6. CÁLCULO Y ANÁLISIS DE FACTURAS
6.6 ANÁLISIS DE FACTURA DE TARIFA 3.1A DH3P CON MEDIDA EN BT
One of the most important principles of finance is that money has a time value. In the most general sense this refers to the fact that a dollar today is worth more than a dollar in one year’s time. One reason for this is that a dollar today can earn interest while waiting for one year. This means that a given sum of money (say a cash flow of $5,500) should be valued differently, depending on when the cash flow is to occur. If the interest rate is 10% per annum, the present value of $5,500 received at the end of the first year is $5,000. This is because, $5,000 today can be invested at 10% to earn $500 interest at the end of the year. If the $5,500 is received at the end of two years from today, its present value is smaller than $5,000; it is approximately $4,546.
Capital budgeting decisions deal with sizable investments in long-lived projects. As discussed in Chapter 2, the cash flows of a project are spread over many years. In many cases, large sums of money are invested in the first year and net operating cash flows are received over a number of years. At the termination of the project, terminal cash flows are realized. In addition to the initial investment in the first year of the project, capital expenditures may occur at later stages of the project, for example, upgrades to the plant and equipment. The cash flows occurring at different times have to be converted to a common denominator to assess if the cash inflows exceed the cash outflows.
This can be done by translating all the cash flows into either their present or future values using a suitable ‘rate’ to represent the time value of money. A suitable proxy for the time value of money can be obtained from risk-free asset returns such as government bond yields or insured banks’ term deposit rates. These rates of return are generally termed ‘risk-free rates’. Using a risk-free rate, the cash flows occurring at different points of time can be converted into their present values at the beginning of the current year or future values at the end of the final year of the project’s planning horizon. The commonly used method in finance is to convert all the values into their present values using an appropriate discount rate (or interest rate) to represent the time value. The other method is to convert all the values into their future values, again using an appropriate interest rate to represent the time value of money. Whilst the present-value approach is the norm in project appraisal, the future-value approach may be useful at times.
In property investment analysis, discussed in Chapter 14, loan calculation formulae are used in the process of arriving at equity cash flows for the investment. These calculations also involve present- and future-value concepts.
This chapter briefly presents the essential formulae involved in project appraisal and mortgage loan calculations, with examples. It is not intended to be a detailed coverage of standard financial mathematics, but rather a quick illustration of the application of essential project appraisal and loan calculation formulae to facilitate the understanding of the project appraisal content of this book. The underlying philosophy of this chapter is to learn by doing. The emphasis is on problem-solving. Readers are urged to make full use of writing pads, pens, calculators and computers (in particular, the Excel workbooks provided on the Web).
Study objectives
After studying this chapter the reader should be able to:
r apply a discounting rate to cash flows occurring at different points in time to translate them into a common measure of value
r calculate present value, net present value (NPV) and internal rate of return (IRR) from a given cash flow series
r calculate monthly loan (mortgage) repayments, their interest and principal components and the loan balance of a mortgage loan
r understand the financial mathematics involved in the discounted cash flow techniques (such as NPV and IRR) and mortgage loans
r apply the relevant annuity formulae for project appraisal
Symbols used
t time period
r discount rate
C cash flow
Ct cash flow at end of period t
COt capital outlays at the beginning of period t
Anr present value of an annuity (of $1 per period) with the payments being made at the end of each period. The number of periods is n and the discount rate is r . (1+r )−t present value of a dollar to be received at the end of period t using a discount
rate of r .
PV present value of a cash flow stream where: P V =
n
t=1
(1+r )−tCt
FV future value of a cash flow stream at the end of period n where: F V =
n
t=1
NPV net present value IRR internal rate of return
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:
r=C1−C0 C0
The symbol, r, used for the discount rate is also employed for the rate of return to alert the reader that the discount rate is a rate of return.
Workbook 5.1
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:
r=1,100−1,000
1,000 =0.1 or 10% per annum.
Note on timing and timing symbols
In Example 5.1, the initial cash outlay occurs at the start of the year, and the cash inflow occurs at the end of the year. This structure reflects the standard timing assumptions within financial analysis. These are that cash flows occur at particular points of time, rather than within time periods, and that the initial investment (or capital outlay) occurs at the beginning of the period and subsequent cash flows occur at the end of the relevant time periods.
