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Chapter 5: Essential Formulae in Project Appraisal

A Coverage of the Formulae and Symbols Used to Evaluate

Investment Projects

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Study objectives After studying this chapter the reader should be

able to: Apply a discounting rate to cash flows occurring at different points

in time to translate them into a common measure of value Calculate present value, net present value (NPV) and internal rate

of return (IRR) from a given cash flow series Calculate monthly loan (mortgage) repayments, their interest and

principal components and the loan balance of a mortgage loan Understand the financial mathematics involved in the discounted

cash flow techniques (such as NPV and IRR) and mortgage loans Apply the relevant annuity formulae for project appraisal

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Fundamentals in Financial Evaluation

Money has a time value: a $ or £ or € today, is worth more than a $ or £ or € next year.

A risk free interest rate may represent the time value of money.

The commonly used method in finance is to convert all the values into their present values using an appropriate discount rate to represent the time value.

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Moving Money Through Time

Investment projects are long lived, so we usually use annual interest rates.

With compound interest rates, money moved forward in time is ‘compounded’, whilst money moved backward in time is ‘discounted’.

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Financial Calculations

Time value calculations in capital budgeting usually assume that interest is annually compounded.

‘Money’ in investment projects is known as ‘cash flows’: the symbol is: Cn Cash flow at end of period n.

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Rate of return

The rate of return (ROR) is a basic concept in finance.

If there are only two cash flows, a cash outlay or an investment at the beginning of the year and a cash inflow or the realization of the investment at the end of the year, the rate of return is usually measured by:

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Rate of return

Example 5.1 Suppose you invest $1,000 at the

beginning of the year and receive a total of $1,100 at the end of the year. The rate of return is:

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What is the payback period?

The number of years required to recover a project’s cost,

or how long does it take to get the business’s money back?

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Payback for Project N’s (Long: Most CFs in out years)

10 8060

0 1 2 3

-100

=

CFt

Cumulative -100 -90 -30 50

PaybackL 2 + 30/80 = 2.375 years

0100

2.4

10

70 2050

0 1 2 3

-100CFt

Cumulative -100 -30 20 40

PaybackS 1 + 30/50 = 1.6 years

100

0

1.6

=

Project M’s (Short: CFs come quickly)

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Strengths of Payback:

1. Provides an indication of a project’s risk and liquidity.

2. Easy to calculate and understand.

Weaknesses of Payback:

1. Ignores the TVM.

2. Ignores CFs occurring after the payback period.

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Future value of a single sum

Future value is calculated using the following formula: FV = PV (1+ r )n

Example 5.2 You place $1 in a term deposit at the beginning

of year 1 for a period of six years at a compound interest rate of 10% per annum. How much will you get at the end of six years? FV =1.00 × (1.10)6 = $1.771561≈ $1.77

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Example 5.3 Instead of $1, suppose you deposited

$2,000 (or incurred a capital expenditure of $2,000). Then, the future value of this investment is:

Future value of a single sum

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Present value of a single sum Present value is the opposite of future value.

The formula for the future value presented previously can be rearranged to calculate the present value of a future cash flow:

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Present value of a single sum

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Present value of a single sum

Example 5.4 What is the present value of $1.771561

(≈$1.77) to be received in six years’ time, if the time value of money (or the appropriate discount rate) is 10% per annum? PV = $1.771561×(1 + 0.10)−6 =

$1.771561×(0.56447393)=$1.00

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What is the present value of $3,543.12 to be received at the end of a six year period, if the time value of money is 10% per annum?

Present value of a single sum

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Financial Calculations: Cash Flow Series A payment series in which cash flows

are Equally sized And Equally timed

is known as an annuity.

There are four types:

1. Ordinary annuities; the cash flows occur at the end of each time period.

2. Annuities due; the cash flows occur at the start of each time period.

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Project analysis normally involves a series of cash flows occurring at different points in time.

The future value of a series of cash flows is calculated using the formula:

Future value of a series of cash flows

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Future value of a series of cash flows

Example 5.6 It is estimated that an investment project will

receive net cash inflows at the end of each of the first five years. They are $10,000, $20,000, $30,000, $45,000 and $60,000. What is the future value of these cash flows at the end of year 5, if the time value of money is 20% per annum?

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Future value of a series of cash flows

FV = 10,000×(1.2)5−1 + 20,000×(1.2)5−2+ 30,000×(1.2)5−3+45, 000×(1.2)5−4+ 60,000×(1.2)5−5

= 10,000(1.2)4 + 20,000(1.2)3 + 30,000(1.2)2

+45,000(1.2)1 + 60,000(1.2)0

= 20,736 + 34,560 + 43,200 + 54,000 + 60,000 = $212,496

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The present value of series of cash

flows is:

Financial Calculations

Present value of a series of cash flows

n

nn

rCFPV

1 )1(

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Example 5.7 What is the present value of the three cash flows

$100, $200 and $600, to be received at EOY 1, EOY 2 and EOY 3, respectively, if the time value of money (or discount rate) is 10% per annum?

