# Reserving

Loss reserving refers to the process of calculating, at a particular time, an amount which represents liabilities associated with a portfolio of insurance policies. Loss reserves are useful to various stakeholders such as internal management, investors and regulators.

## Loss Development

Each portfolio of policies will trigger a number of claims and every such claim has a life of its own from the initial reporting of the claim to its closure/settlement. The insurer will carefully record expenses and payments for claims associated with a portfolio of policies. The actuary can then use historical claims data to establish reserves for various stakeholders. We give a brief overview of some of the key terminology relevant to historical claims development data:

Term Description
Accident Year Refers to the year when a claim triggering incident took place.
Case outstanding Represents an amount, established by a claims professional (typically a claims adjuster), representing the dollar value of the liabilities to the insurer associated with a group of claims.
Reported claims The case outstanding plus the payments made to the insureds for a group of claims.
Ultimate claims The cumulative sum of all payments made to insureds for a group of settled claims i.e., the total cost to the insurer to settle a group of claims.
IBNR The difference between ultimate claims and reported claims.

## The Development Triangle

A development triangle is a common way to organize and display historical claims data for the purposes of actuarial analysis. The most common data sets displayed in a development triangle are case outstanding, reported claims, paid claims and reported claim count.

A development triangle organizes claims data as a table or triangular matrix $C_{i,j}$ with $i = 0,\ldots,I$ and $j = 0,\ldots,I - i$. The rows of the triangle represent accident years and the columns represent valuation dates. For example, in the table below, 43,169,009 represents loss amounts related to claims occurring in accident year 0, valued as of two years.

Reported claims
Accident Year 0 1 2 3 4
0 37,017,487 43,169,009 45,568,919 46,784,558 47,337,318
1 38,954,484 46,045,718 48,882,924 50,219,672
2 41,155,776 49,371,478 52,358,476
3 42,394,069 50,584,112
4 44,755,243

We would like to develop methods to estimate the non-observables $C_{i,j}$ where $j \gt I -i$ especially the unknown ultimate number $C_{i,J}$.

The chain ladder technique, also known as the developmental technique, assumes that historical claims development is indicative or predictive of future claims development. Chain ladder methods rely on historical data found in the #development triangle to predict future claims development.

### Development Factors

We are primarily interested in the progression of the $C_{i,j}$ as $j$ increases. Since $C_{i,j}$ typically represents cumulative quantities such as cumulative reported or paid claims, we expect $C_{i,j}$ to be increasing in $j$.

The development factors are the observed ratios $C_{i,j+1}/C_{i,j}$. They represent a measure of growth of claims development associated with a single accident year. If $I$ is large enough, we should expect the development factors to tend to 1 as all claims associated with an accident year are permanently closed. We calculate the development factors corresponding to the development triangle of reported claims presented above:

Development factors
Accident year 0 1 2 3 To ult
0 1.166 1.056 1.027 1.012 1.00
1 1.182 1.062 1.027
2 1.200 1.061
3 1.193

### Selecting Development Factors

The chain ladder technique assumes that historical deveopment factors are predictive of future development factors; consequently, one is expected to use such historical factors to select development factors that can be used to project future claims development for any given accident year. There are different methods used to select development factors. The simplest approach is to use a particular average of the historical development factors. To exemplify this simple approach, we calculate various averages associated with the table of historical development factors above:

Averages
Averaging Method 0 1 2 3 To ult
Simple average last 3 years 1.192 1.06 1.027 1.012 1.000
Volume weighted last 3 years 1.186 1.059 1.027 1.012 1.000
Selected 1.192 1.06 1.027 1.012 1.000

In this particular case, the development factors used for future development is the average development factor for the last 3 observable years. For instance, the selected development factor for development year 0 is 1.192, the average of 1.193, 1.2 and 1.182.

In the table above, the entry corresponding to row selected and column to ult is a tail factor -- a factor used to deduct the ultimate reported claims value from the latest observable reported claims value. Ideally the value of $I$ in the development triangle is large enough that the tail factor equals 1. In the example above, the tail factor is 1: the insurer expects all claims' development to cease 5 years after the accident year. It's important to note that the tail factor can exceed 1 in so called long tailed lines such as worker's compensation.

### Mack-Method

The Mack chain ladder method is a statistical method to estimate developmental factors in the chain ladder method. The method assumes the following:

• Distinct rows of the array/matrix $C_{i,j}$ are independent.
• $\operatorname{E}[C_{i,j+1} | C_{i,0},\ldots,C_{i,j}] = f_j C_{i,j}$ with $f_j$ a constant.
• $\operatorname{Var}[C_{i,j+1} | C_{i,0},\ldots,C_{i,j}] = \sigma_{j}^2 C_{i,j}$ with $\sigma_j$ a constant.

