r_{j}\) for all \(i,j\), with \(\sum_{j=1}^{n}r_{j}=1\) and \(Y_{i,j}=C_{i,j}-C_{i,j-1}\).

Proof. First, let’s show that @ref(eq:mack 93) \(\Longrightarrow \mathbb{E}\left[ Y_{ij}\right] =c_{i}r_{j}\). By iteratively applying @ref(eq:mack 93), we obtain \[ \mathbb{E}\left[ C_{i,n}\right] =\lambda _{j}\lambda _{j+1}...\lambda _{n-1}.\mathbb{E}\left[ C_{ij}\right], \] and thus, by inversion, \[\begin{eqnarray*} \mathbb{E}\left[ Y_{i,j}\right] &=&\mathbb{E}\left[ C_{i,j}\right] -\mathbb{E}% \left[ C_{i,j-1}\right]\\ &=&\left( \frac{1}{\lambda _{j}\lambda _{j+1}...\lambda _{n-1}}-\frac{1}{\lambda _{j-1}\lambda _{j+1}...\lambda _{n-1}}\right) \mathbb{E}\left[ C_{i,n}\right]. \end{eqnarray*}\]% By defining \[\begin{eqnarray*} c_{i}&=&\mathbb{E}\left[ C_{i,n}\right]\\ r_{1}&=&\left( \prod_{k=1}^{n-1}\lambda _{k}\right) ^{-1}\\ r_{j}&=&\left( \prod_{k=j}^{n-1}\lambda _{k}\right) ^{-1}-\left( \prod_{k=j-1}^{n-1}\lambda _{k}\right) ^{-1}\\ &&\text{ for $j=2,...,n-1$ and $r_{n}=1-1/\lambda _{n-1}$}, \end{eqnarray*}\] we obtain \(\mathbb{E}[ Y_{ij}] =c_{i}r_{j}\) for all \(i,j\).

Reciprocally, let us show that $=c_{i}r_{j}$ (14.6). The expectation of the cumulative sum of payments can be written as \[\begin{equation*} \mathbb{E}\left[ C_{i,j}\right] =\mathbb{E}\left[ Y_{i,1}\right] +...+% \mathbb{E}\left[ Y_{i,j}\right] =c_{i}(r_{1}+...+r_{j}) \end{equation*}\] and thus% \[\begin{equation*} \frac{\mathbb{E}\left[ C_{i,j+1}\right] }{\mathbb{E}\left[ C_{ij}\right] }=% \frac{r_{1}+...+r_{j}+r_{j+1}}{r_{1}+...+r_{j}}=\lambda _{j}\text{ for }% j=1,...,n-1, \end{equation*}\] which completes the proof.

14.4.1.2 Stochastic Version of Chain Ladder

(Mack 1993) proposed a non-parametric model, conditional on the realization of the triangle, allowing for the estimation of errors made during the evaluation of reserves. For this purpose, he assumes line-by-line independence, i.e.,

  • (H1) \(\left( C_{i,j}\right) _{j=1,...,n}\) and $( C_{i^{},j}) _{j=1,…,n} $ are independent for \(i\neq i'\),

and that it is possible to link the conditional expectation of \(C_{i,j+1}\), given the past $C_{i,1},…,C_{i,j} $, to the last observation \(C_{i,j}\), up to a multiplicative factor corresponding to a link ratio, i.e.

  • (H2) \(\mathbb{E}\left[ C_{i,j+1}|C_{i,1},...,C_{i,j}\right] =\lambda _{j}\cdot C_{i,j}\).

Under these two assumptions, (Mack 1993) showed that the induced stochastic model provides exactly the same reserves as the standard Chain Ladder method.

Proposition 14.3 Under assumptions (H1) and (H2), for all \(k>n-i+1\), we have, denoting the historical data ${i}={ C{ij}|i+jn+1} $% \[\begin{equation*} \mathbb{E}\left[ C_{i,k}|\mathcal{H}_{i}\right] =\lambda _{n-i+1}\lambda _{n-i+2}...\lambda _{k-1}\cdot C_{i,n+1-i} \end{equation*}\]

14.4.1.3 Properties of Chain Ladder Estimators

Specifying a stochastic model through H1 and H2 allows us to study the properties of the estimators of the development coefficients of the Chain Ladder method.

Proposition 14.4 Under assumptions (H1) and (H2), the standard Chain Ladder estimators, i.e.% \[\begin{equation*} \widehat{\lambda }_{j}=\frac{\sum_{i=1}^{n-j}C_{i,j+1}}{% \sum_{i=1}^{n-j}C_{i,j}}\text{ for }j=1,...,n-1 \end{equation*}\]% are unbiased and uncorrelated.

Proof. According to (H2), \[ \mathbb{E}\left[ C_{i,j+1}|C_{i,1},...,C_{i,j}\right] =\mathbb{E}\left[ C_{i,j+1}|\mathcal{H}_{j}\right] =\lambda _{j}\cdot C_{ij}, \] and thus \[\begin{equation*} \mathbb{E}\left[ \widehat{\lambda }_{j}|\mathcal{H}_{j}\right] =\frac{\sum_{i=1}^{n-j}% \mathbb{E}\left[ C_{i,j+1}|\mathcal{H}_{j}\right] }{\sum_{i=1}^{n-j}C_{i,j}}=\frac{% \sum_{i=1}^{n-j}\lambda _{j}\cdot C_{ij}}{\sum_{i=1}^{n-j}C_{i,j}}=\lambda _{j}. \end{equation*}\]% Taking the expectation, we obtain \[ \mathbb{E}\left[ \widehat{% \lambda }_{j}\right] =\mathbb{E}\left[ \mathbb{E}\left[ \widehat{\lambda }% _{j}|\mathcal{H}_{j}\right] \right] =\lambda _{j}; \] thus, \(\widehat{\lambda}_{j}\) is an unbiased estimator of \(\lambda _{j}\).

Additionally, for \(j<k\),% \[\begin{eqnarray*} \mathbb{E}\left[ \widehat{\lambda }_{j}\widehat{\lambda }_{k}\right] &=&% \mathbb{E}\left[ \mathbb{E}\left[ \widehat{\lambda }_{j}\widehat{\lambda }% _{k}|\mathcal{H}_{k}\right] \right] \\ &=&\mathbb{E}\left[ \widehat{\lambda }_{j}\mathbb{E}% \left[ \widehat{\lambda }_{k}|\mathcal{H}_{k}\right] \right] \\ &=&\mathbb{E}\left[ \widehat{\lambda }% _{k}\right] \mathbb{E}\left[ \widehat{\lambda }_{j}\right], \end{eqnarray*}\] which completes the proof.

As Mack notes, this lack of correlation between the estimators may seem surprising since the estimators are based on the same observations. However, this lack of correlation is central because it allows us to write% \[\begin{equation*} \mathbb{E}\left[ \widehat{\lambda }_{j}\widehat{\lambda }_{j+1}...\widehat{% \lambda }_{k-1}\widehat{\lambda }_{k}\right] =\lambda _{j}\lambda _{j+1}...\lambda _{k-1}\lambda _{k}. \end{equation*}\]% Therefore, the estimated reserve amount \(\widehat{R}_{i}=\widehat{C}% _{i,n}-C_{i,j}\) is an unbiased estimator of the reserve amount \(R_{i}=C_{i,n}-C_{i,j}\).

14.4.1.4 Prediction Error

Using these estimates, it is possible to study the prediction error more precisely by focusing on the average distance between the estimator \(\widehat{R}_{i}\) and the true value \(R_{i}\). To do this, it is necessary to add a third assumption to H1-H2 formulated above:

  • (H3) \(\mathbb{V}\left[ C_{i,k+1}|C_{i,1},...,C_{i,k}\right] =C_{i,k}\cdot \sigma_{k}^{2}\) for \(i=1,...,n\) and \(k=1,...,n-1\). We are then able to state the following result.

