LP Modeling
Real Estate Investment Planning
Winston-Salem Development Management (WSDM) is trying to complete is
investment plans for the next three years. Currently, WSDM has 2 million
dollars available for investment. At six month intervals over the next three
years, WSDM expects the following income stream from previous investments:
$500,000 (6 months from now), $400,000; $380,000; $360,000; $340,000;
and $300,000 (at the end of third year). There are three development
projects in which WSDM is considering participating. The Foster City
Development would, if WSDM participated fully, have the following cash flow
stream (projected) at six month intervals over the next three years
(negative numbers represent investments, positive numbers represent income):
-$3,000,000; -$1,000,000; -$1,800,000; $400,000; $1,800,000;
$1,800,000; $5,500,000. The last figure is its estimated value at the end
of three years. A second project involves taking over the operation of some
old lower-middle income housing on the condition that certain initial
repairs to it be made and that it be demolished at the end of three years.
The cash flow stream for this project if participated in fully would be:
-$2,000,000; -$500,000; $1,500,000; $1,500,000; $1,500,000; $200,000;
-$1,000,000.
The third project, The Disney-Universe Hotel, would have the following cash
flow stream (six-month intervals) if WSDM participated fully. Again the last
figure is the estimated value at the end of the three years: -$2,000,000;
-$2,000,000; -$1,800,000; $1,000,000; $1,000,000; $1,000,000;
$6,000,000. WSDM can borrow money for half-year intervals at 3.5 percent
interest per half year. At most, 2 million dollars can be borrowed at one
time, i.e., the total outstanding principal can never exceed 2 million. WSDM
can invest surplus funds at 3 percent per half-year.
WSDM may choose to participate in any of the three projects
at a level less than 100 percent.
If WSDM participates in a project at less than 100 percent, all
the cash flows of that project are reduced appropriately. For example,
if WSDM participates in the Disney-Universe Hotel project at the 50 percent
level, then the cash flow stream to WSDM for this project is at the
50 percent level as well yielding the stream -$1,000,000;
-$1,000,000; -$900,000; $500,000; $500,000; $500,000;
$3,000,000.
-
Model the problem of maximizing the return on investment as a linear
program. The key constraints are that the sum of the cash flows from
all sources must be zero in each period.
-
Now assume that there
is a tax rate of 50 percent on profit for any period. If there is a
loss in a period, 80 percent of this loss can be carried forward
to the next period to be counted as part of the losses in that period.
The cash flow streams given above provide no information on the
operating profits for each of the projects. This information is
given in the table below where the
revenues minus expenses for the three projects for each period are listed.
Expenses include depreciation. Remember that
Profit = revenue - expenses,
where in this equation Profit may be negative when a loss is
incurred.
This equation can be refined to separate out losses by writing
Profit - Loss = revenue - expenses,
where we now require both Profit and Loss to be non-negative
with Loss = 0 if Profit >0, and Profit = 0 if
Loss > 0.
| Period
| Project
|
| Foster City
| Lower-Middle Housing
| Disney Universe
|
| 1
| -100,000
| -200,000
| -150,000
|
| 2
| -300,000
| -400,000
| -200,000
|
| 3
| -600,000
| -200,000
| -300,000
|
| 4
| -100,000
| 500,000
| -200,000
|
| 5
| 500,000
| 1,000,000
| 500,000
|
| 6
| 1,000,000
| 100,000
| 800,000
|
| 7
| 4,000,000
| -1,000,000
| 5,000,000
|
Only 20 percent of the cash flow stream from previous investments
(that is the stream $500,000; $400,000; $380,000; $360,000; $340,000; $300,000
listed above)
is taxable in the period in which it is received. That is, only
20 percent of the previous investment cash flows are to be considered
as profit. This gives a profit stream from these investments of
($100,000; $80,000; $76,000; $72,000; $68,000; $60,000).
In this new setting model the problem of maximizing the
return on investment as a linear
program.
The tax structure must be accommodated in your model.
In this scenario the taxes in any period must be subtracted
from the amount that can be invested in that period.
The tax structure requires
that you keep track of the profits and losses from period to
period by using an equation of the form
(*)
Profit - Loss = revenue - expense -.8 (last periods loss),
to separate out profits from losses in each period.
Simply specify that both Profit and Loss are non-negative
variables and don't worry that they both can't positive at the same
time (this will work itself out in the solution).
The revenue minus expenses for each project are already computed
for you in the table above, but don't forget to include the
20% taxable revenue from previous investments, the revenue from
the 3 percent investment
(yielding a profit of 0.03 times the amount invested
per period), and the expenses from
the per period loans (incurring an interest payment of .035 times
the amount borrowed per period).
For example using units of thousands of dollars, at the beginning of
time period 2 if F, M, and D are the percent participation
in the Foster City, Housing Development, and the Disney-Universe Hotel,
respectively, B1 is the amount borrowed at the beginning of period 1,
L1 is the amount
lent at the beginning of period 1,
P2 is the profit seen at the end of period 1 and the beginning of period 2,
C1 is the loss seen at the beginning of period 1,
and C2 is the loss seen at the end of period 1 and the beginning
of period 2, then equation (*) above becomes
P2 - C2 = 100 + .03L1 - 300F - 400M - 200D - .035B1 - .8C1 .
Finally, remember that in this second model the sum of the cash flows
from all source must still be zero. But now the taxed profit
each period is an additional (negative) cash flow to be included in this sum.
This cash flow is not present in first LP model given above.
For example, for the beginning of time period 2
(using the above notation) this yields
the equation
1000F + 500M + 2000D + 1.035B1 + L2 + .5P2
= 500 + 1.03L1 + B2.