Estimate age-to-age factors — chainladder • ChainLadder

Basic chain-ladder function to estimate age-to-age factors for a given cumulative run-off triangle. This function is used by Mack- and MunichChainLadder.

Usage

chainladder(Triangle, weights = 1, delta = 1)

Arguments

Triangle

cumulative claims triangle. A (mxn)-matrix \(C_{ik}\) which is filled for \(k \leq n+1-i; i=1,\ldots,m; m\geq n \), see qpaid for how to use (mxn)-development triangles with m<n, say higher development period frequency (e.g quarterly) than origin period frequency (e.g annual).

weights

weights. Default: 1, which sets the weights for all triangle entries to 1. Otherwise specify weights as a matrix of the same dimension as Triangle with all weight entries in [0; 1], where entry \(w_{i,k}\) corresponds to the point \(C_{i,k+1}/C_{i,k}\). Hence, any entry set to 0 or NA eliminates that age-to-age factor from inclusion in the model. See also 'Details'.

delta

'weighting' parameters. Default: 1; delta=1 gives the historical chain-ladder age-to-age factors, delta=2 gives the straight average of the observed individual development factors and delta=0 is the result of an ordinary regression of \(C_{i,k+1}\) against \(C_{i,k}\) with intercept 0, see Barnett & Zehnwirth (2000).

Please note that MackChainLadder uses the argument alpha, with alpha = 2 - delta, following the original paper Mack (1999)

Details

The key idea is to see the chain-ladder algorithm as a special form of a weighted linear regression through the origin, applied to each development period.

Suppose y is the vector of cumulative claims at development period i+1, and x at development period i, weights are weighting factors and F the individual age-to-age factors F=y/x. Then we get the various age-to-age factors:

Basic (unweighted) linear regression through the origin: lm(y~x + 0)
Basic weighted linear regression through the origin: lm(y~x + 0, weights=weights)
Volume weighted chain-ladder age-to-age factors: lm(y~x + 0, weights=1/x)
Simple average of age-to-age factors: lm(y~x + 0, weights=1/x^2)

Barnett & Zehnwirth (2000) use delta = 0, 1, 2 to distinguish between the above three different regression approaches: lm(y~x + 0, weights=weights/x^delta).

Thomas Mack uses the notation alpha = 2 - delta to achieve the same result: sum(weights*x^alpha*F)/sum(weights*x^alpha) # Mack (1999) notation

Value

chainladder returns a list with the following elements:

Models: linear regression models for each development period
Triangle: input triangle of cumulative claims
weights: weights used
delta: deltas used

References

Thomas Mack. The standard error of chain ladder reserve estimates: Recursive calculation and inclusion of a tail factor. Astin Bulletin. Vol. 29. No 2. 1999. pp.361:366

G. Barnett and B. Zehnwirth. Best Estimates for Reserves. Proceedings of the CAS. Volume LXXXVII. Number 167. November 2000.

Author

Markus Gesmann <markus.gesmann@gmail.com>

Examples

## Concept of different chain-ladder age-to-age factors.
## Compare Mack's and Barnett & Zehnwirth's papers.
x <- RAA[1:9,1]
y <- RAA[1:9,2]

F <- y/x
## wtd. average chain-ladder age-to-age factors
alpha <- 1 ## Mack notation
delta <- 2 - alpha ## Barnett & Zehnwirth notation

sum(x^alpha*F)/sum(x^alpha)
#> [1] 2.999359
lm(y~x + 0 ,weights=1/x^delta)
#> 
#> Call:
#> lm(formula = y ~ x + 0, weights = 1/x^delta)
#> 
#> Coefficients:
#>     x  
#> 2.999  
#> 
summary(chainladder(RAA, delta=delta)$Models[[1]])$coef
#>   Estimate Std. Error  t value   Pr(>|t|)
#> x 2.999359   1.130203 2.653822 0.02908283

## straight average age-to-age factors
alpha <- 0
delta <- 2 - alpha 
sum(x^alpha*F)/sum(x^alpha)
#> [1] 8.206099
lm(y~x + 0, weights=1/x^(2-alpha))
#> 
#> Call:
#> lm(formula = y ~ x + 0, weights = 1/x^(2 - alpha))
#> 
#> Coefficients:
#>     x  
#> 8.206  
#> 
summary(chainladder(RAA, delta=delta)$Models[[1]])$coef
#>   Estimate Std. Error  t value   Pr(>|t|)
#> x 8.206099   4.113487 1.994925 0.08115167

## ordinary regression age-to-age factors
alpha=2
delta <- 2-alpha
sum(x^alpha*F)/sum(x^alpha)
#> [1] 2.217241
lm(y~x + 0, weights=1/x^delta)
#> 
#> Call:
#> lm(formula = y ~ x + 0, weights = 1/x^delta)
#> 
#> Coefficients:
#>     x  
#> 2.217  
#> 
summary(chainladder(RAA, delta=delta)$Models[[1]])$coef
#>   Estimate Std. Error  t value     Pr(>|t|)
#> x 2.217241  0.4112176 5.391893 0.0006522995

