Package 'ChainLadder'

Description

ChainLadder provides methods and models which are typically used in insurance claims reserving.

The package grew out of presentations given at the Stochastic Reserving Seminar at the Institute of Actuaries in 2007 and 2008 and followed by talks at CAS meetings in 2008 and 2010.

More information is available on the project web site https://github.com/mages/ChainLadder

For more financial packages see also CRAN Task View 'Emperical Finance' at https://CRAN.R-project.org/view=Finance.

Author(s)

Maintainer: Markus Gesmann <[email protected]>

References

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

Gerhard Quarg and Thomas Mack. Munich Chain Ladder. Blatter DGVFM 26. Munich. 2004.

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

B. Zehnwirth and G. Barnett. Best Estimates for Reserves. Proceedings of the CAS. Volume LXXXVII. Number 167.November 2000.

Clark, David R., "LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach," CAS Forum, Fall 2003.

Zhang Y. A general multivariate chain ladder model.Insurance: Mathematics and Economics, 46, pp. 588:599, 2010.

Zhang, Y. Likelihood-based and Bayesian Methods for Tweedie Compound Poisson Linear Mixed Models, Statistics and Computing, forthcoming.

Bardis, Majidi, Murphy. A Family of Chain-Ladder Factor Models for Selected Link Ratios. Variance. Pending. Variance 6:2, 2012, pp. 143-160.

Modelling the claims development result for solvency purposes. Michael Merz, Mario V. Wüthrich. Casualty Actuarial Society E-Forum, Fall 2008.

Claims Run-Off Uncertainty: The Full Picture. Michael Merz, Mario V. Wüthrich. Swiss Finance Institute Research Paper No. 14-69. https://www.ssrn.com/abstract=2524352. 2014

Markus Gesmann. Claims Reserving and IBNR. Computational Actuarial Science with R. Chapman and Hall/CRC. 2014

Examples

## Not run: 
  demo(ChainLadder)
  
## End(Not run)

Description

Run-off triangle of a worker's compensation portfolio of a large company

Usage

data(ABC)

Format

A matrix with 11 accident years and 11 development years.

Source

B. Zehnwirth and G. Barnett. Best Estimates for Reserves. Proceedings of the CAS. Volume LXXXVII. Number 167. November 2000.

Examples

ABC
plot(ABC)
plot(ABC, lattice=TRUE)

Description

Given a Triangle in matrix ("wide") format, convert to data.frame ("long") format.

Usage

as.LongTriangle(Triangle, varnames = names(dimnames(Triangle)), 
                value.name = "value", na.rm = TRUE)

Arguments

Details

Unlike the as.data.frame.triangle method, and Unlike the 'melt' method in the 'reshape2' package, this function returns a data.frame where the rownames and colnames of Triangle are stored as factors. This can be a critical feature when the order of the levels of the columns is important. For example, when a Triangle is plotted, the order of the origin and dev dimensions is important. See Examples section.

Value

A data.frame.

Author(s)

Daniel Murphy

See Also

as.data.frame.triangle

Examples

as.LongTriangle(GenIns)
## Not run: 
ggplot(as.LongTriangle(GenIns), 
       aes(x = dev, y = value, group = origin, color = origin)) + geom_line()

## End(Not run)

Description

Calculate the matrix of age-to-age factors (also called "report-to-report" factors, or "link ratios") for an object of class triangle.

Usage

ata(Triangle, NArow.rm = TRUE, colname.sep = "-",
        colname.order=c("ascending","descending"))

Arguments

Details

ata constructs a matrix of age-to-age (ata) factors resulting from a loss "triangle" or a matrix. Simple averages and volume weighted averages are saved as "smpl" and "vwtd" attributes, respectively.

Value

A matrix with "smpl" and "vwtd" attributes.

Author(s)

Daniel Murphy

See Also

summary.ata, print.ata and chainladder

Examples

ata(GenIns)

# Volume weighted average age-to-age factor of the "RAA" data
y <- attr(ata(RAA), "vwtd")
y
# "To ultimate" factors with a 10% tail
y <- rev(cumprod(rev(c(y, 1.1))))
names(y) <- paste(colnames(RAA), "Ult", sep="-")
y

## Label the development columns in "ratio-type" format
ata(RAA, colname.sep=":", colname.order="desc")

Description

Run-off triangles of Personal Auto and Commercial Auto insurance.

Usage

data(auto)

Format

A list of three matrices, paid Personal Auto, incurred Personal Auto and paid Commercial Auto respectively.

Source

Zhang (2010). A general multivariate chain ladder model. Insurance: Mathematics and Economics, 46, pp. 588-599.

Examples

data(auto)
names(auto)

Description

Run-off triangles of Automobile Bodily Injury Liability.

Usage

data(AutoBI)

Format

Portfolio of automobile bodily injury liability for an experience period of 1969 to 1976. Paid Claims, Closed Claims and Reported Claim Counts respectively

Source

Berquist, J.R. and Sherman, R.E., Loss Reserve Adequacy Testing: A Comprehensive, Systematic Approach, Proceedings of the Casualty Actuarial Society, LXIV, 1977, pp.123-184.

Examples

data(AutoBI)
names(AutoBI)
AutoBI$AutoBIPaid
AutoBI$AutoBIClosed
AutoBI$AutoBIReportedCounts

Description

The BootChainLadder procedure provides a predictive distribution of reserves or IBNRs for a cumulative claims development triangle.

Usage

BootChainLadder(Triangle, R = 999, process.distr=c("gamma", "od.pois"), seed = NULL)

Arguments

Details

The BootChainLadder function uses a two-stage bootstrapping/simulation approach. In the first stage an ordinary chain-ladder methods is applied to the cumulative claims triangle. From this we calculate the scaled Pearson residuals which we bootstrap R times to forecast future incremental claims payments via the standard chain-ladder method. In the second stage we simulate the process error with the bootstrap value as the mean and using the process distribution assumed. The set of reserves obtained in this way forms the predictive distribution, from which summary statistics such as mean, prediction error or quantiles can be derived.

Value

BootChainLadder gives a list with the following elements back:

Note

The implementation of BootChainLadder follows closely the discussion of the bootstrap model in section 8 and appendix 3 of the paper by England and Verrall (2002).

Author(s)

Markus Gesmann, [email protected]

References

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

Barnett and Zehnwirth. The need for diagnostic assessment of bootstrap predictive models, Insureware technical report. 2007

See Also

See also summary.BootChainLadder, plot.BootChainLadder displaying results and finally CDR.BootChainLadder for the one year claims development result.

Examples

# See also the example in section 8 of England & Verrall (2002) on page 55.

B <- BootChainLadder(RAA, R=999, process.distr="gamma")
B
plot(B)
# Compare to MackChainLadder
MackChainLadder(RAA)
quantile(B, c(0.75,0.95,0.99, 0.995))

# fit a distribution to the IBNR
library(MASS)
plot(ecdf(B$IBNR.Totals))
# fit a log-normal distribution 
fit <- fitdistr(B$IBNR.Totals[B$IBNR.Totals>0], "lognormal")
fit
curve(plnorm(x,fit$estimate["meanlog"], fit$estimate["sdlog"]), col="red", add=TRUE)

# See also the ABC example in  Barnett and Zehnwirth (2007) 
A <- BootChainLadder(ABC, R=999, process.distr="gamma")
A
plot(A, log=TRUE)

## One year claims development result
CDR(A)

Description

The B-S Paid Claim Development Adjustment methods adjusts paid claims based on the underlying relation between paid and closed claims.

Usage

BS.paid.adj(Triangle.rep.counts = NULL, Triangle.closed, Triangle.paid, 
            ult.counts = NULL, regression.type = "exponential")

Arguments

Details

The importance of recognizing the impact of shifts in the rate of settlement of claims upon historical paid loss data can materially affect the ultimate projections.

This functions adjusts the paid claims based on the numerical method described in the B-S paper.

Berquist and Sherman presented a technique to adjust the paid claim development method for changes in settlement rates. The first step of the paid claims adjustment is to determine the disposal rates by accident year and maturity.

The disposal rate is defined as as the cumulative closed claim counts for each accident year-maturity age cell divided by the selected ultimate claim count for the particular accident year.

If ultimate claim counts have been provided, they will be used to calulate the disposal rates, otherwise ultimate claim counts will be estimated from the cumulative reported claim counts triangle with a standard development method.

The disposal rates along the latest diagonal will be selected as the basis for adjusting the closed claim count triangle, The selected disposal rate for each maturity are multiplied by the ultimate number of claims to determine the adjusted triangle of closed claim counts.

Berquist and Sherman then use regression analysis to identify a mathematical formula that approximates the relationship between the cumulative number of closed claims (X) and cumulative paid claims (Y). The algorithm gives the possibility, through the choice of the 'regression.type' field, to fit an exponential model, $Y = a*e^(bX)$, or a linear model, $Y = a+b*X$.

