Chain-ladder method

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
1224364860728496108120
1998 37,017,48743,169,00945,568,91946,784,55847,337,31847,533,26447,634,41947,689,65547,724,67847,742,304
1999 38,954,48446,045,71848,882,92450,219,67250,729,29250,926,77951,069,28551,163,54051,185,767
2000 41,155,77649,371,47852,358,47653,780,32254,303,08654,582,95054,742,18854,837,929
2001 42,394,06950,584,11253,704,29655,150,11855,895,58356,156,72756,299,562
2002 44,755,24352,971,64356,102,31257,703,85158,363,56458,592,712
2003 45,163,10252,497,73155,468,55157,015,41157,565,344
2004 45,417,30952,640,32255,553,67356,976,657
2005 46,360,86953,790,06156,786,410
2006 46,582,68454,641,339
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 12-2424-3636-4848-6060-7272-8484-9696-108108-120To ult
1998 1.1661.0561.0271.0121.0041.0021.0011.0011.000
1999 1.1821.0621.0271.0101.0041.0031.0021.000
2000 1.2001.0611.0271.0101.0051.0031.002
2001 1.1931.0621.0271.0141.0051.003
2002 1.1841.0591.0291.0111.004
2003 1.1621.0571.0281.010
2004 1.1591.0551.026
2005 1.1601.056
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
12-2424-3636-4848-6060-7272-8484-9696-108108-120To ult
Simple average last 5 years 1.1681.0581.0271.0111.0041.0031.0021.0011.000
Simple average last 3 years 1.1641.0561.0271.0121.0051.0031.0021.0011.000
Volume weighted last 5 years 1.1681.0581.0271.0111.0041.0031.0021.0011.000
Volume weighted last 3 years 1.1641.0561.0271.0121.0051.0031.0021.0011.000
Selected 1.1641.0561.0271.0121.0051.0031.0021.0011.0001.000
Cumulative to ultimate 1.2921.1101.0511.0231.0111.0061.0031.0011.0001.000

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

Estimation of ultimate claims[1]
Accident yearReported claimsDevelopment factor to ultimateProjected ultimate claims
199847,742,3041.00047,742,304
199951,185,7671.00051,185,767
200054,837,9291.00154,892,767
200156,299,5621.00356,468,461
200258,592,7121.00658,944,268
200357,565,3441.01158,198,563
200456,976,6571.02358,287,120
200556,786,4101.05159,682,517
200654,641,3391.11060,651,886
200748,853,5631.29263,118,803
Total543,481,587569,172,456

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]

See also

References

  1. https://www.casact.org/library/studynotes/Friedland_estimating.pdf
  2. Schmidt, Klaus D. (1999). "Chain Ladder Prediction and Asset Liability Management". Blätter der DGVFM. 24: 1–9. doi:10.1007/BF02808592.
  3. "Chain Ladder Method (CLM)".
  4. https://www.casact.org/library/studynotes/Werner_Modlin_Ratemaking.pdf
  5. "Archived copy" (PDF). Archived from the original (PDF) on 2014-03-27. Retrieved 2016-03-13.CS1 maint: archived copy as title (link)
  6. https://www.casact.org/pubs/forum/06fforum/273.pdf
  7. "Understanding Loss Development Factors". 2011-10-03.
  8. http://journals.cambridge.org/article_S0515036100015294
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