## 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)


### 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

### 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

### Author(s)

Markus Gesmann <markus.gesmann@gmail.com>

### 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 ata, predict.ChainLadder MackChainLadder,

### 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)
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
## 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)

## 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)

## 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)))
summary(CL3$Models[[1]])$coef
predict(CL3)