basic finance sheet

8/15/2019 Basic Finance Sheet

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Nottingham University Business School© 2015 David P. Newton

1

BASIC FINANCE TECHNIQUES

This is a handout on some basics. It is not required reading; however, many of you in

the tutorials I gave after Lecture 2 had difficulties with some of the details in questions,

particularly with the one that was apparently the easiest: the first question (that’s why I

set it). It’s the ‘easy’ things that most often catch people out.

I hope this extra handout will help. I guess you might find the section on continuous

compounding useful (note Figure 6).

There is no need to memorise formulae for elementary things such as conversion of rates

between different time periods (often, you can do this in your head); I needed to express

the methods on paper, you need to be able to perform the calculations in whatever

manner you prefer, so long as you can get the right answers.

David Newton

Discounted Cash Flow and Net Present Value

The concept underlying discounted cash flow (DCF), present value (PV) and net present

value (NPV) is one of the most important in basic finance. Mathematically, the processis merely the inverse of that for compounding interest, with a “discount rate” substituted

for an “interest rate”. It is complicated by the difficulty of choosing an appropriate rate

at which to discount. Before you read on, remember that financial calculations can be

no more exact than the numbers fed into the equations; if the numbers are approximate

then so will be the answers from the equations!

People sometimes find themselves able to perform DCF, PV and NPV calculations but

are quietly puzzled by what they have done. Sometimes, this is because they have

forgotten the ideas of simple and compound interest (their textbooks proceed directly todiscounting) and they’re slightly embarrassed by not properly remembering what they

wer e taught at school. Sometimes, they’re happy with interest calculations and with the

mechanics of DCF and NPV but they don’t see why any particular number should be

used as the discount rate - so they use a number they find in a textbook (ten percent is a

convenient figure!) or a number used by most people doing similar calculations in their

company or in their university.

To those who do not fully remember what they were taught about simple and compound

interest at school: don’t worry; it’s here - and with the consolation that maybe there are afew points not taught in school. There is more here than at first appears. For example,

consider some rate of interest on a loan say 0.5 p.a. (i.e. 50% interest per annum), for

six months only; would you prefer to be paid simple interest or compound interest?

Answer: for periods longer than the interest period (here = 1 year) you'd prefer

compounding but for lesser periods simple interest (surprisingly?) pays more! I leave it

to you to read on and check this.


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2

Interest on Investments

Simple Interest

Suppose that today you borrow from me £100 and agree to repay me, one year from

today, with £100 plus one tenth (10%) of that amount as interest. The amount which

you would repay us would be calculated as:

£100 (1 + 0.1) = £110.

Suppose, instead, that I agree to lend you the money for four years with "simple interest"

set at ten percent (10%) each year (in Latin: "per annum" or "p.a.") and that all moneyowed to us should be paid together, after four years. The amount which you owed

would grow like this:

Today 1 Year 2 Years 3 Years 4 Years

£100 £110 £120 £130 £140

After four years had passed, you would repay me with £140. Notice that after one year the

amount owed continues to grow by the same amount (£10) each year; interest is only being

charged on the original sum (£100), not on the amount owed in the previous year. For

example, after two years the amount owed is £110 plus ten percent of £100 (£110 + £10 =

£120) NOT £110 plus ten percent of £110. For this reason, it is called "simple interest".

If we represent time, measured in years, by the symbol t, then the amount owed may berepresented by the equation

Amount owed = £100(1+0.1t)

The £ sign may be dropped because we know our equations refer to financial value and

the particular currency used is irrelevant to our discussion. The interest rate (here: 0.1,

giving £10 interest on £100) can be replaced with a general symbol, r, to represent all

possible rates:

Amount owed = 100(1 + rt)

The "amount owed" could also be called the "future value" or "FV". To distinguish this

from future value using compound interest, which we will consider later, it will be calledFVsimple. The amount lent can be called the "present value" or "PV", allowing us to

write a general formula for the calculation of simple interest:

FVsimple = PV(1 + rt)


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Example

PV = 100 the amount initially lent

r = 0.1 interest rate = 0.1 (as a percentage: ten percent)

t = 4 since repayment will be made four years later

FVsimple = PV(1+rt)

= 100(1+0.4)

= 140 This is the result you have already seen: £100 lent

at simple interest of ten percent returns £140 after

four years.

Periodic versus Steady Accumulation of Simple Interest

Originally, I specified the amounts which you would owe me at the end of each

successive year (£110, £120, £130 and £140). These amounts are represented in Figure

1.

90

100

110

120

130

140

150

Time / years

F V s i m p l e

/ d o l l a r s

0 1 2 3 4

Figure 1Simple Interest on $100

We could agree that the amount which you owe me increases only once per year. This

would be of practical importance if we also agreed that our arrangement could be

terminated by repayment of the amount owed at any time. In this case, the amounts

payable would change as represented in Figure 2. This would mean, in effect, that we

had agreed that the equation for calculating the amount owed would only apply whenwhole numbers of years had passed. In other words, it would be understood that t in the

equation FVsimple = PV(1 + rt) could only be given an integer value (1, 2, 3, ...).

Look at Figure 2 and consider the amount payable if the full repayment

were to be made after one and a half years. If 1.5 is substituted for t in the

equation, the result is 100(1+0.15) = 115. Clearly, the agreement calls for

the true value of t to be rounded down to the nearest integer (1.5 replaced

by 1) so that the amount payable is 100(1+0.1) = 110.


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Do note that Figure 2 could be wrongly interpreted as showing two values

for FV at each integer year (1, 2, 3, 4) but that what is meant is a sudden

change to the upper value (at 1 year, 110 is meant, at 2 years 120 is meant,

etc.)

Alternatively, since the points on Figure 1 clearly all lie on one straight line, we could

agree that the amount which you owe increases steadily, as represented in Figure 3. Now

any value for t can be substituted into the equation FVsimple = PV(1 + rt) and the

correct answer, for steady accumulation of simple interest, is obtained.

In Figure 3, steadily accumulated simple interest at 0.10 p.a. (10% p.a.)

after one and a half years gives FVsimple = 100(1+0.15) = 115.

Now an important point will be emphasised, though it might not seem so at this stage. If

interest is accumulated steadily then future value can be calculated over a total time

which is not an integer multiple of the unit of time used in the interest rate. In plainer

English, using the example, even though the rate is 0.10 per year (10% p.a.), steadily

accumulated interest can be calculated for 1.5 years, or any other value, using the

equation given. Graphically, in Figure 1 only points at integer values of years are shown

but intermediate values can be calculated, as shown in Figure 3. A similar situation

exists for compound interest but in that case there is more possibility for confusion - and

then you may find that returning to consider simple interest may help you with

compound interest.

