## Copyright (C) 2009 Esteban Cervetto
##
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## under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 3 of the License, or (at
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##
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## .
## -*- texinfo -*-
## @deftypefn {Function File} {@var{quotas} =} quotaad (@var{s},@var{v})
## Calculate the cumulative quotas by the Additive method.
##
## @var{s} is a mxn matrix that contains the run-off triangle, where m is the number of accident-years
## and n is the number of periods to final development. @var{s} may contain u = m-n complete years.
## The value @var{s}(i,k), 1<=i<=m, 0<=k<=n-1 represents the cumulative losses from accident-period i
## settled with a delay of at most k years.
## The values @var{s}(i,k) with i + k > m must be zero because is future time.
## @var{v} is an mx1 vector of known volume measures (like premiums or the number of contracts).
##
## The Additive method asumes that exists a development pattern on the incremental loss ratios (IRL).
## This means that the identity
## @group
## @example
## E[Z(i,k) ]
## IRL(k) = ------------
## V(i)
## @end example
## @end group
## holds for all k = {0,...,n-1} and for all i = {1,...,m}.
## Z represents the incremental losses; then losses satisfy
## Z(k) = (S(k) - S(k-1) ),Z(0) = S(0) for all i = {1,...,m}.
##
## @var{quotas} returns a row vector with the cumulative quotas. The transformation
## from incremental loss ratios to cumulative quotas is:
## @group
## @example
## l=k
## E IRL(l)
## l=0
## @var{quotas}(k) = -----------
## l=n-1
## E IRL(l)
## l=0
## @end example
## @end group
##
## @seealso {bferguson, quotald, quotapanning}
## @end deftypefn
## Author: Act. Esteban Cervetto ARG
##
## Maintainer: Act. Esteban Cervetto ARG
##
## Created: jul-2009
##
## Version: 1.1.0
##
## Keywords: actuarial reserves insurance bornhuetter ferguson chainladder
function [quotas] = quotaad (S,V)
[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)
u = m - n; #rows of the upper square
S = fliplr(triu(fliplr(S),-u)); #ensure S is triangular
if (size(V) ~= [m,1])
usage(strcat("volume V must be of size [",num2str(m),",1]" ));
end
# Z triangle
Z = [S(:,1), S(:,2:n)-S(:,1:n-1)];
Z = fliplr(triu(fliplr(Z),-u)); #clean Z
# calc the empirical incremental loss ratios
LRI = Z ./ repmat (V,1,n);
#weights V(i)/sum(1,n-k,V(i))
W = repmat(V,1,n); #numerator
W =fliplr(triu(fliplr(W),-u)); #clean low triangle
a = repmat(sum(W),m,1); #denominator
a = fliplr(triu(fliplr(a),-u)); #clean low triangle
W = W./a; #divide by
W = fliplr(triu(fliplr(W),-u));
# incremental Loss Ratios AD
LRI_AD = diag(LRI' * W)'; #weighted product
quotas = cumsum(porcentual(LRI_AD)); #calc cumulated quota
end