The relation is estimated based on unadjusted closed claim counts and unadjusted paid claims. Once the regression coefficients are estimated, they will be used to adjust paid claims based on such coefficients and the adjusted closed claim counts triangle.

Value

BS.paid.adj returns the adjusted paid claim triangle

Author(s)

Marco De Virgilis [email protected]

References

Berquist, J.R. and Sherman, R.E., Loss Reserve Adequacy Testing: A Comprehensive, Systematic Approach, Proceedings of the Casualty Actuarial Society, LXIV, 1977, pp.123-184.

See Also

See also qpaid for dealing with non-square triangles, inflateTriangle to inflate a triangle based on an inflation rate,

Examples

# Adjust the Triangle of Paid Claims based on Reported Claim Counts

adj_paid <- BS.paid.adj( Triangle.rep.counts = AutoBI$AutoBIReportedCounts,
                         Triangle.closed = AutoBI$AutoBIClosed,
                         Triangle.paid = AutoBI$AutoBIPaid,
                         regression.type = 'exponential' )

# Calculate the IBNR from the standard unadjusted Paid Triangle

std_ibnr <- summary(MackChainLadder(AutoBI$AutoBIPaid))$Totals[4, 1]

# Calculate the IBNR from the adjusted Paid Triangle

adj_ibnr <- summary(MackChainLadder(adj_paid))$Totals[4, 1]

# Compare the two

adj_ibnr
std_ibnr

## For more examples see:
## Not run: 
 demo(BS.paid.adj)

## End(Not run)

Description

Standard deviation of the claims development result after one year for the distribution-free chain-ladder model (Mack) and Bootstrap model.

Usage

CDR(x, ...)
## S3 method for class 'MackChainLadder'
CDR(x, dev=1, ...)
## S3 method for class 'BootChainLadder'
CDR(x, probs=c(0.75, 0.95), ...)
## Default S3 method:
CDR(x, ...)

Arguments

Details

Merz & Wüthrich (2008) derived analytic formulae for the mean square error of prediction of the claims development result for the Mack chain-ladder model after one year assuming:

• The opening reserves were set using the pure chain-ladder model (no tail)

• Claims develop in the year according to the assumptions underlying Mack's model

• Reserves are set after one year using the pure chain-ladder model (no tail)

Value

A data.frame with various IBNR/reserves and one-year statistics of the claims development result.

Note

Tail factors are currently not supported.

Author(s)

Mario Wüthrich and Markus Gesmann with contributions from Arthur Charpentier and Arnaud Lacoume for CDR.MackChainLadder and Giuseppe Crupi and Markus Gesmann for CDR.BootChainLadder.

References

Michael Merz, Mario V. Wüthrich. Modelling the claims development result for solvency purposes. Casualty Actuarial Society E-Forum, Fall 2008.

Michael Merz, Mario V. Wüthrich. Claims Run-Off Uncertainty: The Full Picture. Swiss Finance Institute Research Paper No. 14-69. https://www.ssrn.com/abstract=2524352. 2014

See Also

See also MackChainLadder and BootChainLadder

Examples

# Example from the 2008 Merz, Wuthrich paper mentioned above
MW2008
M <- MackChainLadder(MW2008, est.sigma="Mack")
plot(M)
CDR(M)
# Return all run-off result developments
CDR(M, dev="all")

# Example from the 2014 Merz, Wuthrich paper mentioned above
MW2014
W <- MackChainLadder(MW2014, est.sigma="Mack")
plot(W)
CDR(W)

# Example with the BootChainLadder function, assuming overdispered Poisson model
B <- BootChainLadder(MW2008, process.distr=c("od.pois"))
B
CDR(B)

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

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:

Author(s)

Markus Gesmann <[email protected]>

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)

Description

Check for Year-on-Year Inflation rates down the columns of a run-off triangle

Usage

checkTriangleInflation(Triangle)

Arguments

Details

The sensitivity of projections of ultimate losses based on incurred loss development factors to changes in the adequacy level of case reserves increases significantly for the long-tail lines. In particular, if the adequacy of the case reserve is changing, the estimates of ultimate losses based on reported claims could be severely distorted.

The function fits an exponential inflation model that takes the form of:

$Y=a*(1+b)^x$

where $Y$ represents the inflated claim amount, $a$ represents the claim amount at the beginning of each period (e.g. AY=0), $b$ is the inflation rate and $x$ is the time (e.g. AY).

Fitting such a model on the average level of the case outstanding (or any other average claim amount) for each development period, it is possible to appreciate the inflation rate that has affected the average case reserve.

It is necessary to check the inflation on average amounts, otherwise the estimates may be distorted due to an increase in the number of claims rather than an actual increase in the inflation level.

If the level of inflation is material, it would be necessary to adjust each cell in the triangle.
This is to to have each diagonal in the triangle at the same level as the latest diagonal (i.e. latest valuation). This adjustment would prevent distortions in the estimates caused by inflation and not by actual variations in the claim experience.

Value

checkTriangleInflation returns a list with the following elements

Author(s)

Marco De Virgilis [email protected]

References

Berquist, J.R. and Sherman, R.E., Loss Reserve Adequacy Testing: A Comprehensive, Systematic Approach, Proceedings of the Casualty Actuarial Society, LXIV, 1977, pp.123-184.

See Also

See also qpaid for dealing with non-square triangles, inflateTriangle to inflate a triangle based on an inflation rate,

Examples

# Create a triangle of average outstanding claims as the ratio between O/S Claims 
# and Open Claims (i.e. the number of outstanding claims)
avg <- MedMal$MedMalOutstanding / MedMal$MedMalOpen

# Check the level of average inflation Y-o-Y
test<-checkTriangleInflation(avg)

# Plot the results
# A model of exponential inflation fits quite well the level of average O/S claims
# This is particularly evident for DP 1,2,3
plot(test)

# Get the summary in an analytical way to observe the ratios and the number of points used
summary(test)

# Print the output
print(test) 
# There is an inflation level equal to .15 at the first development period. It would be 
# appropriate to adjust the triangle before proceeding with any estimate method.

Description

Analyze loss triangle using Clark's Cape Cod method.

Usage

ClarkCapeCod(Triangle, Premium, cumulative = TRUE, maxage = Inf, 
        adol = TRUE, adol.age = NULL, origin.width = NULL,
        G = "loglogistic")

Arguments

Details

Clark's "Cape Cod" method assumes that the incremental losses across development periods in a loss triangle are independent. He assumes that the expected value of an incremental loss is equal to the theoretical expected loss ratio (ELR) times the on-level premium for the origin year times the change in the theoretical underlying growth function over the development period. Clark models the growth function, also called the percent of ultimate, by either the loglogistic function (a.k.a., "the inverse power curve") or the weibull function. Clark completes his incremental loss model by wrapping the expected values within an overdispersed poisson (ODP) process where the "scale factor" sigma^2 is assumed to be a known constant for all development periods.

The parameters of Clark's "Cape Cod" method are therefore: ELR, and omega and theta (the parameters of the loglogistic and weibull growth functions). Finally, Clark uses maximum likelihood to parameterize his model, uses the ODP process to estimate process risk, and uses the Cramer-Rao theorem and the "delta method" to estimate parameter risk.

Clark recommends inspecting the residuals to help assess the reasonableness of the model relative to the actual data (see plot.clark below).

Value

A list of class "ClarkLDF" with the components listed below. ("Key" to naming convention: all caps represent parameters; mixed case represent origin-level amounts; all-lower-case represent observation-level (origin, development age) results.)

