Pike Robert, Neal Bill. Corporate finance and investment: decisions and strategy
Подождите немного. Документ загружается.
.
Chapter 3 Present values and financial arithmetic 65
3.5 PRESENT VALUE
An alternative way of assessing the worth of an investment is to invert the compound-
ing process to give the present value of the future cash flows. This process is called
discounting.
The time-value of money principle argues that, given the choice of £100 now or the
same amount in one year’s time, it is always preferable to take the £100 now because
it could be invested over the next year at, say, a 10 per cent interest rate to produce £110
at the end of one year. If 10 per cent is the best available annual rate of interest, then
one would be indifferent to (i.e. attach equal value to) receiving £100 now or £110 in
one year’s time. Expressed another way, the present value of £110 received one year
hence is £100.
We obtained the present value (PV) simply by dividing the future cash flow by 1
plus the rate of interest, i, i.e.
Discounting is the process of adjusting future cash flows to their present values. It is,
in effect, compounding in reverse.
Recall that earlier we specified the future value as:
Dividing both sides by we find the present value:
which can be read as the present value of future cash flow FV receivable in n years’ time
given a rate of interest i. This is the process of discounting future sums to their present
values.
Let us apply the present value formula to compute the present value of £133 receiv-
able three years hence, discounted at 10 per cent:
The message is: do not pay more than £100 today for an investment offering a cer-
tain return of £133 after three years, assuming a 10 per cent market rate of interest.
PV
110%, 3 yrs2
£133
11 0.102
3
£133
1.33
£100
V
o
FV
n
11 i2
n
11 i2
n
FV
n
V
o
11 i2
n
PV
£110
11 0.102
£110
11.12
£100
Calculator tip
Your calculator should have a power function key, usually Try the following steps for
the previous example.
Input 1.1
Press function key
Input 3
Press
Display 1.331
Press 1/x
Multiply 133
Press
Answer 99.9
x
y
x
y
.
present value
The current worth of future
cash flows
discounting
The process of reducing cash
flows to present values
CFAI_C03.QXD 10/26/05 11:11 AM Page 65
.
66 Part I A framework for financial decisions
■ Discount tables
Much of the tedium of using formulae and power functions can be eased by using dis-
count tables or computer-based spreadsheet packages. In the previous example, the
discount factor for £1 for a 10 per cent discount rate in three years’ time is:
This can be found in Appendix C by locating the 10 per cent column and the 3 year
row. We call this the present value interest factor (PVIF) and express it as
or
Multiplying the cash flow of £133 by the discount factor yields the same result as
before:
With a constant annual cash flow, termed an annuity, we can shorten the discount-
ing operation. Appendix D provides the present value interest factor for an annuity
(PVIFA). Thus, if £133 is to be received in each of the next three years, the present value
is:
It is standard practice to write interest factors as: Interest factor(rate, period).
Examples:
is the present-value interest factor at 8 per cent for ten years.
is the present-value interest factor for an annuity at 10 per cent for four
years.
PVIFA
110,42
PVIF
18,102
£133 2.4868 £331
PV £133 PVIFA
110%, 3 yrs2
PV £133 0.751 £100 1subject to rounding2
PVIF
110,32
.
PVIF
110%, 3 yrs2
1
11.102
3
1
1.33
0.751
Self-assessment activity 3.2
Calculate the present value of £623 receivable in eight years’ time plus £1,092 receivable
eight years after that, assuming an interest rate of 7 per cent.
(Answer in Appendix A at the back of the book)
Example of present values: Soldem Pathetic plc
Soldem Pathetic Football Club has recently been bought up by a wealthy business-
man who intends to return the club to its former glory days. He also wants to pay a
good dividend to the shareholders of the newly-formed quoted company by making
sound investments in quality players. One such player the manager would dearly
like in his squad is Bryan Riggs, currently on the market for around £9 million. The
chairman reckons that, quite apart from the extra income at the turnstiles from buy-
ing him, he could be sold for £11 million by the end of the year, given the way trans-
fer prices are moving. Should he bid for Riggs?