0 1 2 3 4

90

100

110

120

130

140

Time / years

Figure 3Simple Interest on $100 (Steady)

0 1 2 3 4

90

100

110

120

130

140

Time / years

Figure 2Simple Interest on $100 (Periodic)


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5

Conversion Of Simple Interest Rates To Different Time Periods

Usually, compound interest is used in finance. However, it is often the case that over

short periods of up to a few months simple interest is used. Although it may appear

logically inconsistent to mix simple and compound interest in a financial calculation, in

practical finance convenience is key; use of simple interest over short periods is little

different from using compound interest. Consequently, it is sometimes necessary to be

able to convert a simple interest over a certain period of time to its equivalent over a

different period.

Rather than immediately giving a formula for the conversion between interest rates over

different time periods, first I’ll show you an example which may make the idea behind

that formula more obvious. The formula for calculating future values was derivedearlier:

FVsimple = PV(1 + rt)

Using this formula, if the simple interest rate per six month period is 10% then a debt of

£100, without repayment, increases as shown in the table below. Taking as an example

the amount owed after two periods of six months, FV = 100(1+0.2) = 120. Clearly, we

can see that this is an annual interest rate of 20%.

6 month 12 month

Period FV Period

1 110

2 120 1

3 130

4 140 2

5 150

6 160 3

If the measurement of time in six month periods is replaced by measurement in years

then the periods 2, 4, 6 become 1, 2, 3 and the appropriate interest rate changes from

10% per period (of six months) to 20% per period (of one year). Repeating this using

two years instead on one, we get FV = 100(1+0.4) = 140. The number 0.4 is obtained

by multiplying 0.2 by 2 when a time period one years, or by multiplying 0.1 by 4 when a

time period is six months.

In converting from time periods measured in multiples of six months to time periods

measured in multiples of twelve months the interest rate changed from 10% per period

to 20% per period. To distinguish between the two rates and two time periods,

subscripts “a” and “b” will be given to r and t.


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The general formula for conversion is:

Formula Example

r ata r bt b (0.2)1= (0.1)2 = 0.2

Remember: r a and ta use a time period different from that used for r b and t b

If three of the variables in this equation are known then the fourth can be calculated.

For example, a rate of 20% p.a. (0.20) is equivalent to a six-month rate, r,

given by:

r 6

120 20 010( . ) . since a

b

a b r

ttr

Similarly, if each month has 30 days then an interest rate of 6% (0.06) over

three months is equivalent to a rate, r over fifteen days, given by:

r 15

900 06 0 01( . ) .

and, conversely, a rate of 1% (0.01) over fifteen days is equivalent to 6%

(0.06) over three months:

r 90

15

0 01 0 06( . ) .

Compound Interest

Suppose that today we agree that I should lend you £100 for four years at 10% interest,

"compounded" annually, and that all money owed to me should be paid together, after

fours years; in other words, the same arrangement as described for simple interest, but in

this instance accumulating "compound" interest. The amount which you owed me

would grow like this:


£100 £110.00 £121.00 £133.10 £146.41

After one year, the amount owed is identical to that owed using simple interest; £100

plus 10% of £100. The difference between simple interest and compound interest is that

simple interest involves accumulating interest on the original sum (PV) only but

compound interest includes interest on the original sum and also interest on the interest

accumulated in earlier periods. Compound interest on £100 accumulates as shown

below:


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Year 0 100

Year 1 10% interest is charged on £100 and so 100 becomes 110

Total: £110

Year 2 100 becomes 100 + 10 (£100 plus interest, as before)

10 becomes 10 + 1 (£10 interest from Year 1, PLUS interest on that)

Total: £121

Year 3 100 becomes 100 + 10 (£100 plus interest, as before)

10 becomes 10 + 1 (£10 interest from Year 1, PLUS interest on that)

10 becomes 10 + 1 (£10 interest from Year 2 PLUS interest on that)

1 becomes 1 + 0.1 (£1 interest from Year 2, PLUS interest on that)Total: £133.1

Year 4

You may easily show that the sum of the figures above, plus 10% interest on

each, is £146.41.

Once you have appreciated how interest is charged on interest from earlier periods, a

more straightforward way of showing the compounding process is:

Time / years Amount owed / dollars

0 100.00

1 100(1 + 0.1) = 110.00

2 110(1 + 0.1) = 121.003 121(1 + 0.1) = 133.10

4 133.1(1 + 0.1) = 146.41

Since the amount for each year is 1.1 times the amount for the preceding year, the

amount for any year in general, "t", is given by

Future value, FV = 100*(1+0.1)t = 100*1.1t

The value today (here £100) is the "present value", PV. Generalising to any interest

rate, r, and any value of t, we have

Future value, FV = PV(1 + r)t

As FVsimple has already been distinguished by its subscript, there is no need to specify

“FVcompound” and so “FV” will be used to represent future value under compound

interest. It is sometimes convenient to distinguish between future values at different

times using a subscript showing the time. For example, future values at one year and

two years could be represented by FV1 and FV2. Thus, in textbooks, you may see the

general equation written as FVt = PV(1 + r)t. , where you are expected to understand


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that compound interest is meant and that the subscript, t, is just a label emphasising the

time for which a particular FV is calculated.

Before electronic calculators and computers were available, “compound interest tables”

were published to help with calculations of future values. These are also known as

tables of future values or tables of “future value factors”. Although such tables are now

practically redundant, you may still encounter them. A limited range of values are

shown in Table 1. You might like to check one or two of these, using a calculator or

computer, setting PV = 1 in the equation for FV.

Table 1

Future value of £1 after t Years (Compound Interest)

Rate: 10% 20% 30% 40% 50%

Years

1 1.1000 1.2000 1.3000 1.4000 1.5000

2 1.2100 1.4400 1.6900 1.9600 2.2500

3 1.3310 1.7280 2.1970 2.7440 3.3750

4 1.4641 2.0736 2.8561 3.8416 5.0625

5 1.6105 2.4883 3.7129 5.3782 7.5938

6 1.7716 2.9860 4.8268 7.5295 11.3906

7 1.9487 3.5832 6.2749 10.5414 17.0859

8 2.1436 4.2998 8.1573 14.7579 25.6289

9 2.3579 5.1598 10.6045 20.6610 38.443410 2.5937 6.1917 13.7858 28.9255 57.6650

For example, £100 invested at 50% p.a. compound interest

for ten years would become £100(57.6650) = £5,766.50.

