Evaluate the Probability in the Upper Tail of the Aggregate Loss Distribution

Description

Evaluate the probability in the upper tail of the aggregate loss distribution using different methods: Panjer recursion, Monte Carlo simulations, Markov bound, Cantelli bound, Moment bound, and Chernoff bound.

Details

The package tailloss contains functions to estimate the exceedance probability curve of the aggregated losses. There are two ‘exact’ approaches: Panjer recursion and Monte Carlo simulations, and four approaches producing upper bounds: the Markov bound, the Cantelli bound, the Moment bound, and the Chernoff bound. The upper bounds are useful and effective when the number of events in the catalogue is large, and there is interest in estimating the exceedance probabilities of exceptionally high losses.

Author(s)

Isabella Gollini <isabella.gollini@bristol.ac.uk>, and Jonathan Rougier.

This work was supported by the Natural Environment Research Council [Consortium on Risk in the Environment: Diagnostics, Integration, Benchmarking, Learning and Elicitation (CREDIBLE); grant number NE/J017450/1]

References

Gollini, I., and Rougier, J. C. (2015), "Rapidly bounding the exceedance probabilities of high aggregate losses", http://arxiv.org/abs/1507.01853.

Examples

data(UShurricane)

# Compress the table to millions of dollars

USh.m <- compressELT(ELT(UShurricane), digits = -6)
s <- seq(1,40)
EPC <- matrix(NA, length(s), 6)
colnames(EPC) <- c("Panjer", "MonteCarlo", "Markov",
 "Cantelli", "Moment", "Chernoff")
EPC[, 1] <- fPanjer(USh.m, s = s)[, 2]
EPC[, 2] <- fMonteCarlo(USh.m, s = s)[, 2]
EPC[, 3] <- fMarkov(USh.m, s = s)[, 2]
EPC[, 4] <- fCantelli(USh.m, s = s)[, 2]
EPC[, 5] <- fMoment(USh.m, s = s)[, 2]
EPC[, 6] <- fChernoff(USh.m, s = s)[, 2]
matplot(s, EPC, type = "l", lwd = 2, xlab = "s", ylim = c(0, 1), lty = 1:6,
  ylab = expression(plain(Pr)(S>=s)), main = "Exceedance Probability Curve")
zoombox(s, EPC, x0 = c(30, 40), y0 = c(0, .1), y1 = c(.3, .6), type = "l",
  lwd = 2, lty = 1:6)
legend("topright", legend = colnames(EPC), lty = 1:6, col = 1:6, lwd = 2)

EPCcap <- matrix(NA, length(s), 6)
colnames(EPCcap) <- c("Panjer", "MonteCarlo", "Markov",
 "Cantelli", "Moment", "Chernoff")
EPCcap[, 1] <- fPanjer(USh.m, s = s, theta = 2, cap = 5)[, 2]
EPCcap[, 2] <- fMonteCarlo(USh.m, s = s, theta = 2, cap = 5)[, 2]
EPCcap[, 3] <- fMarkov(USh.m, s = s, theta = 2, cap = 5)[, 2]
EPCcap[, 4] <- fCantelli(USh.m, s = s, theta = 2, cap = 5)[, 2]
EPCcap[, 5] <- fMoment(USh.m, s = s, theta = 2, cap = 5)[, 2]
EPCcap[, 6] <- fChernoff(USh.m, s = s, theta = 2, cap =  5)[, 2]
matplot(s, EPCcap, type = "l", lwd = 2, xlab = "s", ylim = c(0, 1), lty = 1:6,
  ylab = expression(plain(Pr)(S>=s)), main = "Exceedance Probability Curve")
zoombox(s, EPCcap, x0 = c(30, 40), y0 = c(0, .1), y1 = c(.3, .6), type = "l",
  lwd = 2, lty = 1:6)
legend("topright", legend = colnames(EPC), lty = 1:6, col = 1:6, lwd = 2)

Event Loss Table

Description

Function to create an ELT object

Usage

ELT(X = NULL, Rate = NULL, Loss = NULL, ID = NULL)

Arguments

Value

An object ELT, a data frame with 3 columns. The column ID contains the ID of each event. The column Rate contains the arrival rates of a single occurrence of event. The column Loss contains the expected losses from each single occurrence of event.

See Also

data.frame

Examples

rate <- c(.1, .02, .05)
loss <- c(2, 5, 7)

ELT(Rate = rate, Loss = loss)
# Same as
rl <- data.frame(Rate = rate, Loss = loss)
ELT(rl)

US hurricane data

Description

US hurricane data provided by Peter Taylor and Dickie Whitaker.

