TomSym Financing an Early Retirement Scheme - Revision history

Mbot at 13:33, 8 November 2011

2011-11-08T13:33:29Z

← Older revision		Revision as of 09:33, 8 November 2011
Line 150:		Line 150:

	FuncEv 15 Iter 17		FuncEv 15 Iter 17
	Elapsed time: 0.~~012000~~ sec.		Elapsed time: 0.007000 sec.
	buy 911 bonds of type 1		buy 911 bonds of type 1
	buy 747 bonds of type 2		buy 747 bonds of type 2

Mbot at 21:24, 2 November 2011

2011-11-02T21:24:41Z

← Older revision		Revision as of 17:24, 2 November 2011
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	Amounts required every year		Amounts required every year

			<pre>
	+------------------+----+---+---+---+---+----+---+		+------------------+----+---+---+---+---+----+---+
	\|Year \| 1\| 2\| 3\| 4\| 5\| 6\| 7\|		\|Year \| 1\| 2\| 3\| 4\| 5\| 6\| 7\|
Line 12:		Line 13:
	\|Amount (in 1000 $)\|1000\|600\|640\|480\|760\|1020\|950\|		\|Amount (in 1000 $)\|1000\|600\|640\|480\|760\|1020\|950\|
	+------------------+----+---+---+---+---+----+---+		+------------------+----+---+---+---+---+----+---+
			</pre>

	For the investments, the bank decides to take three different types of bonds, SNCF bonds, Fujitsu bonds, and Treasury bonds. The money that is not invested in these bonds is placed as savings with a guaranteed yield of 3.2%. The table below lists all information concerning the yield and the durations of the bonds and also the value of a bond. In this type of bond, it is only possible to buy an integer number of bonds, and the invested capital remains locked in for the total duration of the bond. Every year, only the interest on the capital is distributed. The person in charge of the retirement plan decides to buy bonds at the beginning of the first year, but not in the following years. How should he organize the investments in order to spend the least amount of money to cover the projected retirement plan?		For the investments, the bank decides to take three different types of bonds, SNCF bonds, Fujitsu bonds, and Treasury bonds. The money that is not invested in these bonds is placed as savings with a guaranteed yield of 3.2%. The table below lists all information concerning the yield and the durations of the bonds and also the value of a bond. In this type of bond, it is only possible to buy an integer number of bonds, and the invested capital remains locked in for the total duration of the bond. Every year, only the interest on the capital is distributed. The person in charge of the retirement plan decides to buy bonds at the beginning of the first year, but not in the following years. How should he organize the investments in order to spend the least amount of money to cover the projected retirement plan?
Line 17:		Line 19:
	Information about loans		Information about loans

			<pre>
	+--------+----------------------+---------+--------+		+--------+----------------------+---------+--------+
	\|Loan \|Value of bonds (in k$)\| Interest\|Duration\|		\|Loan \|Value of bonds (in k$)\| Interest\|Duration\|
Line 24:		Line 27:
	\|Treasury\| 0.5 \| 6.5% \| 6 years\|		\|Treasury\| 0.5 \| 6.5% \| 6 years\|
	+--------+----------------------+---------+--------+		+--------+----------------------+---------+--------+
			</pre>

	==Variables==		==Variables==

			<pre>
	amounts Retirement costs		amounts Retirement costs
	baseinterest Interest from money not invested in bonds		baseinterest Interest from money not invested in bonds
Line 32:		Line 37:
	interest Interest from bonds		interest Interest from bonds
	duration Years to place the money		duration Years to place the money
			</pre>

	==Reference==		==Reference==

Mbot: Created page with "{{Part Of Manual|title=the TomSym Manual|link=TomSym Manual}} ==Problem description== The National Agricultural Bank (NAB) decides to establish an early retirement s..."

2011-11-02T21:01:50Z

Created page with "{{Part Of Manual|title=the TomSym Manual|link=TomSym Manual}} ==Problem description== The National Agricultural Bank (NAB) decides to establish an early retirement s..."

New page

{{Part Of Manual|title=the TomSym Manual|link=[[TomSym|TomSym Manual]]}}

==Problem description==

The National Agricultural Bank (NAB) decides to establish an early retirement scheme for fifteen employees who are taking early retirement. These employees will retire over a period of seven years starting from the next year. To finance this early retirement scheme, the bank decides to invest in bonds for this seven-year period. The necessary amounts to cover the pre-retirement leavers are given in the following table; they have to be paid at the beginning of every year.