Assuming a 10 per cent rate of interest as the reward that the other shareholders
demand for accepting the delayed payoff, the present value (PV) of £11 million receiv-
able one year hence is:
£10 million
PV discount factor future cash flow
1
1.10
£11 million
annuity
A constant annual cash flow
for a prescribed period of time
CFAI_C03.QXD 10/26/05 11:11 AM Page 66
.
Chapter 3 Present values and financial arithmetic 67
In the highly simplified example above, we assumed that the future value was cer-
tain and the interest rate known. Of course, a spectrum of interest rates is listed in the
financial press.
This variety of rates arises predominantly because of uncertainty surrounding the
future and imperfections in the capital market. To simplify our understanding of the
time-value of money concept, let us ‘assume away’ these realities. The lender knows
with certainty the future returns arising from the proposal for which finance is sought,
and can borrow or lend on a perfect capital market. The latter assumes the following:
1 Relevant information is freely available to all participants in the market.
2 No transaction costs or taxes are involved in using the capital market.
3 No participant (borrower or lender) can influence the market price for funds by the
scale of its activities.
4 All participants can lend and borrow at the same rate of interest.
Under such conditions, the corporate treasurer of a major company like Shell can
raise funds no more cheaply than the chairman of Soldem Pathetic. A single market
rate of interest prevails. Borrowers and lenders will base time-related decisions on this
unique market rate of interest. The impact of uncertainty will be discussed in later
chapters; for now, these simplistic assumptions will help us to grasp the basics of
financial arithmetic.
■ The effect of discounting
Figure 3.1 shows how the discounting process affects present values at different rates
of interest between 0 and 20 per cent. The value of £1 decreases very significantly as the
rate and period increase. Indeed, after 10 years, for an interest rate of 20 per cent, the
present value of a cash flow is only a small fraction of its nominal value.
Table 3.3 summarises the discount factors for three rates of interest. It is useful to
develop a ‘feel’ for how money changes with time for these rates of interest. The 15 per
cent discount rate is particularly useful, because investment surveys (e.g. Pike 1988)
suggest that this is a popular discount rate for evaluating capital projects. It also hap-
pens to be easy to remember: every five years the discounted value halves. Thus, with
a 15 per cent discount rate, after five years the value of £1 is 50p, after 10 years 25p, etc.
How much better off will the club be if it buys Riggs? The answer is, in present value
terms:
We call this the
net present value (NPV). The decision to buy the player makes eco-
nomic sense; it promises to create wealth for the club and its shareholders, even
excluding the likely additional gate receipts. Of course, Riggs could break a leg in the
very first game for his new club and never play again. In such an unfortunate situa-
tion, the club would achieve a negative NPV of £9 million, the initial cost.
Alternatively, he could be insured against such injury, in which case there would be
premiums to pay, resulting in a lower net present value.
Another way of looking at this issue is to ask whether the investment offers a return
greater than could have been achieved by investing in financial, rather than human,
assets. The return over one year from acquiring Riggs’ services is:
If the available rate of interest is 10 per cent, the investment in Riggs is a considerably
more rewarding prospect.
Return
Profit
investment
£11 m £9 m
£9 m
100 22.2%.
£10 million £9 million £1 million
net present value
The present value of
the future net benefits
less the initial cost
CFAI_C03.QXD 10/26/05 11:11 AM Page 67
.
68 Part I A framework for financial decisions
Present value of £1
1.00
0%
5%
10%
15%
20%
0246810
Periods
0.75
0.50
0.25
Figure 3.1
The relationship
between present value
of £1 and interest over
time
Table 3.3
Present value of a single
future sum
Year 10% 15% 20%
0 £1.00 £1.00 £1.00
5 0.60 0.50 0.40
10 0.40 0.25 0.16
15 0.24 0.12 0.06
20 0.15 0.06 0.03
25 0.09 0.03 0.01
Self-assessment activity 3.3
Your company is just about to sign a deal to purchase a fleet of lorries for £1 million. The
payment terms are £500,000 down payment and £500,000 at the end of five years. No
one present has a calculator or discount tables to hand. If the cost of capital for the com-
pany is 15 per cent, what is the present value cost of the purchase?