Proposition 14.5 The mean squared error of the reserve amount for year \(i\), \(\widehat{R}_{i}=\widehat{C}_{i,n}-C_{i,j}\), defined as \[ mse\left( \widehat{R}_{i}\right) =\mathbb{E}\left[ \left( \widehat{R}_{i}-R_{i}\right) ^{2}\Big|\mathcal{H}_{i}\right] \] is estimated by% \[\begin{equation*} \widehat{mse}\left( \widehat{R}_{i}\right) =\widehat{C}_{i,n}^{2}% \sum_{k=n-i+1}^{n-1}\frac{\widehat{\sigma }_{k}^{2}}{\widehat{\lambda }% _{k}^{2}}\left( \frac{1}{\widehat{C}_{i,k}}+\frac{1}{\sum_{j=1}^{n-k}C_{j,k}}% \right) , \end{equation*}\]% where \(\widehat{C}_{i,k}=\widehat{\lambda }_{n-i+1}...\widehat{\lambda }% _{k-1}.C_{i,n-i+1}\) for all \(k>n-i+1\), with the convention \(\widehat{C}% _{i,n-i+1}=C_{i,n-i+1}\), and% \[\begin{equation*} \widehat{\sigma }_{k}^{2}=\frac{1}{n-k-1}\sum_{i=1}^{n-k}C_{i,k}\left( \frac{% C_{i,k+1}}{C_{i,k}}-\widehat{\lambda }_{k}\right) ^{2}, \hspace{2mm}k=1,...,n-2. \end{equation*}\]% The value of \(\widehat{\sigma }_{k}^{2}\) for \(k=n-1\) is then extrapolated% d in such a way that% \[ \frac{\widehat{\sigma }_{n-3}^{2}}{\widehat{\sigma }_{n-2}^{2}}=\frac{% \widehat{\sigma }_{n-2}^{2}}{\widehat{\sigma }_{n-1}^{2}} \] which gives \[ \widehat{\sigma }_{n-1}^{2}=\min \left\{ \frac{\widehat{\sigma } _{n-2}^{4}}{\widehat{\sigma } _{n-3}^{2}},\min \left\{ \widehat{\sigma } _{n-3}^{2},\widehat{\sigma } _{n-2}^{2}\right\} \right\} . \]

For the proof of this result, as well as for the proof of the following corollary, we refer to (Mack 1993).

Corollary 14.1 The mean squared error of the total reserve amount, \(\widehat{R}=\widehat{R}% _{2}+...+\widehat{R}_{n}\) can be estimated as follows:% \[ \widehat{mse}\left( \widehat{R}\right) =\sum_{i=2}^{n}\left( \left( mse\left( \widehat{R}_{i}\right) \right) ^{2}+\widehat{C}_{i,n}\left( \sum_{j=i+1}^{n}\widehat{C}_{j,n}\right)\right. \left.\sum_{k=n-i+1}^{n-1}\frac{\frac{2\widehat{% \sigma }_{k}^{2}}{\widehat{\lambda }_{k}^{2}}}{\sum_{j=1}^{n-k}C_{j,k}% }\right). \]

::: {.example name=“Continuation of Example ??”] The following estimates are obtained for \(\widehat{\sigma }% _{k}^{2}\) and \(\sqrt{\widehat{mse}\left( \widehat{R_{k}}\right)}\): \[\begin{equation*} \tiny{ \vline% \begin{array}{rccccccc} \hline \text{k \ }\vline & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \widehat{\sigma }_{k}^{2}\text{ \ }\vline & \multicolumn{1}{r}{2} & \multicolumn{1}{r}{35} & \multicolumn{1}{r}{151} & \multicolumn{1}{r}{95} & \multicolumn{1}{r}{258} & \multicolumn{1}{r}{95} & \\ \sqrt{\widehat{mse}\left( \widehat{R_{k}}\right)} \text{ \ }\vline & \multicolumn{1}{r}{} & \multicolumn{1}{r}{{5,145}} & \multicolumn{1}{r}{% {7,911}} & \multicolumn{1}{r}{{9,362}} & \multicolumn{1}{r}{{14,222}} & \multicolumn{1}{r}{{15,494}} & {19,936} \\ \hline \end{array}% \vline } \end{equation*}\]% Therefore, finally \[\begin{equation*} {\tiny {\vline% \begin{array}{rcccccc} \hline \text{Year \ }\vline & 1996 & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline \widehat{R_{i}}\text{ \ }\vline & \multicolumn{1}{r}{3,068} & \multicolumn{1}{r}{7,475} & \multicolumn{1}{r}{15,991} & \multicolumn{1}{r}{ 46,087} & \multicolumn{1}{r}{88,249} & \multicolumn{1}{r}{162,501} \\ \sqrt{\widehat{mse}\left( \widehat{R_{i}}\right)} \text{ \ }\vline & \multicolumn{1}{r}{5,145} & \multicolumn{1}{r}{7,911} & \multicolumn{1}{r}{ 9,362} & \multicolumn{1}{r}{14,222} & \multicolumn{1}{r}{15,494} & \multicolumn{1}{r}{19,936} \\ \text{\% }% \text{ \ }\vline & \multicolumn{1}{r}{167.7\%} & \multicolumn{1}{r}{105.8\%} & \multicolumn{1}{r}{58.5\%} & \multicolumn{1}{r}{30.9\%} & \multicolumn{1}{r}{17.6\%} & \multicolumn{1}{r}{12.3\%} \\ \hline \end{array}% \vline}} \end{equation*}\] :::

14.4.1.5 Verification of Assumptions H1-H2-H3

As noted by (Mack 1993), this method relies on three fundamental assumptions that need to be tested. If these three assumptions are not met, the model is not valid. Different empirical approaches have been proposed by (Mack 1993) to verify the validity of H1-H2-H3.

Hypothesis H1 assumes independence between different occurrence years. In practice, this assumption may not hold for several reasons:

  • Changes in the claims management team, which may, for example, result in faster payments.
  • Inflation factor (which, although not occurring on a calendar basis by line, can render the vectors non-independent).

Hypothesis H2 implies that, at a given \(j\), the points \(\left( C_{i,j},C_{i,j+1}\right)\) should be roughly aligned on a line with slope \(\widehat{\lambda}_{j}\) passing through the origin. Similarly, a graphical interpretation of hypothesis H3 can be given: at a given \(j\), the points \(\left( C_{i,j},D_{i,j}\right)\), where

\[\begin{equation*} D_{i,j}=\frac{C_{i,j+1}-\widehat{\lambda}_{j}.C_{i,j}}{\sqrt{C_{i,j}}} \end{equation*}\]

correspond to the residuals of a least squares estimation, should be unstructured.

Mack proposed a method to test the validity of the assumptions underlying his model. We refer, for example, to (Pitrebois et al. 2002) for an example.

14.4.1.6 Inclusion of a Tail Factor

Since the development of claims occurring in accident year \(i\) is not necessarily completed after \(n\) years of development, a tail factor \(\hat{\lambda}_{ult}>1\) is used to estimate the ultimate amount \(C_{i,ult}\) by \[ \hat{C}_{i,ult}= \hat{C}_{in}\, \hat{\lambda}_{ult}. \]

We can take \[ \hat{\lambda}_{ult}=\displaystyle\prod_{k=n}^{\infty} \hat{\lambda}_k, \] where the future \(\hat{\lambda}_k\) values are estimated through linear extrapolation of \(\ln (\hat{\lambda}_k -1)\). However, it is essential to ensure that this tail factor is plausible and consistent with past experience regarding the future development of claims.

14.4.1.7 Error Formulas for Loss Ratios

Here, we are interested in the loss ratio, which is of great importance for insurers. Let \(C_{ik}\) represent the insurer’s claim amounts. For the occurrence year \(i\), we define the corresponding loss ratio as \((S/P)_i=\displaystyle\frac{C_{in}}{\Pi_i}\), which is the final amount of claims for year \(i\) divided by the premium income relative to year \(i\). This ratio is estimated as \((\hat{S/P})_i=\displaystyle\frac{\hat{C}_{in}}{\Pi_i}\), and the estimation error is directly obtained as \[ \widehat{mse[(\hat{S/P})_i]}=(\frac{1}{\Pi_i})^2 \widehat{mse(\hat{C}_{in})}, \] where \(\widehat{mse(\hat{C}_{in})}\) has been obtained earlier.

Example 14.10 (Continuation of Example ??)

The errors on the reserves are given in the following table: {

} For given annual premiums, we obtain the following results: {\[ \begin{tabular}{|c|c|c|c|c|c|} \hline $i$ & $\hat{C}_{in}$ & $\Pi_i$ & $\widehat{S/P}$ & $\sqrt{\widehat{mse(\hat{S/P})}}$ & $\sqrt{\widehat{mse(\hat{S/P})}}$ as \% of $\hat{S/P}$ \\ \hline $2$ & $45.01$ & $64$ & $0.7033$ & $0.0036$ & $0.51\%$ \\ $3$ & $51.05$ & $77$ & $0.6630$ & $0.0068$ & $1.03\%$ \\ $4$ & $57.38$ & $78$ & $0.7357$ & $0.0078$ & $1.06\%$ \\ $5$ & $64.16$ & $85$ & $0.7548$ & $0.0089$ & $1.18\%$\\ \hline \end{tabular} \]}

To get an idea of the expected loss ratio, we often work with an average S/P ratio calculated over the recent occurrence years. Let’s define \[ (S/P)_m = \displaystyle\frac{1}{n-m+1} \sum_{i=m}^n \frac{C_{in}}{\Pi_i}, \] which is the average S/P ratio from occurrence years \(m\) (\(1\leq m\leq n\)) to \(n\). We estimate it simply by replacing \(C_{in}\) with \(\hat{C}_{in}\). This gives us \[\begin{eqnarray*} \widehat{mse}[(\hat{S/P})_m] &=& \frac{1}{(n-m+1)^2} \sum_{i=m}^{n} \left\{\frac{mse(\hat{C}_{in})}{(\Pi_i)^2}\right.\\ &&+ \left.\frac{\hat{C}_{in}}{\Pi_i} \left( \sum_{j=i+1}^{n}\frac{\hat{C}_{jn}}{\Pi_j} \right) \sum_{k=n+1-i}^{n-1} \frac{2 \hat{\sigma}_k^2 / \hat{\lambda}_k^2}{\sum_{j=1}^{n-k} C_{jk}} \right\}. \end{eqnarray*}\]