Format

Data frame with 32060 rows and 3 columns

Details

• EventID. ID of 32060 events.

• Rate. Annual rate of occurrence.

• Loss. Loss associated to each event measured in $.

Compress the event loss table

Description

Function to merge losses of the same amount adding up their corresponding occurrence rates, and to round the losses to the 10^digits integer value.

Usage

compressELT(ELT, digits = 0)

Arguments

Value

Data frame containg two numeric columns. The column Loss contains the expected losses from each single occurrence of event. The column Rate contains the arrival rates of a single occurrence of event.

Examples

data(UShurricane)

# Compress the table to thousands of dollars

USh.k <- compressELT(ELT(UShurricane), digits = -3)
summary(USh.k)

# Compress the table to millions of dollars

USh.m <- compressELT(ELT(UShurricane), digits = -6)
summary(USh.m)

Cantelli Bound.

Description

Function to bound the total losses via the Cantelli inequality.

Usage

fCantelli(ELT, s, t = 1, theta = 0, cap = Inf)

Arguments

Details

Cantelli's inequality states:

\Pr( S \geq s) \leq \frac{\sigma^2}{\sigma^2 + (s - \mu)^2} \quad \mbox{for } s \geq \mu,

where \mu = E[S] and \sigma^2 = Var[S] < \infty are the mean and the variance of the distribution of S.

Value

A numeric matrix, containing the pre-specified losses s in the first column and the upper bound for the exceedance probabilities in the second column.

Examples

data(UShurricane)

# Compress the table to millions of dollars

USh.m <- compressELT(ELT(UShurricane), digits = -6)
EPC.Cantelli  <- fCantelli(USh.m, s = 1:40)
plot(EPC.Cantelli, type = "l", ylim = c(0, 1))
# Assuming the losses follow a Gamma with E[X] = x, and Var[X] = 2 * x
EPC.Cantelli.Gamma  <- fCantelli(USh.m, s = 1:40, theta = 2, cap = 25)
EPC.Cantelli.Gamma
plot(EPC.Cantelli.Gamma, type = "l")
# Compare the two results:
plot(EPC.Cantelli, type = "l", main = "Exceedance Probability Curve", ylim = c(0, 1))
lines(EPC.Cantelli.Gamma, col = 2, lty = 2)
legend("topright", c("Dirac Delta", expression(paste("Gamma(",
alpha[i] == 1 / theta^2, ", ", beta[i] ==1 / (x[i] * theta^2), ")", " cap =", 5))),
lwd = 2, lty = 1:2, col = 1:2)

Chernoff Bound.

Description

Function to bound the total losses via the Chernoff inequality.

Usage

fChernoff(ELT, s, t = 1, theta = 0, cap = Inf, nk = 1001,
  verbose = FALSE)

Arguments

Details

Chernoff's inequality states:

\Pr(S \geq s) \leq \inf_{k > 0} e^{-k s} M_S(k)

where M_S(k) is the Moment Generating Function (MGF) of the total loss S. The fChernoff function optimises the bound over a fixed set of nk discrete values.

Value

A numeric matrix, containing the pre-specified losses s in the first column and the upper bound for the exceedance probabilities in the second column.

Examples

data(UShurricane)

# Compress the table to millions of dollars

USh.m <- compressELT(ELT(UShurricane), digits = -6)
EPC.Chernoff <- fChernoff(USh.m, s = 1:40)
EPC.Chernoff
plot(EPC.Chernoff, type = "l", ylim = c(0, 1))
# Assuming the losses follow a Gamma with E[X] = x, and Var[X] = 2 * x
EPC.Chernoff.Gamma <- fChernoff(USh.m, s = 1:40, theta = 2, cap = 5)
EPC.Chernoff.Gamma
plot(EPC.Chernoff.Gamma, type = "l", ylim = c(0, 1))
# Compare the two results:
plot(EPC.Chernoff, type = "l", main = "Exceedance Probability Curve", ylim = c(0, 1))
lines(EPC.Chernoff.Gamma, col = 2, lty = 2)
legend("topright", c("Dirac Delta", expression(paste("Gamma(",
alpha[i] == 1 / theta^2, ", ", beta[i] ==1 / (x[i] * theta^2), ")", " cap =", 5))),
lwd = 2, lty = 1:2, col = 1:2)

Markov Bound.