The goal of the Mack-method is to estimate the factors $f_j$ using the observable $C_{i,j}$. The estimators, denoted $\hat{f}_j$, then become the selected development factors. The estimators are defined as follows:

[$] \begin{equation} \hat{f}_j = \frac{\sum_{i=0}^{I-j}C_{i,j+1}}{\sum_{i=0}^{I-j}C_{i,j}}. \end{equation} [$]

They have the following desirable properties:

• $\hat{f_j}$ is an unbiased estimator for $f_j$: $\operatorname{E}[\hat{f_j}] = f_j$.
• The estimator $\hat{f_j}$ is a minimum variance estimator in the following sense:

[$] \hat{f_j} = \underset{X \in S_j}{\operatorname{argmin}} \operatorname{Var}[ X | A_j ],\, S_j = \{\sum_{i=0}^{I-j}w_i C_{i,j+1}/C_{i,j} | \sum_{i=0}^{I-j}w_i = 1\} [$]

with $A_j = \cup_{i=0}^{I-j}\{C_{i,0},\ldots,C_{i,j}\}$ the claims information contained in the first $j$ periods.

Applying the method to the (reported claims) data given above, we obtain the following selected factors:

0123
1.1861.0591.0271.012

### Project Ultimate Claims

The key to projecting ultimate claims is to calculate cumulative claim development factors (CDF) which are defined as follows:

[$] \begin{equation} \label{cdf} \hat{F_i} = \prod_{j = I- i}^{J-1}\hat{f_j} \end{equation} [$]

with $\hat{f}_j$ denoting the selected development factors. The projection for ultimate claims for accident year $i$ equals

[$] \begin{equation} \label{puc} C_{i,I-i}\hat{F_i}. \end{equation} [$]

Using formula \ref{puc}, we can calculate the projected ultimate reported claims for the example above (with a selected tail factor of 1):

Estimation of ultimate claims
Accident year Reported claims CDF (3 year avg / Mack-Method) Projected ultimate claims (3 year avg / Mack-Method)
0 47,337,318 1.000 / 1.000 47,337,318 / 47,337,318
1 50,219,672 1.012 / 1.012 50,822,308 / 50,822,308
2 52,358,476 1.039 / 1.039 54,400,457 / 54,400,457
3 50,584,112 1.102 / 1.101 55,743,691 / 55,693,107
4 44,755,243 1.313 / 1.305 58,763,634 / 50,595,802
Total 245,659,807 267,472,394 / 259,253,978

Here is a summary on how to project ultimate claims:

Project Ultimate Claims Using the Chain Ladder Method
1. Create development triangle
2. Select development factors using historical development factors (averages, Mack-method, etc.)
3. Select tail factor
4. Calculate CDFs using formula \ref{cdf}
5. Calculate ultimate claims using formula \ref{puc}

## The Claim Ratio Method

The claim ratio method, or loss ratio method, is one of the simplest methods to project ultimate unpaid claims. The method projects ultimate claims for an accident year by multiplying a predetermined exposure level by a selected claim ratio representing claims per unit of exposure:

Projection for ultimate claims = Exposure * Selected claim ratio.

The claim ratio method's accuracy depends on the ability to establish a proper exposure level associated with an accident year and then establishing an estimate for the ultimate claim per unit of exposure. The projection for ultimate claims associated with the chain ladder method is derived by multiplying the reported claims for an accident year by the selected CDF whereas the claim ratio method multiplies a selected exposure level associated with an accident year by a selected claim ratio. In other words, the chain ladder technique depends on the initial claim experience of an accident year whereas the claims ratio method depends on an exposure level which is a much more natural measure of size for a portfolio of policies.

A common type of exposure is earned premium. Earned premium is the portion of an insurance premium which is considered "earned" by the insurer, based on the part of the policy period that the insurance has been in effect, and during which the insurer has been exposed to loss. For instance, if a 365-day policy with a full premium payment at the beginning of the term has been in effect for 120 days, 120/365 of the premium is considered earned. Earned premium will not be returned to the insured if the policy is cancelled.

### Example

Claim ratio method
Accident year Reported claims at 12/31/08 Initial selected ultimate claims On-level earned premium Trend factor at 07/01/2008 Trended adjusted ultimate claims Trended adjusted claim ratio
2000 10,000,000 10,012,500 24,000,000 2.954 29,576,925 83.0%
2001 8,000,000 8,220,000 18,000,000 2.580 21,207,600 79.0%
2002 9,400,000 9,591,000 19,000,000 2.253 21,608,523 76.0%
2003 15,600,000 13,845,000 23,000,000 1.968 27,246,960 79.0%
2004 16,500,000 19,700,000 32,000,000 1.719 33,864,300 79.0%
2005 18,500,000 25,700,000 47,000,000 1.501 38,575,700 82.0%
2006 16,500,000 29,850,000 50,000,000 1.311 39,133,350 78.0%
2007 14,000,000 42,800,000 57,000,000 1.145 49,006,000 86.0%
2008 8,700,000 51,150,000 62,000,000 1.000 51,150,000 83.0%