Example 14.11 (Continuation of Example ??) With \((S/P)_m\) being the average S/P ratio calculated over the years \(i\) to \(n\) (i.e., \(m=i\)), this gives us: { \[\begin{tabular}{|c|c|c|c|c|c|} \hline $i$ & $\hat{C}_{in}$ & $\Pi_i$ & $\widehat{(S/P)}_m$ & $\sqrt{\widehat{mse[(\hat{S/P})_m]}}$ & $\sqrt{\widehat{mse[(\hat{S/P})_m]}}$ as \% of $(\hat{S/P})_m$ \\ \hline $1$ & $40.16$ & $60$ & $0.7052$ & $0.0039$ & $0.55\%$ \\ $2$ & $45.01$ & $64$ & $0.7142$ & $0.0049$ & $0.68\%$ \\ $3$ & $51.05$ & $77$ & $0.7178$ & $0.0060$ & $0.84\%$ \\ $4$ & $57.38$ & $78$ & $0.7452$ & $0.0070$ & $0.93\%$ \\ $5$ & $64.16$ & $85$ & $0.7548$ & $0.0089$ & $1.18\%$\\ \hline \end{tabular}\] }

14.4.2 The Christophides Log-Linear Model

This model is directly inspired by the De Vylder model. We assume that the increments satisfy \(Y_{i,j}=r_{j}\cdot p_{i}\), where \(p_{i}\) corresponds to the ultimate loss for claims occurring in year \(i\), and \(r_{j}\) is the proportion of the amount \(p_{i}\) paid in year \(j\) (which depends only on \(j\)). By taking the logarithm, we can consider the following log-linear model, \[\begin{equation*} X_{i,j}=\log Y_{i,j}=\alpha _{i}+\beta _{j}+\varepsilon _{i,j}, \end{equation*}\] where \(\varepsilon _{i,j}\) represents the error terms. The identifiability condition for the model is \(\beta _{0}=0\). We assume that the errors are independent and Gaussian, $_{i,j}or( 0,^{2}) $. Consequently, \(X_{ij}\sim\mathcal{N}or(\alpha_i+\beta_j,\sigma^2)\), independently.

In particular, it can be noted that \(X_{i,j}\) is also Gaussian, with \(\mathbb{E}\left[ X_{i,j}\right] =\alpha _{i}+\beta _{j}\) and \(\mathbb{V}[X_{i,j}] =\sigma ^{2}\). This model, being a simple regression model, is easily implemented.

Example 14.12 (Continuation of Example ??) The regression coefficients are given by \[ \tiny{ \begin{tabular}{|cccccccc|} \hline & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ \\ \hline \multicolumn{1}{|l}{$\alpha _{i}$} & \multicolumn{1}{r}{$10.10$} & \multicolumn{1}{r}{$10.38$} & \multicolumn{1}{r}{$10.22$} & \multicolumn{1}{r}{$10.31$} & \multicolumn{1}{r}{$10.75$} & \multicolumn{1}{r}{$10.80$} & \multicolumn{1}{r|}{$10.95$} \\ \multicolumn{1}{|l}{$\beta _{j}$} & \multicolumn{1}{r}{} & \multicolumn{1}{r}{$0.08$} & \multicolumn{1}{r}{$-0.19$} & \multicolumn{1}{r}{$-0.77$} & \multicolumn{1}{r}{$-1.38$} & \multicolumn{1}{r}{$-1.78$} & \multicolumn{1}{r|}{$-2.35$} \\ \hline \end{tabular}% } \] with \(\widehat{\sigma}=4.22\).

Remark. As we explained in Chapter 9, log-linear models are not GLM models, but they are easy to use. Among the disadvantages of these models, note that if increments are negative, they cannot be used. Furthermore, the estimation of reserves increases with the variance of the data.

14.4.3 The Renshaw and Verrall Poisson Model

14.4.3.1 Model

(Renshaw and Verrall 1998) used a GLM model to replicate the results obtained by the standard Chain Ladder method. We consider the non-cumulative payments, i.e., the variables \(Y_{i,j}\). We assume that the sums by column are positive, i.e., \(\sum_{i=1}^{n-j+1}Y_{ij}\geq 0\) for all \(j=1,...,n\). This assumption, however, does not prevent having negative payments in some years.

We assume that for all \(i,j\), \(Y_{ij}\sim\mathcal{P}oi(\mu_{ij})\), independently, with \[\begin{equation*} \ln \mu_{ij}=\mu +\alpha _{i}+\beta _{j}, \end{equation*}\] and the identification constraints \(\alpha _{1}=\beta _{1}=0\). We then numerically seek the maximum likelihood estimators of \(\alpha _{i}\), \(\beta _{j}\), and $$. Once these are obtained, we define% \[\begin{equation} \widehat{C}_{i,n}=C_{i,n-i+1}+\sum_{j=n-i+2}^{n}\exp \left( \widehat{\mu }+% \widehat{\alpha }_{i}+\widehat{\beta }_{j}\right), \tag{14.7} \end{equation}\] where \(i=2,3,...,n.\)

14.4.3.2 Conditional Maximum Likelihood

It is possible to obtain these estimators in another way using conditional likelihood. Let \(p_{i|j}\) represent the probability that a claim occurring in year \(i\) is reported after \(j\) years. Under a “stationarity” assumption, we can assume that the reporting time does not depend on the year of occurrence. Then,

\[\begin{equation*} p_{j|i} = \frac{p_{j}}{\sum_{k=1}^{n-i+1}p_{k}}, \end{equation*}\]

where \(p_{j}\) is the (non-conditional) probability that a claim is reported after \(j\) years. We assume that all claims are closed in at most \(n\) years, i.e., \(\sum_{k=1}^{n}p_{k}=1\).

Proposition 14.6 The estimators for \(p_{j}\) can be obtained recursively as follows:

\[\begin{equation} \widehat{p}_{j} = \frac{Y_{1,j}+Y_{2,j}+\ldots+Y_{n-j+1,j}}{C_{1,n}+(1-\widehat{p}_{n})^{-1}C_{2,n-1}+\ldots+(1-\widehat{p}_{j+1}-\ldots-\widehat{p}_{n})^{-1}C_{n-j+1,j}}, \end{equation}\]

These estimators coincide with the maximum likelihood (conditional) estimators.

Proof. The conditional likelihood is given by:

\[\begin{equation*} \mathcal{L}_{C}=\prod_{i=1}^{n}\frac{C_{i,n-i+1}!}{\prod_{j=1}^{n-i+1}Y_{i,j}!}\prod_{j=1}^{n-i+1}p_{j|i}, \end{equation*}\]

By using the fact that, conditional on \(i\), the reporting time follows a multinomial distribution, we obtain the log-conditional likelihood:

\[\begin{eqnarray*} \log \mathcal{L}_{C} &=& \sum_{i=1}^{n}\sum_{j=1}^{n-i+1}Y_{ij}\log p_{j|i}\\ &=& \sum_{i=1}^{n}\sum_{j=1}^{n-i+1}Y_{ij}\left( \log p_{j}-\log \sum_{k=1}^{n-i+1}p_{k}\right). \end{eqnarray*}\]

Note that the term \(p_{n}\) only appears once, for \(i=1\). Also,

\[\begin{equation*} \frac{\partial \log \mathcal{L}_{C}}{\partial p_{n}} = \frac{Y_{1,n}}{p_{n}} - \sum_{j=1}^{n}\frac{Y_{1,j}}{\sum_{k=1}^{n}p_{k}}, \end{equation*}\]

and \(\partial \log \mathcal{L}_{C}/\partial p_{n}=0\) implies

\[\begin{equation*} \widehat{p}_{n} = \frac{Y_{1,n}}{\sum_{k=1}^{n}Y_{1,k}} = \frac{Y_{1,n}}{C_{1,n}}. \end{equation*}\]

Assuming that at orders \(u=j+1,j+2,\ldots,n\),

\[\begin{equation*} \widehat{p}_{u} = \left( \sum_{i=1}^{n-u+1}Y_{i,u}\right) \left( \sum_{i=1}^{n-u+1}\frac{C_{i,n-i+1}}{\sum_{k=1}^{n-i+1}p_{k}}\right) ^{-1}, \end{equation*}\]

we can note that at step \(j\), the likelihood equation can be written as:

\[\begin{eqnarray*} \frac{\partial \log \mathcal{L}_{C}}{\partial p_{j}} &=& \sum_{i=1}^{n-j+1}\left(\frac{Y_{i,j}}{p_{j}}-\sum_{j=1}^{n-i+1}\frac{Y_{i,j}}{\sum_{k=1}^{n-i+1}p_{k}}\right)\\ &=& \frac{\sum_{i=1}^{n-j+1}Y_{i,j}}{p_{j}}-\sum_{i=1}^{n-j+1}\frac{C_{i,n-i+1}}{\sum_{k=1}^{n-i+1}p_{k}}. \end{eqnarray*}\]

Then, \(\partial \log \mathcal{L}_{C}/\partial p_{j}=0\) implies

\[\begin{equation} \widehat{p}_{j} = \left[ \sum_{i=1}^{n-j+1}Y_{i,j}\right] \left[ \sum_{i=1}^{n-j+1}\frac{C_{i,n-i+1}}{\sum_{k=1}^{n-i+1}p_{k}}\right] ^{-1}. \tag{14.8} \end{equation}\]

By defining

\[\begin{equation} \widehat{C}_{i,n} = \frac{C_{i,n-i+1}}{1-\sum_{j=n-i+2}^{n}\widehat{p}_{j}}, \end{equation}\]

for \(i=2,3,\ldots,n\), we can rewrite equation (14.8) in the form of equation (??), as announced.