Note that, to four decimal places, (1 + 0.50)10 = 57.6650.

In some textbooks, in which the numbers in the table are called

“future value factors”, or FVF, a pair of subscripts is used to

identify the interest rate per period and number of periods:

FVFi,n. Thus, FVF0.50,10 = 57.6650.

Reinforcing what will doubtless be asked in class: yes, do use a calculator with

"scientific functions"; in particular logarithm, exponential, power and roots (= 1/power).

A memory for equations which can be recalled & used is useful too.... but that’ll get you

in trouble with university rules in examinations...


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9

Periodic versus Steady Accumulation of Compound Interest

Compound interest was introduced using an interest rate of 0.10 p.a. (10% p.a.). This is

a convenient rate for both simple and compound interest because the numbers arising in

calculations quickly become familiar, making it easy to concentrate on finance rather

than arithmetic. Suppose we agree that the £100 we lent you should be charged at 50%

p.a. interest instead of 10%; a handsome arrangement for me! Also, it will make the

graphs I want to show you clearer. £100 at 10 % p.a. and 50% p.a. grows like this:


At 10% p.a. £100 £110.00 £121.00 £133.10 £146.41

At 50% p.a. £100 £150.00 £225.00 £337.50 £506.25

These numbers can be calculated from FV = PV(1 + r)t

or read directly from Table 1 (multiplying by 100).

The amounts for 50% p.a. are represented in Figure 2-4 (the straight line for steady

simple interest at 50% is included, for comparison).

0

100

200

300

400

500

600

Time / years

F V / d o l l a r s

0 1 2 3 4

Figure 4

Compound Interest at 50% on $100

steady s impleinterest at50%,for

You have seen how simple interest can be accumulated periodically or steadily. A

similar argument can be followed here. We could agree that the amount which you owe

increases only once per year, in which case, the amounts payable would change in step-

wise fashion, as shown by one of the lines in Figure 5. This is periodic accumulation.

When simple interest was considered, the next stage in the argument, in effect,concerned the smooth joining of the dots in the graph (Figure 1)! This did not seem

difficult, since the dots all fell on one straight line and the straight line was represented

by the equation for FVsimple. In Figure 4, the dots clearly do not lie on a straight line

but we do have an equation for the curve which joins them:

Future value, FV = PV(1 + r)t


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This equation gives the smooth curve, for steady accumulation of compound interest,

shown in Figure 5. This is an important result. Even though interest is “compounded

annually”, compound interest can be accumulated and calculated for times which are not

integer numbers of years.

In Figure 5, steadily accumulated compound interest at 0.50 p.a. (50% p.a.)

after one and a half years gives FV = 100(1+0.50)1.5 = 183.71.

Alternatively, if the agreement were for periodic accumulation, with interest

added only at integer numbers of years, then the amount owed would have

been as shown by the “stepped” line (the amount owed “jumps”

discountinuously, as in Figure 2 for simple interest, but in order to make the

diagram clearer where this occurs, dotted lines have been drawn in this

case). The true value of t would be rounded down to the nearest integer(1.5 replaced by 1) and the amount owed would then be given by

100(1+0.50)1 = 150. These results hold equally well if another time period

is used; for example, half-years instead of years. After a year and nine

months, FV = 100(1+0.50)1.75 = 203.31 but, with stepping every six

months, the true value of t would be rounded down to the nearest 0.5 (1.75

replaced by 1.5) and the amount owed would then be given by

100(1+0.50)1.5

= 183.71. Notice that at some level of accuracy, “stepping”

will, in effect, be used; compound interest may be charge to the nearest

month or to the nearest day but will not commonly be charged more

accurately.

Figure 5

Compound Interest at 50% on $100

Showing steady and periodic interest for both conpound and simple interest

0

200

400

600

800

Time / years

F V /

D o l l a r s

0 1 2 3 4 5

DPN: these lines should comeout smooth, not dotted. Sorry...poor drawing package!


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11

Conversion Of Compound Interest Rates To Different Time Periods

Suppose you have an interest rate of 0.21 (21%) over two years and you want to know

its equivalent compound rate per annum. These are familiar numbers; the first example

we looked at for compound interest was 0.10 p.a. (10% p.a.), which compounds to 0.21

(21%) over two years.

0.21 per two-year period is equivalent to 0.10 per one-year period

Consider what has been done here. The formula for future value, FV = PV(1 + r) t, has

been used twice to give expressions for FV which must be equal:

FV = PV(1 + r)t = PV(1 + 0.21)

1

r is per two-year period; t = one periodFV = PV(1 + r)t = PV(1 + 0.10)2 r is per one-year period; t = two periods

PV(1 + 0.21)1 = PV(1 + 0.10)2 PV appears on both sides and cancels:

(1 + 0.21)1 = (1 + 0.10)2 (which is true!)

Given only one of 0.21 and 0.10, the other could have been calculated

The general formula for conversion is easily derived; the difficult part is in carefully

defining what is meant by the symbols in it. Since the future value formula is used

twice, subscripts will be given to r and t in order to distinguish two pairs of values: r a

and ta; r b and t b where r a is the interest rate per period of length ta; r b is the interest rate

per period of length t b.

Example: r a = 0.10 per year (p.a.) ta = two yearsr b = 0.21 per “two-years” t b = one “two-year” (a and b are only labels, so it doesn’t matter

which “t, b” pair takes which value).

The trick is to use the future value formula twice, for the same total period of time but

measured in different units of time, then equate the two results. The general formula for

conversion is then quickly found:

Equations Example

PV r a ta PV r b

t b( ) ( )1 1 PV PV( . ) ( . )1 0 21 1 010 2

( ) ( )1 1r a

ta r b

t b ( . ) ( . )1 021 1 010 2

If three of the variables in this formula are known then the fourth can be calculated.

