MackChainLadder {ChainLadder} | R Documentation |
The Mack-chain-ladder model forecasts future claims developments based on a historical cumulative claims development triangle and estimates the standard error around those.
MackChainLadder(Triangle, weights = 1, alpha=1, est.sigma="log-linear", tail=FALSE, tail.se=NULL, tail.sigma=NULL)
Triangle |
cumulative claims triangle. Assume columns are the development
period, use transpose otherwise.
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] |
alpha |
'weighting' parameter.
Default: 1 for all development periods; alpha=1 gives the historical chain
ladder age-to-age factors, alpha=0 gives the straight average of the
observed individual development factors and alpha=2 is the result of
an ordinary regression of C_{i,k+1} against C_{i,k} with
intercept 0, see also Mack's 1999 paper and chainladder |
est.sigma |
defines how to estimate sigma_{n-1}, the variability of the individual age-to-age factors at development time n-1. Default is "log-linear" for a log-linear regression, "Mack" for Mack's approximation from his 1999 paper. Alternatively the user can provide a numeric value. If the log-linear model appears to be inappropriate (p-value > 0.05) the 'Mack' method will be used instead and a warning message printed. |
tail |
can be logical or a numeric value.
If tail=FALSE no tail factor will be applied, if tail=TRUE
a tail factor will be estimated via a linear extrapolation of
log(chain ladder factors - 1), if tail is a numeric
value than this value will be used instead. |
tail.se |
defines how the standard error of the tail factor is estimated. Only needed if
a tail factor > 1 is provided. Default is NULL . If
tail.se is NULL , tail.se is estimated via
"log-linear" regression, if tail.se is a numeric
value than this value will be used instead. |
tail.sigma |
defines how to estimate individual tail variability. Only needed if
a tail factor > 1 is provided.
Default is NULL . If tail.sigma is NULL ,
tail.sigma
is estimated via "log-linear" regression, if tail.sigma is a numeric
value than this value will be used instead |
Following Mack's 1999 paper let C_{ik} denote the cumulative loss amounts of origin period (e.g. accident year) i=1,...,m, with losses known for development period (e.g. development year) k <= n+1-i. In order to forecast the amounts C_{ik} for k > n+1-i the Mack chain-ladder-model assumes:
CL1: E[ F_ik| C_i1,C_i2,...,C_ik ] = f_k with F_ik=C_{i,k+1}/C_ik
CL2: Var( C_{i,k+1}/C_ik | C_i1, C_i2, ... ,C_ik ) = sigma_k^2/( w_ik C^alpha_ik)
CL3: { C_i1, ... ,C_in}, { C_j1, ... ,C_jn}, are independent for origin period i != j
with w_{ik} in [0;1], alpha in {0,1,2}. If these assumptions are hold, the Mack-chain-ladder-model gives an unbiased estimator for IBNR (Incurred But Not Reported) claims.
The Mack-chain-ladder model can be regarded as a weighted linear regression
through the origin for each development period:
lm(y ~ x + 0, weights=w/x^(2-alpha))
,
where y
is the vector of claims at development period
k+1 and x
is the vector of claims at development period
k.
MackChainLadder returns a list with the following elements
call |
matched call |
Triangle |
input triangle of cumulative claims |
FullTriangle |
forecasted full triangle |
Models |
linear regression models for each development period |
f |
chain-ladder age-to-age factors |
f.se |
standard errors of the chain-ladder age-to-age factors f (assumption CL1) |
F.se |
standard errors of the true chain-ladder age-to-age factors F_{ik} (square root of the variance in assumption CL2) |
sigma |
sigma parameter in CL2 |
Mack.ProcessRisk |
variability in the projection of future losses not explained by the variability of the link ratio estimators (unexplained variation) |
Mack.ParameterRisk |
variability in the projection of future losses explained by the variability of the link-ratio estimators alone (explained variation) |
Mack.S.E |
total variability in the projection of future losses by the chain ladder method; the square root of the mean square error of the chain ladder estimate: Mack.S.E.^2 = Mack.ProcessRisk^2 + Mack.ParameterRisk^2 |
Total.Mack.S.E |
total variability of projected loss for all origin years combined |
weights |
weights used. |
alpha |
alphas used. |
tail |
tail factor used. If tail was set to TRUE the output will include the linear model used to estimate the tail factor |
Additional references for further reading:
England, PD and Verrall, RJ. Stochastic Claims Reserving in General Insurance (with discussion), British Actuarial Journal 8, III. 2002
Murphy, Daniel M. Unbiased Loss Development Factors. Proceedings of the Casualty Actuarial Society Casualty Actuarial Society - Arlington, Virginia 1994: LXXXI 154-222.
Barnett and Zehnwirth. Best estimates for reserves. Proceedings of the CAS, LXXXVI I(167), November 2000.