14.4.3.3 Relationship with the Chain Ladder Method

In the case of a Poisson model, the Chain Ladder method is a simple approach to obtain maximum likelihood conditional estimators.

Proposition 14.7 By defining the following recursive relationships:

\[\begin{equation} \widehat{\lambda }_{j}=\frac{1}{1-\widehat{p}_{j-1}\cdot \widehat{\lambda }_{j+1}\widehat{\lambda }_{j+2}\ldots\widehat{\lambda }_{n-1}}, \tag{14.9} \end{equation}\]

the estimators \(\widehat{C}_{i,n}\) coincide with the estimators of the standard Chain Ladder model, assuming a Poisson model.

Proof. With the \(widehat{lambda }_{j}\) defined by (14.9), we have \[ \widehat{\lambda }_{j}\widehat{\lambda }_{j+1}...\widehat{\lambda }_{n-1}=\left( 1-\widehat{p}_{j+1}-...-\widehat{p}_{n}\right) ^{-1}, \] such that \[\begin{eqnarray*} \widehat{\lambda }_{j-1}\widehat{\lambda }_{j}\widehat{\lambda }_{j+1}...%}. \widehat{\lambda }_{n-1} &=&\left( 1-\widehat{p}_{j}-\widehat{p}_{j+1}-...-% \widehat{p}_{n}\right) ^{-1} \\ &=&\left( \widehat{p}_{j}-\left( \widehat{lambda }_{j}\widehat{lambda }% _{j+1}...\widehat{\lambda }_{n-1}\right) ^{-1}\right) ^{-1}, \end{eqnarray*}\]% and therefore, (??) can be rewritten as%. \[\begin{equation*} \widehat{C}_{n-j+1,n}=\frac{C_{n-j+1,j}}{1-\sum_{k=j+1}^{n}\widehat{p}_{k}}% =C_{n-j+1,j}\cdot \widehat{\lambda }_{j}\widehat{\lambda }_{j+1}...\widehat{% \lambda }_{n-1} \end{equation*}\] as announced.

14.5 GLM and Reserving

14.5.1 Principle

In the previous section, we examined the Poisson model with a logarithmic link, which allows us to obtain an estimated amount equal to that obtained by the Chain-Ladder method. This has a significant practical advantage (as the Chain Ladder method is almost always used as a reference), but it is not the only approach. More generally, we can consider the class of Generalized Linear Models (GLMs).

Let’s assume that the annual payments \(Y_{i,j}\) are independent and belong to the exponential family with the density (??), i.e.,

\[\begin{equation*} f\left( y_{i,j}|\theta _{i,j},\phi \right) =\exp \left(\frac{y_{i,j}\theta _{i,j}-b\left( \theta _{i,j}\right) }{\phi/\omega_{ij} }+c\left( y_{i,j},\phi \right) \right), \end{equation*}\]

where the mean and variance are given by:

\[\begin{equation*} \left\{ \begin{array}{l} \mu _{i,j}=\mathbb{E}\left[ Y_{i,j}\right] =b^{\prime }\left(\theta _{i,j}\right) \\ \mathbb{V}\left[ Y_{i,j}\right] =b^{\prime \prime }\left(\theta _{i,j}\right)\frac{\phi}{\omega_{ij}} =V\left( \mu _{i,j}\right) \frac{\phi}{\omega_{ij}} .% \end{array}% \right. \end{equation*}\]

Most reserving models are based on a linear predictor of the form \(\eta _{i,j}=\alpha _{i}+\beta _{j}\). The link function relates the expectation \(\mu _{i,j}\) to the linear predictor, ${i,j}=g( {i,j}) $, where \(g\) is a monotone and differentiable function.

Example 14.13 In the model by (Renshaw and Verrall 1998), the (deterministic) Chain Ladder model predicts exactly the same amounts as the Poisson model with a logarithmic link, \(Y_{i,j}\sim \mathcal{P}oi\left( \alpha _{i}\beta _{j}\right)\) independently. To make the model identifiable, it is also assumed that \(\beta _{1}+...+\beta _{n}=1\), so that the \(\beta _{j}\) indicate the fraction of claims settled in development year \(j\), and \(\alpha _{i}\) represents a volume effect. The maximum likelihood estimation of \(\alpha _{i}\) and \(\beta _{j}\) coincides with the so-called “margins” method. Indeed, the likelihood can be written as:

\[\begin{equation*} \mathcal{L}=\prod_{i,j}\exp \left( -\alpha _{i}\beta _{j}\right)\frac{\left( \alpha _{i}\beta _{j}\right) ^{Y_{i,j}}}{Y_{i,j}!} \end{equation*}\]

This allows us to express the log-likelihood as:

\[\begin{equation*} \log \mathcal{L}=\sum_{i,j}\left( -\alpha _{i}\beta _{j}+Y_{i,j}\log \left( \alpha _{i}\beta _{j}\right) -\log \left( Y_{i,j}!\right) \right). \end{equation*}\]

The maximum likelihood is obtained by calculating:

\[\begin{equation*} \frac{\partial \log \mathcal{L}}{\partial \alpha _{i}}=\sum_{j}\left( -\beta _{j}+Y_{i,j}\frac{\beta _{j}}{\alpha _{i}\beta _{j}}\right) \end{equation*}\]

which equals zero for:

\[\begin{equation*} \widehat{\alpha }_{i}=\frac{\sum_{j}Y_{i,j}}{\sum_{j}\beta _{j}}. \end{equation*}\]

Example 14.14 Separation methods can also be expressed using GLM models. For the arithmetic separation model, \(Y_{i,j}\sim \mathcal{P}oi\left( \beta _{j}\gamma _{i+j-1}\right)\) independently (with the constraint \(\beta _{1}+...+\beta _{n}=1\)). For the geometric separation model, \(Y_{i,j}\sim \mathcal{LN}or\left( \log \left( \beta _{j}\gamma _{i+j-1}\right) ,\sigma ^{2}\right)\) independently (with the constraint \(\beta _{1}\times ...\times \beta _{n}=1\)).

Example 14.15 In the De Vylder model, \(\alpha _{i}\) and \(% \beta _{j}\) are determined by minimizing:

\[\begin{equation*} \sum_{i,j}\left( Y_{i,j}-\alpha _{i}\beta _{j}\right) ^{2} \end{equation*}\]

Alternatively, \(\alpha _{i}\) and \(\beta _{j}\) can be obtained as maximum likelihood estimators in the model:

\[\begin{equation*} Y_{i,j}\sim \mathcal{N}or\left( \alpha _{i}\beta _{j},\sigma^{2}\right) \text{ independently}. \end{equation*}\]

14.5.2 Tweedie Models

By revisiting the Tweedie models (as presented in Section 10.9.14.2 on a priori pricing), we obtain a relatively wide family of exponential distributions for which the variance function is an exponential function.

Example 14.16 (Continuation of Example ??) Here, we obtain \(\widehat{\gamma}\approx 1.5\) (Figure ?? shows the likelihood as a function of \(\gamma\)). The suggested underlying distribution here falls between the Poisson distribution (\(\gamma=1\)) and the Gamma distribution (\(\gamma=2\)).

Recall that the GLM model is characterized by three components: the variance function \(V\), the link function \(g\), and the covariates \(\mathbf{X}\). More formally, we need to specify \(V(\mu_{i,j})\), where \(\mu_{i,j}=\mathbb{E}[Y_{i,j}]\), and \(g(\mu_{i,j})=\eta_{i,j}\), where \(\eta_{i,j}=\mathbf{X}\boldsymbol{\beta}=\alpha_i+\beta_j\), assuming that only row and column effects need to be considered. A relatively large family of GLM models is characterized by power-type variance and link functions, i.e., \(V(x)=x^\gamma\) and \(g(x)=x^\theta\).