Author(s)

Daniel Murphy

References

Clark, David R., "LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach", Casualty Actuarial Society Forum, Fall, 2003 https://www.casact.org/sites/default/files/database/forum_03fforum_03ff041.pdf

See Also

ClarkLDF

Examples

X <- GenIns
colnames(X) <- 12*as.numeric(colnames(X))
CC.loglogistic  <- ClarkCapeCod(X, Premium=10000000+400000*0:9, maxage=240)
CC.loglogistic

# Clark's "CapeCod method" also works with triangles that have  
# more development periods than origin periods. The Premium
# is a contrived match to the "made up" 'qincurred' Triangle.
ClarkCapeCod(qincurred, Premium=1250+150*0:11, G="loglogistic")

# Method also works for a "triangle" with only one row:
# 1st row of GenIns; need "drop=FALSE" to avoid becoming a vector.
ClarkCapeCod(GenIns[1, , drop=FALSE], Premium=1000000, maxage=20)

# If one value of Premium is appropriate for all origin years
# (e.g., losses are on-level and adjusted for exposure)
# then only a single value for Premium need be provided.
ClarkCapeCod(GenIns, Premium=1000000, maxage=20)

# Use of the weibull function generates a warning that the parameter risk 
# approximation results in some negative variances. This may be of small 
# concern since it happens only for older years with near-zero 
# estimated reserves, but the warning should not be disregarded 
# if it occurs with real data.
Y <- ClarkCapeCod(qincurred, Premium=1250+150*0:11, G="weibull")

# The plot of the standardized residuals by age indicates that the more
# mature observations are more loosely grouped than the less mature, just
# the opposite of the behavior under the loglogistic curve.
# This suggests that the model might be improved by analyzing the Triangle 
# in two different "blocks": less mature vs. more mature. 
# The QQ-plot shows that the tails of the empirical distribution of
# standardized residuals are "fatter" than a standard normal. 
# The fact that the p-value is essentially zero says that there is 
# virtually no chance that the standardized residuals could be 
# considered draws from a standard normal random variable.
# The overall conclusion is that Clark's ODP-based CapeCod model with 
# the weibull growth function does not match up well with the qincurred 
# triangle and these premiums.
plot(Y)

Description

Analyze loss triangle using Clark's LDF (loss development factor) method.

Usage

ClarkLDF(Triangle, cumulative = TRUE, maxage = Inf, 
        adol = TRUE, adol.age = NULL, origin.width = NULL,
        G = "loglogistic")

Arguments

Details

Clark's "LDF method" assumes that the incremental losses across development periods in a loss triangle are independent. He assumes that the expected value of an incremental loss is equal to the theoretical expected ultimate loss (U) (by origin year) times the change in the theoretical underlying growth function over the development period. Clark models the growth function, also called the percent of ultimate, by either the loglogistic function (a.k.a., "the inverse power curve") or the weibull function. Clark completes his incremental loss model by wrapping the expected values within an overdispersed poisson (ODP) process where the "scale factor" sigma^2 is assumed to be a known constant for all development periods.

The parameters of Clark's "LDF method" are therefore: U, and omega and theta (the parameters of the loglogistic and weibull growth functions). Finally, Clark uses maximum likelihood to parameterize his model, uses the ODP process to estimate process risk, and uses the Cramer-Rao theorem and the "delta method" to estimate parameter risk.

Clark recommends inspecting the residuals to help assess the reasonableness of the model relative to the actual data (see plot.clark below).

Value

A list of class "ClarkLDF" with the components listed below. ("Key" to naming convention: all caps represent parameters; mixed case represent origin-level amounts; all-lower-case represent observation-level (origin, development age) results.)

Author(s)

Daniel Murphy

References

Clark, David R., "LDF Curve-Fitting and Stochastic Reserving: A Maximum Likelihood Approach", Casualty Actuarial Society Forum, Fall, 2003 https://www.casact.org/sites/default/files/database/forum_03fforum_03ff041.pdf

See Also

ClarkCapeCod

Examples

X <- GenIns
ClarkLDF(X, maxage=20)

# Clark's "LDF method" also works with triangles that have  
# more development periods than origin periods
ClarkLDF(qincurred, G="loglogistic")

# Method also works for a "triangle" with only one row:
# 1st row of GenIns; need "drop=FALSE" to avoid becoming a vector.
ClarkLDF(GenIns[1, , drop=FALSE], maxage=20)

# The age of the first evaluation may be prior to the end of the origin period.
# Here the ages are in units of "months" and the first evaluation 
# is at the end of the third quarter.
X <- GenIns
colnames(X) <- 12 * as.numeric(colnames(X)) - 3
# The indicated liability increases from 1st example above, 
# but not significantly.
ClarkLDF(X, maxage=240)
# When maxage is infinite, the phase shift has a more noticeable impact:
# a 4-5% increase of the overall CV.
x <- ClarkLDF(GenIns, maxage=Inf)
y <- ClarkLDF(X, maxage=Inf)
# Percent change in the bottom line CV:
(tail(y$Table65$TotalCV, 1) - tail(x$Table65$TotalCV, 1)) / tail(x$Table65$TotalCV, 1)

Description

This function finds the values of delta, one for each development period, such that the model coefficients resulting from the 'chainladder' function with delta = solution are consistent with an input vector of 'selected' development age-to-age factors.

Usage

CLFMdelta(Triangle, selected, tolerance = .0005, ...)

Arguments

Details

For a given input Triangle and vector of selected factors, a search is conducted for chainladder models that are "consistent with" the selected factors. By "consistent with" is meant that the coefficients of the chainladder function are equivalent to the selected factors. Not all vectors of selected factors can be considered consistent with a given Triangle; feasibility is subject to restrictions on the 'selected' factors relative to the input 'Triangle'. See the References.

The default average produced by the chainladder function is the volume weighted average and corresponds to delta = 1 in the call to that function; delta = 2 produces the simple average; and delta = 0 produces the "regression average", i.e., the slope of a regression line fit to the data and running through the origin. By convention, if the selected value corresponds to the volume-weighted average, the simple average, or the regression average, then the value returned will be 1, 2, and 0, respectively.

Other real-number values for delta will produce a different average. The point of this function is to see if there exists a model as determined by delta whose average is consistent with the value in the selected vector. That is not always possible. See the References.

It can be the case that one or more of the above three "standard averages" will be quite close to each other (indistinguishable within tolerance). In that case, the value returned will be, in the following priority order by convention, 1 (volume weighted average), 2 (simple average), or 0 (regression average).

Value

A vector of real numbers, say delta0, such that coef(chainladder(Triangle, delta = delta0)) = selected within tolerance. A logical attribute 'foundSolution' indicates if a solution was found for each element of selected.

Author(s)

Dan Murphy

References

Bardis, Majidi, Murphy. A Family of Chain-Ladder Factor Models for Selected Link Ratios. Variance. Pending. Variance 6:2, 2012, pp. 143-160.

Examples

x <- RAA[1:9,1]
y <- RAA[1:9,2]
F <- y/x
CLFMdelta(RAA[1:9, 1:2], selected = mean(F)) # value is 2, 'foundSolution' is TRUE
CLFMdelta(RAA[1:9, 1:2], selected = sum(y) / sum(x)) # value is 1, 'foundSolution' is TRUE

selected <- mean(c(mean(F), sum(y) / sum(x))) # an average of averages
CLFMdelta(RAA[1:9, 1:2], selected) # about 1.725
CLFMdelta(RAA[1:9, 1:2], selected = 2) # negative solutions are possible

# Demonstrating an "unreasonable" selected factor.
CLFMdelta(RAA[1:9, 1:2], selected = 1.9) # NA solution, with warning

Description

Extract residuals of a MackChainLadder model by origin-, calendar- and development period.

Usage

## S3 method for class 'ChainLadder'
coef(object, ...)

Arguments

Value

The function returns a vector of the single-parameter coefficients – also called age-to-age (ATA) or report-to-report (RTR) factors – of the models produced by running the 'chainladder' function.

Author(s)

Dan Murphy

See Also

See Also chainladder

Examples

coef(chainladder(RAA))

Description

Functions to convert between cumulative and incremental triangles

Usage

incr2cum(Triangle, na.rm=FALSE)
cum2incr(Triangle)

Arguments

Details

incr2cum transforms an incremental triangle into a cumulative triangle, cum2incr provides the reserve operation.

Value

Both functions return a triangle.

Author(s)

Markus Gesmann, Christophe Dutang

See Also

See also as.triangle

Examples

# See the Taylor/Ashe example in Mack's 1993 paper

#original triangle
GenIns

#incremental triangle
cum2incr(GenIns)

#original triangle
incr2cum(cum2incr(GenIns))

# See the example in Mack's 1999 paper

#original triangle
Mortgage
incMortgage <- cum2incr(Mortgage)
#add missing values
incMortgage[1,1] <- NA
incMortgage[2,1] <- NA
incMortgage[1,2] <- NA

#with missing values argument
incr2cum(incMortgage, na.rm=TRUE)

#compared to 
incr2cum(Mortgage)

Description

One of the three basic assumptions underlying the chain ladder method is the independence of the accident years. The function tests this assumption.

Usage

cyEffTest(Triangle, ci = 0.95)

Arguments

Details

The main reason why this independence can be violated in practice is the fact that there could be certain calendar year effects such as major changes in claims handling or in case reserving or external influences such as substantial changes in court decisions or inflation.

As described by the Mack's 1994 paper a procedure is designed to test for calendar year influences.

The procedure returns a summary statistic $Z$ which is assumed to be Normally Distributed. It is therefore possible to define a confidence interval threshold in order to evaluate the outcome of the test.