## Compare different models
CL0 <- chainladder(RAA)
## age-to-age factors
sapply(CL0$Models, function(x) summary(x)$coef["x","Estimate"])
#> [1] 2.999359 1.623523 1.270888 1.171675 1.113385 1.041935 1.033264 1.016936
#> [9] 1.009217
## f.se
sapply(CL0$Models, function(x) summary(x)$coef["x","Std. Error"])
#> [1] 1.130203277 0.135836119 0.090498216 0.025389927 0.035376679 0.022577813
#> [7] 0.004881918 0.015055851         NaN
## sigma
sapply(CL0$Models, function(x) summary(x)$sigma)
#> [1] 166.983470  33.294538  26.295300   7.824960  10.928818   6.389042   1.159062
#> [8]   2.807704        NaN
predict(CL0)
#>       dev
#> origin    1         2         3        4        5        6        7        8
#>   1981 5012  8269.000 10907.000 11805.00 13539.00 16181.00 18009.00 18608.00
#>   1982  106  4285.000  5396.000 10666.00 13782.00 15599.00 15496.00 16169.00
#>   1983 3410  8992.000 13873.000 16141.00 18735.00 22214.00 22863.00 23466.00
#>   1984 5655 11555.000 15766.000 21266.00 23425.00 26083.00 27067.00 27967.34
#>   1985 1092  9565.000 15836.000 22169.00 25955.00 26180.00 27277.85 28185.21
#>   1986 1513  6445.000 11702.000 12935.00 15852.00 17649.38 18389.50 19001.20
#>   1987  557  4020.000 10946.000 12314.00 14428.00 16063.92 16737.55 17294.30
#>   1988 1351  6947.000 13112.000 16663.88 19524.65 21738.45 22650.05 23403.47
#>   1989 3133  5395.000  8758.905 11131.59 13042.60 14521.43 15130.38 15633.68
#>   1990 2063  6187.677 10045.834 12767.13 14958.92 16655.04 17353.46 17930.70
#>       dev
#> origin        9       10
#>   1981 18662.00 18834.00
#>   1982 16704.00 16857.95
#>   1983 23863.43 24083.37
#>   1984 28441.01 28703.14
#>   1985 28662.57 28926.74
#>   1986 19323.01 19501.10
#>   1987 17587.21 17749.30
#>   1988 23799.84 24019.19
#>   1989 15898.45 16044.98
#>   1990 18234.38 18402.44

CL1 <- chainladder(RAA, delta=1)
## age-to-age factors
sapply(CL1$Models, function(x) summary(x)$coef["x","Estimate"])
#> [1] 2.999359 1.623523 1.270888 1.171675 1.113385 1.041935 1.033264 1.016936
#> [9] 1.009217
## f.se
sapply(CL1$Models, function(x) summary(x)$coef["x","Std. Error"])
#> [1] 1.130203277 0.135836119 0.090498216 0.025389927 0.035376679 0.022577813
#> [7] 0.004881918 0.015055851         NaN
## sigma
sapply(CL1$Models, function(x) summary(x)$sigma)
#> [1] 166.983470  33.294538  26.295300   7.824960  10.928818   6.389042   1.159062
#> [8]   2.807704        NaN
predict(CL1)
#>       dev
#> origin    1         2         3        4        5        6        7        8
#>   1981 5012  8269.000 10907.000 11805.00 13539.00 16181.00 18009.00 18608.00
#>   1982  106  4285.000  5396.000 10666.00 13782.00 15599.00 15496.00 16169.00
#>   1983 3410  8992.000 13873.000 16141.00 18735.00 22214.00 22863.00 23466.00
#>   1984 5655 11555.000 15766.000 21266.00 23425.00 26083.00 27067.00 27967.34
#>   1985 1092  9565.000 15836.000 22169.00 25955.00 26180.00 27277.85 28185.21
#>   1986 1513  6445.000 11702.000 12935.00 15852.00 17649.38 18389.50 19001.20
#>   1987  557  4020.000 10946.000 12314.00 14428.00 16063.92 16737.55 17294.30
#>   1988 1351  6947.000 13112.000 16663.88 19524.65 21738.45 22650.05 23403.47
#>   1989 3133  5395.000  8758.905 11131.59 13042.60 14521.43 15130.38 15633.68
#>   1990 2063  6187.677 10045.834 12767.13 14958.92 16655.04 17353.46 17930.70
#>       dev
#> origin        9       10
#>   1981 18662.00 18834.00
#>   1982 16704.00 16857.95
#>   1983 23863.43 24083.37
#>   1984 28441.01 28703.14
#>   1985 28662.57 28926.74
#>   1986 19323.01 19501.10
#>   1987 17587.21 17749.30
#>   1988 23799.84 24019.19
#>   1989 15898.45 16044.98
#>   1990 18234.38 18402.44