Rearranging to show conversion from, say, r a to r b:

r b r a ta t b( ) /1 1 Formula for compound rate conversion

For example, a rate of 21% p.a. (0.21) is equivalent to a six-month rate, r,


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given by:

r ( . ) / . .1 021 1 2 1 11 1 01 since r b r a ta t b( ) /1 1

Similarly, if each month has 30 days then an interest rate of 6% (0.06) over

three months is equivalent to a rate, r, compounded every fifteen days, given

by:

r ( . ) / . / .1 006 15 90 1 1061 6 00098 (to four decimal places)

and, conversely, a rate of 0.976% (0.00976) compounded every fifteen days

is equivalent to 6% (0.06) over three months:

r ( . ) .1 00098 6 1 0060 (to three decimal places)

A use closer to home is in credit card interest charged monthly; interest of

2% per month allowed to accumulate for twelve months would amount to

24% if simple interest were charged. However, compound interest is

charged and the equivalent annual rate is (1+0.02)12-1= 0.2682 (to four

decimal places) or 26.82% p.a.. If you are given the annual rate and want

to calculate the monthly rate then:

r b r a ta t b( ) / ( . ) /1 1 1 0 2682 1 12 = 0.02 per month (2% per month)

The application of compound interest rates converted between different units of time

becomes problematic if cash is withdrawn within the total period under consideration.Here’s why:

Suppose you have two bonds which give cash payments (called “coupons”)

every six months and one year respectively. This will happen for several

years (the exact number does not matter, for our purposes), starting with a

payment from one bond six months from now, followed by both bonds one

year from now. Payments are expressed as percentages of a fixed sum

(£1,000): 5% per six-month period and 10.25% per annum. These rates are

equivalent, since 1.052 - 1 = 0.1025. However, the bonds are not

necessarily equally attractive investments; their worth depends on other

interest rates. Consider a one-year period. After six months, one bond pays

5% (£50). Now, if this is re-invested somewhere at 5% per six-month

period then it will earn 5% of 5% (£2.50) and after one year the owner of

the bond will have £50 (first coupon) + £2.50 (interest on first coupon) +£50 (second coupon) = £102.50, which is 10.25% of £1,000 and, hence,

there is no difficulty. However, if the first coupon (£50) is re-invested at

more or less than 5% then clearly the outcome after a year will differ from

that from holding the 10.25% p.a. bond. Investors in bonds would not

consider the bonds equivalent just because their compound interest rates are

equivalent. The equivalence of the rates is based on the assumption that

interest is accumulated during each year.


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Thus, compound interest rates can be converted readily to equivalent rates for different

time periods, but if cash payments are made within the longest time span covered by any

of the rates they are no longer fully comparable.

Nominal, Actual and Effective Interest Rates

Up to this point it has been emphasised that compound interest rates cannot be converted

to equivalent rates over shorter periods by simple division. Thus, a rate of 10% p.a. is

equivalent to 4.88% per six-month period; it is not equivalent to 5% per six-month

period (as would be the case if simple interest were being applied). However, it is

common practice to quote compound interest rates in a way which looks just like this!

Example: interest payments (called “coupons”) received by owners of

bonds may be quoted as “10% p.a., semi-annual”. This is understood to

mean 5% every six months! Worse: it may be that someone writing

or talking about a well-known type of bond which pays interest semi-

annually, such as a US Treasury bond, may assume that you know that“10% coupon” means 5% paid every six months. Note:

(1+0.10)1/2

- 1 = 0.0488 (to four decimal places).

(1+0.05)2 - 1 = 0.1025; 5% every six months is equivalent to 10.25% p.a.

A nominal interest rate quoted as 10% p.a. semi-annually is actually a rate of 5% per

six-month period. This is not a conversion to an equivalent rate as described previously;

the amount actually paid is 5% every six months. However, the actual rate can be

converted to an equivalent annual rate, as before:

Nominal rate “10% p.a. semi-annually”

Actual rate 5% semi-annually (i.e. 5% per six-months)

The equivalent rate can be had from the equation r b r a ta t b( ) /1 1

Substituting 0.05 for r a, 2 for ta (two six-month periods) and 1 for t b:

Equivalent rate = ( . )1 005 2 1 = 0.1025 p.a. (10.25% p.a.)

This rate is known as the effective interest rate and, if it is an annual rate, it is sometimes

called the annual percentage rate or APR. The nominal rate may alternatively be called

the stated interest rate. As long as you know how to calculate the effective rate, there is

no problem in quoting practically a nominal rate (assure yourself of this; it will bereferred to later when “continuous compounding” is considered)

A nominal rate of 10% p.a., with actual compounding periods of six months, three

months and one month converts into three different effective rates as follows:

semi-annual 1 01

2

2.

= 1.1025 Effective rate = 10.25% p.a.


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quarterly 1 01

4

4.

= 1.1038* Effective rate = 10.38% p.a.

monthly 1 01

12

12.

= 1.1047* Effective rate = 10.47% p.a.

* To four decimal places

Looking at these three results, it is easy to deduce the general equation for converting

nominal rates to effective rates. Taking a nominal interest rate which is actually divided

into parts (division by the integer m, say) then applied m times:

1 1r

m

mtr no al effective

min

r r

m

mt

effectiveno al1 1min

Continuous Compounding

We now have a formula for converting a nominal rate to an effective rate.

Mathematically, a time period can be sliced into as many shorter periods as you wish.

Imagine what happens when m is made larger and larger. This increases the power towhich the number in parentheses is raised, tending to produce a larger result but at the

same time the number in parentheses becomes smaller, which tends to produce a smaller

result. The net effect of these opposing tendencies is demonstrated below, using t = 1

and r = 0.1. The results were calculated, to eight decimal places, using a spreadsheet.


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This is called “continuous compounding”. It follows that future value with continuous

compounding at a nominal rate, r per period, is given by the equation:

FV PVert where r is understood to be the nominal rate

In continuous compounding, the time between successive compounding operations is

infinitesimally small; the number of compounding operations over the period of a

nominal rate is infinite. These are not practical operations! Yet we can comprehend the

summation of an infinite number of infinitesimally small compounding operations and,

more to the point practically, there is a formula for the result. Graphically, lines

representing compounding using progressively shorter periods shift to higher values of

FV but move no higher than the limit of the line for continuous compounding. This is

demonstrated in Figure 6.

0

200

400

600

800

Time / years

FV / pounds

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Figure6$100, r = 0.10 p.a. compound interest

Compounding period decreased:4years, 2 years, 1 year, 6 months, 3 monthsand, finally, continuous compounding 4

2

1

Continuous compounding is extremely useful in the derivation of formulae in advanced

finance; for example, in the pricing of “options” and other “derivative instruments”.

These have become so important in finance that you should not avoid understanding

continuous compounding.

