1## Copyright (C) 2009 Esteban Cervetto <estebancster@gmail.com> 2## 3## Octave is free software; you can redistribute it and/or modify it 4## under the terms of the GNU General Public License as published by 5## the Free Software Foundation; either version 3 of the License, or (at 6## your option) any later version. 7## 8## Octave is distributed in the hope that it will be useful, but 9## WITHOUT ANY WARRANTY; without even the implied warranty of 10## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 11## General Public License for more details. 12## 13## You should have received a copy of the GNU General Public License 14## along with Octave; see the file COPYING. If not, see 15## <http://www.gnu.org/licenses/>. 16 17## -*- texinfo -*- 18## @deftypefn {Function File} {@var{quotas} =} quotaad (@var{s},@var{v}) 19## Calculate the cumulative quotas by the Mack method. 20## 21## @var{s} is a mxn matrix that contains the run-off triangle, where m is the number of accident-years 22## and n is the number of periods to final development. @var{s} may contain u = m-n complete years. 23## The value @var{s}(i,k), 1<=i<=m, 0<=k<=n-1 represents the cumulative losses from accident-period i 24## settled with a delay of at most k years. 25## The values @var{s}(i,k) with i + k > m must be zero because is future time. 26## @var{v} is a mx1 vector of known volume measures (like premiums or the number of contracts). 27## 28## The Mack method asumes that exists a vector @var{v} and a vector P(i) 1<=i<=m of parameters 29## such that holds for all i = {1,...,m} the next identity: 30## 31## @group 32## @example 33## ultimate(i) = V(i)*P(i) 34## @end example 35## @end group 36## 37## where 38## 39## @group 40## @example 41## l=n-1 42## P(i)= O_mack(i) * E IRL_Mack(l) 43## l=0 44## @end example 45## @end group 46## 47## , 48## 49## @group 50## @example 51## l=n-k-1 52## E Z(j,k) 53## j=0 54## IRL_Mack(i) = --------------------- 55## l=n-k-1 56## E V(i)*O_Mack(l) 57## l=0 58## @end example 59## @end group 60## 61## and 62## 63## @group 64## @example 65## l=n-i-1 66## E Z(i,l) 67## l=0 68## O_Mack(i) = ------------------ 69## l=n-1 70## E V(i)*IRL(l) (see IRL definition in quotaad function) 71## l=0 72## @end example 73## @end group 74## 75## Z represents the incremental losses; then losses satisfy 76## Z(k) = (S(k) - S(k-1) ),Z(0) = S(0) for all i = {1,...,m}. 77## 78## @var{quotas} returns a row vector with the cumulative quotas. The formula is: 79## @group 80## @example 81## l=k 82## E IRL_Mack(l) 83## l=0 84## @var{quotas}(k) = ------------------ 85## l=n-1 86## E IRL_Mack(l) 87## l=0 88## @end example 89## @end group 90## 91## @seealso {bferguson, quotald, quotapanning, quotaad} 92## @end deftypefn 93 94## Author: Act. Esteban Cervetto ARG <estebancster@gmail.com> 95## 96## Maintainer: Act. Esteban Cervetto ARG <estebancster@gmail.com> 97## 98## Created: jul-2009 99## 100## Version: 1.1.0 101## 102## Keywords: actuarial reserves insurance bornhuetter ferguson chainladder 103 104function [quotas] = quotamack (S,V) 105 106[m,n] = size (S); #triangle with m years (i=1,2,u,...u+1,u+2,....m) and n periods (k=0,1,2,...n-1) 107u = m - n; #rows of the upper square 108S = fliplr(triu(fliplr(S),-u)); #ensure S is triangular 109 110if (size(V) ~= [m,1]) 111 usage(strcat("volume V must be of size [",num2str(m),",1]" )); 112end 113 114# Z triangle 115Z = [S(:,1), S(:,2:n)-S(:,1:n-1)]; 116Z = fliplr(triu(fliplr(Z),-u)); #clean Z 117 118# calculate empirical individual loss ratios 119a = repmat (V,1,n); 120LRI = Z ./ a; 121 122# weights V(i)/sum(1,n-k,V(i)) 123num =fliplr(triu(fliplr(a),-u)); #numerator and clean low triangle 124den = repmat(sum(num),m,1); #denominator 125den = fliplr(triu(fliplr(den),-u)); #clean low triangle 126W = num./den; #divide by 127W = fliplr(triu(fliplr(W),-u)); 128 129# incremental Loss Ratios AD 130LRI_AD = diag(LRI' * W)'; #weighted product 131 132if (u==0) 133b = (diag(fliplr(S),-u) ./ flipud(cumsum(LRI_AD)') ) ./ V; 134else 135b = ([S(1:u,n); diag(fliplr(S),-u)] ./ [sum(LRI_AD)*ones(1,u);flipud(cumsum(LRI_AD)')] ) ./ V; 136end 137 138sZ = sum (Z); #sum of Z 139sb = repmat(b,1,n); 140sb = fliplr(triu(fliplr(sb),-u)); 141sV = repmat(V,1,n); 142sV = fliplr(triu(fliplr(sV),-u)); 143 144LRI_Mack = sZ ./ (diag(sb'*sV))'; 145quotas = cumsum(porcentual(LRI_Mack)); #calculate cumulated quota 146 147end 148