(Answer in Appendix A at the back of the book)
3.6 PRESENT VALUE ARITHMETIC
We have seen that the present value of a future cash flow is found by multiplying the
cash flow by the present value interest factor. The present value concept is not difficult
to apply in practice. This section explains the various present value formulae, and illus-
trates how they can be applied to investment and financing problems. Throughout, we
shall use the symbol X to denote annual cash flow in pounds and i to denote the inter-
est, or discount, rate (expressed as a percentage). Recall that PVIF is the present value
interest factor and PVIFA is the PVIF for an annuity.
■ Present value
We know that the present value of X receivable in n years is calculated from the
expression:
X times PVIF
1i,n2
PV
1i,n2
X
n
11 i2
n
CFAI_C03.QXD 10/26/05 11:11 AM Page 68
.
Chapter 3 Present values and financial arithmetic 69
Example
Calculate the present value of £1,000 receivable in 10 years’ time, assuming a discount
rate of 14 per cent:
Alternatively, the table in Appendix C provides PVIF of 0.26974 for and
per cent:
The present value of £1,000 receivable ten years hence, discounted at 14 per cent, is
thus £269.74.
PV £1,000 0.26974 £269.74
i 14
n 10
PVIF
114%, 10 yrs2
1
11.142
10
0.26974
Self-assessment activity 3.4
Calculate the present value of £1,000 receivable 12 years hence, assuming the discount
rate is 12 per cent.
(Answer in Appendix A at the back of the book)
■ Valuing perpetuities
Frequently, an investment pays a fixed sum each year for a specified number of years.
A series of annual receipts or payments is termed an annuity. The simplest form of
annuity is the infinite series or perpetuity. For example, certain government stocks offer
a fixed annual income, but there is no obligation to repay the capital. The present value
of such stocks (called irredeemables) is found by dividing the annual sum received by
the annual rate of interest:
Example
Uncle George wishes to leave you in his will an annual sum of £10,000 a year, starting
next year. Assuming an interest rate of 10 per cent, how much of his estate must be set
aside for this purpose? The answer is:
Suppose that your benevolent uncle now wishes to compensate for inflation, esti-
mated to be at 5 per cent per annum. The formula can be adjusted to allow for growth
at the rate of g per cent p.a. in the annual amount. (The derivation of the present value
of a growing perpetuity is found in Appendix II at the end of the chapter.)
As long as the growth rate is less than the interest rate, we can compute the present
value required:
This formula plays a key part in analysing financial decisions and will be developed
further in Chapter 4, when we consider the valuation of assets, shares and companies.
PV
£10,000
0.10 0.05
£200,000
PV
X
i g
PV perpetuity
£10,000
0.10
£100,000
PV perpetuity
X
i
perpetuity
A constant annual cash flow
for an infinite period of time
CFAI_C03.QXD 10/26/05 11:11 AM Page 69
.
70 Part I A framework for financial decisions
■ Valuing annuities
An annuity is an investment paying a fixed sum each year for a specified period of
time. Examples of annuities are many credit agreements and house mortgages.
The life of an annuity is less than that of a perpetuity, so its value will also be some-
what less. In fact, the formula for calculating the present value of an annuity of £A is
found by calculating the present value of a perpetuity and deducting the present value
of that element falling beyond the end of the annuity period. This gives the somewhat
complicated formula (see Appendix II at the end of chapter for the derivation) for the
present value of an annuity (PVA):
In words, the present value of an annuity for n years at i per cent is the annual sum
multiplied by the appropriate present value interest factor for an annuity.