Present value of a series of cash flows

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Present value when the discount rate varies

Generally, the cash flows of future years are converted to present values by applying a single discount rate to all the cash flows.

There are circumstances where the use of different rates for different periods is justified .

For example, when the discount rate is adjusted to incorporate a risk premium and if the degree of uncertainty of annual cash flows varies from year to year, the use of different discount rates for different periods may be warranted

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Present value when the discount rate varies

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Then the present value is calculated as follows:

Present value when the discount rate varies

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Financial Calculations: Cash Flow Series

3. Deferred annuities; the first cash flow occurs later than one time period into the future

4. Perpetuities; the cash flows begin at the end of the first period, and go on forever.

Annuities: types 3 and 4.

α

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Present value of an ordinary annuity

Annuity formulae are useful in NPV calculations in which the value of the cash flows is the same for a number of years.

To use the ordinary annuity formula, the following conditions should be satisfied: the value of the cash flows in each period is the

same; the period or the interval for the cash flows remains

unchanged the receipt/payment of the cash flows should occur at

the end of each regular period

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Present value of an ordinary annuity

Annuity formulae are useful in NPV calculations in which the value of the cash flows is the same for a number of years.

r

rCF

n)+(1

11

= ANPV

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Present value of an ordinary annuity

Example 5.10 A project is expected to have an economic life of

five years. The value of this project’s net cash inflows is estimated to be $2,000 for each year and this is to be received at the end of each year. The appropriate discount rate is 15% per annum. What is the present value of this project’s cash inflows?

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Present value of an ordinary annuity

$6,704.40=15.0

5)15.0+(1

11

2000 = ANPV

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Present value of a deferred annuity

Deferred annuity means that the annuity starts not from the end of the first year (or period) but from a few years (or periods) later

The present value of such an annuity can be found by a two-stage process. First, the present-value concept is applied to find the

value of the annuity at the beginning of the first annuity period.

Then this single sum is discounted to the present, using the formula introduced in the earlier section, titled ‘Present value of a single sum’.

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Example 5.11 There is an annuity payment of $60 per

year for twenty years, but the first payment is at the end of year 10. The interest rate is 10% per annum. What is the present value of these twenty payments?

Present value of a deferred annuity

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Present value of a deferred annuity

6.2169)1.0+(1

1

1.0

20)1.0+(1

11

60 = ANPV

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Example 5.12 A project’s annual net cash inflows (to be

received at the end of each year) are estimated as follows. For the first nine years the project does not generate any cash inflow. For the next eleven years (i.e. from the tenth to the twentieth years inclusive), it generates $60 per year.

The discount rate is 10% per annum. What is the present value of this project’s cash inflows?

Present value of a deferred annuity

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3.1659)1.0+(1

1

1.0

11)1.0+(1

11

60 = ANPV

Present value of a deferred annuity

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Perpetuity

A perpetuity is a special case of an annuity in which the number of equal cash flows is infinite. Because the cash flows go on forever, we cannot calculate the future value of a perpetuity; a perpetuity does not have an end of life. Hence, a future value cannot be calculated. The formula for the present value of a perpetuity is:

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Net present value

The net present value (NPV) of a project is calculated by subtracting the present value of the capital outlays from the present value of the cash inflows.

If the capital outlay occurs only at the beginning of year 1, that is, at EOY 0, The formula for the NPV in such situations is:

If capital outlays occur in different years, the relevant formula is:

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Example 5.13 Net cash inflows of a project are estimated as:

EOY 1 $3,000; EOY 2 $4,000; and EOY 3 $8,000. Capital outlays for the project will occur during the first and second years and they are estimated as $2,000 and $1,500, respectively. They are assumed to occur at the beginning of each period. The timing notation for the correct discounting of these will be: EOY 0 $2,000; and EOY 1 $1,500. The discount rate is 10% per annum.

Net present value

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Net present value

Example 5.13

Solution

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Net present value

ExampleGiven the expected cash flow forecasts for Otobai Company’s Motor Scooter project, listed on the next slide, determine the NPV of the project given changes in the cash flow components using a 10% cost of capital. Assume that all variables remain constant, except the one you are changing.

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Net present value

315-Flow CashNet

3.0flow cash Operating

1.5taxafter Profit

1.550% @ .Taxes

3profitPretax

1.5onDepreciati

3Costs Fixed

30Costs Variable

37.5Sales

15-Investment

10-1 Years0Year

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Net present value

Net present value =

4.181.0

10)1.0+(1

11

3 = ANPV

NPV= 18.4-15= 3.43 billion Yen

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What’s Franchise S & L NPV?

Net present value

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Net present value

Net present value for project (S)

= 78.82 Net present value for project (L)

= 49.18

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Rationale for the NPV Method

NPV = PV inflows - Cost= Net gain in wealth.

Accept project if NPV > 0.

Choose between mutually exclusive projects on basis ofhigher NPV. Adds most value.

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Using NPV method, which franchise(s) should be accepted?

If Franchise S and L are mutually exclusive, accept S because NPVs > NPVL .

If S & L are independent, accept both; NPV > 0.