The effective rate can be calculated for continuous compounding:

ert r effective1

r erteffective 1 where r is understood to be the nominal rate


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In the previous section I wrote that “as long as you know how to calculate

the effective rate, there is no problem in quoting practically a nominal rate”.

This was probably clear to you. A nominal rate of 10% p.a. semi-annually

gives an effective rate of 10.25% p.a. and FV = PV(1+0.1025). Practically,

therefore, you have the rate quoted and a formula for calculating future

value. Likewise, with continuous compounding you have a nominal rate

and a formula for calculating future value: FV = PVe rt. Therefore,

continuous compounding can be used practically.

The nominal rates used in continuous compounding can be converted to equivalent

nominal rates, continuously compounded over different time periods, and to equivalent

actual rates over finite periods (“discontinuous” rates, if you will!). Conversion between

continuously compounded rates is particularly convenient, and the equation reduces tothe same simple result as found for simple interest:

er ata er bt b and so r ata r bt b

For example, 20% per two-years, continuously compounded, is equivalent

to 10% annually, continuously compounded. Note that time periods in this

example are measured in units of 2 years and units of 1 year. You may

prefer to switch out of using t (retaining this for use with units of 1 year

only) and replace it with e.g. "n" periods, "m" periods, etc.

For conversion between continuously compounded rates and “discontinuous”

compounding rates the formula is:

er ata r b t b( )1

r br a tat b

exp 1 switching to notation whereby exp(x) = ex

For example, 10% p.a., continuously compounded, is equivalent to

10.52% p.a. compounded annually (i.e. “10.52 % p.a”.), since ta = t b = 1

and e0.1 - 1 = 0.1052 (to four decimal places). It is also equivalent to 5.13%

per semi-annual period compounded semi-annually (i.e. just “5.13% semi-

annually”) since, in this case, ta = 1, t b = 2 and e

0.05

- 1 = 0.0513 (to four decimal places). Notice that if t b = 1, the formula for calculation of the

effective rate is obtained.


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Discounting: DCF and NPV

Discounting is the process of calculating the present value of future cash flows and is a

fundamental financial technique. The description “discounted cash flow” is often

abbreviated to DCF. To an investor, a nominal amount of cash does not have the same

value at different times; generally, a dollar next year is worth less than a dollar today,

since a dollar could be invested today to repay more than a dollar next year. DCF is

important because it allows the values of future cash flows to be adjusted to a common

time. Bringing all values to a common time allows them to be summed, so that the total

present value of a series of cash flows, to be received at many different times, can be

calculated. This allows alternative investments to be compared.

Example. Would you prefer to receive one million pounds nowor ten years from now? If you take the money now, you can invest it.

If you are a cautious investor you bank the money and receive interest.

For simplicity, suppose you receive 10% p.a. compound interest, fixed

over ten years. If you leave all money in the bank, after ten years you

have a million pounds times (1+0.10)10

, which is 2,593,742 pounds.

Clearly, your financial preference should be to receive one million

pounds now. If, instead, you are promised one million pounds now and

one million pounds ten years from now it would not be sensible to say

that your gain in value today is two million pounds. A better way would

be to determine how much money, put in the bank today, would yield a

million pounds after ten years, then add that amount to the million pounds

received today. A million pounds received ten years from now is theequivalent of an amount in the bank today of one million divided by

(1+0.10)10

, which is 385,543 pounds - making the value of the two receipts,

measured today, 1, 385,543 pounds. This is the basis of DCF.

When an initial cash investment is made (a “negative cash flow”), followed by cash

flows in later periods (preferably positive!), the sum of the discounted cash flows is

known as the net present value or NPV. NPV provides a useful way of evaluating

investments. The superiority of NPV over other methods for evaluation of investments

is covered in elementary textbooks as is its use making investment decisions.

The idea behind NPV follows from DCF and can be shown in a simple

example. Suppose you are considering an investment of one million pounds

today in a project. You expect the project to return a series of cash flowsannually for ten years, after which the project ends. You use the method of

discounted cash flow (DCF) to calculate the present value (PV) of the future

cash flows. You then compare this PV with your initial investment and, if the

PV is greater than your initial investment, you consider that the investment

would be profitable. Using NPV to express this comparison, you would write

NPV = -1,000,000 + PV. Then, the exact equivalent of the comparison is to

say that if NPV of the project is greater than zero (positive) you consider that

the investment would be profitable.


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Although, in principle, DCF could involve using the mathematics of either compound

interest or simple interest, in practice the mathematics of compound interest is always

used. We already have formulae relating future value to present value. For

“discontinuous” compounding this is:

FV PV r tt ( )1

For a single future value, the formula showing discounting is easily obtained by

rearranging this formula:

PV FV r t FV

r tt

t

( )( )

11

Notice that the reciprocal of a number raised

to a power may be shown as a negative power

A crucial feature of both PV and FV is that present values and future values for the same

time are additive. If a series of expected cash flows are discounted back to the present

then the present value of the series is the sum of the present values of the individual cash

flows. This is important because it allows projects, each with cash flows expected at

many different times and in different amounts to be valued and, hence, compared. The

additivity of present values is easily demonstrated:

A cash flow of £110 one year from now, discounted at 0.10 p.a. (10% p.a.)is worth £100 today. Similarly, £121 two years from now is also worth

£100 today. Therefore, the series of cash flows, starting today and arriving

in consecutive years, £100, £110, £121, discounted at 10% p.a. is worth

£300

PV 100 110

1 010

121

1 01 2 100 100 100 300

( . ) ( . )

There is only one present value for the series but more future values; one for each future

time. Cash flows are discounted back from future times to more recent times and

compounded forward to more distant times. For the series of cash flows in the example,

the future value after one year is £330 and the future value after two years is £363:

FV year 1 100 1 01 110 121

1 01110 110 110 330( . )

( . )

Here, 100 has been compounded forward, 110 does not need to be adjusted

and 121 has been discounted back.


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FV years2 100 1 01 2 110 1 01 121 121 121 121 363( . ) ( . )

Here, 100 has been compounded forward two years, 110 has been

compounded forward one year and 121 did not need to be adjusted.

Notice how consistent and convenient PV and FV calculations are: PV and FV values

can be inter-converted before addition or afterwards, with no difference. The only rule

when moving between values at different times is that you must convert all cash amounts

to the same time before adding them.