Value

cyEffTest returns a list with the following elements

Note

Additional references for further reading:

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

Author(s)

Marco De Virgilis [email protected]

References

Mack, T., Measuring the Variability of Chain Ladder Reserve Estimates, Casualty Actuarial Society Forum, Spring 1994

See Also

See also qpaid for dealing with non-square triangles, dfCorTest for the test for correlations between subsequent development factors, chainladder for the chain-ladder method, summary.cyEffTest, plot.cyEffTest

Examples

# Before actually applying the Chain Ladder technique it is necessary to check
# wether the triangle has Calendar Year Effect

# Apply the function to the triangle and save the output into the variable test
test <- cyEffTest(RAA)

# Plot the confidence interval and the test metric
plot(test)

# The metric is within the confidence interval, therefore the triangle doesn't
# have Calendar Year Effect

# Print the summary table
summary(test)

# Print only the main outcomes
print(test)
# The test has returned a negative outcome. This means that the triangle is 
# not affected by Caledar Year Effect and therefore the chain ladder method 
# can be applied.

Description

One of the main assumptions underlying the chain ladder method is the uncorrelation of subsequest development factor. The function tests this assumption.

Usage

dfCorTest(Triangle, ci = .5)

Arguments

Details

As described by the Mack's 1994 paper a procedure is designed to test for calendar year influences.

The usual test for uncorrelatedness requires that we have identically distributed pairs of observations which come from a Normal distribution. Both conditions are usually not fulfilled for adjacent columns of development factors. Spearman's correlation coefficient is therefore used.

The metric calulated by the procudeure described return a statistic $T$ that it is assumed to be Normally Distributed. It is therefore possible to define a confidence interval threshold in order to evaluate the outcome of the test.

Value

dfCorTest returns a list with the following elements

Note

Additional references for further reading:

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

Venter, G.G., Testing the Assumptions of Age-to-Age Factors, Proceedings of the Casualty Actuarial Society LXXXV, 1998, pp. 807-847

Author(s)

Marco De Virgilis [email protected]

References

Mack, T., Measuring the Variability of Chain Ladder Reserve Estimates, Casualty Actuarial Society Forum, Spring 1994

See Also

See also qpaid for dealing with non-square triangles, cyEffTest for the test for calendar year effect, chainladder for the chain-ladder method, summary.dfCorTest, plot.dfCorTest

Examples

# Before actually applying the Chain Ladder technique it is necessary to check
# whether the Development Factors are correlated

# Apply the function to the triangle and save the output into the variable test
test <- dfCorTest(RAA)

# Plot the confidence interval and the test metric
plot(test)

# The metric is within the confidence interval, therefore the Development Factors are nor correlated

# Print the summary table
summary(test)

# Print only the main outcomes
print(test)
# The test has returned a negative outcome. This means that the triangle is 
# not affected by Development Factor Correlation and therefore the chain ladder method 
# can be applied.

Description

Run off triangle of accumulated general insurance claims data. GenInsLong provides the same data in a 'long' format.

Usage

GenIns

Format

A matrix with 10 accident years and 10 development years.

Source

TAYLOR, G.C. and ASHE, F.R. (1983) Second Moments of Estimates of Outstanding Claims. Journal of Econometrics 23, 37-61.

References

See table 1 in: Distribution-free Calculation of the Standard Error of Chain Ladder Reserve Estimates, Thomas Mack, 1993, ASTIN Bulletin 23, 213 - 225

Examples

GenIns
plot(GenIns)

plot(GenIns, lattice=TRUE)


head(GenInsLong)

## Convert long format into triangle
## Triangles are usually stored as 'long' tables in data bases
as.triangle(GenInsLong, origin="accyear", dev="devyear", "incurred claims")

Description

Return most recent values for all origin periods of a cumulative development triangle.

Usage

getLatestCumulative(Triangle, na.values = NULL)

Arguments

Value

A vector of most recent non-'NA' (and synonyms, if appropriate) values of a triangle for all origin periods. The names of the vector equal the origin names of the Triangle. The vector will have additional attributes: "latestcol" equalling the index of the column in Triangle corresponding to the row's rightmost entry; "rowsname" equalling the name of the row dimension of Triangle, if any; "colnames" equalling the corresponding column name of Triangle, if any; "colsname" equalling the name of the column dimension of Triangle, if any.

Author(s)

Ben Escoto, Markus Gesmann, Dan Murphy

See Also

See also as.triangle.

Examples

RAA
getLatestCumulative(RAA)
Y <- matrix(c(1,  2,  3,
              4,  5,  0, 
              6, NA, NA), byrow=TRUE, nrow=3)
getLatestCumulative(Y) # c(3, 0, 6) 
getLatestCumulative(Y, na.values = 0) # c(3, 5, 6)

Description

This function implements loss reserving models within the generalized linear model framework. It takes accident year and development lag as mean predictors in estimating the ultimate loss reserves, and provides both analytical and bootstrapping methods to compute the associated prediction errors. The bootstrapping approach also generates the full predictive distribution for loss reserves.

Usage

glmReserve(triangle, var.power = 1, link.power = 0, cum = TRUE, 
      mse.method = c("formula", "bootstrap"),  nsim = 1000, nb = FALSE, ...)

Arguments

Details

This function takes an insurance loss triangle, converts it to incremental losses internally if necessary, transforms it to the long format (see as.data.frame) and fits the resulting loss data with a generalized linear model where the mean structure includes both the accident year and the development lag effects. The distributions allowed are the exponential family that admits a power variance function, that is, $V(\mu)=\mu^p$. This subclass of distributions is usually called the Tweedie distribution and includes many commonly used distributions as special cases.

This function does not allow the user to specify the GLM options through the usual family argument, but instead, it uses the tweedie family internally and takes two arguments, var.power and link.power, through which the user still has full control of the distribution forms and link functions. The argument var.power determines which specific distribution is to be used, and link.power determines the form of the link function.

When the Tweedie compound Poisson distribution 1 < p < 2 is to be used, the user has the option to specify var.power = NULL, where the variance power p will be estimated from the data using the cplm package. The bcplm function in the cplm package also has an example for the Bayesian compound Poisson loss reserving model. See details in tweedie, cpglm and bcplm.

glmReserve allows certain measures of exposures to be used in an offset term in the underlying GLM. To do this, the user should not use the usual offset argument in glm. Instead, one specifies the exposure measure for each accident year through the exposure attribute of triangle. Make sure that these exposures are in the original scale (no log transformations for example). If the vector is named, make sure the names coincide with the rownames/origin of the triangle. If the vector is unnamed, make sure the exposures are in the order consistent with the accident years, and the character rownames of the Triangle must be convertible to numeric. If the exposure attribute is not NULL, the glmReserve function will use these exposures, link-function-transformed, in the offset term of the GLM. For example, if the link function is log, then the log of the exposure is used as the offset, not the original exposure. See the examples below. Moreover, the user MUST NOT supply the typical offset or weight as arguments in the list of additional arguments .... offset should be specified as above, while weight is not implemented (due to prediction reasons).

Two methods are available to assess the prediction error of the estimated loss reserves. One is using the analytical formula (mse.method = "formula") derived from the first-order Taylor approximation. The other is using bootstrapping (mse.method = "bootstrap") that reconstructs the triangle nsim times by sampling with replacement from the GLM (Pearson) residuals. Each time a new triangle is formed, GLM is fitted and corresponding loss reserves are generated. Based on these predicted mean loss reserves, and the model assumption about the distribution forms, realizations of the predicted values are generated via the rtweedie function. Prediction errors as well as other uncertainty measures such as quantiles and predictive intervals can be calculated based on these samples.

Value

The output is an object of class "glmReserve" that has the following components:

Note

The use of GLM in insurance loss reserving has many compelling aspects, e.g.,

• when over-dispersed Poisson model is used, it reproduces the estimates from Chain Ladder;

• it provides a more coherent modeling framework than the Mack method;

• all the relevant established statistical theory can be directly applied to perform hypothesis testing and diagnostic checking;

However, the user should be cautious of some of the key assumptions that underlie the GLM model, in order to determine whether this model is appropriate for the problem considered:

• the GLM model assumes no tail development, and it only projects losses to the latest time point of the observed data. To use a model that enables tail extrapolation, please consider the growth curve model ClarkLDF or ClarkCapeCod;

• the model assumes that each incremental loss is independent of all the others. This assumption may not be valid in that cells from the same calendar year are usually correlated due to inflation or business operating factors;

• the model tends to be over-parameterized, which may lead to inferior predictive performance.

To solve these potential problems, many variants of the current basic GLM model have been proposed in the actuarial literature. Some of these may be included in the future release.

Support of the negative binomial GLM was added since version 0.2.3.

Author(s)

Wayne Zhang [email protected]

References

England P. and Verrall R. (1999). Analytic and bootstrap estimates of prediction errors in claims reserving. Insurance: Mathematics and Economics, 25, 281-293.