Remark. Remember that if \(\gamma\) tends to 0, 1, or 2, we recover the Normal, Poisson, or Gamma models, respectively. And if \(\theta\) tends to -1, 0, 1, or 2, we obtain inverse, logarithmic, identity, or quadratic link functions.

Note that the use of deviance \(D(\gamma, \theta)\) is not relevant here because two deviances for different variance functions (and hence different \(\gamma\) values) are not comparable. In this case, we can use the extended quasi-likelihood method (\(EQL\)) or the pseudo-likelihood method (\(PL\)), which we briefly present below.

14.5.2.1 Extended Quasi-Likelihood Method

This method adds a correction term to the deviance (to correct for deviations from the variance function), namely:

\[ EQL = \sum_{i,j\in \Delta}D_{i,j} + \sum_{i,j\in \Delta}\log\left(2\pi\phi V(Y_{i,j})\right) \]

where

\[ D_{i,j} = 2\int_{\mu_{i,j}}^{Y_{i,j}}\frac{Y_{i,j}-s}{V(s)}ds \]

However, note that the number of parameters \(p\) does not appear here. It is possible to account for the degrees of freedom by considering:

\[ EQL^* = \sum_{i,j\in \Delta}D_{i,j} + \frac{n-p}{n}\sum_{i,j\in \Delta}\log\left(2\pi\phi V(\mu_{i,j})\right) \]

14.5.2.2 Pseudo-Likelihood

Here, we seek to minimize:

\[ PL = \sum_{i,j\in \Delta}X_{i,j}^2 + \frac{n-p}{n}\sum_{i,j\in \Delta}\log\left(2\pi\phi V(\mu_{i,j})\right) \]

where

\[ X_{i,j}^2 = \frac{(Y_{i,j}-\mu_{i,j})^2}{V(\mu_{i,j})} \]

This distance allows us to determine the “best” model within the subclass of GLM models by considering the following two optimization programs:

\[\begin{align*} (1) &: (\beta^*(\theta,\gamma), \phi^*(\theta,\gamma)) = \underset{\beta, \phi}{\text{argmin}}\{PL(\beta, \phi, \theta, \gamma)\} \\ (2) &: (\theta^*, \gamma^*) = \underset{\gamma, \theta}{\text{argmin}}\{PL(\beta^*(\theta,\gamma), \phi^*(\theta,\gamma), \theta, \gamma)\} \end{align*}\]

14.5.3 Which Factorial Model to Choose?

In the model we have presented, we assumed that it was possible to express the mean \(\mu_{i,j}\) as a linear combination of a row effect and a column effect,

\[ \mu_{i,j} = \mathbb{E}[Y_{i,j}] = g^{-1}(\mu+\alpha_i+\beta_j). \]

However, this model can be simplified in some cases by considering, for example, a linear effect based on the year of occurrence,

\[ \mu_{i,j} = g^{-1}(\mu+\alpha \cdot i+\beta_j), \]

or a linear effect based on the development year,

\[ \mu_{i,j} = g^{-1}(\mu+\alpha_i+\beta\cdot j), \]

or even linear effects in both variables,

\[ \mu_{i,j} = g^{-1}(\mu+\alpha\cdot i+\beta\cdot j). \]

Different transformations of the two time scales are also possible. Thus, in log-linear models, one can use explanatory variables such as \(\ln i\) or \(\ln j\), which correspond to “power” effects of the occurrence years or development years.

14.6 Choosing the Reserving Method

In this overview of reserving methods, we have seen that a vast number of modeling approaches can be proposed. Since there is no universal method that can apply to all insurance companies, all claims handling practices, or all types of risks, the actuary’s choice can be challenging. This problem becomes even more significant for lines of business with a slow development process, such as liability insurance or construction insurance.

Reserving is subject to numerous constraints, including tax, managerial, and, most importantly, regulatory constraints (on this last point, (Tosetti et al. 2000) provide an overview for the French market). It’s worth noting that the dilemma between prudential (or regulatory) rules, which require that the technical reserves must be “sufficient” to cover the commitments made, and tax rules is far from simple. All reserving schemes ignore tax rules, which can occasionally lead to conflicts.

In practice, faced with the multitude of possible estimates, it can be challenging to find the “right” level of reserving. However, some ideas can be used to choose between different models. For example, one can study how the loss triangle has developed in the past and see which model provided the best estimates. We have a triangle consisting of \(n\) accident years, and we are interested in the first sub-triangle of \(m\) development years:

[Diagram not included in the text]

Three sets can be distinguished in the triangle:

  1. The upper triangle (black points \(\bullet\)), corresponding to the observed triangle \(n-m\) years ago.
  2. The complement of the upper triangle (white points \(\circ\)), corresponding to what was predicted using the upper triangle.
  3. The complement (in white within the triangle), including both the most recent accident years (the last \(n-m\) years), which were not considered when constructing the upper triangle, and the last \(n-m\) development years.

A natural idea might be to use the proposed model on the upper triangle (1), and then compare the values predicted by this model for the lower triangle (2) with the actual values. The “best” model would then be the one that minimizes the sum of squared errors.

Note that this approach can only be used if you have a large number of years of observation, which is not always the case.

14.7 Practical Case Studies

14.7.1 Automobile Insurance

14.7.1.1 Data

The following statistics are taken from the motor liability policies of a German company. German company. We have data on the amounts paid out and reserve amounts for claims incurred between between 1985 and 1998. These data have been analyzed in detail in (Pitrebois et al. 2002).

Tables 14.2 and 14.3 show payments and reserves data respectively. respectively. We examine the amounts without taking inflation inflation, i.e. we make the implicit assumption of constant inflation over the entire observation period. We work with either payments or total amounts (payments plus reserves).

Table 14.2: Cumulated Payments
Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1985 31499 43711 45509 46312 46786 47204 47404 47768 47963 48168 48428 48557 48863 49081
1986 36822 49591 51733 52841 53605 54156 54857 55256 55433 55791 56014 56416 56640
1987 40962 53307 55310 56594 57359 58096 58525 58886 59416 59721 60031 60472
1988 43350 56043 57981 58942 59844 60463 61006 61349 61715 61934 62267
1989 42638 55788 58168 59980 60944 62208 63360 64064 64617 65018
1990 44666 60675 63281 64662 65543 66268 66797 67314 68541
1991 58291 83957 87690 90437 92102 94021 95751 96794
1992 69050 93642 97694 100042 101654 103027 104706
1993 68513 91377 95537 98251 100020 100857
1994 63337 85106 88755 91226 93105
1995 62555 81700 84782 86785
1996 59407 78481 81517
1997 61091 79892
1998 74211
Table 14.3: Reserves
Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1985 30298 17875 9181 6324 5280 4390 4086 3724 3176 2748 2711 2441 1790 1160
1986 31562 19374 12314 9068 7352 6098 5492 4436 4095 4006 3932 3935 3716
1987 30975 17637 13143 9940 8285 7141 6381 6132 5500 4430 4330 3977
1988 30663 15453 10983 7834 6309 5608 4597 4113 3558 3510 3312
1989 34256 22821 16002 12937 10224 8722 7798 7001 6452 4982
1990 33768 20393 16368 11470 9336 8534 8808 8444 6500
1991 52319 31160 24682 20420 18519 17992 15417 14928
1992 43031 25069 22502 21257 20804 20306 14730
1993 52012 26968 21766 19263 19601 17641
1994 48321 24106 23127 21173 23077
1995 48787 24120 24334 23765
1996 47640 23303 22950
1997 54561 28220
1998 83807

14.7.1.2 Payments

The results obtained for payments using the Chain Ladder method and the Projected Case Estimate method are shown in Table 14.4. The Projected Case Estimate method yields higher results than the Chain Ladder method as we approach the more recent accident years. This is especially true for the amount corresponding to the year 1998, which is due to a coefficient \(\hat{h}_2\) (see Table 14.5) and a particularly high initial reserve.

Table 14.4: Cumulative final payments by accident year
Year Chain Ladder Projected Case Estimate
1985 49081105 49081105
1986 58892686 57092631
1987 61048730 61221169
1988 63232335 63149034
1989 66354966 66688925
1990 70310892 70849125
1991 100146339 102722924
1992 109235417 111178780
1993 106563521 109038895
1994 99674656 104711187
1995 94416888 99791030
1996 90899256 94394931
1997 92784165 96358740
1998 115382193 137137105

Table 14.5 presents the development factors of the Chain Ladder method, \(\hat{\lambda}_k\), and their corresponding \(\hat{\sigma}_k\), as well as the development coefficients of the reserve model (\(\widehat{k}_{j+1}\)) and the payment model (\(h_{j+1}\)) of the Projected Case Estimate method.

