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ChainLadder (version 0.2.19)

chainladder: Estimate age-to-age factors

Description

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)

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

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)

Author

Markus Gesmann <markus.gesmann@gmail.com>

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

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.

See Also

See also ata, predict.ChainLadder MackChainLadder,

Examples

Run this code
## 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)
lm(y~x + 0 ,weights=1/x^delta)
summary(chainladder(RAA, delta=delta)$Models[[1]])$coef

## straight average age-to-age factors
alpha <- 0
delta <- 2 - alpha 
sum(x^alpha*F)/sum(x^alpha)
lm(y~x + 0, weights=1/x^(2-alpha))
summary(chainladder(RAA, delta=delta)$Models[[1]])$coef

## ordinary regression age-to-age factors
alpha=2
delta <- 2-alpha
sum(x^alpha*F)/sum(x^alpha)
lm(y~x + 0, weights=1/x^delta)
summary(chainladder(RAA, delta=delta)$Models[[1]])$coef

## Compare different models
CL0 <- chainladder(RAA)
## age-to-age factors
sapply(CL0$Models, function(x) summary(x)$coef["x","Estimate"])
## f.se
sapply(CL0$Models, function(x) summary(x)$coef["x","Std. Error"])
## sigma
sapply(CL0$Models, function(x) summary(x)$sigma)
predict(CL0)

CL1 <- chainladder(RAA, delta=1)
## age-to-age factors
sapply(CL1$Models, function(x) summary(x)$coef["x","Estimate"])
## f.se
sapply(CL1$Models, function(x) summary(x)$coef["x","Std. Error"])
## sigma
sapply(CL1$Models, function(x) summary(x)$sigma)
predict(CL1)

CL2 <- chainladder(RAA, delta=2)
## age-to-age factors
sapply(CL2$Models, function(x) summary(x)$coef["x","Estimate"])
## f.se
sapply(CL2$Models, function(x) summary(x)$coef["x","Std. Error"])
## sigma
sapply(CL2$Models, function(x) summary(x)$sigma)
predict(CL2)

## 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)))
CL3 <- chainladder(RAA, weights=weights)
summary(CL3$Models[[1]])$coef
predict(CL3)

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