See Also

See also glm, glm.nb, tweedie, cpglm and MackChainLadder.

Examples

data(GenIns)
GenIns <- GenIns / 1000

# over-dispersed Poisson: reproduce ChainLadder estimates
(fit1 <- glmReserve(GenIns))
summary(fit1, type = "model")   # extract the underlying glm

# which:
# 1 Original triangle 
# 2 Full triangle 
# 3 Reserve distribution
# 4 Residual plot
# 5 QQ-plot

# plot original triangle
plot(fit1, which = 1, xlab = "dev year", ylab = "cum loss")

# plot residuals 
plot(fit1, which = 4, xlab = "fitted values", ylab = "residuals")

# Gamma GLM:
(fit2 <- glmReserve(GenIns, var.power = 2))

# compound Poisson GLM (variance function estimated from the data):
(fit3 <- glmReserve(GenIns, var.power = NULL))

# Now suppose we have an exposure measure
# we can put it as an offset term in the model
# to do this, use the "exposure" attribute of the 'triangle'
expos <- (7 + 1:10 * 0.4) * 100
GenIns2 <- GenIns
attr(GenIns2, "exposure") <- expos
(fit4 <- glmReserve(GenIns2))
# If the triangle's rownames are not convertible to numeric,
# supply names to the exposures
GenIns3 <- GenIns2
rownames(GenIns3) <- paste0(2007:2016, "-01-01")
names(expos) <- rownames(GenIns3)
attr(GenIns3, "exposure") <- expos
(fit4b <- glmReserve(GenIns3))

# use bootstrapping to compute prediction error
## Not run: 
set.seed(11)
(fit5 <- glmReserve(GenIns, mse.method = "boot"))

# compute the quantiles of the predicted loss reserves
t(apply(fit5$sims.reserve.pred, 2, quantile, 
        c(0.025, 0.25, 0.5, 0.75, 0.975)))
        
# plot distribution of reserve
plot(fit5, which = 3)

## End(Not run)

# alternative over-dispersed Poisson: negative binomial GLM
(fit6 <- glmReserve(GenIns, nb = TRUE))

Description

Inflate the amounts of a Triangle from the latest diagonal based on an Inflation Rate

Usage

inflateTriangle(Triangle, rate)

Arguments

Details

The sensitivity of projections of ultimate losses based on incurred loss development factors to changes in the adequacy level of case reserves increases significantly for the long-tail lines. In particular, if the adequacy of the case reserve is changing, the estimates of ultimate losses based on reported claims could be severely distorted. The function deflates the amounts of latest diagonal to each diagonal of the triangle according to the inflation rate provided, considering an exponential model. The purpose of restating the amounts is to have each diagonal in the triangle at the same level as the latest diagonal (i.e. latest valuation). Ideally the metrics that should be restated are average O/S or average claim paid.

Value

inflateTriangle returns the inflated triangle according to the provided rate

Author(s)

Marco De Virgilis [email protected]

References

Berquist, J.R. and Sherman, R.E., Loss Reserve Adequacy Testing: A Comprehensive, Systematic Approach, Proceedings of the Casualty Actuarial Society, LXIV, 1977, pp.123-184.

See Also

See also qpaid for dealing with non-square triangles, checkTriangleInflation to check Y-o-Y Triangle Inflation Rates,

Examples

# Create a Triangle of Average Case O/S

avg <- MedMal$MedMalOutstanding / MedMal$MedMalOpen

# Select a rate of 15% and inflate the average =/S Triangle

inflated_tr <- inflateTriangle(Triangle = avg, rate = .15) 

# Multiply it by open claims and add paymnets to calulate the adjusted Reported Claims Trinagle

adj_reported <- inflated_tr * MedMal$MedMalOpen + MedMal$MedMalPaid

# Calculate the IBNR from the unadjusted Triangle

std_ibnr <- summary(MackChainLadder(MedMal$MedMalReported))$Totals[4, 1]

# Calculate the IBNR from the adjusted Triangle

adj_reported_ibnr <- summary(MackChainLadder(adj_reported))$Totals[4, 1]

# Compare the two

std_ibnr - adj_reported_ibnr

Description

This function is created to facilitate the fitting of the multivariate functions when specifying different models in two different development periods, especially when separate chain-ladder is used in later periods.

Usage

Join2Fits(object1, object2)

Arguments

Details

The inputs must be of class "MultiChainLadder" because this function depends on the model slot to determine what kind of object is to be created and returned. If both objects have "MCL", then an object of class "MCLFit" is created; if one has "GMCL" and one has "MCL", then an object of class "GMCLFit" is created, where the one with "GMCL" is assumed to come from the first development periods; if both have "GMCL", then an object of class "GMCLFit" is created.

Author(s)

Wayne Zhang [email protected]

See Also

See also MultiChainLadder

Description

This function combines first momoent estimation from fitted regression models and second moment estimation from Mse method to construct an object of class "MultiChainLadder", for which a variety of methods are defined, such as summary and plot.

Usage

JoinFitMse(models, mse.models)

Arguments

Author(s)

Wayne Zhang [email protected]

See Also

See also MultiChainLadder.

Description

Run-off triangles of General Liability and Auto Liability.

Usage

data(auto)

Format

A list of two matrices, General Liability and Auto Liability respectively.

Source

Braun C (2004). The prediction error of the chain ladder method applied to correlated run off triangles. ASTIN Bulletin 34(2): 399-423

Examples

data(liab)
names(liab)

Description

This calculates the link ratio function per the CLFM paper.

Usage

LRfunction(x, y, delta)

Arguments

Details

Calculated the link ratios resulting from a chainladder model over a development period indexed by (possibly vector valued) real number delta. See formula (5) in the References.

Value

A vector of link ratios.

Author(s)

Dan Murphy

References

Bardis, Majidi, Murphy. A Family of Chain-Ladder Factor Models for Selected Link Ratios. Variance. Pending. 2013. pp.tbd:tbd

Examples

x <- RAA[1:9,1]
y <- RAA[1:9,2]
delta <- seq(-2, 2, by = .1)
plot(delta, LRfunction(x, y, delta), type = "l")

Description

Run off triangle of simulated incremental claims data

Usage

data(M3IR5)

Format

A matrix with simulated incremental claims of 14 accident years and 14 development years.

Source

Appendix A7 in B. Zehnwirth. Probabilistic Development Factor Models with Applications to Loss Reserve Variability, Prediction Intervals, and Risk Based Capital. Casualty Actuarial Science Forum. Spring 1994. Vol. 2.

Examples

M3IR5
plot(M3IR5)
plot(incr2cum(M3IR5), lattice=TRUE)

Description

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

Usage

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

Arguments

Details

Following Mack's 1999 paper let $C_{ik}$ denote the cumulative loss amounts of origin period (e.g. accident year) $i=1,\ldots,m$, with losses known for development period (e.g. development year) $k \le n+1-i$. In order to forecast the amounts $C_{ik}$ for $k > n+1-i$ the Mack chain-ladder-model assumes:

$\mbox{CL1: } E[ F_{ik}| C_{i1},C_{i2},\ldots,C_{ik} ] = f_k \mbox{ with } F_{ik}=\frac{C_{i,k+1}}{C_{ik}}$

$\mbox{CL2: } Var( \frac{C_{i,k+1}}{C_{ik}} | C_{i1},C_{i2}, \ldots,C_{ik} ) = \frac{\sigma_k^2}{w_{ik} C^\alpha_{ik}}$

$\mbox{CL3: } \{ C_{i1},\ldots,C_{in}\}, \{ C_{j1},\ldots,C_{jn}\},\mbox{ are independent for origin period } i \neq j$

with $w_{ik} \in [0;1]$, $\alpha \in \{0,1,2\}$. If these assumptions hold, the Mack chain-ladder gives an unbiased estimator for IBNR (Incurred But Not Reported) claims.

Here $w_{ik} are the \code{weights} from above.$

The Mack chain-ladder model can be regarded as a special form of a weighted linear regression through the origin for each development period: lm(y ~ x + 0, weights=weights/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 $k$.

It is necessary, before actually applying the model, to check if the main assumptions behind the model (i.e. Calendar Year Effect and Correlation between subsequent Accident Years, see dfCorTest, cyEffTest) are verified.

Value

MackChainLadder returns a list with the following elements

Note

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.

Author(s)

Markus Gesmann [email protected]

References

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

See Also

See also qpaid for dealing with non-square triangles, chainladder for the underlying chain-ladder method, dfCorTest to check for Calendar Year Effect, cyEffTest to check for Development Factor Correlation, summary.MackChainLadder, quantile.MackChainLadder, plot.MackChainLadder and residuals.MackChainLadder displaying results, CDR.MackChainLadder for the one year claims development result.