Suppose an annuity of £1,000 is issued for 20 years at 10 per cent. Using the table in
Appendix D, we find the present value as follows:
£1,000 8.5136 £8,513.60
PVA
110%, 20 yrs2
£1,000 PVIFA
110,202
A PVIFA
1i,n2
PVA
1i,n2
A¢
1
i
1
i11 i2
n
≤
Self-assessment activity 3.5
Calculate the present value of £250 receivable annually for 21 years plus £1,200 receiv-
able after 22 years, assuming an interest rate of 11 per cent.
(Answer in Appendix A at the back of the book)
■ Calculating interest rates
Sometimes, the present values and future cash flows are known, but the rate of interest
is not given. A credit company may offer to lend you £1,000 today on condition that you
repay £1,643 at the end of three years. To find the compound rate of interest on the loan,
we solve the present value formula for i:
Rearranging the formula,
Turning to the tables in Appendix C and looking for 0.6086 under the year 3 col-
umn, we find the rate of interest is 18 per cent. As we shall see in Chapter 5, this cal-
culation is fundamental to investment and finance decisions and is termed the internal
rate of return.
It is also possible to solve the present value formula for i:
i 1FV>PV2
1>n
1
11 i2
n
FV>PV
PV
FV
11 i2
n
PVIF
1i,32
PV
FV
£1,000
£1,643
0.60864
PV
1i,n2
PVIF
1i,n2
FV
internal rate of return
The rate of return that equates
the present value of future cash
flows with initial investment
outlay
CFAI_C03.QXD 10/26/05 11:11 AM Page 70
.
Chapter 3 Present values and financial arithmetic 71
In the above example:
i 11,643>1,0002
1>3
1 0.18 or 18%
3.7 VALUING BONDS
DCF has long been used to help value financial securities. We consider share valuation
in the next chapter, but deal here with the valuation of bonds. A bond is a long-term
loan which promises to pay interest and repay the loan in accordance with the agreed
terms. Governments, local authorities, companies and other organisations frequently
seek to raise funds by issuing bonds, sometimes termed loan stock.
Once issued, bonds are traded in the secondary bond markets. Although a bond has
a par, or nominal, value – typically £100 – its actual value will vary according to the
cash flows it pays (interest and repayments) and the prevailing rate of interest for this
type of bond. The fair price is the present value of the future interest and repayments.
Bondo Ltd issues a two-year bond with a 10 per cent coupon rate and interest
payable annually. The bond is priced at its face value of £100:
The bond value above includes the present value of the first year’s interest plus the
present value of the two elements of the Year 2 cash flow (i.e. interest and redemption
value).
Assume that the interest rate unexpectedly rises to 12 per cent. The bond is now
priced in the secondary market at a discount at the lower value of £96.62, reflecting the
fact that the 10 per cent interest rate is now less attractive to investors:
Assume now that the interest rates fall to 8 per cent. The bond would now be
viewed as more attractive and lead it to be priced at a premium:
From the above we may conclude that bonds will sell:
■ at a discount where the coupon rate is below the market interest rate, and
■ at a premium where the coupon rate is above the market interest rate.
£103.57
£10
1.08
£10 £100
11.082
2
£96.62
£10
1.12
£10 £100
11.122
2
£100
£10
1.10
£10 £100
11.102
2
V
o
PV 1interest payments2 PV 1redemption value2
Who wants to be a millionaire?
An advertisement in the financial press read: ‘How to become a millionaire? Invest £9,138
in the M&G Recovery unit trust in 1969 and wait for 25 years.’ So, for those of us who
missed out on this investment, let us grudgingly calculate its annual return:
By investing in a unit trust earning an annual rate of return of around 21 per cent,
£9,138 turns you into a millionaire in 25 years’ time. All you have to do is find an
investment giving 21 per cent for 25 years!
