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## Copyright (C) 2009 Esteban Cervetto <estebancster@gmail.com>
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##
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## Octave is free software; you can redistribute it and/or modify it
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## under the terms of the GNU General Public License as published by
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## the Free Software Foundation; either version 3 of the License, or (at
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## your option) any later version.
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##
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## Octave is distributed in the hope that it will be useful, but
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## WITHOUT ANY WARRANTY; without even the implied warranty of
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## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
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## General Public License for more details.
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##
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## You should have received a copy of the GNU General Public License
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## along with Octave; see the file COPYING.  If not, see
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## <http://www.gnu.org/licenses/>.
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## -*- texinfo -*-
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## @deftypefn {Function File} {@var{quotas} =} quotald (@var{s})
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## Calculate the cumulative quotas by the Loss Development (Chainladder) method.
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##
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## @var{s} is a mxn matrix that contains the run-off triangle, where m is the number of accident-years
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## and n is the number of periods to final development. @var{s} may contain u = m-n complete years.
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## The value @var{s}(i,k), 1<=i<=m, 0<=k<=n-1 represents the cumulative losses from accident-period i
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## settled with a delay of at most k years. 
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## The values @var{s}(i,k) with i + k > m must be zero because is future time. 
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##
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## The LD method asumes that exists a development pattern on the individual factors.
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## This means that the identity 
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## @group
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## @example
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##             E[S(i,k) ]
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## LDI(k) =   -------------
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##            E[S(i,k-1) ]
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## @end example
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## @end group
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## holds for all k = {0,...,n-1} and for all i = {1,...,m}.
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##
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## @var{quotas} returns a row vector with the cumulative quotas. The transformation
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## from individual factors to cumulative quotas is:
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## @group
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## @example
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##                    l=n-1    1
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## @var{quotas}(k) =  II    -------
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##                    l=k+1    LDI(l) 
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## @end example
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## @end group
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##
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## @seealso {bferguson, ultimateld, quotapanning, quotaad, quotamack}
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## @end deftypefn
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## Author: Act. Esteban Cervetto ARG <estebancster@gmail.com>
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##
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## Maintainer: Act. Esteban Cervetto ARG <estebancster@gmail.com>
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##
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## Created: jul-2009
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##
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## Version: 1.1.0 
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##
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## Keywords: actuarial reserves insurance bornhuetter ferguson chainladder
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function quotas = quotald(S)
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[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)
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u = m - n;                                     #rows of the upper square
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S = fliplr(triu(fliplr(S),-u));                   #ensure S is triangular  
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# calculate the triangle of individual development factors (LDI).
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LDI = [ones(m,1), S(:,2:n)./S(:,1:n-1)];
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LDI = fliplr(triu(fliplr(LDI),-u));
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LDI (m,1) = 0;                     #last row element without partner
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# weights
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W =  fliplr(triu(fliplr(S),1-u));  #get T values to use
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W =  shift (W,1,2);                #redim k = k-1, 
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W = porcentual(W,1);
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#individual development factors (LDI) or Chainladder factors
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LDI_CL  = diag(LDI' * W)';                 #weighted product
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quotas = 1./cumprod(fliplr(LDI_CL));       #calcs cumulated quota
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quotas (n) = 1;                            #last value is 1
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quotas = fliplr(shift(quotas,1));
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end

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