Description

Function to bound the total losses via the Markov inequality.

Usage

fMarkov(ELT, s, t = 1, theta = 0, cap = Inf)

Arguments

Details

Cantelli's inequality states:

\Pr( S \geq s) \leq \frac{E[S]}{s}

Value

A numeric matrix, containing the pre-specified losses s in the first column and the upper bound for the exceedance probabilities in the second column.

Examples

data(UShurricane)

# Compress the table to millions of dollars

USh.m <- compressELT(ELT(UShurricane), digits = -6)
EPC.Markov <- fMarkov(USh.m, s = 1:40)
plot(EPC.Markov, type = "l", ylim = c(0, 1))
# Assuming the losses follow a Gamma with E[X] = x, and Var[X] = 2 * x
EPC.Markov.Gamma <- fMarkov(USh.m, s = 1:40, theta = 2, cap = 5)
EPC.Markov.Gamma
plot(EPC.Markov.Gamma, type = "l", ylim = c(0, 1))
# Compare the two results:
plot(EPC.Markov, type = "l", main = "Exceedance Probability Curve", ylim = c(0,1))
lines(EPC.Markov.Gamma, col = 2, lty = 2)
legend("topright", c("Dirac Delta", expression(paste("Gamma(",
alpha[i] == 1 / theta^2, ", ", beta[i] ==1 / (x[i] * theta^2), ")", " cap =", 5))),
lwd = 2, lty = 1:2, col = 1:2)

Moment Bound.

Description

Function to bound the total losses via the Moment inequality.

Usage

fMoment(ELT, s, t = 1, theta = 0, cap = Inf, verbose = FALSE)

Arguments

Details

Moment inequality states:

\Pr(S \geq s) \leq \min_{k = 1, 2 \dots} \frac{E(S^k)}{s^k}

where E(S^k) is the k-th moment of the total loss S distribution.

Value

A numeric matrix, containing the pre-specified losses s in the first column and the upper bound for the exceedance probabilities in the second column.

Examples

data(UShurricane)

# Compress the table to millions of dollars

USh.m <- compressELT(ELT(UShurricane), digits = -6)
EPC.Moment <- fMoment(USh.m, s = 1:40)
EPC.Moment
plot(EPC.Moment, type = "l", ylim = c(0, 1))
# Assuming the losses follow a Gamma with E[X] = x, and Var[X] = 2 * x
EPC.Moment.Gamma <- fMoment(USh.m, s = 1:40, theta = 2, cap = 5)
EPC.Moment.Gamma
plot(EPC.Moment.Gamma, type = "l", ylim = c(0, 1))
# Compare the two results:
plot(EPC.Moment, type = "l", main = "Exceedance Probability Curve", ylim = c(0, 1))
lines(EPC.Moment.Gamma, col = 2, lty = 2)
legend("topright", c("Dirac Delta", expression(paste("Gamma(",
alpha[i] == 1 / theta^2, ", ", beta[i] ==1 / (x[i] * theta^2), ")", " cap =", 5))),
lwd = 2, lty = 1:2, col = 1:2)

Monte Carlo Simulations.

Description

Function to estimate the total losses via the Monte Carlo simulations.

Usage

fMonteCarlo(ELT, s, t = 1, theta = 0, cap = Inf, nsim = 10000,
  verbose = FALSE)

Arguments

Value

If verbose = FALSE the function returns a numeric matrix, containing in the first column the pre-specified losses s, and the estimated exceedance probabilities in the second column. If verbose = TRUE the function returns a numeric matrix containing four columns. The first column contains the losses s, the second column contains the estimated exceedance probabilities, the other columns contain the 95% confidence bands. The attributes of this matrix are a vector simS containing the simulated losses.

Examples

data(UShurricane)

