fChernoff: Chernoff Bound.

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

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

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.
nk	Number of optimisation points.
verbose	Logical. If TRUE attaches the minimising index. The default is verbose = FALSE.

Details

Chernoff's inequality states:

Pr(S ≥ s) ≤ 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)

Example output

Loading required package: MASS
       s Upper Pr[S>=s]
 [1,]  1    1.000000000
 [2,]  2    1.000000000
 [3,]  3    1.000000000
 [4,]  4    1.000000000
 [5,]  5    1.000000000
 [6,]  6    1.000000000
 [7,]  7    0.985087693
 [8,]  8    0.942110629
 [9,]  9    0.880481138
[10,] 10    0.807590507
[11,] 11    0.729247718
[12,] 12    0.649803465
[13,] 13    0.572377239
[14,] 14    0.499003006
[15,] 15    0.431120582
[16,] 16    0.369417774
[17,] 17    0.314232879
[18,] 18    0.265438219
[19,] 19    0.222818456
[20,] 20    0.185932996
[21,] 21    0.154357376
[22,] 22    0.127503946
[23,] 23    0.104779232
[24,] 24    0.085728478
[25,] 25    0.069850533
[26,] 26    0.056690646
[27,] 27    0.045837807
[28,] 28    0.036911505
[29,] 29    0.029622453
[30,] 30    0.023698690
[31,] 31    0.018906111
[32,] 32    0.015034063
[33,] 33    0.011914006
[34,] 34    0.009421061
[35,] 35    0.007429420
[36,] 36    0.005839889
[37,] 37    0.004583932
[38,] 38    0.003584876
[39,] 39    0.002799363
[40,] 40    0.002180146
       s Upper Pr[S>=s]
 [1,]  1      1.0000000
 [2,]  2      1.0000000
 [3,]  3      1.0000000
 [4,]  4      1.0000000
 [5,]  5      1.0000000
 [6,]  6      1.0000000
 [7,]  7      0.9643082
 [8,]  8      0.9543254
 [9,]  9      0.9444459
[10,] 10      0.9346687
[11,] 11      0.9249927
[12,] 12      0.9154169
[13,] 13      0.9059402
[14,] 14      0.8965617
[15,] 15      0.8872802
[16,] 16      0.8780948
[17,] 17      0.8690045
[18,] 18      0.8600083
[19,] 19      0.8511052
[20,] 20      0.8422943
[21,] 21      0.8335747
[22,] 22      0.8249452
[23,] 23      0.8164052
[24,] 24      0.8079535
[25,] 25      0.7995893
[26,] 26      0.7913117
[27,] 27      0.7831198
[28,] 28      0.7750127
[29,] 29      0.7669896
[30,] 30      0.7590494
[31,] 31      0.7511915
[32,] 32      0.7434150
[33,] 33      0.7357189
[34,] 34      0.7281025
[35,] 35      0.7205650
[36,] 36      0.7131055
[37,] 37      0.7057232
[38,] 38      0.6984173
[39,] 39      0.6911871
[40,] 40      0.6840317

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