Examples

## 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)

## Only use the last 5 diagonals, i.e. the last 5 calendar years
calPeriods <- (row(GenIns) + col(GenIns) - 1)
(weights <- ifelse(calPeriods <= 5, 0, ifelse(calPeriods > 10, NA, 1)))
MackChainLadder(GenIns, weights=weights, est.sigma = "Mack")

## 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)

Description

Run-off triangles based on a fire portfolio

Usage

data(MCLpaid)
data(MCLincurred)

Format

A matrix with 7 origin years and 7 development years.

Source

Gerhard Quarg and Thomas Mack. Munich Chain Ladder. Blatter DGVFM. 26, Munich, 2004.

Examples

MCLpaid
MCLincurred
op=par(mfrow=c(2,1))
plot(MCLpaid)
plot(MCLincurred)
par(op)

Description

Run-off triangles of Medical Malpractice Data insurance.

Usage

data(MedMal)

Format

U.S. medical malpractice insurance for an experience period of 1969 to 1976. Reported Claims, Paid Claims, Case Outstanding and Open Claims (i.e. the number of outstanding claims) respectively

Source

Berquist, J.R. and Sherman, R.E., Loss Reserve Adequacy Testing: A Comprehensive, Systematic Approach, Proceedings of the Casualty Actuarial Society, LXIV, 1977, pp.123-184.

Examples

data(MedMal)
names(MedMal)
MedMal$MedMalReported
MedMal$MedMalPaid
MedMal$MedMalOutstanding
MedMal$MedMalOpen

Description

Development triangle of a mortgage guarantee business

Usage

data(Mortgage)

Format

A matrix with 9 accident years and 9 development years.

Source

Competition Presented at a London Market Actuaries Dinner, D.E.A. Sanders, 1990

References

See table 4 in: Distribution-free Calculation of the Standard Error of Chain Ladder Reserve Estimates, Thomas Mack, 1993, ASTIN Bulletin 23, 213 - 225

Examples

Mortgage
Mortgage
plot(Mortgage)
plot(Mortgage, lattice=TRUE)

Description

Mse is a generic function to calculate mean square error estimations in the chain-ladder framework.

Usage

Mse(ModelFit, FullTriangles, ...)

## S4 method for signature 'GMCLFit,triangles'
Mse(ModelFit, FullTriangles, ...)
## S4 method for signature 'MCLFit,triangles'
Mse(ModelFit, FullTriangles, mse.method="Mack", ...)

Arguments

Details

These functions calculate the conditional mean square errors using the recursive formulas in Zhang (2010), which is a generalization of the Mack (1993, 1999) formulas. In the GMCL model, the conditional mean square error for single accident years and aggregated accident years are calcualted as:

$\hat{mse}(\hat{Y}_{i,k+1}|D)=\hat{B}_k \hat{mse}(\hat{Y}_{i,k}|D) \hat{B}_k + (\hat{Y}_{i,k}' \otimes I) \hat{\Sigma}_{B_k} (\hat{Y}_{i,k} \otimes I) + \hat{\Sigma}_{\epsilon_{i_k}}.$

$\hat{mse}(\sum^I_{i=a_k}\hat{Y}_{i,k+1}|D)=\hat{B}_k \hat{mse}(\sum^I_{i=a_k+1}\hat{Y}_{i,k}|D) \hat{B}_k + (\sum^I_{i=a_k}\hat{Y}_{i,k}' \otimes I) \hat{\Sigma}_{B_k} (\sum^I_{i=a_k}\hat{Y}_{i,k} \otimes I) + \sum^I_{i=a_k}\hat{\Sigma}_{\epsilon_{i_k}} .$

In the MCL model, the conditional mean square error from Merz and Wüthrich (2008) is also available, which can be shown to be equivalent as the following:

$\hat{mse}(\hat{Y}_{i,k+1}|D)=(\hat{\beta}_k \hat{\beta}_k') \odot \hat{mse}(\hat{Y}_{i,k}|D) + \hat{\Sigma}_{\beta_k} \odot (\hat{Y}_{i,k} \hat{Y}_{i,k}') + \hat{\Sigma}_{\epsilon_{i_k}} +\hat{\Sigma}_{\beta_k} \odot \hat{mse}^E(\hat{Y}_{i,k}|D) .$

$\hat{mse}(\sum^I_{i=a_k}\hat{Y}_{i,k+1}|D)=(\hat{\beta}_k \hat{\beta}_k') \odot \sum^I_{i=a_k+1}\hat{mse}(\hat{Y}_{i,k}|D) + \hat{\Sigma}_{\beta_k} \odot (\sum^I_{i=a_k}\hat{Y}_{i,k} \sum^I_{i=a_k}\hat{Y}_{i,k}') + \sum^I_{i=a_k}\hat{\Sigma}_{\epsilon_{i_k}} +\hat{\Sigma}_{\beta_k} \odot \sum^I_{i=a_k}\hat{mse}^E(\hat{Y}_{i,k}|D) .$

For the Mack approach in the MCL model, the cross-product term $\hat{\Sigma}_{\beta_k} \odot \hat{mse}^E(\hat{Y}_{i,k}|D)$in the above two formulas will drop out.

Value

Mse returns an object of class "MultiChainLadderMse" that has the following elements:

Author(s)

Wayne Zhang [email protected]

References

Zhang Y (2010). A general multivariate chain ladder model.Insurance: Mathematics and Economics, 46, pp. 588-599.

Zhang Y (2010). Prediction error of the general multivariate chain ladder model.

See Also

See also MultiChainLadder.

Description

The function MultiChainLadder implements multivariate methods to forecast insurance loss payments based on several cumulative claims development triangles. These methods are multivariate extensions of the chain-ladder technique, which develop several correlated triangles simultaneously in a way that both contemporaneous correlations and structural relationships can be accounted for. The estimated conditional Mean Square Errors (MSE) are also produced.

Usage

MultiChainLadder(Triangles, fit.method = "SUR", delta = 1,
  int = NULL, restrict.regMat = NULL, extrap = TRUE, 
  mse.method = "Mack", model = "MCL", ...)
    
MultiChainLadder2(Triangles, mse.method = "Mack", last = 3, 
  type = c("MCL", "MCL+int", "GMCL-int", "GMCL"), ...)

Arguments

Details

This function implements multivariate loss reserving models within the chain-ladder framework. Two major models are included. One is the Multivariate chain-ladder (MCL) model proposed by Prohl and Schmidt (2005). This is a direct multivariate generalization of the univariate chain-ladder model in that losses from different triangles are assumed to be correlated but the mean development in one triangle only depends on its past values, not on the observed values from other triangles. In contrast, the other model, the General Multivariate chain-ladder (GMCL) model outlined in Zhang (2010), extends the MCL model by allowing development dependencies among different triangles as well as the inclusion of regression intercepts. As a result, structurally related triangles, such as the paid and incurred loss triangles or the paid loss and case reserve triangles, can be developed together while still accounting for the potential contemporaneous correlations. While the MCL model is a special case of the GMCL model, it is programmed and listed separately because: a) it is an important model for its own sake; b) different MSE methods are only available for the MCL model; c) extrapolation of the residual variance estimation can be implemented for the MCL model, which is considerably difficult for the GMCL model.

We introduce some details of the GMCL model in the following. Assume N triangles are available. Denote $Y_{i,k}=(Y^{(1)}_{i,k}, \ldots ,Y^{(N)}_{i,k})$ as an $N \times 1$ vector of cumulative losses at accident year i and development year k, where (n) refers to the n-th triangle. The GMCL model in development period k (from development year k to year k+1) is:

$Y_{i,k+1}=A_k + B_k \cdot Y_{i,k}+\epsilon_{i,k},$

where $A_k$ is a column of intercepts and $B_k$ is the $N \times N$ development matrix. By default, MultiChainLadder sets $A_k$ to be zero. This behavior can be changed by appropriately specifying the int argument. Assumptions for this model are:

$E(\epsilon_{i,k}|Y_{i,1},\ldots,Y_{i,I+1-k})=0.$

$cov(\epsilon_{i,k}|Y_{i,1},\ldots,Y_{i,I+1-k})=\Sigma_{\epsilon_{i,k}}=D(Y_{i,k}^{\delta/2})\Sigma_k D(Y_{i,k}^{\delta/2}).$

$\mbox{losses of different accident years are independent}.$

$\epsilon_{i,k} \, \mbox{are symmetrically distributed}.$

The GMCL model structure is generally over-parameterized. Parameter restrictions are usually necessary for the estimation to be feasible, which can be specified through the restrict.regMat argument. We refer the users to the documentation for systemfit for details and the demo of the present function for examples.