Table 14.5: Standard distortion functions
Index \(\lambda_k\) \(\sigma_k\) PCE \(k_{j+1}\) PCE \(h_{j+1}\)
1 1.3388 296.1864 0.9803 0.4294
2 1.0415 27.4872 0.9391 0.1289
3 1.0250 39.8119 0.9418 0.1010
4 1.0162 24.0810 1.0056 0.0836
5 1.0132 40.7404 0.9921 0.0799
6 1.0128 44.4508 0.9427 0.0884
7 1.0083 17.9956 0.9987 0.0710
8 1.0086 43.0330 0.9551 0.0900
9 1.0051 9.5186 0.9290 0.0653
10 1.0050 4.9988 1.0486 0.0765
11 1.0059 19.0237 1.0323 0.0886
12 1.0051 11.9030 0.9468 0.0832
13 1.0045 7.4477 0.7700 0.1218

Table 14.6 presents the results of the calculation standard errors on reserves estimated by Chain Ladder. The relative standard error decreases with each year of occurrence. years of occurrence.

Table 14.6: Reserves uncertainty
Name Reserves \(R_i\) \(\sqrt{\text{mse}(Ri)}\) \(\sqrt{\text{mse}(Ri)}\) en % de \(R_i\)
1986 252683 82361 33%
1987 576893 145563 25%
1988 965571 232266 24%
1989 1337211 244398 18%
1990 1769736 269468 15%
1991 3352433 598863 18%
1992 4529328 667898 15%
1993 5706261 830105 15%
1994 6569621 912313 14%
1995 7631816 919035 12%
1996 9382503 988059 11%
1997 12891799 1040287 8%
1998 41170897 3336963 8%
Total 96136752 5158558 5%

14.7.1.3 Charges

The Table~ref{incurred} shows the results for expenses, i.e. the sum of payments made and reserves built up.

Note that for the Chain Ladder method, the results are obtained by directly working with the triangle of total amounts, which is obtained by summing the payment and reserve triangles. For the Projected Case Estimate method, the amounts are obtained by separately summing the completed payment and reserve triangles. If we separately apply the Chain Ladder method to the payment and reserve triangles and then sum them, we get lower amounts (for example, €116,315,139 in 1998).

14.7.1.4 Testing the Assumptions of the Mack Model

If we test the validity of assumptions H1 and H2 underlying the stochastic version of the Chain-Ladder method proposed by Mack on the current payment triangle (of size \(n=14\)), we find that the tests lead to rejecting the non-correlation of successive development factors but not the absence of a calendar year effect. Examining the individual development factors presented in Table ??, we observe that more recent calendar years do not behave exactly like older years; the factors are lower in the first year and higher in the following year. This leads us to consider that the triangle is actually composed of several homogeneous blocks.

We separate the triangle into two groups: we consider the first 8 years of occurrence as one group and the last 6 years as another group. Therefore, we obtain two triangles to be completed separately and on which to perform tests. The assumptions underlying the Mack model are not rejected this time (at 95%).

14.7.1.5 Adaptation of Error Formulas

The measures of variability of reserves estimated by the Chain Ladder method (given in Table ??) were performed for development factors calculated in the standard way, using all available data.

In practice, when working with a significant number of occurrence years and noticing, as in our example, a break in the data triangle (i.e., claims occurring in more recent years do not develop in the same way as those in previous years), the development factors are estimated from the most recent years.

We, therefore, work with the following data triangle, consisting of \(n\) occurrence years and \(n\) development years.

\[ \left( \begin{array}{cccccccccc} \cline{7-10} & C_{11} & C_{12} & \ldots & C_{1,n-m} & \vline & C_{1,n-m+1} & \ldots & C_{1,n-1} & C_{1n} \\ & C_{21} & C_{22} & \ldots & C_{2,n-m} & \vline & C_{2,n-m+1} & \ldots & C_{2,n-1} & \\ & \vdots & \vdots & & \vdots & \vline & \vdots & \rotatebox{45}{...} & & \\ & C_{m1} & C_{m2} & \ldots & C_{m,n-m} & \vline & C_{m,n-m+1} & & & \\ \cline{2-5} \vline & C_{m+1,1} & C_{m+1,2} & \ldots & C_{m+1,n-m} & & & & & \\ \vline & \vdots & \vdots & \rotatebox{45}{...} & & & & & & \\ \vline & C_{n-1,1} & C_{n-1,2} & & & & & & & \\ \vline & C_{n1} & & & & & & & & \end{array} \right) \]

We assume that the first \(m\) occurrence years form one block, meaning they have a similar development pattern, and the last \(n-m\) years form another block with a different development pattern from the first block.

We use the Chain Ladder method to obtain ultimate claims amounts. Therefore, we consider that the data satisfies the three assumptions H1-H2-H3. The estimators \(\hat{\lambda}_k\) of \(\lambda_k\) and \(\hat{\sigma}_k^2\) of \(\sigma_k^2\) will have the same form as before but will differ depending on \(k\): - For \(k\) ranging from \(1\) to \(n-m-1\), the coefficients are calculated from the most recent occurrence years, i.e., from \(m+1\) to \(n\). - For \(k\) ranging from \(n-m\) to \(n-1\), the coefficients are calculated as before, from occurrence years \(1\) to \(m\).

The development coefficients \(\hat{\lambda}_k\) are, therefore, still the result of weighted linear regressions by \(\frac{1}{C_{ik}}\), but on a smaller dataset. It’s as if we are working on two different triangles to which we separately apply the Chain Ladder method. Note that as long as we work on the two triangles separately, we do not need \(\hat{\lambda}_{n-m}\) and \(\hat{\sigma}_{n-m}^2\), coefficients that allow us to switch from one triangle to the other. Additionally, the three assumptions H1-H2-H3 must be verified on both separated triangles, not on the entire initial dataset.

The upper right triangle, \(\{C_{ij}\) for \(1\leq i\leq m\) and \(n-m+1\leq j\leq n+1-i\}\) yields the ultimate amounts \(C_{1n}, \hat{C}_{2n}, \ldots, \hat{C}_{mn}\), while the lower left triangle \(\{C_{ij}\) for \(m+1\leq i\leq n\) and \(1\leq j\leq n+1-i\}\) yields the ultimate amounts \(C_{m+1,n-m}, \hat{C}_{m+2,n-m}, \ldots, \hat{C}_{n,n-m}\), as shown in the following table: \[ \left( \begin{array}{ccccccccc} C_{11} & C_{12} & \ldots & C_{1,n-m} & \vline & C_{1,n-m+1} & \ldots & C_{1,n-1} & C_{1n} \\ C_{21} & C_{22} & \ldots & C_{2,n-m} & \vline & C_{2,n-m+1} & \ldots & C_{2,n-1} & \hat{C}_{2n}\\ \vdots & \vdots & & \vdots & \vline & \vdots & & \vdots & \vdots \\ C_{m1} & C_{m2} & \ldots & C_{m,n-m} & \vline & C_{m,n-m+1} & \ldots & \hat{C}_{m,n-1} & \hat{C}_{mn}\\ \hline C_{m+1,1} & C_{m+1,2} & \ldots & C_{m+1,n-m} & \vline & & & & \\ \vdots & \vdots & & \vdots & \vline & & & & \\ C_{n-1,1} & C_{n-1,2} & \ldots & \hat{C}_{n-1,n-m} & \vline & & & &\\ C_{n1} & \hat{C}_{n2} & \ldots & \hat{C}_{n,n-m} & \vline & & & &\\ \end{array} \right). \]

Let’s write down the explicit formulas for the coefficients \(\hat{\lambda}_k\) and \(\hat{\sigma}_k^2\). The estimators for the development coefficients are given by: \[\begin{equation} \renewcommand{\arraystretch}{3.5} \begin{array}{lcl} \hat{\lambda}_{k} &=& \frac{\displaystyle\sum_{i=m+1}^{n-k}C_{i,k+1}} {\displaystyle\sum_{i=m+1}^{n-k}C_{ik}} \text{ for }k= 1,\ldots, n-m-1, \\ \hat{\lambda}_{k} &=& \frac{\displaystyle\sum_{i=1}^{n-k}C_{i,k+1}} {\displaystyle\sum_{i=1}^{n-k}C_{ik}} \text{ for }k=n-m,\ldots, n-1. \end{array} \tag{14.10} \end{equation}\] The \(\hat{\lambda}_k\) are unbiased estimators of \(\lambda_k\). The estimators for \(\sigma_k\) are: \[\begin{equation} \renewcommand{\arraystretch}{2.5} \begin{array}{lcl} \hat{\sigma}_k^2 &=& \frac{1}{(n-k)-m-1}\displaystyle\sum_{i=m+1}^{n-k} C_{ik} \, \left(\frac{C_{i,k+1}}{C_{ik}}-\hat{\lambda}_k\right)^2 \text{ for }k=1,\ldots, n-m-2, \\ \hat{\sigma}_k^2 &=& \frac{1}{n-k-1}\displaystyle\sum_{i=1}^{n-k} C_{ik} \, \left(\frac{C_{i,k+1}}{C_{ik}}-\hat{\lambda}_k\right)^2 \text{ for }k= n-m,\ldos, n-2. \end{array} \tag{14.11} \end{equation}\] These are unbiased estimators of \(\sigma_k^2\). We still need to estimate \(\sigma_{n-m-1}^2\) and \(\sigma_{n-1}^2\), which can be done by extrapolation based on the previous \(\hat{\sigma}_k^2\) or by a formula similar to the one used in Proposition 14.5.

