chainladder: Estimate age-to-age factors
In edalmoro/ChainLadderQuantileV1: Statistical Methods and Models for Claims Reserving in General Insurance

Description Usage Arguments Details Value Author(s) References See Also Examples

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.

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

`Triangle`	cumulative claims triangle. A (mxn)-matrix C_{ik} which is filled for k ≤q n+1-i; i=1,…,m; m≥q 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 accident years).
`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}.
`delta`	'weighting' parameters, either 0,1 or 2. 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 Mack (1999) used the notation of alphas, with alpha=2-delta.

The key idea is to see the chain ladder algorithm as 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, w are weighting factors and F the individual age-to-age factors F=y/x, than we get the various age-to-age factors for different deltas (alphas) as:

sum(w*x^alpha*F)/sum(w*x^alpha) # Mack (1999) notation

delta <- 2-alpha

lm(y~x + 0 ,weights=w/x^delta) # Barnett & Zehnwirth (2000) notation

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

Markus Gesmann <markus.gesmann@gmail.com>

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,

## 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]

weights <- RAA
weights[!is.na(weights)] <- 1
w <- weights[1:9,1]

F <- y/x
## wtd. average chain ladder age-to-age factors
alpha <- 1
delta <- 2-alpha

sum(w*x^alpha*F)/sum(w*x^alpha)
lm(y~x + 0 ,weights=w/x^delta)
summary(chainladder(RAA, weights=weights, delta=delta)$Models[[1]])$coef

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

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

## Change weights

weights[2,1] <- 0.5
w <- weights[1:9,1] 

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

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

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

## Model review
CL0 <- chainladder(RAA, weights=weights, delta=0)
## 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)

CL1 <- chainladder(RAA, weights=weights, 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)

CL2 <- chainladder(RAA, weights=weights, 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)


## Forecasting

predict(CL0)
predict(CL1)
predict(CL2)