In particular, if one restricts the development matrix to be diagonal, the GMCL model will reduce to the MCL model. When non-diagonal development matrix is used and the GMCL model is applied to paid and incurred loss triangles, it can reflect the development relationship between the two triangles, as described in Quarg and Mack (2004). The full bivariate model is identical to the "double regression" model described by Mack (2003), which is argued by him to be very similar to the Munich chain-ladder (MuCL) model. The GMCL model with intercepts can also help improve model adequacy as described in Barnett and Zehnwirth (2000).

Currently, the GMCL model only works for trapezoid data, and only implements mse.method = "Mack". The MCL model allows an additional mse estimation method that assumes independence among the estimated parameters. Further, the MCL model using fit.method = "OLS" will be equivalent to running univariate chain-ladders separately on each triangle. Indeed, when only one triangle is specified (as a list), the MCL model is equivalent to MackChainLadder.

The GMCL model allows different model structures to be specified in each development period. This is generally achieved through the combination of the int argument, which specifies the periods that have intercepts, and the restrict.regMat argument, which imposes parameter restrictions on the development matrix.

In using the multivariate methods, we often specify separate univariate chain-ladders for the tail periods to stabilize the estimation - there are few data points in the tail and running a multivariate model often produces extremely volatile estimates or even fails. In this case, we can use the subset operator "[" defined for class triangles to split the input data into two parts. We can specify a multivariate model with rich structures on the first part to reflect the multivariate dependencies, and simply apply multiple univariate chain-ladders on the second part. The two models are subsequently joined together using the Join2Fits function. We can then invoke the predict and Mse methods to produce loss predictions and mean square error estimations. They can further be combined via the JoinFitMse function to construct an object of class MultiChainLadder. See the demo for such examples.

To facilitate such a split-and-join process for most applications, we have created the function MultiChainLadder2. This function splits the data according to the last argument (e.g., if last = 3, the last three periods go into the second part), and fits the first part according to the structure indicated in the type argument. See the 'Arguments' section for details.

Value

MultiChainLadder returns an object of class MultiChainLadder with the following slots:

Note

When MultiChainLadder or MultiChainLadder2 fails, the most possible reason is that there is little or no development in the tail periods. That is, the development factor is 1 or almost equal to 1. In this case, the systemfit function may fail even for fit.method = "OLS", because the residual covariance matrix $\Sigma_k$ is singular. The simplest solution is to remove these columns using the "[" operator and fit the model on the remaining part.

Also, we recommend the use of MultiChainLadder2 over MultiChainLadder. The function MultiChainLadder2 meets the need for most applications, is relatively easy to use and produces more stable but very similar results to MultiChainLadder. Use MultiChainLadder only when non-standard situation arises, e.g., when different parameter restrictions are needed for different periods. See the demo for such examples.

Author(s)

Wayne Zhang [email protected]

References

Buchwalder M, Bühlmann H, Merz M, Wüthrich M.V (2006). The mean square error of prediction in the chain-ladder reserving method (Mack and Murphy revisited), ASTIN Bulletin, 36(2), 521-542.

Prohl C, Schmidt K.D (2005). Multivariate chain-ladder, Dresdner Schriften zur Versicherungsmathematik.

Mack T (1993). Distribution-free calculation of the standard error, ASTIN Bulletin, 23, No.2.

Mack T (1999). The standard error of chain-ladder reserve estimates: recursive calculation and inclusion of a tail factor, ASTIN Bulletin, 29, No.2, 361-366.

Merz M, Wüthrich M (2008). Prediction error of the multivariate chain ladder reserving method, North American Actuarial Journal, 12, No.2, 175-197.

Zhang Y (2010). A general multivariate chain-ladder model.Insurance: Mathematics and Economics, 46, pp. 588-599.

Zhang Y (2010). Prediction error of the general multivariate chain ladder model.

See Also

See also MackChainLadder, MunichChainLadder, triangles, MultiChainLadder, summary,MultiChainLadder-method and plot,MultiChainLadder,missing-method.

Examples

# This shows that the MCL model using "OLS" is equivalent to 
# the MackChainLadder when applied to one triangle 

data(GenIns)
(U1 <- MackChainLadder(GenIns, est.sigma = "Mack"))
(U2 <- MultiChainLadder(list(GenIns), fit.method = "OLS"))

# show plots 
parold <- par(mfrow = c(2, 2))
plot(U2, which.plot = 1:4)
plot(U2, which.plot = 5)
par(parold)


# For mse.method = "Independence", the model is equivalent 
# to that in Buchwalder et al. (2006)

(B1 <- MultiChainLadder(list(GenIns), fit.method = "OLS", 
    mse.method = "Independence"))

# use the unbiased residual covariance estimator 
# in Merz and Wuthrich (2008)
(W1 <- MultiChainLadder2(liab, mse.method = "Independence", 
    		control = systemfit::systemfit.control(methodResidCov = "Theil"))) 

## Not run: 
# use the iterative residual covariance estimator
for (i in 1:5){
  W2 <- MultiChainLadder2(liab, mse.method = "Independence", 
      control = systemfit::systemfit.control(
      methodResidCov = "Theil", maxiter = i))
  print(format(summary(W2)@report.summary[[3]][15, 4:5], 
          digits = 6, big.mark = ","))    
}

# The following fits an MCL model with intercepts for years 1:7
# and separate chain-ladder models for the rest periods
f1 <- MultiChainLadder2(auto, type = "MCL+int")

# compare with the model without intercepts through residual plots
f0 <- MultiChainLadder2(auto, type = "MCL")

parold <- par(mfrow = c(2, 3), mar = c(3, 3, 2, 1))
mt <- list(c("Personal Paid", "Personal Incured", "Commercial Paid"))
plot(f0, which.plot = 3, main = mt)
plot(f1, which.plot = 3, main = mt)
par(parold)

## summary statistics
summary(f1, portfolio = "1+3")@report.summary[[4]]


# model for joint development of paid and incurred triangles 
da <- auto[1:2]
# MCL with diagonal development
M0 <- MultiChainLadder(da)
# non-diagonal development matrix with no intercepts
M1 <- MultiChainLadder2(da, type = "GMCL-int")
# Munich chain-ladder
M2 <- MunichChainLadder(da[[1]], da[[2]])
# compile results and compare projected paid to incurred ratios
r1 <- lapply(list(M0, M1), function(x){
          ult <- summary(x)@Ultimate
          ult[, 1] / ult[, 2]
      })
names(r1) <- c("MCL", "GMCL")
r2 <- summary(M2)[[1]][, 6]
r2 <- c(r2, summary(M2)[[2]][2, 3])
print(do.call(cbind, c(r1, list(MuCl = r2))) * 100, digits = 4)


## End(Not run)

# To reproduce results in Zhang (2010) and see more examples, use:
## Not run: 
 demo(MultiChainLadder)

## End(Not run)

Description

This class includes the first and second moment estimation result using the multivariate reserving methods in chain-ladder. Several primitive methods and statistical methods are also created to facilitate further analysis.

Objects from the Class

Objects can be created by calls of the form new("MultiChainLadder", ...), or they could also be a result of calls from MultiChainLadder or JoinFitMse.

Slots

model:

Object of class "character". Either "MCL" or "GMCL".

Triangles:

Object of class "triangles". Input triangles.

models:

Object of class "list". Fitted regression models using systemfit.

coefficients:

Object of class "list". Estimated regression coefficients.

coefCov:

Object of class "list". Estimated variance-covariance matrix of coefficients.

residCov:

Object of class "list". Estimated residual covariance matrix.

fit.method:

Object of class "character". Could be values of "SUR" or "OLS".

delta:

Object of class "numeric". Parameter for weights.

int:

Object of class "NullNum". Indicator of which periods have intercepts.

mse.ay:

Object of class "matrix". Conditional mse for each accident year.

mse.ay.est:

Object of class "matrix". Conditional estimation mse for each accident year.

mse.ay.proc:

Object of class "matrix". Conditional process mse for each accident year.

mse.total:

Object of class "matrix". Conditional mse for aggregated accident years.

mse.total.est:

Object of class "matrix". Conditional estimation mse for aggregated accident years.

mse.total.proc:

Object of class "matrix". Conditional process mse for aggregated accident years.

FullTriangles:

Object of class "triangles". Completed triangles.

restrict.regMat:

Object of class "NullList"

Extends

Class "MultiChainLadderFit", directly. Class "MultiChainLadderMse", directly.

Methods

$

signature(x = "MultiChainLadder"): Method for primitive function "$". It extracts a slot of x with a specified slot name, just as in list.

[[

signature(x = "MultiChainLadder", i = "numeric", j = "missing"): Method for primitive function "[[". It extracts the i-th slot of a "MultiChainLadder" object, just as in list. i could be a vector.