So, applying Chain Ladder to the two triangles separately allows us to use the formula from Proposition 14.5 to estimate the variability of reserves as follows:

  • For the upper right block: \[ \widehat{mse(\hat{C}_{in})}=\hat{C}_{in}^2 \sum_{k=n+1-i}^{n-1} \frac{\hat{\sigma}_k^2}{\hat{\lambda}_k^2} \, \left(\frac{1}{\hat{C}_{ik}} + \frac{1}{\sum_{j=1}^{n-k}C_{jk}} \right) \] for \(i=2,\ldots, m\);
  • For the lower left block: \[ \widehat{mse(\hat{C}_{i,n-m})}=\hat{C}_{i,n-m}^2 \sum_{k=n+1-i}^{n-m-1} \frac{\hat{\sigma}_k^2}{\hat{\lambda}_k^2} \, \left(\frac{1}{\hat{C}_{ik}} + \frac{1}{\sum_{j=m+1}^{n-k}C_{jk}} \right) \] for \(i= m+2,\ldos, n\).

We would still like to estimate the ultimate amounts \(\hat{C}_{in}\) for \(i=m+1,\ldos, n\) and their variability. To do this, we first need to make an assumption about how we will complete the remaining block, namely the lower right block. We will consider three different assumptions.

First Hypothesis

We apply the Chain Ladder method with the estimated coefficients \(\lambda_k\) on accident years \(1\) to \(m\). Therefore, we assume that the block to be completed satisfies the assumptions for applying Chain Ladder and develops as in the earlier accident years (similar to the upper right block). We obtain, for \(i= m+1,\ldots,n\): \[ \widehat{mse(\hat{C}_{in})}=\hat{C}_{in}^2 \left( \sum_{k=n+1-i}^{n-m-1} \frac{\hat{\sigma}_k^2}{\hat{\lambda}_k^2} \, \left(\frac{1}{\hat{C}_{ik}} + \frac{1}{\sum_{j=m+1}^{n-k}C_{jk}} \right)\right. \] \[ \left.+ \sum_{k=n-m}^{n-1} \frac{\hat{\sigma}_k^2}{\hat{\lambda}_k^2} \, \left(\frac{1}{\hat{C}_{ik}} + \frac{1}{\sum_{j=1}^{n-k}C_{jk}} \right) \right], \] where the \(\hat{\lambda}_k\) values are calculated using (14.10), and the \(\hat{\sigma}_k^2\) values are calculated using (14.11). For the proof of this formula, we refer to (Pitrebois et al. 2002).

Second Hypothesis

Using development factors calculated for the first \(m\) years to complete the lower part of the matrix, when we split it in the first place because the development in recent years was different from earlier years, may not be very logical.

Another hypothesis is to assume that the total development from year \(1\) to year \(n\) is the same for all accident years (from \(1\) to \(n\)). Therefore, we assume that \(C_{in}=\lambda_{tot} \cdot C_{i1}\) for all \(i\). We estimate \(\lambda_{tot}\) as \(\hat{\lambda}_{tot}\) over the first \(m\) accident years: \[ \hat{\lambda}_{tot}=\frac{\displaystyle\sum_{i=1}^{m}\hat{C}_{in}} {\displaystyle\sum_{i=1}^{m}C_{i1}}. \] We then examine the total development already experienced by claims occurring in years \(m+1\) to \(n\) and estimate it as: \[ \hat{\lambda}_{int}=\frac{\displaystyle\sum_{i=m+1}^{n}\hat{C}_{i,n-m}} {\displaystyle\sum_{i=m+1}^{n}C_{i1}}. \] By calculating the ratio \(\hat{\lambda}_{tot} / \hat{\lambda}_{int}\), we obtain the factor by which to multiply the estimated claim amounts in development year \((n-m)\) to obtain the amounts in development year \(n\). This factor is comparable to a : \[ \hat{\lambda}_{ult}=\frac{\hat{\lambda}_{tot}}{\hat{\lambda}_{int}}. \]

Of course, other definitions of \(\hat{\lambda}_{ult}\) can be used to complete the lower triangle.

Third Hypothesis

The issue with the second hypothesis is that we do not know the portion of the claim paid during the first development year. Therefore, it is challenging to estimate the total development factor to reach ultimate amounts.

A third way to complete the lower right block, using only the data from the lower left block, is to start with the estimated development factors for development years \(1\) to \(n-m-1\) and extrapolate to obtain the factors for subsequent development years. For example, one can fit a negative exponential curve. (Sherman 1984) shows that a better fit is obtained using a curve of the form \(\lambda_t=1+at^{-b}\), where \(t\) represents the development year, and \(\lambda_t\) is the development factor for year \(t\). The function can be rewritten as \(\ln (\lambda_t-1)=\ln a +b \ln \frac{1}{t}\), and the parameters \(a\) and \(b\) can be obtained through linear regression of \(\ln (\lambda_t-1)\) on \(\ln \frac{1}{t}\).

The operation of these three hypotheses can be summarized as follows. Suppose in recent years, claims develop more rapidly at the beginning than in the past. Hypothesis 1 assumes that these claims will subsequently follow the same development pattern as claims in earlier years. Hypothesis 2 assumes that, overall, these claims develop as in the past, meaning that if they develop faster at the beginning, they will develop more slowly later. Finally, Hypothesis 3 assumes that if claims develop more rapidly at the beginning, they will continue to do so later. Only a deep understanding of the portfolio and the reserving process allows for the choice of the hypothesis closest to reality.

14.7.1.6 Application to German Data

The lower left block and the upper right block are completed separately using the Chain Ladder method. Let’s examine the three hypotheses considered for completing the lower right block.

  1. First Hypothesis

The lower right block is completed using the coefficients from the upper right block. The results are presented in Table ??.

  1. Second Hypothesis

The lower right block is completed using a . The total development after \(14\) years experienced by claims occurring between \(1985\) and \(1992\) averages \(\hat{\lambda}_{tot}=1.5691\). The development already undergone in the first \(6\) years by claims occurring between \(1993\) and \(1998\) averages \(\hat{\lambda}_{int}=1.4536\). Therefore, we need to multiply the amounts obtained in the sixth year of development for the lower left block by \(\hat{\lambda}_{ult}=\frac{\hat{\lambda}_{tot}}{{\lambda}_{int}}=1.0795\) to obtain the ultimate amounts (after \(14\) years of development). %We have \(\hat{\lambda}_1\geq \hat{\lambda}_{ult}\geq \hat{\lambda}_2\), in %fact, \(1.3228\geq 1.0795 \geq 1.0414\), and therefore, we choose %\(mse(\hat{\lambda}_1)\geq mse(\hat{\lambda}_{ult})\geq mse(\hat{\lambda}_2)\), as well as %\(mse(\lambda_{i1})\geq mse(\lambda_{i,ult})\geq mse(\lambda_{i2})\) for some \(i\). We then obtain the results presented in Table ??.

  1. Third Hypothesis

We start with the coefficients estimated using the Chain Ladder method on the lower left triangle, specifically the \(\widehat{\lambda}_k\) for \(k=1,\ldots,5\) given in Table ??.

We then fit a power-inverse curve by linear regression of \(\ln (\lambda_k-1)\) on \(\ln (1/k)\), resulting in the model \[ \hat{\lambda}_k=1+0.2671 \, k^{-2.1038}, \] which leads to the coefficients in Table ??.

We then add these coefficients to the bottom right-hand block to obtain the final amounts shown in Table~ref{hyp3}.

\end{enumerate}

In summary, Hypothesis 1 assumes that more recent claims, which developed more slowly at the beginning, will subsequently follow the same development as claims in earlier years. Hypothesis 2 assumes that, overall, these claims develop as in the past, and therefore, if they develop more slowly at the beginning, they will develop more quickly later. Hypothesis 3 assumes that if claims develop more slowly at the beginning, they will continue to do so later. It leads to lower final amounts.

14.7.2 Medical Malpractice

14.7.2.1 Context

The second example is taken from (Denuit, Maréchal, and Closon 2005) and concerns medical malpractice liability insurance. The statistics used, which will be presented in detail later in this section, pertain to the entire Belgian medical malpractice insurance market. These are global figures without a breakdown of indemnities by type or detail of damage suffered.

In general, statistics related to the claims experience in the medical malpractice insurance branch exhibit significant fluctuations, even at the market level. This instability makes modeling long-term trends challenging. Claim reporting and settlement rates vary significantly among market participants, making it impossible to extrapolate one company’s experience to the entire market. Additionally, aggregated data can be challenging to interpret because it combines highly heterogeneous statistics.