The results in the example can be inter-converted by noting that:

300 1 01 330( . ) 330 1 0 1 300/ ( . )

330 1 01 363( . ) 363 1 0 1 330/ ( . )

It will be convenient to use continuous compounding and discounting when option

pricing theory is considered, in Part 2, and so we’ll include the coresponding formulae

for continuous discounting. We’ve slipped in a subscript, t, for FV, to fit with the 1 year

and two year examples, above, but you can drop it if you prefer (as we will do in

Powerpoint slides):

FV PVertt continuous compounding

PV FV e rt FV

ertt

t continuous discounting

The annual cash flows 100, 110, 121 can be continuously discounted at

10% p.a. as follows (to two decimal places)

PVe e

100 110

01

121

0 2 100 9953 99 07 29860

. . . . .

It is convenient at this point to follow the convention in textbooks and switch from using

FV to C when representing a future cash flow, since the emphasis will now switch from

compounding forward (PV to FV) as the inverse of discounting back (FV to PV) to

discounting alone. It is helpful to be able to understand the algebraic representation of

the equations (in order to appreciate finance textbooks but also for their descriptiveconvenience). For all cases other than continuous discounting:

PV FV

r tt

( )1 becomes PV

C

r tt

( )1

and a general equation for calculation of present value, PV, by discounted

cash flow (DCF) from year 1 to year “T” is:


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PV C

r

C

r

C

r

C

r

C

r TT1 2 3 4

1 1 2 1 3 1 4 1( ) ( ) ( ) ( ).....

( )

Another way of representing this is to use the mathematical symbol for

summation, . This is simply a shorthand way of writing the previous

equation:

PV Ct

r tt

t T

( )11

Although this equation could easily be made to include the cash flow, C0 at

time, t=0 (now!), it is conventional to name the present value including this

term net present value (NPV). The slightly modified equations are:

NPV C C

r

C

r

C

r

C

r

C

r TT

01 1 2 1 3 1 4 1

1 2 3 4

( ) ( ) ( ) ( ).....

( )

NPV C Ct

r tt

t T

011 ( )

Since (1+r)

0

= 1, this can be represented more simply:

NPV Ct

r tt

t T

( )10

A numerical example is already available in the form of the series of cash

flows introduced earlier:

PV 100 110

1 0 10

121

1 01 2 100 100 100 300

( . ) ( . )

Since the immediate cash flow, C0 = 100, is positive (an attractive“investment”!) it is a moot but unimportant point whether or not the

description NPV should have been used here.

The corresponding equations for continuous compounding are similar:

PV FV

ertt becomes PV

C

ertt


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and the general equation for calculation of present value is:

PV C

er

C

e r

C

e r

C

e r

C

erTT1 2 3 4

2 3 4 .....

NPV C C

ertt

t

t T

01

Again, since e0 = 1, this can be represented more simply:

NPV C

ertt

t

t T

0

The mechanics of discounting have been explained, but how is the discount rate chosen?

If all that concerned us were money safely deposited in a bank (one which will not

default), discounting would be an uncomplicated process, using the same rate for

discounting as used for money on deposit. Discounting is far more useful. It can be

used in valuing projects, assemblies of projects or even companies; but these do not

have interest rates specifically describing them. In the next section, an overview of

discount rate selection will be given.

Choosing The Discount Rate and Understanding NPV

In order to understand NPV it is important to appreciate the way in which the discount

rate is related to risk. It is not essential to know the detailed mathematical formulations

of risk and rate of return before gaining an insight.

An important characteristic of NPV is the combination of cash flows from different

times into a single valuation. Since discounting over several periods is not the main

concern in this section, however, simple examples will be given using only cash flows

today and one year from today, in order to make the examples clearer

The discount rate for a particular investment is chosen by considering what else might

be done with the cash instead of committing it to that investment. For a sensiblecomparison, the appropriate discount rate is the best rate of return available, chosen

from alternative investments of comparable risk. You can reach this conclusion by

thinking about different types of investment. First, suppose you have cash which you

could invest in one of several banks for one year. You ask the banks what rate of

interest each would pay if you deposited your cash with them over this period. These

particular banks offer you identical services and you believe they are all equally unlikely

to fail but they offer you different interest rates on your cash. Sensibly, if you deposited


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your cash you would do so at the highest rate. Likewise, you would use the highest

interest rate as the discount rate which you would apply to investments which you

consider to have the same risk as depositing your money in the bank. The NPV of

investing cash in the bank giving the best rate would be zero (of course, since you

discount at that rate!) and the NPV of investment in one of the other banks would be

negative. NPV in this situation is a measure of the value of investing your cash in a

bank compared with investment in the best bank.

For example, suppose you have £1,000,000 in cash, the best rate offered is

0.10 p.a. (10% p.a.). and another, inferior rate offered is 0.0975 p.a. (9.75%

p.a.). If you deposit your cash at the inferior rate then after one year you

would have £1,097,500. The net present value of this investment would be

calculated like this (to the nearest £):

NPV £1, , £1, ,

( . )£1, , £997, £2,000 000

097 500

1 0 10000 000 727 272

This is a negative NPV and shows that you should not invest at the inferior

rate

Now suppose that an alternative investment becomes available. You might decide to use

the same discount rate to value that investment:

Suppose the new investment is for one year and offers a return of

£1,600,000 on £1,000,000 invested.

NPV £1, , £1, ,

( . )£454,000 000

600 000

1 010545

This means that if you consider the alternative investment to be as safe as depositing

your money in the best bank, you would confidently expect to receive, after one year,

the same amount which you would have received from the bank had you invested

£1,454,545. In other words, you would expect the alternative investment to bring you

additional present value of £454,545 and so you would consider it a very good

investment.

This is obvious if you split the numbers as follows:

£1,000,000(1+0.10) =£1,100,000

£454,545(1+0.10) = £500,000

The first equation is equivalent to investment in the bank, the second shows

the extra value acquired by investing in the alternative proposition, which

has a present value of £454,545 and a future value of £500,000.


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If the alternative investment really is only as risky as investment in the bank, then the

NPV analysis above is correct. However, what should you do if you think the

alternative investment is more risky? Presumably, you would no longer think it worth an

extra £454,545? In general, the more risky the investment the higher the rate of return

required by investors. Suppose, for the moment, you are able in some way to classify

investments according to their risk (we need to introduce portfolio theory and capital

asset pricing). It would be reasonable to associate the same discount rate with all

investments having the same risk. Suppose the risk of the alternative investment is

classed with investments requiring a discount rate of 0.30 p.a. (30% p.a.). This would

mean that you would deem a return of 30% to be just sufficient, no more and no less, to

compensate you for taking the risk of the alternative investment.

