The chain-ladder or development[1] method is a prominent[2][3] actuarial loss reserving technique. The chain-ladder method is used in both the property and casualty[1][4] and health insurance[5] fields. Its intent is to estimate incurred but not reported claims and project ultimate loss amounts.[5] The primary underlying assumption of the chain-ladder method is that historical loss development patterns are indicative of future loss development patterns.[1][3][4]

## Methodology

According to Jacqueline Friedland's "Estimating Unpaid Claims Using Basic Techniques," there are seven steps to apply the chain-ladder technique:

1. Compile claims data in a development triangle
2. Calculate age-to-age factors
3. Calculate averages of the age-to-age factors
4. Select claim development factors
5. Select tail factor
6. Calculate cumulative claim development factors
7. Project ultimate claims

Age-to-age factors, also called loss development factors (LDFs) or link ratios, represent the ratio of loss amounts from one valuation date to another, and they are intended to capture growth patterns of losses over time. These factors are used to project where the ultimate amount losses will settle.

## Example

First, losses (either reported or paid) are compiled into a triangle, where the rows represent accident years and the columns represent valuation dates. For example, 43,169,009 represents loss amounts related to claims occurring in 1998, valued as of 24 months.

Reported claims[1]
Valuation
date

Accident year
12 24 36 48 60 72 84 96 108 120
1998 37,017,487 43,169,009 45,568,919 46,784,558 47,337,318 47,533,264 47,634,419 47,689,655 47,724,678 47,742,304
1999 38,954,484 46,045,718 48,882,924 50,219,672 50,729,292 50,926,779 51,069,285 51,163,540 51,185,767
2000 41,155,776 49,371,478 52,358,476 53,780,322 54,303,086 54,582,950 54,742,188 54,837,929
2001 42,394,069 50,584,112 53,704,296 55,150,118 55,895,583 56,156,727 56,299,562
2002 44,755,243 52,971,643 56,102,312 57,703,851 58,363,564 58,592,712
2003 45,163,102 52,497,731 55,468,551 57,015,411 57,565,344
2004 45,417,309 52,640,322 55,553,673 56,976,657
2005 46,360,869 53,790,061 56,786,410
2006 46,582,684 54,641,339
2007 48,853,563

Next, age-to-age factors are determined by calculating the ratio of losses at subsequent valuation dates. From 24 months to 36 months, accident year 1998 losses increased from 43,169,009 to 45,568,919, so the corresponding age-to-age factor is 45,568,919 / 43,169,009 = 1.056. A "tail factor" is selected (in this case, 1.000) to project from the latest valuation age to ultimate.

Age-to-age factors[1]
Accident year 12-24 24-36 36-48 48-60 60-72 72-84 84-96 96-108 108-120 To ult
1998 1.166 1.056 1.027 1.012 1.004 1.002 1.001 1.001 1.000
1999 1.182 1.062 1.027 1.010 1.004 1.003 1.002 1.000
2000 1.200 1.061 1.027 1.010 1.005 1.003 1.002
2001 1.193 1.062 1.027 1.014 1.005 1.003
2002 1.184 1.059 1.029 1.011 1.004
2003 1.162 1.057 1.028 1.010
2004 1.159 1.055 1.026
2005 1.160 1.056
2006 1.173
2007

Finally, averages of the age-to-age factors are calculated. Judgmental selections are made after observing several averages. The age-to-age factors are then multiplied together to obtain cumulative development factors.

Averages[1]
Month range

Averaging method
12-24 24-36 36-48 48-60 60-72 72-84 84-96 96-108 108-120 To ult
Simple average last 5 years 1.168 1.058 1.027 1.011 1.004 1.003 1.002 1.001 1.000
Simple average last 3 years 1.164 1.056 1.027 1.012 1.005 1.003 1.002 1.001 1.000
Volume weighted last 5 years 1.168 1.058 1.027 1.011 1.004 1.003 1.002 1.001 1.000
Volume weighted last 3 years 1.164 1.056 1.027 1.012 1.005 1.003 1.002 1.001 1.000
Selected 1.164 1.056 1.027 1.012 1.005 1.003 1.002 1.001 1.000 1.000
Cumulative to ultimate 1.292 1.110 1.051 1.023 1.011 1.006 1.003 1.001 1.000 1.000

The cumulative development factors multiplied by the reported (or paid) losses to project ultimate losses.

Estimation of ultimate claims[1]
Accident year Reported claims Development factor to ultimate Projected ultimate claims
1998 47,742,304 1.000 47,742,304
1999 51,185,767 1.000 51,185,767
2000 54,837,929 1.001 54,892,767
2001 56,299,562 1.003 56,468,461
2002 58,592,712 1.006 58,944,268
2003 57,565,344 1.011 58,198,563
2004 56,976,657 1.023 58,287,120
2005 56,786,410 1.051 59,682,517
2006 54,641,339 1.110 60,651,886
2007 48,853,563 1.292 63,118,803
Total 543,481,587 569,172,456

Incurred but not reported can be obtained by subtracting reported losses from ultimate losses, in this case, 569,172,456 - 543,481,587 = 25,690,869.[6] [7] [8]

## Limitations

The chain-ladder technique is only accurate when patterns of loss development in the past can be assumed to continue in the future.[1][3][4] In contrast to other loss reserving methods such as the Bornhuetter-Ferguson method, it relies only on past experience to arrive at an incurred but not reported claims estimate.

When there are changes to an insurer's operations, such as a change in claims settlement times, changes in claims staffing, or changes to case reserve practices, the chain-ladder method will not produce an accurate estimate without adjustments.[1]

The chain-ladder method is also very responsive to changes in experience, and as a result, it may be unsuitable for very volatile lines of business.[5]