chainladder {ChainLadder} | R Documentation |

## 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 |

`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 |

`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 Please note that |

### 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

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
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)
```

*ChainLadder*version 0.2.18 Index]