Transcript CHAPTER 6: LINEAR PROGRAMMING

CHAPTER12:
CAPITAL
INVESTMENT
DECISIONS
INTRODUCTION
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Linear programming models
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company has finite production capacity (machinery
resource constraints)
If the demand suddenly increase, the company would be
unable to meet this extra demand without increasing the
amount of machine time that is available for production
One way of meeting the additional product demand is for
the company to buy a piece of machinery with greater
production capacity.
Buying new equipment involves decisions making over
future planning time periods
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Break-Even model
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Production capacity of the company is limited (250
units of output per production time period).
If the demand for the company's product is greater than
250 units, then the company would be faced with the
dilemma of how to handle this excess demand.
For a short-term increased demand
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For a long-term increased demand
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Introducing overtime working
Raising the price of the product
Expanding the existing production facilities
Building a bigger scale production plant.
Company is faced with a decision which involves the
costs and benefits that will accrue to the company over
some future time horizon.
COMPOUNDING
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Notation for different time periods (yearly basis)
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t0 --- to stands for time period zero and represents right now;
t1 ---to stands for time period one and represents 1 year
into the future;
t2 ---to stands for time period two and represents 2 years
into the future;
tn ---to stands for time period n and represents n years into
the future , where n can take on any value from 0,1,2...
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Initial capital or lump sum
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A financial investor has a sum of money to
invest in time period t0, for example £100
If the investor deposits his £100 in an interest
bearing bank account, how much will he have
after 1 year.
Suppose: the going rate of interest is 10%.
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For year 1, the value of the investment to
the investor:
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(what he starts with) + (the interest on what he starts
with)
£100 + 10%x£100 = £100(1+10%) = £110
t0—10%—>t1
100 —> 100+10% of 100
= 100(1+10%)
=£110
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For year 2, the value of the investment to the
investor:
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(what he starts with at the beginning of year 2) + (the
interest earned over year 2)
£110 +10%x£l 10 = £110(1+10%) = £121
t0 —10% —> t1 —10% —> t2
100 —> 100+10%xl00
=100(1+10%)
=110
—> 110+10%x l10
= 110(1+10%)
=£121
110(1+10%) = 100(l+10%)(l+10%)
= 100(l+10%)2
= £121
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For year 3, the value of the investment to the
investor:
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t0 —10%—> t1 —10%—> t2 —10%—> t3
100 —> 100+10%xl00
=100(1+10%)
= 110 —> 110+10%xll0
= 110(1+10%)
=100(1+10%)2
=121
—> 121+10%xl21
=121(1+10%)
=100(l+10%)3
=£133.1
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The value of the investment at different
time periods can be summarised as
follows:
t0 ----------> t1 -------------> t2 -------------> t3
100
100(l+10%)1 100(l+10%)2 100(l+10%)3
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For year n, General compounding idea as
follows:
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For a given initial lump sum A
For a given term of investment n
For a given rate of interest i%
Future value (FV) of the investment is given by:
FV = A(l+i%)n
DISCOUNTING
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Discounting is the reverse side of the coin to compounding
With discounting the time direction is reversed.
For the value of a sum of money in the future, we want to know what
this future sum is worth to us right now.
Considering the position of a money lender, A client would like to
borrow some money in order to finance some immediate
expenditure, however the client does have an asset that will be
available , not to-day, but in 1 years time. At this time the asset will
have a value of £100. Thus the client has an asset - £100 available
in one years time -but unfortunately for him he wants money NOW.
The client can thus pose the following question to the money lender:
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How much will you be prepared to lend me
right now given that I can pay you back £100 in
one year time?
For money lender, it’s a compounding
problem for 1 year, which can be
represented as:
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t0—10%—>t1
? —>
?(1+10%) =100
?=100/(1+10%) =£90.91
£90.91 is called the discounted value or the
present value (PV) of £100 in 1 year.
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The inverse relationship between the present
value of a future sum of money and the going
rate of interest
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If the interest rate increases, the denominator in the
expression for ? increases, and results in a fall in
the PV.
If the rate of interest falls, the denominator falls , and
results in the PV of a future sum of money will rise.
For 2 years,
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t0 ------> t1-----> t2
?-----------------> ?(l+10%)2 = 100
?=100/(1+10%)2 = £ 82.65
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t0 <—10%— t1 <—10%—t2
90.91 < --- 100
82.65 < --------------------100
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£100 in 1 year is worth £90.91 to-day
£100 in 2 years is worth £82.65 to-day.
General rule for discounting can be
represented as follows:
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t0 < --------------------t1 < --------------t2
100 /(l+10%)1 <---- 100
100 / (l+10%)2 < ----------------------100
For 3 year, PV = 100/(l+10%)3
For 4 year, PV = 100/(l+10%)4
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For a given sum of money A, due to be paid n
years into the future, and a going rate of
interest of i% per year, then the PRESENT
VALUE of this future amount is given by:
A
PV 
(1  i %)n
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i) numerator --- the fixed amount of money that is
being promised in the future.
ii) denominator ---one plus the going rate of interest
in brackets; and the brackets are raised to a power
determined by how far in the future the money is
promised.
NET PRESENT VALUE
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an investment project
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A large amount of money is spent RIGHT NOW
Operational benefits are spread out over a number of years
into the future.