A return of £1,600,000 on an investment of £1,000,000, discounted at0.30 p.a. (30% p.a.) gives an NPV of:

NPV £1, , £1, ,

( . )£230,000 000

600 000

1 0 30769

The NPV is positive; the investment is a good one, though not as

attractive as it would have been if its risk had been as low as an

investment in the bank! You would expect to receive £1,300,000 for

an investment of comparable risk. For this alternative investment you

expect to receive £1,300,000 plus the equivalent of investing an extra

£230,769 at the same level of risk, since:

£1,000,000(1+0.30) = £1,300,000

£230,769(1+0.30) = £300,000

NPV is a tool for decision making. The discount rate is chosen either by direct

comparison with another investment yielding a known rate of return and considered to

be of comparable risk, or by statistical methods using data from many investments to

define risk and relate it to expected rates of return. A positive NPV is value in excess of

that which you ought to expect, given the level of risk implied by the discount rate. By

choosing a discount rate you are, knowingly or unknowingly, also choosing to compare

the investment with some other investment (discount rate is sometimes called the

“opportunity cost of capital”, expressing the idea of a return foregone by investing in an

alternative project). The comparison could be with a particular investment, such as a

similar project already undertaken within a company, or it could be with a theoretical

project of the same risk. The combination of many investments of differing risk andreturn in order to determine a relationship between risk and return are considered in

textbooks under the Capital Asset Pricing Model but need not concern us here.

Suppose you use the bank deposit rate as the discount rate and obtain a positive NPV for

a project. If it is of equally low risk then the project is a better investment than

depositing with the bank; if not, then it merely promises a better return but no account

has been taken of the greater likelihood that its promised cash flows might not arrive in

full or on time! In the same way, if you use the rate of return from another project as the


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discount rate, then NPV shows whether the new investment is inferior (negative NPV),

equal (NPV is zero) or superior (positive NPV) to that project if the risks are equal. This

means the project discount rate must be chosen with care. If, for example, a poorly

performing project were chosen for comparison then many later projects would be

judged favourably but if one were chosen which had delivered an exceptionally high rate

of return then few later projects would have positive values of NPV! Rather than

comparing a single past project, therefore, decision makers may set a “benchmark” rate

(a minimum acceptable return) as a practical way of using NPV to rank projects.

Compared with this benchmark, positive NPV projects will be acceptable; the higher the

NPV the more valuable the investment

In order to set a benchmark rate as close as possible to a rate truly reflecting the risk of a

company project, it is necessary to look outside the company. It may be that thecompany takes on higher (or lower) risk on projects at the benchmark rate of return than

is generally the case for other companies in its industry. If an industry-wide estimate of

appropriate return for a given level of risk can be determined, then it can be used as the

discount rate, in preference to a company-specific rate. However, financial details of

companies’ projects may be hard to come by, making it impossible to determine the

relationship. Stock price data, though, are readily available for many large companies,

and it is possible to determine the appropriate return for a given level of risk for

investment in other companies’ shares in the same industry on a stockmarket at a

particular time.

Why does it matter that the rate be as close as possible to a rate truly reflecting the risk

of a company project? Isn’t a rate typical for the company perfectly acceptable or even

more appropriate? An example will show why it is important to use discount ratesreflecting risk as closely as possible:

Two projects are under evaluation. They are equally risky and the

correct discount rate, to take this level of risk into account, is 0.20 p.a.

(20% p.a.). You are considering the effects of setting a benchmark rate

at either 0.10 p.a. (10% p.a.) or 0.30 p.a. (30% p.a.). The cash flows and

discounted cash flows are as follows:


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PROJECT 1

Year: 0 1 2 3 4 5

Cash flow: -250 336 0 0 0 49.77 NPV

Discounted at 10% -250 305.45 0 0 0 30.90 86.36

Discounted at 20% -250 280.00 0 0 0 20.00 50.00

Discounted at 30% -250 258.46 0 0 0 13.40 21.87

PROJECT 2

Year: 0 1 2 3 4 5

Cash flow: -250 0 0 0 0 746.50 NPV

Discounted at 10% -250 0 0 0 0 463.52 213.52

Discounted at 20% -250 0 0 0 0 300.00 50.00

Discounted at 30% -250 0 0 0 0 201.05 -48.95

When the correct discount rate is applied, the projects are assessed as having equal

value (50.00). However, if the company used a benchmark rate of ten percent per

annum Project 2 would be assessed as considerably more valuable than Project 1

(213.52 contrasted with 86.36). Conversely, if the company used a benchmark rate of

thirty percent per annum Project 2 would be rejected (NPV = -48.95) while Project 1

would still offer positive value compared with the benchmark (NPV = 21.87).

Assuming that managers discount at rates properly chosen to reflect risk, company

shareholders can delegate operations to them with a simple instruction: maximise NPV!

The central idea which has been discussed in this section is that for each level of risk

there is, in principle, an appropriate discount rate; the greater the risk, the greater the

discount rate. Viewed in this way, investors assess the level of risk and are seeking

extra value at this level of risk. If the NPV is negative, the investment is not sufficiently

attractive to justify the risk taken. If the NPV is zero then the investment gives no better

return than appropriate for the risk taken. If the NPV is positive, an investment has been

found which gives a better return than necessary to compensate for its risk and is,

therefore, attractive to an investor. One consequence is sometimes not understood.

Consider two projects, both lasting one year, discounted at different rates in order to

reflect their different levels of risk. It so happens that the projects have the same

positive NPV. Therefore, they are equally valuable now to an investor. The projects

start and finish at the same times. Suppose they are both completed successfully with

cash flows as expected. They are not then equally valuable. Here are the data:


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NPV analysis at the start

Project 1 Project 2

Discount rate p.a. 0.1 0.6

Initial investment £100,000 £100,000

Received at end £132,000 £192,000

NPV £20,000 £20,000

Outcome if cash flows are as expected

Rate of return 32.00% 92.00%

The projects required the same initial investment, were for the same period and had the

same NPV; and yet one gave a return of 32% while the other gave a return of 92%.

How do we reconcile these rates of return with the equal values of NPV?

