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+++ b/inst/quotamack.m
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+## Copyright (C) 2009 Esteban Cervetto <estebancster@gmail.com>
+##
+## Octave is free software; you can redistribute it and/or modify it
+## under the terms of the GNU General Public License as published by
+## the Free Software Foundation; either version 3 of the License, or (at
+## your option) any later version.
+##
+## Octave is distributed in the hope that it will be useful, but
+## WITHOUT ANY WARRANTY; without even the implied warranty of
+## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
+## General Public License for more details.
+##
+## You should have received a copy of the GNU General Public License
+## along with Octave; see the file COPYING.  If not, see
+## <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {@var{quotas} =} quotaad (@var{s},@var{v})
+## Calculate the cumulative quotas by the Mack 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 a mx1 vector of known volume measures (like premiums or the number of contracts).
+##  
+## The Mack method asumes that exists a vector @var{v} and a vector P(i) 1<=i<=m of parameters 
+## such that holds for all i = {1,...,m} the next identity:
+##
+## @group
+## @example
+## ultimate(i) = V(i)*P(i)
+## @end example
+## @end group
+##
+## where
+## 
+## @group
+## @example
+##                   l=n-1            
+## P(i)= O_mack(i) *   E   IRL_Mack(l)
+##                    l=0             
+## @end example
+## @end group
+##
+## ,
+##
+## @group
+## @example
+##                   l=n-k-1             
+##                     E     Z(j,k)      
+##                    j=0                
+## IRL_Mack(i)   =  ---------------------
+##                   l=n-k-1                 
+##                     E   V(i)*O_Mack(l)    
+##                    l=0                
+## @end example
+## @end group
+##
+## and
+##
+## @group
+## @example
+##                   l=n-i-1                                                      
+##                     E     Z(i,l)                                               
+##                    l=0                                                         
+## O_Mack(i)     =  ------------------                                            
+##                   l=n-1                                                         
+##                     E   V(i)*IRL(l)    (see IRL definition in quotaad function)
+##                    l=0                                                               
+## @end example
+## @end group
+## 
+## 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 formula is:
+## @group
+## @example
+##                    l=k               
+##                     E   IRL_Mack(l)  
+##                    l=0                
+## @var{quotas}(k) =  ------------------
+##                   l=n-1              
+##                     E   IRL_Mack(l)  
+##                    l=0               
+## @end example
+## @end group
+## 
+## @seealso {bferguson, quotald, quotapanning, quotaad}
+## @end deftypefn
+
+## Author: Act. Esteban Cervetto ARG <estebancster@gmail.com>
+##
+## Maintainer: Act. Esteban Cervetto ARG <estebancster@gmail.com>
+##
+## Created: jul-2009
+##
+## Version: 1.1.0 
+##
+## Keywords: actuarial reserves insurance bornhuetter ferguson chainladder
+
+function [quotas] = quotamack (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
+
+# calculate empirical individual loss ratios
+a = repmat (V,1,n);
+LRI = Z ./ a;
+
+# weights V(i)/sum(1,n-k,V(i)) 
+num =fliplr(triu(fliplr(a),-u));            #numerator and clean low triangle
+den = repmat(sum(num),m,1);                 #denominator
+den = fliplr(triu(fliplr(den),-u));         #clean low triangle
+W = num./den;                               #divide by
+W = fliplr(triu(fliplr(W),-u));
+
+# incremental Loss Ratios AD
+LRI_AD  = diag(LRI' * W)';                  #weighted product
+
+if (u==0)
+b = (diag(fliplr(S),-u) ./ flipud(cumsum(LRI_AD)') ) ./ V;
+else
+b = ([S(1:u,n); diag(fliplr(S),-u)] ./ [sum(LRI_AD)*ones(1,u);flipud(cumsum(LRI_AD)')] ) ./ V;
+end
+
+sZ = sum (Z);                              #sum of Z
+sb = repmat(b,1,n);
+sb = fliplr(triu(fliplr(sb),-u));
+sV = repmat(V,1,n);
+sV = fliplr(triu(fliplr(sV),-u));
+
+LRI_Mack = sZ ./ (diag(sb'*sV))';
+quotas = cumsum(porcentual(LRI_Mack));     #calculate cumulated  quota  
+
+end