As medical malpractice is a branch with particularly long development (it often takes about twenty years before all claims related to a financial year are finally settled), the available statistics often cover periods that are too short to fully capture the development of claims. To avoid underestimating costs for insurers, it is crucial to also consider the amounts provisioned (using methods such as Projected Case Estimate, for example). However, balance sheet provisions do not enjoy the same objectivity as actual payments made by insurers. Factors like tax considerations can distort the reserved amounts, which are also subject to evaluation errors by managers. The method described here allows for the evaluation of reserves solely based on payments made. We will use an approach based on Generalized Linear Models (GLMs), which will enable us to combine two payment triangles to leverage a longer history and avoid the use of the reserve triangle (as in the Projected Case Estimate method).

14.7.2.2 Available Statistics

The numbers in Table ?? are taken from (Denuit, Maréchal, and Closon 2005) and pertain to the entire Belgian medical malpractice insurance market. They represent payments made by companies practicing medical malpractice insurance, broken down by the year of occurrence of the claim and the development year. In addition to the triangle in Table ??, covering the period 1995-2003, another triangle for the period 1977-1994 is provided in Table ??. The use of GLMs allows us to combine these two triangles and thus study much longer developments (up to 18 years in our example).

14.7.2.3 Analysis Method

The model used for payments in the settlement triangles is of the form: \[\begin{equation} Y_{ij}\sim\mathcal{G}am(\mu_{ij},\nu) \text{ independently,} \tag{14.12} \end{equation}\] where \(\mu_{ij}=\mu\alpha_i\beta_j\gamma_c\), and \(c=i+j-1\) represents the year in which payment \(Y_{ij}\) was made.

The occurrence year and development year are treated as factors, and their influence on payments \(Y_{ij}\) is quantified by the parameters \({\alpha}_1=1\), \({\alpha}_2\), \(\ldots\), \({\alpha}_n\) for the occurrence year, and \({\beta}_1=1\), \({\beta}_2\), \(\ldots\), \({\beta}_n\) for the development year. Similarly, the influence of year \(c\) in which the payment was made is taken into account as a factor, with associated parameters \({\gamma}_1=1\), \({\gamma}_2\), \(\ldots\), \({\gamma}_n\). Note that the triple constraint \({\alpha}_1={\beta}_1={\gamma}_1=1\) ensures that \(\mathbb{E}[C_{11}]=\mu\), making this parameter interpretable as the average cost of a claim paid during the first development year of the first occurrence year.

14.7.2.4 Analysis Results

The gamma regression approach allows us to simultaneously consider the two triangles presented above.

A preliminary analysis involves the occurrence year and development pace as two explanatory factors for the payments, as well as constant inflation. This inflation is considered non-significant (p-value of 67%, and the diagonal effect is likely taken into account by both the row and column factors). This led us to re-adjust the model without explicitly including inflation (which leaves the AIC unchanged, as the model is over-parameterized). The estimated parameters are shown in Figure ??. The estimated \(\alpha_i\) values suggest a logarithmic trend over time, which is confirmed by a least squares fitting model, as shown in Figure ??.

We fit a new Gamma regression model to the payments, logarithmically entering the occurrence year on the score scale. This provides new estimates of the \(\beta_j\) parameters, as seen in Figure ??. We observe significant variability in the estimated \(\beta_j\) values, and no clear structure emerges from Figure ??. Therefore, we adjust these parameters using a parametric model with a cautious approach, and we smooth them using a loess (locally weighted least squares) procedure. The result is visible in Figure ??.

In the cautious approach, we only retain the highest \(\widehat{\beta}_j\) values (excluding developments 8, 10, 11, 12, and 14), and we adjust them using a model that includes the absolute difference between the development year and the 9th year (the shape of the \(\widehat{\beta}_j\) in Figure ?? suggesting a peak at \(j=9\)). In this case, the estimated cost of claims occurring in year \(i\) and paid in year \(i+j-1\) is estimated as \[\begin{equation} \widehat{C}_{ij}= \exp\Big(9.53810+1.05577\times\ln(i-1976)+1.06755-0.13519\times |j-9|\Big). \tag{14.13} \end{equation}\] Note that this way, the model extrapolates payments for developments beyond 18.

In the second approach, we simply smooth the \(\widehat{\beta}_j\) values represented in Figure ??. To do this, we use the loess method. The result is visible in Figure ??. In this second approach, the estimated cost of claims occurring in year \(i\) and paid in year \(i+j-1\) is given by \[\begin{equation} \exp\Big(9.53810+1.05577\times\ln(i-1976)+\widehat{\beta}_j^{\text{smoothed}}\Big) (\#eq:Eq13.16) \end{equation}\] where the values of \(\widehat{\beta}_j^{\text{smoothed}}\) are listed in Table ??.

14.7.2.5 Reconstruction of Global Cash Flows

The reconstruction for the year 1977 using (14.13), with the longest development, is excellent since the model predicts a total cost (over 20 years of development) of €440,456.90, while the observed cost is €415,577. However, as seen in Figure ??, despite the good reconstruction of the total claims cost, significant differences remain between some annual cash flows. This is due to the instability of payments related to medical liability insurance.

The reconstruction obtained using the smoothed coefficients \(\beta_j\) with the loess method is also visible in Figure \(ref?)(fig:FluxFinanciers1977). Using (??), we obtain a total cost of €294,223.2 for the year 1977, which is significantly less than the sum of payments made. This is a common phenomenon in non-life insurance, where smoothing techniques often lead to an underestimation of costs.

14.8 Bibliographical Notes

A portion of this chapter is derived from (Pitrebois et al. 2002) and (Magis 2003).

Books addressing non-life insurance provisioning are relatively rare. However, noteworthy mentions include (Taylor 2012), which provides a relatively broad overview of the techniques used, and (Kaas et al. 2008), which presents provisioning as an application of GLM models.

The article by (England and Verrall 2002) clearly presents the issue within the general framework of GLM and GAM models.

Finally, (Schmidt 1999) offers a very comprehensive bibliography of works dedicated to provisioning throughout the 20th century.

Postface

Astesan, Eugène. 1938. “Les réserves Techniques Des Sociétés d’assurances Contre Les Accidents d’automobiles.” Collection d’études Sur Le Droit Des Assurances.
Denuit, Michel, Xavier Maréchal, and Jean-Pierre Closon. 2005. “Etude Relative Aux Coûts Potentiels Liés à Une éventuelle Modification Des règles Du Droit de La Responsabilité médicale. Phase II: Développement d’un Modèle Actuariel Et Premières Estimations.” Health Services Research (HSR). Bruxelles: Centre fédéral d’expertise Des Soins de Santé (KCE).
England, Peter D, and Richard J Verrall. 2002. “Stochastic Claims Reserving in General Insurance.” British Actuarial Journal 8 (3): 443–518.
Henry, M. 1937. “Etude Sur Le Coût Moyen Des Sinistres En Responsabilité Civile Automobile.” Bulletin Trimestriel de l’Institut Des Actuaires Françmais 43: 113–78.
Kaas, Rob, Marc Goovaerts, Jan Dhaene, and Michel Denuit. 2008. Modern Actuarial Risk Theory: Using r. Vol. 128. Springer.
Mack, Thomas. 1993. “Distribution-Free Calculation of the Standard Error of Chain Ladder Reserve Estimates.” ASTIN Bulletin: The Journal of the IAA 23 (2): 213–25.
Magis, Denuit, C. 2003. “Au-Delà de Chain-Ladder: Les Méthodes Stochastiques de Réservation.” Actu-L 3: 31–58.
Pitrebois, Sandra, Michel Denuit, Jean-François Walhin, and Philippe De Longueville. 2002. “Etude de Techniques IBNR Modernes.” Actu-L 2: 29.
Reid, DH. 1986. “Discussion of Methods of Claim Reserving in Non-Life Insurance.” Insurance: Mathematics and Economics 5 (1): 45–56.
Renshaw, Arthur E, and Richard J Verrall. 1998. “A Stochastic Model Underlying the Chain-Ladder Technique.” British Actuarial Journal 4 (4): 903–23.
Schmidt, Klaus D. 1999. “A Bibliography on Loss Reserving.”
Sherman, Richard E. 1984. “Extrapolating, Smoothing and Interpolating Development Factors.” In Proceedings of the Casualty Actuarial Society, 71:122–55. 136.
Taylor, Gregory. 1977. “Separation of Inflation and Other Effects from the Distribution of Non-Life Insurance Claim Delays.” ASTIN Bulletin: The Journal of the IAA 9 (1-2): 219–30.
———. 2012. Loss Reserving: An Actuarial Perspective. Vol. 21. Springer.
Tosetti, Alain, Thomas Béhar, Michel Fromenteau, and Stéphane Ménart. 2000. Assurance: Comptabilité réglementation Actuariat. Economica.