fMarkov: Markov Bound.

In tailloss: Estimate the Probability in the Upper Tail of the Aggregate Loss Distribution

Description

Function to bound the total losses via the Markov inequality.

Usage

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

Arguments

ELT	Data frame containing 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.
s	Scalar or numeric vector containing the total losses of interest.
t	Scalar representing the time period of interest. The default value is t = 1.
theta	Scalar containing information about the variance of the Gamma distribution: sd[X] = x * theta. The default value is theta = 0: the loss associated to an event is considered as a constant.
cap	Scalar representing the financial cap on losses for a single event, i.e. the maximum possible loss caused by a single event. The default value is cap = Inf.

Details

Cantelli's inequality states:

Pr( S ≥ s) ≤ 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)

Example output

Loading required package: MASS
Warning message:
In ES/s :
  Recycling array of length 1 in array-vector arithmetic is deprecated.
  Use c() or as.vector() instead.

Warning message:
In ES/s :
  Recycling array of length 1 in array-vector arithmetic is deprecated.
  Use c() or as.vector() instead.

       s Upper Pr[S>=s]
 [1,]  1     1.00000000
 [2,]  2     1.00000000
 [3,]  3     1.00000000
 [4,]  4     0.86068333
 [5,]  5     0.68854667
 [6,]  6     0.57378889
 [7,]  7     0.49181905
 [8,]  8     0.43034167
 [9,]  9     0.38252593
[10,] 10     0.34427333
[11,] 11     0.31297576
[12,] 12     0.28689444
[13,] 13     0.26482564
[14,] 14     0.24590952
[15,] 15     0.22951556
[16,] 16     0.21517083
[17,] 17     0.20251373
[18,] 18     0.19126296
[19,] 19     0.18119649
[20,] 20     0.17213667
[21,] 21     0.16393968
[22,] 22     0.15648788
[23,] 23     0.14968406
[24,] 24     0.14344722
[25,] 25     0.13770933
[26,] 26     0.13241282
[27,] 27     0.12750864
[28,] 28     0.12295476
[29,] 29     0.11871494
[30,] 30     0.11475778
[31,] 31     0.11105591
[32,] 32     0.10758542
[33,] 33     0.10432525
[34,] 34     0.10125686
[35,] 35     0.09836381
[36,] 36     0.09563148
[37,] 37     0.09304685
[38,] 38     0.09059825
[39,] 39     0.08827521
[40,] 40     0.08606833

tailloss documentation built on May 2, 2019, 4:18 a.m.