20.66%
1£1 million>£9,1382
1>25
1
i 1FV>PV2
1>n
1
discount
The amount below the face
value of a financial instrument
at which it sells
Coupon rate
The nominal annual rate of
interest expressed as a per-
centage of the principal value
premium
The amount above the face
value of a financial instrument
at which it sells
CFAI_C03.QXD 10/26/05 11:11 AM Page 71
.
72 Part I A framework for financial decisions
In the above example, the market interest was known. It may be that we know the bond
prices and wish to calculate the yield to maturity. Here we use the same formula but
the unknown is the interest rate. Thus, in the above where the market price is £103.57
we solve the equation (using a computer or trial and error) to find that 8 per cent is the
yield to maturity. The bond has a 10 per cent coupon and is priced at £103.57 to yield 8
per cent.
■ Valuing a bond in Millie Meter plc
Some time ago you purchased an 8 per cent bond in the fashion chain, Millie Meter.
Today, it has a par value of £100 and two years to maturity. Interest is payable half-
yearly. What is it worth?
Assuming the current comparable rate of interest is 8 per cent, the value should
equal the par value of £100.
Notice that because payments are made half-yearly, both the interest and discount
rate are half the annual figures.
In reality, the required rate of return demanded by investors may be different from
the original coupon rate. Let us say it is 10 per cent. As this is higher than the coupon
rate, the bond value for Millie Meter will fall below its par value:
This example shows that an investor would have to pay £96.45 for a bond offering
a 4 per cent coupon rate (i.e. based on the par value of £100) plus the redemption
value in two years’ time, assuming that the market rate of interest for this security is
10 per cent.
For actively traded bonds there is little need to value them in this way because, if
the bond market is efficient, it has already done it for you. All you need do is to look
at the latest quoted price. However, the required rate of return is less easy to obtain.
Who says, in the above example, that 10 per cent is the return expected by the market
for this type of bond? The answer is simple. If we know the current bond price, we put
this in the above equation to find that discount rate which equates price with the dis-
counted future cash flows – 10 per cent in the previous example.
V
o
4
11.052
4
11.052
2
4
11.052
3
4
11.052
4
100
11.052
4
£96.45
V
o
4
11.042
4
11.042
2
4
11.042
3
4
11.042
4
100
11.042
4
£100
Back to the future
FT
‘Tis the season to be jolly but there’s
always someone to cry ‘Humbug’.
According to Guy Monson from Saracen
Investment Fund in London, things are
pretty much as they were back in 1843
when Charles Dickens gave the world
Ebenezer Scrooge, miser extraordinaire,
in his novel A Christmas Carol.
Interest rates, government bond yields
and inflation are all within a whisker of
where they stood 141 years ago. There’s
also much living beyond one’s means:
that exercised Scrooge then and worries
analysts now.
If that wasn’t enough, some things have
actually got worse since the days of poverty
that Dickens so savagely chronicled. Back
then, income tax stood at just 5 per cent.
As old Ebenezer so charmingly put it:
‘Every idiot who goes around with Merry
Christmas on his lips should be buried
with his own pudding.
’
Source: Financial Times, 23 December 2004, p. 12
yield to maturity
The interest rate at which the
present value of the future
cash flows equals the current
market price
CFAI_C03.QXD 10/26/05 11:11 AM Page 72
.
Chapter 3 Present values and financial arithmetic 73
■
Factors affecting interest rates
It is common in financial management to talk about the interest rate ruling in the
money market. However, it is important to realise that there is never a single prevail-
ing rate. At any time, there is a spectrum of interest rates on offer – along this spectrum
the rates depend on the identity of the borrower e.g. firm or government, and hence,
the degree of risk faced by the lender, the amount lent or borrowed and the period over
which the loan is made available. The last of these aspects is referred to as the term
structure of interest rates. This shows how the yields offered for loans of different
maturities vary as the term of the loan increases. We discuss this, together with the
Yield Curve, in Appendix I to this chapter.