The answer lies in the incorporation of risk into NPV. At the end of the project the

outcome is certain, at the start it was not. At the start, the NPV calculations took into

account the different levels of risk of the projects. At the start of the projects when the NPVs were calculated, “cash flows as expected” was only one of a large number of

different possible outcomes. The discount rates for the projects reflected the

possibilities that the projects would not give the expected outcomes; the amounts which

would actually be received and when they might be received were both uncertain.

Project 2 was deemed more risky than Project 1 and, hence, was assigned a higher

discount rate. The NPV of Project 1 shows that the project was equivalent to having an

extra £20,000 to invest at 10% p.a. at its level of risk. The NPV of Project 2 was also

equivalent to having an extra £20,000 to invest but at 60% p.a. and the appropriate (butdifferent) level of risk.

You now know the principle behind the selection of appropriate discount rates and the

interpretation of NPV.

Special Cases of PV Formulae: Annuities, Perpetuities and Growing Perpetuities

There are several well-known formulae which simplify the calculation of PV and NPV

in special cases. They are worth knowing and are especially important in bond price

calculations. The formulae will be described in this section, and their use in simple

valuation of stocks and bonds are addressed in many textbooks. A good book for those

who fancy themselves as investment bankers or consultants in the 21st

century is the oneused by Rothschilds for their graduate recruits:

Corporate Financial Strategy, Keith Ward, Butterworth Heinemann, ISBN 0

7506 0657 7.

It is often the case that cash payments are made as a fixed amount at regular intervals;

for example, £10,000 on January 1st every year for fifteen years. If the discount rate is


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8% and the first payment is to be received one year from now (so today is January 1st!)

then the present value of the expected future cash flows is the sum of fifteen terms:

PV 10 000

108

10 000

1082

10 000

10815 10 000

1

108

1

1082

1

10815

,

.

,

.....

,

.,

. .....

.

A series of equal cash flows equally spaced over time, such as this, is called an annuity.

There is no need to add all the terms because there is an elementary formula which gives

the same result more easily:

PV

10 000

0 08 1

1

108 15 85595

,

. ( . ) ,

In words: at a discount rate of 8% p.a., an annuity of £10,000 p.a. for fi fteen years, withthe first payment to be made one year from now, is worth £85,595 (to the nearest pound)

The general form of this equation is:

PVannuityC

r r T1

1

1( )

where C is the cash flow paid each time period (for example, each year)

r is the discount rate, as a decimal, for the period between paymentsT is the total time over which the annuity will be paid

You should have no difficulty applying this formula as long as you remember to use the

same time measurement for each of C, r and T. If the cash flows, C, are paid annually

then the discount rate should be per annum and the time, T, should be measured in years.

This was done in the example of a fifteen-year annuity, above. If, however, payments

are made monthly then r should be a monthly discount rate and T should be expressed in

months.

For example, an annuity of £100 p.a. for ten years at a discount rate of

0.10 p.a. (10% p.a.) is worth £614.46, since:

PV C

r r T1

1

1

100

011

1

1 01 10 614 46

( ) . ( . ).

A perpetuity is a special case of annuity in which payments are supposed to be made

forever. On reflection, you may find this idea surprising; but, nevertheless, perpetuities

have been issued. In practice, the issuer may buy back the perpetuity after many years,

using the same formula for valuation as when the perpetuity was issued. The classic


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example of a perpetuity is the issue of “consols” by the British Government, to help

finance war against Napoleon. Consols were bonds which were purchased from the

government and which paid a fixed sum at regular periods without end.

The general equation for valuing perpetuities is:

PV perpetuityC

r

If you are comfortable with the idea of infinity in mathematics then look at the equation for

the present value of an annuity, given earlier. Imagine T becoming larger. As T increases sothe second term in brackets becomes smaller. Taken to the limit of infinite T, the second term

vanishes to zero, leaving the formula for a perpetuity shown above.

For example, a perpetuity of £100 p.a. at a discount rate of 10% (0.10) is

worth £1,000, since:

PV C

r

100

011000

.,

The value of cash paid either from an annuity or from a perpetuity decreases as time

passes, since the cash amount is fixed. Elementary formulae are also available for

valuing annuities and perpetuities whose payments increase each period to (1+g) times

the amount in the preceding period (in other words, payments increase by 100g% each

period). A general formula for the present value of the series of growing cash amounts

is given below. The series terminates after a finite number of terms if it is a growingannuity or contains an infinite number of terms if it is a growing perpetuity:

PV C

r

C g

r

C g

r etc

( )

( )

( )

( )

( )..... .

1

1

1 2

1 2

1 3

The formulae for valuing an annuity and a perpetuity are summarised, below, with those

for a growing annuity and a growing perpetuity:


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Constant Payment Payment Grows By g Each Period

Annuity

C

r r

C

r g

g

r T

T

T1

1

11

1

1( )

( )

( )

PerpetuityC

r

C

r g

Notice that the formulae including growth reduce to those for constant payment if g is

made equal to zero.

One pitfall must be avoided in using the formula for a growing perpetuity: the formula is

invalid unless the growth rate, g, is less than the discount rate, r. Think what this means.

The series can be written as:

PV r C C g

r

C g

r forever ( )

( )

( )

( )

( )..... (!)1

1

1

1 2

1 2

By taking a factor of (1+r) to the left hand side of the equation, the effect of changing

the relative sizes of g and r on the terms in the series becomes easier to understand. If g

is less than r, the terms become progressively smaller and they add up to a finite result.

However, if g is greater than r, the terms become progressively larger and so the sum of

the series increases without limit - it becomes infinite! If g is equal to r, the terms stayconstant and so, again, the sum of the series increases without limit. If you set g equal to

r in the formula for the present value of a growing perpetuity then division by zero

results. To a mathematician this could imply an infinite result, which is correct.

However, if you set g greater than r, you obtain a negative result which is not the correct

answer (the correct answer is infinite!). Beware the second pitfall of believing that the

invalidity of the formula when g is greater than r can be proven using the formula itself.

The derivation of the formula includes the limitation that g must be less than r and so the

formula cannot be invoked to prove its own invalidity outside that condition!

A perpetuity of £100 p.a. at a discount rate of 10% (0.10) and a growth rate

of 15% (0.15) is worth an infinite amount! British government consols

proved to be reliable investments; you might guess that either the issuer of

this perpetuity would not continue to make the payments .... or no one couldafford to buy it, if it were properly valued! Now contrast the result given

by the usual formula:

PV C

r g( ) ( . . )

100

010 015200

This is incorrect; the formula does not apply!

basic finance sheet

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