MackChainLadder: Mack-Chain-Ladder Model
In edalmoro/ChainLadderEDM: Statistical Methods and Models for Claims Reserving in General Insurance

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

The Mack-chain-ladder model forecasts future claims developments based on a historical cumulative claims development triangle and estimates the standard error around those.

1 2	MackChainLadder(Triangle, weights = 1, alpha=1, est.sigma="log-linear", tail=FALSE, tail.se=NULL, tail.sigma=NULL, mse.method="Mack", Quantil=FALSE)

`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. Similarly, if Triangle is so small that log-linear regression is being attempted on a vector of only one non-NA average link ratio, 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
`mse.method`	method used for the recursive estimate of the parameter risk component of the mean square error. Value "Mack" (default) coincides with Mack's formula; "Independence" includes the additional cross-product term as in Murphy and BBMW. Refer to References below.
`Quantil`	quantile at which the reserves are estimated. If quantile is not given, 0.5 is used as default. Refer to References (Probability of sufficiency) below for quantile estimation.

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.

In this version, the skewness of the overall reserve risk distribution and of each accident year is provided. For details of the underlying calculations, see https://ssrn.com/abstract=2344297

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
`Total.ProcessRisk`	vector of process risk estimate of the total of projected loss for all origin years combined by development period
`Total.ParameterRisk`	vector of parameter risk estimate of the total of projected loss for all origin years combined by development period
`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
`Skewness`	Skewness of the distribution
`Quantile`	Reserves at the desired quantile using the Cornish-Fisher expansion

Additional references for further reading:

England, PD and Verrall, RJ. Stochastic Claims Reserving in General Insurance (with discussion), British Actuarial Journal 8, III. 2002

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

Murphy, Daniel M. Unbiased Loss Development Factors. Proceedings of the Casualty Actuarial Society Casualty Actuarial Society - Arlington, Virginia 1994: LXXXI 154-222

Buchwalder, Bühlmann, Merz, and Wüthrich. The Mean Square Error of Prediction in the Chain Ladder Reserving Method (Mack and Murphy Revisited). Astin Bulletin Vol. 36. 2006. pp.521:542

Dal Moro, Krvavych. Probability of sufficiency of Solvency II Reserve risk margins: Practical approximations. ASTIN Bulletin, 47(3), 737-785

See also qpaid for dealing with non-square triangles, chainladder for the underlying chain-ladder method, summary.MackChainLadder, plot.MackChainLadder and residuals.MackChainLadder displaying results and finally CDR.MackChainLadder for the one year claims development result.

## 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)

## Tail
## See the example in Mack's 1999 paper
Mortgage
m <- MackChainLadder(Mortgage)
round(summary(m)$Totals["CV(IBNR)",], 2) ## 26% in Table 6 of paper
plot(Mortgage)
# Specifying the tail and its associated uncertainty parameters
MRT <- MackChainLadder(Mortgage, tail=1.05, tail.sigma=71, tail.se=0.02, est.sigma="Mack")
MRT
plot(MRT, lattice=TRUE)
# Specify just the tail and the uncertainty parameters will be estimated
MRT <- MackChainLadder(Mortgage, tail=1.05)
MRT$f.se[9] # close to the 0.02 specified above
MRT$sigma[9] # less than the 71 specified above
# Note that the overall CV dropped slightly
round(summary(MRT)$Totals["CV(IBNR)",], 2) ## 24%
# tail parameter uncertainty equal to expected value 
MRT <- MackChainLadder(Mortgage, tail=1.05, tail.se = .05)
round(summary(MRT)$Totals["CV(IBNR)",], 2) ## 27%

## Parameter-risk (only) estimate of the total reserve = 3142387
tail(MRT$Total.ParameterRisk, 1) # located in last (ultimate) element
#  Parameter-risk (only) CV is about 19%
tail(MRT$Total.ParameterRisk, 1) / summary(MRT)$Totals["IBNR", ]

## Three terms in the parameter risk estimate
## First, the default (Mack) without the tail
m <- MackChainLadder(RAA, mse.method = "Mack")
summary(m)$Totals["Mack S.E.",]
## Then, with the third term
m <- MackChainLadder(RAA, mse.method = "Independence")
summary(m)$Totals["Mack S.E.",] ## Not significantly greater

## One year claims development results
M <- MackChainLadder(MW2014, est.sigma="Mack")
CDR(M)

## For more examples see:
## Not run: 
 demo(MackChainLadder)

## End(Not run)