chapter 6 - elsevieri an investment typically consists of an initial outlay (and possible other...

CONTENTS PAGE MATERIAL SUMMARY

Essential Topic: Project appraisal techniquesChapter 6

Mathematics of Finance: A Deterministic Approachby S. J. Garrett


CONTENTS PAGE

MATERIAL

Net Present ValueNPV as an appraisal techniqueDiscounted payback periodInternal rate of returnPutting it together

SUMMARY


NET PRESENT VALUE

I An investment typically consists of an initial outlay (andpossible other outlays in the future) followed by receipts.Cash flows at time t could be positive or negative andeither discrete (ct) or continuous (ρ(t)).

I The net present value at interest rate i is denoted NPV(i)

NPV(i) =∑

ctνt +

∫ T

0ρ(t)νtdt

I If an investor can lend and borrow money at rate i1, aproject will be profitable if and only if

NPV(i1) > 0

I Further, if the project ends at time T, the profit at that timeis

NPV(i1)× (1 + i1)T


NPV AS AN APPRAISAL TECHNIQUE

I NPVs can be used to compare the profitability of two ormore alternative projects.

I Under the NPV measure, Project A is more profitable thanProject B if

NPVA(i1) > NPVB(i1)

I If an investor is able to compare projects on the basis ofoverall profitability alone, the NPV can be used as adecision tool.

I The NPV measure is sensitive to the investor’s i1.


EXAMPLE

Three projects have the following cash flowsA Initial outlay of £10,000 in return for an annual income of

£1,000 paid 6-monthly in arrears for 12 years.B Initial outlay of £1,000 in return for an annual income of

£200 at the end of each of the next 20 years.C Initial outlay of £5,000 in return for a continuous payment

stream of £2,000 per annum for 3 years, deferred for 2years.

If the investor can borrow and invest money at 5% per annum,rank the projects under the NPV measure at this rate.


EXAMPLE

Answer

The NPVs are computed as follows

NPVA(i) = −10, 000 + 1, 000a(2)12

NPVB(i) = −1, 000 + 200a20

NPVC(i) = −5, 000 + 2|2, 000a3

At i1 = 5%

NPVA(i1) = −£1, 027.26NPVB(i1) = £1, 492.44NPVC(i1) = £62.55

and soNPVB(i1) > NPVC(i1) > NPVA(i1)

Project B is therefore the best under the NPV measure ati = 5%. Project A is not profitable at this rate.


DISCOUNTED PAYBACK PERIOD

I In many practical problems the net cash flow changes signonly once, from negative to positive. In this case, thebalance in the investor’s account will change fromnegative to positive at a unique time t1 or it will always benegative (in which case the project is not viable).

I If t1 exists, it is called the discounted payback period (DPP)and is defined as the smallest value of t1 such that theaccumulation of all prior cash flows is first greater thanzero.

I Although the DPP is defined on the accumulation, it canequivalently be obtained by finding t1 such that thepresent value of prior cash flows is first greater than zero.

I A project with a lower DPP starts to generate a profit morequickly and this can be important in project appraisals.


EXAMPLE





If the investor can borrow and invest money at 5% per annum,rank the projects under the DPP measure at this rate.


EXAMPLE

Answer

The DPPs are obtained from t1 such that

A −10, 000 + 1, 000a(2)t1= 0

B −1, 000 + 200at1= 0

C −5, 000 + 2|2, 000at1−2 = 0At i1 = 5%

A t1 = 13.96 i.e. DPP is 14 years which is greater than theterm of the project. Project A is therefore not viable ati = 5%.

B t1 = 5.90 i.e. DPP is 6 years (since the payments are atinteger times).

C t1 = 4.96 i.e. DPP is 4.96 years.Project C is therefore the best under the DPP measure at i = 5%.


INTERNAL RATE OF RETURN

I The internal rate of return (IRR) of a project y is theannualized yield obtained over the project’s lifetime.

I It can be calculated by solving the equation of value for therate y ∑

ctνt +

∫ T

0ρ(t)νtdt = 0

I The IRR gives the % return on each £1 invested in theproject.

I The IRR is therefore an additional measure of theprofitability of a project and can be incorporated intoproject appraisals.


EXAMPLE





Assuming that the investor is able to fund the initial outlayfrom existing funds, calculate the IRR in each case.


EXAMPLE

Answer

The IRRs are computed by solving the following equations ofvalue

A −10, 000 + 1, 000a(2)12

= 0B −1, 000 + 200a20 = 0C −5, 000 + 2|2, 000a3 = 0

These can be solved using trial and error (or Goalseek) to showA IRRA = 3.05% per annumB IRRB = 19.43% per annumC IRRC = 5.38% per annum

Project B is therefore the best under the IRR measure.


PUTTING IT TOGETHER

I Neither of the three measures can be considered inisolation.

I For example, Project B has the greatest IRR (and NPV at5%), however its DPP is 6 years.

I Project C has a lower IRR (and NPV at 5%), however itsDPP is only 4.96 years.

I For an investor with unlimited funds, the profitability ofhis investments will be the primary concern. The IRR andNPV measures are more suitable and Project B preferred.

I However, for an investor with limited funds, the time afterwhich the project actually starts to generate a positivereturn is important. Project C might therefore be preferred.

I Further considerations (existing expertise, diversification,strategic fit) are always needed in addition to the financialmeasures.


SUMMARY

I Financial and business projects can be appraised usingthree measures related to compound interest theory:

I net present valueI discounted payback periodsI internal rate of return

I The NPV and IRR give a measure of the overallprofitability of the project.

I The DPP gives a measure of the time to profitability of theproject.

I None of these measures should be considered in isolation.I Other business considerations are required in addition to

these financial measures, for example existing expertiseand strategic fit.

chapter 6 - elsevieri an investment typically consists of an initial outlay (and possible other...

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