How long will the investment project last?
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A 'new* PC ---life of 3 years
A 'new' football star---future of 8 years
An offshore oil rig--- for 40 years.
In general terms we can model a project that has a life span
of N years, that is:
t0 ----> t1----> t2----> t3---> t4 ---> tN
Project Costs
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Initial set-up cost or CAPITAL
EXPENDITURE
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a single payment to get the project started
construction of a factory.
Operational costs
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Labour costs
Material costs
and son on
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In general terms we can indicate this as
follows:
Time t0
t1
t2
t3
...
tN0
Costs C0 C1 C2 C3 ...
CN
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Co ---costs in time period 0
C1 ---costs in time period 1
C2 ---costs in time period 2
CN, the costs involved in the final year of the capital
project.
Project Benefits
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Project revenues are called as BENEFITS
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Generally, the benefits in period 0 will be 0
Bi--- Benefit in period I
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Grevs---Gross Revenues
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Profit Taxes---T represents the tax rate on
company profits.
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Payment of tax is CONDITIONAL
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GRevs > 0, pay taxes
GRevs <= 0, pay no taxes
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Notation (Bt-Ct)+
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If GRevs is positive for any particular period ,
then tax has to be paid. hence NRev will be
less than GRevs.
If GRevs is zero or negative, then no tax is
paid. NRev will be the same as GRev.
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NET REVENUE (NRevs)
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NRevs = GRevs – Tax
NRevs = (Bn-Cn) - T*(Bn-Cn), for n =
1,2,3...,N
NRevs = (Bn-Cn) - T*(Bn-Cn)+ = [(BnCn)+]*(1-T)
Sensitivity Issues
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Costs
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Inverse relationship between OPERATIONAL
COSTS and project PROFITABILITY
For example: Labour costs.
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If the cost of labour rises, GRevs and hence NRevs will
fall, the project will become less profitable.
If a decrease in labour costs, NRevs will rise thus
making the project more profitable.
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Revenues
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a higher price can be charged for the same
output, revenue will rise; an increase in Grevs
and Nrevs, hence an increase in the
profitability of the project.
Tax
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If the tax rate goes up, NRevs will decrease;
and lead to a reduction in the overall
profitability of the project.
THE NET PRESENT VALUE
IDEA .
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In order to work out the value of a future
amount of money in terms of the base period
t0, we have to DISCOUNT or find the
PRESENT VALUE of that future sum.
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The idea of NET PRESENT VALUE can be
written as THE FUNDAMENTAL NPV
FORMULA :
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The NPV value is a measure of the overall
profitability of the investment project. The
NPV figure can be positive or negative. In
terms of a decision making rule we have
the following:
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If NPV > 0, then the project is a worthwhile
investment opportunity.
If NPV < 0 , then the project is NOT a
worthwhile investment opportunity.
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The interpretation of NPV: a net
profitability figure what the project will
actually earn for the company after all
costsinitial capital costs:
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future interest payment costs
yearly operating costs
yearly tax bills
have been paid in full.
The higher is the NPV value then the more
profitable, and hence more desirable, is
the capital investment project.
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Important input; the rate of interest (i%)
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If i% rises, NPV will fall.
If i% falls, NPV will rise.
An inverse relationship between the level of
the rate of interest and the NPV of an
investment project.
A PRACTICAL EXAMPLE
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A company is considering making a line
extension to its range of products.
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Lump Sum: £165,000.
A market life of 5 years.
Benefits
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£105,000 after 1 year
After year 1, to grow at 20% per annum.
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Operating costs
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Tax rate
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Budget in the first year: £25,000
After year 1, to grow annually at 4.5%.
Current: Pay tax of 30% on any profits
In future, in the range (27% - 33% )
The rates of interest
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Current: 12%
In future, in the range ( 10% - 14% )
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Conceptual Paper Worksheet
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IF function
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IF( CONDITION, A, B)
IF(D4 > 0, A2*D4,0)
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A2*D4 is the Tax Rate * G-REV and is the tax
payable if G-REV is >0
0 is the tax payable if G-REV is NOT > 0
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NPV function
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NPV1=NPV(A5, E6:I6)
NPV= NPV1 + D6
For this project the Net Present Value is £145,130, so is
positive. By the NPV decision rule this project is viable.
MODELLING THE UNCERTAINTY IN
THE PROBLEM
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Tax Rate Sensitivity
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If the Tax-Rate varies between 27% and 33% in step 1%,
How does NPV change?
Tool—Table submenu
CWP for sensitivity
Interpretation: For all Tax-Rates the NPV remains
positive, so that for Tax-Rates in the range 27% to 33% the
project remains viable.
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Interest Rate Sensitivity
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Interest rates varies between 10% and 14% in
steps of 0.5% , how does NPV changes?
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Combined sensitivity in Tax and
Interest Rates
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A two way table can be set up quite easily, with
the Interest Rates along the row and the TaxRates down the columns as shown in the
partial CPW below:
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Interpretation:
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At all combinations of Tax-Rates and Interest-Rates the
NPV remains positive, by the NPV decision rule the
project is viable at all combinations of Tax-Rate and
Interest-Rate within the ranges 27%-33% and 10.0% to
14.0%. Further since NPV is quite large this suggests a
robust project