3.8 NET PRESENT VALUE
We have assumed that the paramount objective of the firm is to create as much wealth as
possible for its owners through the efficient use of existing and future resources. To cre-
ate wealth, the present value of all future cash inflows must exceed the present value of
all anticipated cash outflows. Quite simply, an investment with a positive net present value
increases the owners’ wealth. The elements of investment appraisal are shown in Figure 3.2.
Most decisions involve both costs and benefits. Usually, the initial expenditure
incurred on an investment undertaken is clear-cut: it is what we pay for it. This
includes the cash paid to the supplier of the asset plus any other costs involved in
making the project operational. The problems start in measuring the worth of the
investment project. What an asset is worth may have little to do with what it cost or
what value is placed on it in the firm’s Balance Sheet. A machine standing in the firm’s
books at £20,000 may be worth far more if it is essential to the manufacture of a high-
ly profitable product, or far less than this if rendered obsolete through the advent of
new technology. To measure its worth, we need to consider the value of the current and
term structure of interest
rates
Pattern of interest rates on
bonds of the same risk with
different lengths of time to
maturity
Increase shareholder value
Annual cash flow
(
X
t
)
Cost of capital
(
K
)
Discount cash flows at the cost of capital
to find net present value
ACCEPT REJECT
–+
Goal
Inputs
Financial analysis
NPV signal
Decision outcome
Figure 3.2
Investment appraisal
elements
CFAI_C03.QXD 10/26/05 11:11 AM Page 73
.
74 Part I A framework for financial decisions
The net present value rule
Wealth is maximised by accepting all projects that offer positive net present values
when discounted at the required rate of return for each investment.
Most of the main elements in the NPV formula are largely externally determined.
For example, in the case of investment in a new piece of manufacturing equipment,
management has relatively little influence over the price paid, the life expectancy or
the discount rate. These elements are determined, respectively, by the price of capital
goods, the rate of new technological development and the returns required by the cap-
ital market. Management’s main opportunity for wealth creation lies in its ability to
implement and manage the project so as to generate positive net cash flows over the
project’s economic life.
■ An NPV example: Gazza Ltd
The management of Gazza Ltd is currently evaluating an investment in hair dye prod-
ucts costing £10,000. Anticipated net cash inflows are £6,000 received at the end of year
1 and a further £6,000 at the end of year 2. Assuming a discount rate of 10 per cent, cal-
culate the project’s net present value.
We can compute the NPV for Gazza using three different approaches, all of which
will be employed in later chapters.
Self-assessment activity 3.6
Define the main elements in the capital investement decision.
(Answer in Appendix A at the back of the book)
future benefits less costs arising from the investment. Wherever possible, these benefits
should be expressed in terms of cash flows. Sometimes (as will be discussed later) it is
impossible to quantify benefits so conveniently. Typically, investment decisions
involve an initial capital expenditure followed by a stream of cash receipts and dis-
bursements in subsequent periods. The net present value (NPV) method is applied to
evaluate the desirability of investment opportunities. NPV is defined as:
which may be summarised as:
where is the net cash flow arising at the end of year t, I is the initial cost of the invest-
ment, n is the project’s life, and k the minimum required rate of return on the invest-
ment (or discount rate). (The Greek letter or sigma, denotes the sum of all values in
a particular series.)
Note that we have introduced a subtle change in notation, replacing i, which denot-
ed the general market rate of interest, by k, which refers to the rate of return that must
be achieved by the firm in question. As we shall see, k may vary significantly from firm
to firm.
A project’s net present value (NPV) is determined by summing the net annual cash
flows, discounted at a rate that reflects the cost of an investment of equivalent risk on
the capital market, and deducting the initial outlay.
©,
X
t
NPV
a
n
t 1
X
t
11 k2
t
I
NPV
X
1
11 k2
X
2
11 k2
2
X
3
11 k2
3
p
X
n
11 k2
n
I
CFAI_C03.QXD 10/26/05 11:11 AM Page 74