The length of the time period adopted is related to the analysis in hand. In a capital budgeting analysis, cash flows are assumed to occur yearly, as the project will usually extend over many years. In a short-term financial analysis, such as that of liquid cash management, the timing period may be shortened to a day or a week.
In stating when a particular flow is to occur, the convention is to use EOY as an acronym for ‘end of year’, although the symbol Y is also commonly used. Similarly EOM or M is used to denote ‘end of month’. The symbol used will be appropriate to the context. Each of these acronyms has a trailing digit which identifies the period. For example, EOY 1, or Y1, is ‘end of year 1’, and EOY 2, or Y2, is ‘end of year 2’.
It is important to note that cash flows occurring at the start of a project are denoted by EOY 0, or Y0. This timing structure allows all cash flows, including the present one, to be expressed as ‘end of year’ flows. The EOY 0 concept is that year 0 has just come to an end
today, and that year 1 will start tomorrow. Thus EOY 0 may be interpreted as the beginning of year 1.
In project analysis it is particularly important to identify and assign the cash flows to their relevant periods accurately. This means that the analyst must become adept at using the appropriate timing symbols. Correct nomenclature is vitally important in setting the analysis up for computer calculation, because the various computational routines, as in Excel for example, require particular classification of the cash flows. The workbooks accompanying the examples in this chapter demonstrate how to do this.
Future value of a single sum
The future value is the amount to which a present sum (such as a fixed term deposit placed with a bank) will grow at a future date, through the operation (or the add-on effect) of interest. Suppose you invest $1,000 today (EOY 0) for a period of two years at an interest rate of 10% per annum, with interest paid at the end of each year. At the end of the first year (EOY 1), you would have accumulated $1,100 (i.e. the original sum of $1,000 plus $100 (1,000×10%) interest). Assume that this interest is reinvested, thus increasing the level of investment to $1,100 at EOY 1. At EOY 2, the $1,100 would have increased to $1,210 (i.e. $1,100 plus $110 (1,100×10%) interest). The $1,210 is the future value of $1,000 invested at 10% per annum for two years.
The interest has been compounded annually. Compound interest simply means ‘interest earned on interest’. In the example above, $100 interest has been received at EOY 1. This $100 has been reinvested to receive $10 (100×10%) interest at EOY 2 thus bringing the interest for year 2 to $110 (1,000×10%+100×10%). Compound interest is the norm in financial calculations and compound interest is assumed throughout this book, unless otherwise stated.
Future value is calculated using the following formula: FV=PV (1+r )n
Here, PV may be interpreted as the principal (or initial amount) invested in, for example, a term deposit. For simplicity, r may be viewed as the interest rate. For example, if the interest rate is 10% per annum, r is equal to 0.10.
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?
Workbook 5.2
The same question can be stated in the context of capital budgeting as follows. The capital outlay of a project incurred at the beginning of year 1 (that is, at EOY 0), is $1. The economic life of the project is six years. Assuming an annual return of 10%, what is
the future value of this $1 investment at the end of year 6? The answer is: FV=1.00×(1.10)6=$1.771561≈$1.77
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:
Workbook 5.3
FV=2,000×(1.10)6=2,000×(1.771561)=$3,543.12
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:
PV= FV
(1+r )n
If we use symbol Cn (instead of FV ) to denote the cash flows at the end of period n, then: PV= Cn (1+r )n Recall that: 1 (1+r )n =(1+r ) −n Then: PV=Cn(1+r )−n 1 (1+r )n or (1+r )−
nis called the ‘present value factor’.
Example 5.4
In Example 5.2, the future value of $1 in six years’ time, at an interest rate of 10%, was approximately $1.77. Now, let us ask the question the other way round: 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?
Workbook 5.4
PV =$1.771561×(1+0.10)−6=$1.771561×(0.56447393)=$1.00 We now end up with $1.00. Note that rounding errors can make a small difference.
The present value factor in this example is(1+1r )6 = 1 (1.1)6 =
1
Example 5.5
The future-value question in Example 5.3 can be turned around to convert it to a question of present value.
Workbook 5.5
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?
PV=3,543.12×(1.1)−6= 3,543.12
(1.1)6 =$1,999.999=$2,000
This $2,000 was the initial investment in Example 5.3, which grew to a future value of $3,543.12 in six years.
