| Library: | insurance |
| See also: | INSpremgam INSpremwei INSprempareto INSpremlogn INSpremburr |
| Quantlet: | INSpremnorm | |
| Description: | returns the values of premium in the case of normal distribution of losses. The premium is as an approximation for some cummulated claim. |
| Usage: | y = INSpremnorm (mu, sigma, param, ind) | |
| Input: | ||
| mu | scalar, mean of the normal distribution | |
| lambda | scalar, standard deviation of the normal distribution | |
| param | array, if pure risk premium with security loading: relative security loading, if premium with variance or standard deviation: the coefficient, if quantile premium: quantile parameter, if exponential utility function: risk aversion parameter | |
| ind | scalar, if ind=1, pure risk premium if ind=2, safety loaded premium if ind=3, variance loaded premium if ind=4, standard deviation loaded premium if ind=5, quantile premium if ind=6, exponential premium | |
| Output: | ||
| y | array, value of the premium for normal distribution of loss | |
library("insurance")
x =(0:10)
INSpremnorm(3, 1, x, 4)
Contents of y [ 1,] 3 [ 2,] 4 [ 3,] 5 [ 4,] 6 [ 5,] 7 [ 6,] 8 [ 7,] 9 [ 8,] 10 [ 9,] 11 [10,] 12 [11,] 13
library("distribs")
library("insurance")
library("plot")
numberOfPolices = 500
probOfClaim = 0.05
gammaAlpha = 0.9185
gammaBeta = 5.6870e-09
gammaExp = gammaAlpha / gammaBeta
gammaVar = gammaAlpha / gammaBeta^2
aggregatedExpected = gammaAlpha/gammaBeta * numberOfPolices*probOfClaim
aggregetedVariance = numberOfPolices*(gammaExp^2*probOfClaim*(1-probOfClaim) + probOfClaim*gammaVar)
;Quantile premium for simulated data and the normal approximation
xaxis =(1:80)*1e-11
;Exponential premium for the approximation
p = INSpremnorm(aggregatedExpected, sqrt(aggregetedVariance), xaxis, 6) /1e9
; Exponential premium for the sum of gammas
r = numberOfPolices*log(1-probOfClaim + probOfClaim*(1-1/gammaBeta.*xaxis)^-gammaAlpha)./xaxis /1e9
s1 = setmask(xaxis~r,"line", "blue")
s2 = setmask(xaxis~p,"line", "red", "dashed")
plotdisplay = createdisplay(1,1)
show(plotdisplay,1,1,s1,s2)
setgopt(plotdisplay,1,1,"yvalue",0|1)
setgopt(plotdisplay,1,1,"xvalue",0|1e-10)
setgopt(plotdisplay,1,1,"xlabel","Risk aversion parameter","ylabel","Exponential utility premuim(USD billion)")
setgopt(plotdisplay,1,1, "border", 0)
Plot of exponential premium in a cummulative risk model (exact and through normal approximation) for the whole portfolio.