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} =} quotald (@var{s}) 19## Calculate the cumulative quotas by the Loss Development (Chainladder) 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## 27## The LD method asumes that exists a development pattern on the individual factors. 28## This means that the identity 29## @group 30## @example 31## E[S(i,k) ] 32## LDI(k) = ------------- 33## E[S(i,k-1) ] 34## @end example 35## @end group 36## holds for all k = {0,...,n-1} and for all i = {1,...,m}. 37## 38## @var{quotas} returns a row vector with the cumulative quotas. The transformation 39## from individual factors to cumulative quotas is: 40## @group 41## @example 42## l=n-1 1 43## @var{quotas}(k) = II ------- 44## l=k+1 LDI(l) 45## @end example 46## @end group 47## 48## @seealso {bferguson, ultimateld, quotapanning, quotaad, quotamack} 49## @end deftypefn 50 51## Author: Act. Esteban Cervetto ARG <estebancster@gmail.com> 52## 53## Maintainer: Act. Esteban Cervetto ARG <estebancster@gmail.com> 54## 55## Created: jul-2009 56## 57## Version: 1.1.0 58## 59## Keywords: actuarial reserves insurance bornhuetter ferguson chainladder 60 61function quotas = quotald(S) 62 63[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) 64u = m - n; #rows of the upper square 65S = fliplr(triu(fliplr(S),-u)); #ensure S is triangular 66 67# calculate the triangle of individual development factors (LDI). 68LDI = [ones(m,1), S(:,2:n)./S(:,1:n-1)]; 69LDI = fliplr(triu(fliplr(LDI),-u)); 70LDI (m,1) = 0; #last row element without partner 71 72# weights 73W = fliplr(triu(fliplr(S),1-u)); #get T values to use 74W = shift (W,1,2); #redim k = k-1, 75W = porcentual(W,1); 76 77#individual development factors (LDI) or Chainladder factors 78LDI_CL = diag(LDI' * W)'; #weighted product 79quotas = 1./cumprod(fliplr(LDI_CL)); #calcs cumulated quota 80quotas (n) = 1; #last value is 1 81quotas = fliplr(shift(quotas,1)); 82 83end 84