[[

signature(x = "MultiChainLadder", i = "character", j = "missing"): Method for primitive function "[[". It extracts the slots of a "MultiChainLadder" object with names in i, just as in list. i could be a vector.

coef

signature(object = "MultiChainLadder"): Method for function coef, to extract the estimated development matrix. The output is a list.

fitted

signature(object = "MultiChainLadder"): Method for function fitted, to calculate the fitted values in the original triangles. Note that the return value is a list of fitted valued based on the original scale, not the model scale which is first divided by $Y_{i,k}^{\delta/2}$.

names

signature(x = "MultiChainLadder"): Method for function names, which returns the slot names of a "MultiChainLadder" object.

plot

signature(x = "MultiChainLadder", y = "missing"): See plot,MultiChainLadder,missing-method.

residCov

signature(object = "MultiChainLadder"): S4 generic function and method to extract residual covariance from a "MultiChainLadder" object.

residCor

signature(object = "MultiChainLadder"): S4 generic function and method to extract residual correlation from a "MultiChainLadder" object.

residuals

signature(object = "MultiChainLadder"): Method for function residuals, to extract residuals from a system of regression equations. These residuals are based on model scale, and will not be equivalent to those on the original scale if $\delta$ is not set to be 0. One should use rstandard instead, which is independent of the scale.

resid

signature(object = "MultiChainLadder"): Same as residuals.

rstandard

signature(model = "MultiChainLadder"): S4 generic function and method to extract standardized residuals from a "MultiChainLadder" object.

show

signature(object = "MultiChainLadder"): Method for show.

summary

signature(object = "MultiChainLadder"): See summary,MultiChainLadder-method.

vcov

signature(object = "MultiChainLadder"): Method for function vcov, to extract the variance-covariance matrix of a "MultiChainLadder" object. Note that the result is a list of Bcov, that is the variance-covariance matrix of the vectorized $B$.

Author(s)

Wayne Zhang [email protected]

See Also

See also MultiChainLadder,summary,MultiChainLadder-method and plot,MultiChainLadder,missing-method.

Examples

# example for class "MultiChainLadder"
data(liab)
fit.liab <-  MultiChainLadder(Triangles = liab)
fit.liab

names(fit.liab)
fit.liab[[1]]
fit.liab$model
fit.liab@model

do.call("rbind",coef(fit.liab))
vcov(fit.liab)[[1]]
residCov(fit.liab)[[1]]
head(do.call("rbind",rstandard(fit.liab)))

Description

"MultiChainLadderFit" is a virtual class for the fitted models in the multivariate chain ladder reserving framework, "MCLFit" is a result from the interal call .FitMCL to store results in model MCL and "GMCLFit" is a result from the interal call .FitGMCL to store results in model GMCL. The two classes "MCLFit" and "GMCLFit" differ only in the presentation of $B_k$ and $\Sigma_{B_k}$, and different methods of Mse and predict will be dispatched according to these classes.

Objects from the Class

"MultiChainLadderFit" is a virtual Class: No objects may be created from it. For "MCLFit" and "GMCLFit", objects can be created by calls of the form new("MCLFit", ...) and new("GMCLFit", ...) respectively.

Slots

Triangles:

Object of class "triangles"

models:

Object of class "list"

B:

Object of class "list"

Bcov:

Object of class "list"

ecov:

Object of class "list"

fit.method:

Object of class "character"

delta:

Object of class "numeric"

int:

Object of class "NullNum"

restrict.regMat:

Object of class "NullList"

Extends

"MCLFit" and "GMCLFit" extends class "MultiChainLadderFit", directly.

Methods

No methods defined with class "MultiChainLadderFit" in the signature.

For "MCLFit", the following methods are defined:

Mse

signature(ModelFit = "MCLFit", FullTriangles = "triangles"): Calculate Mse estimations.

predict

signature(object = "MCLFit"): Predict ultimate losses and complete the triangles. The output is an object of class "triangles".

For "GMCLFit", the following methods are defined:

Mse

signature(ModelFit = "GMCLFit", FullTriangles = "triangles"): Calculate Mse estimations.

predict

signature(object = "GMCLFit"): Predict ultimate losses and complete the triangles. The output is an object of class "triangles".

Author(s)

Wayne Zhang [email protected]

See Also

See also Mse.

Examples

showClass("MultiChainLadderFit")

Description

This class is used to define the structure in storing the MSE results.

Objects from the Class

Objects can be created by calls of the form new("MultiChainLadderMse", ...), or as a result of a call to Mse.

Slots

mse.ay:

Object of class "matrix"

mse.ay.est:

Object of class "matrix"

mse.ay.proc:

Object of class "matrix"

mse.total:

Object of class "matrix"

mse.total.est:

Object of class "matrix"

mse.total.proc:

Object of class "matrix"

FullTriangles:

Object of class "triangles"

Methods

No methods defined with class "MultiChainLadderMse" in the signature.

Author(s)

Wayne Zhang [email protected]

See Also

See Also MultiChainLadder and Mse.

Examples

showClass("MultiChainLadderMse")

Description

This class stores the summary statistics from a "MultiChainLadder" object. These summary statistics include both model summary and report summary.

Objects from the Class

Objects can be created by calls of the form new("MultiChainLadderSummary", ...), or a call from summary.

Slots

Triangles:

Object of class "triangles"

FullTriangles:

Object of class "triangles"

S.E.Full:

Object of class "list"

S.E.Est.Full:

Object of class "list"

S.E.Proc.Full:

Object of class "list"

Ultimate:

Object of class "matrix"

IBNR:

Object of class "matrix"

S.E.Ult:

Object of class "matrix"

S.E.Est.Ult:

Object of class "matrix"

S.E.Proc.Ult:

Object of class "matrix"

report.summary:

Object of class "list"

coefficients:

Object of class "list"

coefCov:

Object of class "list"

residCov:

Object of class "list"

rstandard:

Object of class "matrix"

fitted.values:

Object of class "matrix"

residCor:

Object of class "matrix"

model.summary:

Object of class "matrix"

portfolio:

Object of class "NullChar"

Methods

$

signature(x = "MultiChainLadderSummary"): Method for primitive function "$". It extracts a slot of x with a specified slot name, just as in list.

[[

signature(x = "MultiChainLadderSummary", i = "numeric", j = "missing"): Method for primitive function "[[". It extracts the i-th slot of a "MultiChainLadder" object, just as in list. i could be a vetor.

[[

signature(x = "MultiChainLadderSummary", i = "character", j = "missing"): Method for primitive function "[[". It extracts the slots of a "MultiChainLadder" object with names in i, just as in list. i could be a vetor.

names

signature(x = "MultiChainLadderSummary"): Method for function names, which returns the slot names of a "MultiChainLadder" object.

show

signature(object = "MultiChainLadderSummary"): Method for show.

Author(s)

Wayne Zhang [email protected]

See Also

See also summary,MultiChainLadder-method, MultiChainLadder-class

Examples

showClass("MultiChainLadderSummary")

Description

The Munich-chain-ladder model forecasts ultimate claims based on a cumulative paid and incurred claims triangle. The model assumes that the Mack-chain-ladder model is applicable to the paid and incurred claims triangle, see MackChainLadder.

Usage

MunichChainLadder(Paid, Incurred, 
                  est.sigmaP = "log-linear", est.sigmaI = "log-linear", 
                  tailP=FALSE, tailI=FALSE, weights=1)

Arguments

Value

MunichChainLadder returns a list with the following elements

Author(s)

Markus Gesmann [email protected]

References

Gerhard Quarg and Thomas Mack. Munich Chain Ladder. Blatter DGVFM 26, Munich, 2004.

See Also

See also summary.MunichChainLadder, plot.MunichChainLadder , MackChainLadder

Examples

MCLpaid
MCLincurred
op <- par(mfrow=c(1,2))
plot(MCLpaid)
plot(MCLincurred)
par(op)

# Following the example in Quarg's (2004) paper:
MCL <- MunichChainLadder(MCLpaid, MCLincurred, est.sigmaP=0.1, est.sigmaI=0.1)
MCL
plot(MCL)
# You can access the standard chain-ladder (Mack) output via
MCL$MackPaid
MCL$MackIncurred

# Input triangles section 3.3.1
MCL$Paid
MCL$Incurred
# Parameters from section 3.3.2
# Standard chain-ladder age-to-age factors
MCL$MackPaid$f
MCL$MackIncurred$f
MCL$MackPaid$sigma
MCL$MackIncurred$sigma
# Check Mack's assumptions graphically
plot(MCL$MackPaid)
plot(MCL$MackIncurred)

MCL$q.f
MCL$rhoP.sigma
MCL$rhoI.sigma

MCL$PaidResiduals
MCL$IncurredResiduals

MCL$QinverseResiduals
MCL$QResiduals

MCL$lambdaP
MCL$lambdaI
# Section 3.3.3 Results
MCL$MCLPaid
MCL$MCLIncurred