CHAPTER12:CAPITAL INVESTMENT DECISIONS

INTRODUCTION

Linear programming models 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

Break-Even model 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 Introducing overtime working Raising the price of the product

For a long-term increased demand 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

Notation for different time periods (yearly basis) t0 --- to stands for time period zero and represents right now; t1 ---to stands for time period one and represents 1 year int

o the future; t2 ---to stands for time period two and represents 2 years in

to 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...

Initial capital or lump sum 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%.

For year 1, the value of the investment to the investor: (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

For year 2, the value of the investment to the investor: (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

For year 3, the value of the investment to the investor: 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

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

For year n, General compounding idea as follows: 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

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:

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: 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.

The inverse relationship between the present value of a future sum of money and the going rate of interest 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, t0 ------> t1-----> t2

?-----------------> ?(l+10%)2 = 100

?=100/(1+10%)2 = £ 82.65

t0 <—10%— t1 <—10%—t2

90.91 < --- 100

82.65 < --------------------100 £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: 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

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:

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.

ni

APV

%)1(

NET PRESENT VALUE

an investment project 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?

A 'new* PC ---life of 3 years A 'new' football star---future of 8 years An off shore oil rig--- for 40 years. In general terms we can model a project that has a life span o

f N years, that is:t0 ----> t1----> t2----> t3---> t4 ---> tN

Project Costs

Initial set-up cost or CAPITAL EXPENDITURE a single payment to get the project started construction of a factory.

Operational costs Labour costs Material costs and son on

In general terms we can indicate this as follows:

Time t0 t1 t2 t3 ... tN0

Costs C0 C1 C2 C3 ... CN

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

Project revenues are called as BENEFITS Generally, the benefits in period 0 will be 0 Bi--- Benefit in period I

Grevs---Gross Revenues

Profit Taxes---T represents the tax rate on company profits.

Payment of tax is CONDITIONAL GRevs > 0, pay taxes GRevs <= 0, pay no taxes

Notation (Bt-Ct)+

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.

NET REVENUE (NRevs) NRevs = GRevs – Tax NRevs = (Bn-Cn) - T*(Bn-Cn), for n = 1,2,3...,

N NRevs = (Bn-Cn) - T*(Bn-Cn)

+ = [(Bn-Cn)+]*

(1-T)

Sensitivity Issues

Costs Inverse relationship between OPERATIONAL CO

STS and project PROFITABILITY For example: Labour costs.

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.

Revenues a higher price can be charged for the same out

put, revenue will rise; an increase in Grevs and Nrevs, hence an increase in the profitability of the project.

Tax If the tax rate goes up, NRevs will decrease; a

nd lead to a reduction in the overall profitability of the project.

THE NET PRESENT VALUE IDEA . 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.

The idea of NET PRESENT VALUE can be written as THE FUNDAMENTAL NPV FORMULA :

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: If NPV > 0, then the project is a worthwhile

investment opportunity. If NPV < 0 , then the project is NOT a

worthwhile investment opportunity.

The interpretation of NPV: a net profitability figure what the project will actually earn for the company after all costsinitial capital costs: 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.

Important input; the rate of interest (i%) 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

A company is considering making a line extension to its range of products. Lump Sum: £165,000. A market life of 5 years.

Benefits £105,000 after 1 year After year 1, to grow at 20% per annum.

Operating costs Budget in the first year: £25,000 After year 1, to grow annually at 4.5%.

Tax rate Current: Pay tax of 30% on any profits In future, in the range (27% - 33% )

The rates of interest Current: 12% In future, in the range ( 10% - 14% )

Conceptual Paper Worksheet

IF function IF( CONDITION, A, B) IF(D4 > 0, A2*D4,0)

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

NPV function

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 Tax Rate Sensitivity

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.

Interest Rate Sensitivity Interest rates varies between 10% and 14% in

steps of 0.5% , how does NPV changes?

Combined sensitivity in Tax and Interest Rates A two way table can be set up quite easily, with

the Interest Rates along the row and the Tax-Rates down the columns as shown in the partial CPW below:

Interpretation: 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