Suppose one wishes to project ultimate claims for accident year 2008 using the claim ratio method applied to the information contained in the table above. The first step is to select the exposure level associated with accident year 2008 which, in this scenario, is given by 62,000,000, the last entry in column labelled on-level earned premium. The second step is to select a claims ratio based on the observed historical claims ratios represented by the last column in the table above. Here we use a simple approach: the claims ratio is set to 79%, the average of the historical claims ratios for accident years 2002 to 2006. We finally obtain the projected ultimate claims for accident year 2008:

Accident year Reported claims at 12/31/08 On-level earned premium Selected claim ratio (avg 2002-2006) Projected ultimate claims
2008 8,700,000 62,000,000 79.0% 48,980,000

## The Bornhuetter-Ferguson method

Unlike the chain ladder method, the Bornhuetter-Ferguson method doesn't rely exclusively on historical claims development to project ultimate claims. The projected ultimate claims under this method is equal to the current reported claims plus the expected unreported claims:

Ultimate claims = Actual Reported Claims + Expected Unreported Claims

The expected unreported claims is calculated by multiplying the expected claims multiplied by an estimate of the % yet to be reported:

Ultimate claims = Actual Reported Claims + Expected Claims * % Unreported

The estimate for the percentage yet to be reported for accident year $i$ is based on the developmental method and is set to $1 - \hat{F_i}^{-1}$. Mathematically the projected ultimate claims for accident year $i$ equals

[$] \begin{equation} \label{bf-method-main} C_{i,I-i} + (1 - \hat{F_{i}}^{-1})\mu_i = Z_i \hat{F_i}C_{i,I-i} + (1 - Z_i)\mu_i \end{equation} [$]

with $\mu_i$ the expected ultimate claims for accident year $i$ and $Z_i = \hat{F_i}^{-1}$. Examining \ref{bf-method-main}, we notice that $\hat{F_i}C_{i,I-i}$ is the projection for ultimate claims given by the developmental technique; consequently, we can view the projection for ultimate claims using the Bornhuetter-Ferguson method as a kind of credibility estimator with credibility weight $Z_i$.

### The Bornhuetter-Ferguson method as MMSE

If we impose specific conditions on the variables $C_{i,j}$ (the developmental triangle), then the Bornhuetter-Ferguson projection for ultimate claims is a minimum mean square estimator.

Suppose the following holds:

• If $C_{i,J}$ denotes the ultimate claims, then $\mu_i = \operatorname{E}[C_{i,J}]$ is known for all $1 \leq i \leq I$
• $C_{i,J} - C_{i,j}$ is independent of $C_{i,j}$ for any $1 \leq j \leq I$
• $F_{i,j} = \mu_i / \operatorname{E}[C_{i,j}]$ are known
• $\operatorname{Var}[C_{i,j}F_{i,j}] = F_{i,j}\operatorname{Var}[C_{i,J}]$

Then the the Bornhuetter-Ferguson estimator for $C_{i,J}$ equals

[$] \hat{C}_{i,J} = Z_i F_iC_{i,I-i} + (1 - Z_i)\mu_i. [$]

with the shorthand $F_i = F_{i,J}$. It is clear from the assumptions above that $\hat{C}_{i,J}$ is an unbiased estimator of $C_{i,J}$. Furthermore, the estimator minimizes variance among all weighted averages of $F_iC_{i,I-i}$ and $\mu_i$:

[$] \hat{C}_{i,J} = \underset{X \in S_i}{\operatorname{argmin}} \operatorname{E}[ (C_{i,J} - X)^2 ],\, S_i = \{ ZF_iC_{i,I-i} + (1-Z)\mu_i \}. [$]

## Average Cost of Claim Method

The average cost of claim method projects costs per claim and claim count to project ultimate claims. The developmental technique is used to project both the ultimate cost per claim and the ultimate claim count.

In order to implement this method, we need to define three development processes:

• $C_{i,j}$ represents the developmental triangle for total aggregate claim costs
• $N_{i,j}$ represents the developmental triangle for aggregate claim count
• $S_{i,j}$ represents the developmental triangle for the average claim cost

The projection for ultimate claims, denoted by $\hat{C}_{i,J}$, equals the projection for ultimate claim, denoted by $\hat{N}_{i,J}$, multiplied by the projection for ultimate average claim cost, denoted by $\hat{S}_{i,J}$.

Average Cost of Claim Method
1. Build developmental triangles $N_{ij}$, $S_{ij}$ for claim count and for average claim cost, respectively.
2. Use the developmental technique to project the ultimate average claim cost $\hat{S}_{i,J}$ and the ultimate claim count $\hat{N}_{i,J}$.
3. The projection for ultimate claims is the product of the projection for ultimate cost per claim and ultimate claim count: $\hat{C}_{i,J} = \hat{S}_{i,J} \hat{N}_{i,J}.$