# Compress the table to millions of dollars

USh.m <- compressELT(ELT(UShurricane), digits = -6)
EPC.MonteCarlo <- fMonteCarlo(USh.m, s = 1:40, verbose = TRUE)
EPC.MonteCarlo
par(mfrow = c(1, 2))
plot(EPC.MonteCarlo[, 1:2], type = "l", ylim = c(0, 1))
matlines(EPC.MonteCarlo[, -2], ylim = c(0, 1), lty = 2, col = 1)
# Assuming the losses follow a Gamma with E[X] = x, and Var[X] = 2 * x and cap = 5m
EPC.MonteCarlo.Gamma <- fMonteCarlo(USh.m, s = 1:40, theta = 2, cap = 5, verbose = TRUE)
EPC.MonteCarlo.Gamma
plot(EPC.MonteCarlo.Gamma[, 1:2], type = "l", ylim = c(0, 1))
matlines(EPC.MonteCarlo.Gamma[, -2], ylim = c(0,1), lty = 2, col = 1)
# Compare the two results:
par(mfrow = c(1, 1))
plot(EPC.MonteCarlo[, 1:2], type = "l", main = "Exceedance Probability Curve",
ylim = c(0, 1))
lines(EPC.MonteCarlo.Gamma[, 1:2], col = 2, lty = 2)
legend("topright", c("Dirac Delta", expression(paste("Gamma(",
alpha[i] == 1 / theta^2, ", ", beta[i] ==1 / (x[i] * theta^2), ")", " cap =", 5))),
lwd = 2, lty = 1:2, col = 1:2)

Panjer Recursion.

Description

Function to calculate the total losses via the Panjer recursion.

Usage

fPanjer(ELT, s, t = 1, theta = 0, cap = Inf, nq = 10, verbose = FALSE)

Arguments

Value

A numeric matrix containing the pre-specified losses s in the first column and the exceedance probabilities in the second column.

References

Panjer, H.H. (1980), ‘The aggregate claims distribution and stop-loss reinsurance,’ Transactions of the Society of Actuaries, 32, 523-545.

Examples

data(UShurricane)

# Compress the table to millions of dollars

USh.m <- compressELT(ELT(UShurricane), digits = -6)

EPC.Panjer <- fPanjer(USh.m, s = 1:40, verbose = TRUE)
EPC.Panjer
plot(EPC.Panjer, type = "l", ylim = c(0,1))
# Assuming the losses follow a Gamma with E[X] = x, and Var[X] = 2 * x and cap = 5m

EPC.Panjer.Gamma <- fPanjer(USh.m, s = 1:40, theta = 2, cap = 5, verbose = TRUE)
EPC.Panjer.Gamma
plot(EPC.Panjer.Gamma, type = "l", ylim = c(0,1))

# Compare the two results:

plot(EPC.Panjer, type = "l", main = 'Exceedance Probability Curve',
ylim = c(0, 1))
lines(EPC.Panjer.Gamma, col = 2, lty = 2)
legend("topright", c("Dirac Delta", expression(paste("Gamma(",
alpha[i] == 1 / theta^2, ", ", beta[i] ==1 / (x[i] * theta^2), ")", " cap =", 5))),
lwd = 2, lty = 1:2, col = 1:2)

Summary statistics for class ELT.

Description

Summary statistics for class ELT.

Usage

## S3 method for class 'ELT'
summary(object, theta = 0, cap = Inf, t = 1, ...)

Arguments

Value

A list containing the data summary, and the means and the standard deviations of N, Y, and S.

Examples

data(UShurricane)
summary(ELT(UShurricane))

Function for zooming onto a matplot(x, y, ...).

Description

Function for zooming onto a matplot(x, y, ...).

Usage

zoombox(x, y, x0, y0 = c(0, 0.05), y1 = c(0.1, 0.6), ...)

Arguments

See Also

matplot, plot

Examples

data(UShurricane)

# Compress the table to millions of dollars

USh.m <- compressELT(ELT(UShurricane), digits = -6)
s <- seq(1,40)
EPC <- matrix(NA, length(s), 6)
colnames(EPC) <- c("Panjer", "MonteCarlo", "Markov",
 "Cantelli", "Moment", "Chernoff")
EPC[, 1] <- fPanjer(USh.m, s = s)[, 2]
EPC[, 2] <- fMonteCarlo(USh.m, s = s)[, 2]
EPC[, 3] <- fMarkov(USh.m, s = s)[, 2]
EPC[, 4] <- fCantelli(USh.m, s = s)[, 2]
EPC[, 5] <- fMoment(USh.m, s = s)[, 2]
EPC[, 6] <- fChernoff(USh.m, s = s)[, 2]
matplot(s, EPC, type = "l", lwd = 2, xlab = "s", ylim = c(0, 1), lty = 1:6,
  ylab = expression(plain(Pr)(S>=s)), main = "Exceedance Probability Curve")
zoombox(s, EPC, x0 = c(30, 40), y0 = c(0, .1), y1 = c(.3, .6), type = "l", lwd = 2, lty = 1:6)
legend("topright", legend = colnames(EPC), lty = 1:6, col = 1:6, lwd = 2)