Future value of a series of cash flows
So far, the calculations have involved a single sum. Project analysis normally involves a series of cash flows occurring at different points in time. Therefore, an understanding of the calculation of future and present values of a series of cash flows is desirable. The future value of a series of cash flows is calculated using the formula:
FV= n t=1 Ct(1+r )n−t 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? Workbook 5.6 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
Note that a number raised to power 0 is equal to 1. So (1.2)0equals 1. The timing of the cash flows is important in setting up the correct powers in this example. The first cash flow occurs at EOY 1. It has then only four years to run until the end of the project. The power value is thus 4. The power value of 4 for the first cash flow is shown in the calculation as (5−1) to demonstrate that, even though the project is for five years overall, the cash flow occurs when the project has only four years to run.
Present value of a series of cash flows
Project analysis normally estimates the present value of a series of future cash flows in the process of computing the project’s net present value. The present value of a series of future cash flows is calculated using the formula:
PV= n t=1 Ct (1+r )t = n t=1 Ct(1+r )−t 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? Workbook 5.7 PV= 100 (1.1)+ 200 (1.1)2 + 600 (1.1)3 =90.91+165.29+450.79=$706.99 Example 5.8
What is the present value of three cash flows $100, $200 and $600, to be received at EOY 1, EOY 3 and EOY 6, respectively, if the time value of money is 10% per annum?
Workbook 5.8 PV= 100 (1.1)+ 200 (1.1)3 + 600 (1.1)6 =90.91+150.26+338.68=$579.85
In these examples, the first cash flow occurs at EOY 1. As its present value is being calculated, it must be discounted for one full year. Its power value is 1. The other power values are applied accordingly.
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, however, 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. It must be noted that when the discount rate includes a risk premium for the uncertainty of the cash flows, that discount rate is greater than the interest rate used to represent just the time value of money. These distinctions are considered in detail in Chapter 7. In this chapter, the focus is on illustrating computation formulae.
Example 5.9
Assume cash flows at the end of years 1, 2, 3, 4 and 5 are $100, $300, $400, $500 and $10, respectively.
The relevant interest rates (or discount rates) for different periods are: Year 1: 10%
Year 2: 5% Year 3: 10% Year 4: 12% Year 5: 11%
Then the present value is calculated as follows: Workbook 5.9 PV= 100 (1.1) + 300 (1.1)(1.05) + 400 (1.1)(1.05)(1.1) + 500 (1.1)(1.05)(1.1)(1.12) + 10 (1.1)(1.05)(1.1)(1.12)(1.11) =$1,023.20
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: r the value of the cash flows in each period is the same; for example,
$500 at end of year 1 $500 at end of year 2 $500 at end of year 3, etc.
r the period or the interval for the cash flows remains unchanged; for example, if they are annual, they have to remain annual; if they are six-monthly, then they have to remain as six-monthly periods; if they are monthly, then they have to remain as monthly etc. r the receipt/payment of the cash flows should occur at the end of each regular period; for
example, end of each year or end of each month, as the case may be. The present value of a $1 annuity is:
Anr=1−(1+r )
−n r
The present value of an annuity of C dollars thus becomes: PV=C×Anr=C
1−(1+r )−n r
This can also be written as: PV=C r 1− 1 (1+r )n 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?
Workbook 5.10 PV=2,000× 1−(1.15)−5 0.15 =2,000× 1− 1 (1.15)5 0.15 =2,000× 1−0.49717 0.15 =2,000×[3.3522] =$6,704.40 A note on financial tables
Many corporate finance textbooks produce tables giving the present values of ordinary annuities, along with present values and future values of $1 at different interest rates for
different time periods. These tables are useful quick references for common interest rates and common time periods. The ranges that they present are, however, limited. These financial tables are artefacts of past times when electronic calculators and spreadsheets, which allow rapid and easy calculation of these factors, were not available. In this text, such tables are not included.
Present value of a deferred annuity
This is also useful in reducing the number of steps in calculating the NPV in project eva- luations.
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’.
PV=C× 1−(1+r )−n r ×(1+r )−t 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? Workbook 5.11 PV=60×1−(1.1) −20 0.1 ×1.1 −9
The index t has the value 9 because the normal present-value annuity formula has already calculated the present value to the beginning of the tenth year, that is, to the end of the ninth year.
PV=60×8.5136×0.4241=$216.64
Next, we will give another example, repeating some of these figures, but applying them