Markus Gesmann markus.gesmann@gmail.com
Thomas Mack. Distribution-free calculation of the standard error of chain ladder reserve estimates. Astin Bulletin. Vol. 23. No 2. 1993. pp.213:225
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
See also qpaid
,
chainladder
,
summary.MackChainLadder
,
plot.MackChainLadder
,
residuals.MackChainLadder
,
MunichChainLadder
,
BootChainLadder
,
# See the Taylor/Ashe example in Mack's 1993 paper GenIns plot(GenIns) plot(GenIns, lattice=TRUE) GNI <- MackChainLadder(GenIns, est.sigma="Mack") GNI$f GNI$sigma^2 GNI # compare to table 2 and 3 in Mack's 1993 paper plot(GNI) plot(GNI, lattice=TRUE) # Different weights # Using alpha=0 will use straight average age-to-age factors MackChainLadder(GenIns, alpha=0)$f # You get the same result via: apply(GenIns[,-1]/GenIns[,-10],2, mean, na.rm=TRUE) # See the example in Mack's 1999 paper Mortgage plot(Mortgage) MRT <- MackChainLadder(Mortgage, tail=1.05, tail.sigma=71, tail.se=0.02, est.sigma="Mack") MRT plot(MRT, lattice=TRUE) # Table 1 in the above paper f <- c(11.10, 4.092, 1.708, 1.276, 1.139, 1.069, 1.026, 1.023, 1.05) f.se <- c(2.24, 0.517, 0.122, 0.051, 0.042, 0.023, 0.015, 0.012, 0.02) F.se3 <- c(7.38, 1.89, 0.357, 0.116, 0.078, 0.033, 0.015, 0.007, 0.03) sig <- c(1337, 988.5, 440.1, 207, 164.2, 74.60, 35.49, 16.89,71) # test output from MackChainLadder MRT$f MRT$f.se MRT$F.se[3,] MRT$sigma plot(MRT) # We observe trends along calendar years. # Table 2 in the above paper MRT$FullTriangle[,9]/1000 ## C_{i9} MRT$FullTriangle[,10]/1000 ## C_{i,ult} MRT$Mack.S.E[,9]/1000 ## s.e.(C_{i9}) # Access process risk error MRT$Mack.ProcessRisk # Access parameter risk error MRT$Mack.ParameterRisk # Total risk MRT$Mack.S.E op <- par(mfrow=c(2,1)) plot(with(summary(MRT)$ByOrigin, Mack.S.E/Ultimate),t="l", ylab="CV(Ultimate)", xlab="origin period") plot(summary(MRT)$ByOrigin[["CV(IBNR)"]], t="l", ylab="CV(IBNR)", xlab="origin period") par(op) # This data set is discussed in many papers, e.g. England and Verrall (2000), # see Table 1 just there RAA plot(RAA) R <- MackChainLadder(RAA) R plot(R) plot(R, lattice=TRUE) # Table 12 in England and Verrall (2000) R$f R$sigma^2 # Table 13 in England and Verrall (2000) # Please note the different indexing of sigma MackChainLadder(RAA, est.sigma=R$sigma[7]) # Table 14 in England and Verrall (2000) MackChainLadder(RAA, est.sigma=R$sigma[8]) # Let's investigate the Mack model in more detail R[["Models"]][[1]] # Model for first development period summary( R[["Models"]][[1]]) # Look at the model stats op <- par(mfrow=c(2,2)) # plot residuals plot( R[["Models"]][[1]]) par(op) # Let's include an intercept in our model newModel <- update(R[["Models"]][[1]], y ~ x+1, weights=1/R[["Triangle"]][1:9,1], data=data.frame(x=R[["Triangle"]][1:9,1], y=R[["Triangle"]][1:9,2]) ) # View the new model summary(newModel) op <- par(mfrow=c(2,2)) plot( newModel ) par(op) # Change the model for dev. period one to the newModel R2 <- R R2[["Models"]][[1]] <- newModel predict(R2) # predict the full triangle with the new model #(only the last origin year will be affected) R2[["FullTriangle"]] <- predict(R2) R2[["FullTriangle"]] R2 # Std. Errors have not been re-estimated! # Plot the result plot(R2, title="Changed R Model") ## Suppose you have a long table with claims development by line of ## business and would like to apply the MackChainLadder on all triangles ## in one go. ## First lets create a table similar to what you would get from a 'real' data base myList <- list("General Liabilty" = RAA/1e3, "General Insurance" = GenIns/1e3, "Workers Comp"=ABC/1e3, "Mortgage Guarantee"=Mortgage/1e3) myData <- do.call("rbind" , lapply(names(myList), function(x) as.data.frame(myList[[x]],lob=x ,na.rm=TRUE))) ## Let's plot a nice summary, but first lets normalise the origin years myData <- do.call("rbind", by(myData, list(lob=myData$lob), function(x) {org=as.numeric(as.character(x$origin)) x$origin <- org-min(org)+2000;x} )) rownames(myData) <- NULL head(myData) ## Does this look familiar? ;-) xyplot(value ~ dev | lob, groups=factor(origin), data=myData, t="l", scales="free", auto.key=list(space="right", points=FALSE, lines=TRUE)) ## Lets create triangles again and apply MackChainLadder for each lob: myResults <- by(myData, list(lob=myData$lob), function(x) MackChainLadder(as.triangle(x), est.sigma="Mack")) ## That's it, lets look at the output myResults ## Summarise all results by origin period in one data frame: by.origin <- function(x){ data.frame(lob=x, origin=dimnames(myResults[[x]]$Triangle)$origin, summary(myResults[[x]])$ByOrigin) } ByOrigin <-do.call("rbind", lapply(names(myResults) , by.origin)) ByOrigin ## Similar for the totals Totals <- do.call("rbind", lapply(names(myResults) , function(x) data.frame(LOB=x, t(summary(myResults[[x]])$Totals)))) Totals require(lattice) barchart(Latest + IBNR ~ factor(origin) | lob, stack=TRUE, data=ByOrigin, scale="free", auto.key=TRUE, as.table=TRUE, xlab="origin")