CL2 <- chainladder(RAA, delta=2)
## age-to-age factors
sapply(CL2$Models, function(x) summary(x)$coef["x","Estimate"])
#> [1] 8.206099 1.695894 1.314510 1.182926 1.126962 1.043328 1.034355 1.017995
#> [9] 1.009217
## f.se
sapply(CL2$Models, function(x) summary(x)$coef["x","Std. Error"])
#> [1] 4.113487235 0.167616428 0.119849168 0.027269226 0.033389333 0.025122915
#> [7] 0.004953969 0.015093015         NaN
## sigma
sapply(CL2$Models, function(x) summary(x)$sigma)
#> [1] 12.340461705  0.474090850  0.317091093  0.066795689  0.074660819
#> [6]  0.050245830  0.008580526  0.021344747          NaN
predict(CL2)
#>       dev
#> origin    1        2         3        4        5        6        7        8
#>   1981 5012  8269.00 10907.000 11805.00 13539.00 16181.00 18009.00 18608.00
#>   1982  106  4285.00  5396.000 10666.00 13782.00 15599.00 15496.00 16169.00
#>   1983 3410  8992.00 13873.000 16141.00 18735.00 22214.00 22863.00 23466.00
#>   1984 5655 11555.00 15766.000 21266.00 23425.00 26083.00 27067.00 27996.90
#>   1985 1092  9565.00 15836.000 22169.00 25955.00 26180.00 27314.32 28252.71
#>   1986 1513  6445.00 11702.000 12935.00 15852.00 17864.61 18638.64 19278.97
#>   1987  557  4020.00 10946.000 12314.00 14566.55 16415.95 17127.21 17715.62
#>   1988 1351  6947.00 13112.000 17235.86 20388.74 22977.34 23972.89 24796.49
#>   1989 3133  5395.00  9149.351 12026.92 14226.95 16033.23 16727.91 17302.61
#>   1990 2063 16929.18 28710.107 37739.73 44643.30 50311.31 52491.18 54294.53
#>       dev
#> origin        9       10
#>   1981 18662.00 18834.00
#>   1982 16704.00 16857.95
#>   1983 23888.27 24108.44
#>   1984 28500.70 28763.38
#>   1985 28761.12 29026.20
#>   1986 19625.90 19806.78
#>   1987 18034.42 18200.63
#>   1988 25242.70 25475.36
#>   1989 17613.97 17776.31
#>   1990 55271.56 55780.98

## Set 'weights' parameter to use only the last 5 diagonals, 
## i.e. the last 5 calendar years
calPeriods <- (row(RAA) + col(RAA) - 1)
(weights <- ifelse(calPeriods <= 5, 0, ifelse(calPeriods > 10, NA, 1)))
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#>  [1,]    0    0    0    0    0    1    1    1    1     1
#>  [2,]    0    0    0    0    1    1    1    1    1    NA
#>  [3,]    0    0    0    1    1    1    1    1   NA    NA
#>  [4,]    0    0    1    1    1    1    1   NA   NA    NA
#>  [5,]    0    1    1    1    1    1   NA   NA   NA    NA
#>  [6,]    1    1    1    1    1   NA   NA   NA   NA    NA
#>  [7,]    1    1    1    1   NA   NA   NA   NA   NA    NA
#>  [8,]    1    1    1   NA   NA   NA   NA   NA   NA    NA
#>  [9,]    1    1   NA   NA   NA   NA   NA   NA   NA    NA
#> [10,]    1   NA   NA   NA   NA   NA   NA   NA   NA    NA
CL3 <- chainladder(RAA, weights=weights)
summary(CL3$Models[[1]])$coef
#>   Estimate Std. Error  t value   Pr(>|t|)
#> x  3.47986   1.060538 3.281222 0.04638316
predict(CL3)
#>       dev
#> origin    1        2        3        4        5        6        7        8
#>   1981 5012  8269.00 10907.00 11805.00 13539.00 16181.00 18009.00 18608.00
#>   1982  106  4285.00  5396.00 10666.00 13782.00 15599.00 15496.00 16169.00
#>   1983 3410  8992.00 13873.00 16141.00 18735.00 22214.00 22863.00 23466.00
#>   1984 5655 11555.00 15766.00 21266.00 23425.00 26083.00 27067.00 27967.34
#>   1985 1092  9565.00 15836.00 22169.00 25955.00 26180.00 27277.85 28185.21
#>   1986 1513  6445.00 11702.00 12935.00 15852.00 17435.13 18166.26 18770.54
#>   1987  557  4020.00 10946.00 12314.00 14259.49 15683.57 16341.26 16884.83
#>   1988 1351  6947.00 13112.00 16600.64 19223.37 21143.20 22029.83 22762.62
#>   1989 3133  5395.00 10318.43 13063.80 15127.75 16638.55 17336.28 17912.95
#>   1990 2063  7178.95 13730.40 17383.58 20130.01 22140.38 23068.83 23836.18
#>       dev
#> origin        9       10
#>   1981 18662.00 18834.00
#>   1982 16704.00 16857.95
#>   1983 23863.43 24083.37
#>   1984 28441.01 28703.14
#>   1985 28662.57 28926.74
#>   1986 19088.45 19264.38
#>   1987 17170.80 17329.05
#>   1988 23148.14 23361.48
#>   1989 18216.33 18384.22
#>   1990 24239.88 24463.29