Investment Strategy using Operations Research
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This document was written by Paul Edwards, around 1995-01-30.
It is released to the public domain.
This document details the calculations required to determine
whether it is better to pay of a home loan, or start negative
gearing. This could have been solved using Operations Research
techniques, but the solution becomes obvious without using
them.
WARNING - following the advice in this document is GUARANTEED
to send you BROKE. DON'T DO IT. Having said that...
Joe earns $50,000 a year. He pays tax of 44.4% for the
amount above $38,000. He pays tax of 35.4% for the amount
above $20,700. He pays tax of 21.4% for the amount above
$5,400. He pays tax of 1.4% for the amount above $0.
He gets paid his entire wage at the beginning of the year with
interest and return from investments all taken into account
and realised on the same day.
1. He has a debt of $100,000 which costs 10% per year to service,
with no tax deduction (the debt could have been incurred by
buying a house, or buying a Porsche, it is irrelevant).
2. He could borrow money to buy shares, which return 3% pa in
income (taxable), and 8% pa capital growth (no tax if realised
in smallish amounts whilst retired). The interest payable on
the loan (at 10% pa rate) is tax-deductible.
Formulate this problem as a linear programming problem, and
work out the optimum strategy for investment between the two
investment areas for the first year.
This problem should be broken up into 4 separate incomes and
borrowings.
(a) There is an income of $12,000 taxed at 44.4%.
(b) There is an income of $17,300 taxed at 35.4%
(c) There is an income of $15,300 taxed at 21.4%
(d) There is an income of $5,400 taxed at 1.4%
Let A = first investment loan, interest offset against (a)
Let B = second investment loan, interest offset against (b)
Let C = third investment loan, interest offset against (c)
Let D = fourth investment loan, interest offset against (d)
Then gain will be F = (A+B+C+D)*.08
And paying off debt will be E = (12000-A*.07)*(1-.444)
+ (17300-B*.07)*(1-.354)
+ (15300-C*.07)*(1-.214)
+ (5400-D*.07)*(1-.014)
= 12000*0.556 + 17300*0.646 + 15300*0.786 + 5400*0.986 +
-0.03892A + -0.04522B + -0.05502C + -0.06902D
= 35198 - 0.03892A - 0.04522B - 0.05502C - 0.06902D
So total wealth will be Z = (-100000 + E) * 1.1 + F
Z = -110000 + 1.1E + F
= -110000 + 1.1(35198 - 0.03892A - 0.04522B - 0.05502C - 0.06902D)
+ .08(A+B+C+D)
= -110000 + 38717.8 - 0.042812A - 0.049742B - 0.060522C - 0.075922D
+ 0.08A + 0.08B + 0.08C + 0.08D
= -71282.2 + 0.037188A + 0.030258B + 0.019478C + 0.004078D
subject to
A/.07 <= 12000, ie A <= 171428
B/.07 <= 17300, ie B <= 247142
C/.07 <= 15300, ie C <= 218571
D/.07 <= 5400, ie D <= 77142
and it's obvious for anyone with any mental horsepower, that
this means borrow 171428+247142+218571+77142 = $714,283 for a
worth of
-71282.2 + 6375.06 + 7478.02 + 4257.33 + 314.59 = -$52857.20
However, the last amount, $314.59 is trivial, so it would be
better not to bother with the last loan. However, there is an
obligation to pay $10,000 interest each year (for the house/
Porsche), ie another constraint
35198 - 0.03892A - 0.04522B - 0.05502C - 0.06902D <= 10000
So we have
maximize Z = -71282.2 + 0.037188A + 0.030258B + 0.019478C + 0.004078D
subject to
-0.03892A - 0.04522B - 0.05502C - 0.06902D + G = -25198
A + H = 171428
B + I = 247142
C + J = 218571
D + K = 77142
A, B, C, D, G, H, I, J, K >= 0
which we can solve using the simplex algorithm...
Of course the answer is already plainly obvious, which is
A = 171428, B = 247142, C = 133591, by working via the obvious
route. So the optimum strategy is to only pay off the minimum
on the house/Porsche, whilst the investment loan becomes
171428+247142+133591 = $552,161
1st year 552161*1.08 = 596334
2 596334*1.08 = 644041
3 644041*1.08 = 695564
4 695564*1.08 = 751209
5 751209*1.08 = 811306
6 811306*1.08 = 876210
7 876210*1.08 = 946307
8 946307*1.08 = 1022011
ie capital gain of 1022011-552161 = $469,850
where interest would be 552161*.1 = $55,216
and dividends would be 1022011*.03 = $30,660
and capital would need to be sold off to repay the debt. A
diversified portfolio would allow the shares to be sold with
little capital gain so that the debt could be reduced ASAP,
and Joe could live off the dividends of the $469,850, which
should be reduced to $369,850 ASAP to pay off the home/Porsche
loan. Thus the dividends would be $11095, or $213 per week.
However, there's not much difference if he borrows less and pays
off the house/Porsche instead.
ie borrow 171428+247142 = $418,570
The amount paid off the debt would be the remainder,
15300*(1-.214) + 5400*(1-.014) = 17350
So in first year (-100000+17350)*1.1 = -90915
2 (-90915+17350)*1.1 = -80921
3 (-80921+17350)*1.1 = -69928
4 (-69928+17350)*1.1 = -57836
5 (-57836+17350)*1.1 = -44535
6 (-44535+17350)*1.1 = -29903
7 (-29903+17350)*1.1 = -13809
8 (-13809+17350) = paid off.
My investment over that same period would be
1st year 418570*1.08 = 452055
2 452055*1.08 = 488220
3 488220*1.08 = 527277
4 527277*1.08 = 569459
5 569459*1.08 = 615016
6 615016*1.08 = 664217
7 664217*1.08 = 717355
8 717355*1.08 = 774743
ie capital gain of 774743-418570 = $356173
where interest would be 418570*.1 = $41,857
and dividends would be 774743*.03 = $23,242
and he would need to sell capital to pay it off. A diversified
portfolio would allow him to sell things with little capital gain
so that he can reduce his debt ASAP, and live off the dividends of
the $356173, which would give him $10685, or $200 per week.
Basically, borrowing as much as you can afford for investment
purposes is recommended, if the returns, interest rate, etc
pan out as per the assumptions above. This information would
probably be well represented in spreadsheet format. At this
point, please read the disclaimer at the top of this document.
@EOT:
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* Origin: Kludging up the works (3:711/934.9)
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