Amounts required every year

+------------------+----+---+---+---+---+----+---+
|Year | 1| 2| 3| 4| 5| 6| 7|
+------------------+----+---+---+---+---+----+---+
|Amount (in 1000 $)|1000|600|640|480|760|1020|950|
+------------------+----+---+---+---+---+----+---+

For the investments, the bank decides to take three different types of bonds, SNCF bonds, Fujitsu bonds, and Treasury bonds. The money that is not invested in these bonds is placed as savings with a guaranteed yield of 3.2%. The table below lists all information concerning the yield and the durations of the bonds and also the value of a bond. In this type of bond, it is only possible to buy an integer number of bonds, and the invested capital remains locked in for the total duration of the bond. Every year, only the interest on the capital is distributed. The person in charge of the retirement plan decides to buy bonds at the beginning of the first year, but not in the following years. How should he organize the investments in order to spend the least amount of money to cover the projected retirement plan?

Information about loans

+--------+----------------------+---------+--------+
|Loan |Value of bonds (in k$)| Interest|Duration|
+--------+----------------------+---------+--------+
|SNCF | 1.0 | 7.0% | 5 years|
|Fujitsu | 0.8 | 7.0% | 4 years|
|Treasury| 0.5 | 6.5% | 6 years|
+--------+----------------------+---------+--------+

==Variables==

amounts Retirement costs
baseinterest Interest from money not invested in bonds
values Values of a bond
interest Interest from bonds
duration Years to place the money

==Reference==

Applications of optimization... Gueret, Prins, Seveaux

<source lang="matlab">
% Marcus Edvall, Tomlab Optimization Inc, E-mail: tomlab@tomopt.com
% Copyright (c) 2005-2009 by Tomlab Optimization Inc., $Release: 7.2.0$
% Written Oct 7, 2005. Last modified Apr 8, 2009.
</source>

==Problem setup==

<source lang="matlab">
amounts = [1000 600 640 480 760 1020 950]'*1000;
baseinterest = 3.2;
values = [1000 800 500]';
interest = [7.0 7.0 6.5]';
duration = [5.0 4.0 6.0]';

b = length(values); % Bonds to buy
t = length(amounts)-1; % Invest occasions

buy = tom('buy',b,1,'int');
invest = tom('invest',t,1);
capital = tom('capital',1,1);

% Some variables are integer.
bnds = {capital >= 0, invest >= 0, buy >= 0};

% First year constr
con1 = {capital - sum(values.*buy) - invest(1) == amounts(1)};

% 2nd - 4th year constr.
con2 = cell(3,1);
for i=1:3
con2{i} = {values'.*interest'/100*buy + ...
invest(i)*(1+baseinterest/100) - invest(i+1) == ...
amounts(i+1)};
end

% 5th - 6th year constr.
con3 = cell(2,1);
for i=1:2
idx1 = find(duration == i+3);
idx2 = find(duration >= i+4);
con3{i} = {values(idx1)'.*(1+interest(idx1)'/100)*buy(idx1) + ...
values(idx2)'.*interest(idx2)'/100*buy(idx2) + ...
(1+baseinterest/100)*invest(i+3) - invest(i+4) == ...
amounts(i+4)};
end

% 7th year constr.
idx = find(duration == 6);
con4 = {values(idx)'.*(1+interest(idx)'/100)*buy(idx) + ...
(1+baseinterest/100)*invest(6) == amounts(7)};

% Objective
objective = capital;

constraints = {bnds, con1, con2, con3, con4};
options = struct;
options.solver = 'cplex';
options.name = 'Financing an Early Retirement Scheme';
Prob = sym2prob('mip',objective,constraints,[],options);
Prob.MIP.cpxControl.EPGAP = 1e-10;

PriLev = 1;
Result = tomRun('cplex', Prob, PriLev);
sol = getSolution(Result);

if PriLev > 0
buy_bonds = sol.buy;
buy_other = sol.invest;

for i = 1:length(buy_bonds),
if buy_bonds(i) ~= 0,
disp(['buy ' num2str(buy_bonds(i)) ' bonds of type ' ...
num2str(i)])
end
end

for i = 1:length(buy_other),
if buy_other(i) ~= 0,
disp(['buy "other things" for $ ' ...
num2str(buy_other(i)) ' year ' num2str(i)])
end
end
end

% MODIFICATION LOG
%
% 051201 med Created.
% 060116 per Added documentation.
% 060125 per Moved disp to end
% 090308 med Converted to tomSym
</source>

<pre>
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Financing an Early Retirement Scheme f_k 4548916.886277667200000000
sum(|constr|) 0.000000000000000509
f(x_k) + sum(|constr|)4548916.886277667200000000
f(x_0) 0.000000000000000000

Solver: CPLEX. EXIT=0. INFORM=101.
CPLEX Branch-and-Cut MIP solver
Optimal integer solution found

FuncEv 15 Iter 17
Elapsed time: 0.012000 sec.
buy 911 bonds of type 1
buy 747 bonds of type 2
buy 1760 bonds of type 3
buy "other things" for $ 1160316.8863 year 1
buy "other things" for $ 760249.0266 year 2
buy "other things" for $ 307378.9955 year 3
buy "other things" for $ 17.1233 year 4
buy "other things" for $ 419.6713 year 5
buy "other things" for $ 12403.1008 year 6

</pre>