CHAPTER

Time Value of Money

F etc.inancialinvolve
decisions like capital budgeting, cost of capital, bond and stock valuations, devidends
cash flows occurring at different time periods. Such decisions require that the
cash flow which a firm is expected to receive and give up over periods should be logically
comparable. The absolute cash flows which differ in timing and risk are not directly comparable.
Cash flows became logically comparable when they are adjusted for their differences in timing and
risk. The understanding and recognising the time value of money and risk is extremely significant
in financial decision-making to attain its objective of maximising the owners' welfare. The welfare
of owners would be maximised when net wealth or net present value is created from making a
financial decision. The net present value is a time value concept, hence let us understand the
concept of time value of money. This value of money concept will be grouped into twO areas
" Future value; and
. Present value.
Future value describes the process of finding what an investment today will grow to in the
future. Present value describes the process of determining what a cash flow to be received in the
future is worth in today'srupees.
TIME VALUE OF MONEY : CONCEPT AND RATIONALE
The first rule of finance states that "money today is worth more than money received tomorrow."
If you are offered Rs. 1,000 today and another option of Rs. 1,000 after one year, which optton wit
you prefer? Obviously, the option to receive the money today is to be exercised because if it is
reinvested, say, at 10% it will become Rs. 1,100 after one year. Conversely, the present value of
Re. 1 tobe received after one year is Re. .909 (Rs. 1,000 + 1,100) today, taking an interest rate of
10 per cent per annum. This proves that money received immediately is of greater value than an
equivalent sum to be received at future date. Recognition of this principle amounts to taking care of
time value of money in finance. Thus, time value of money may be defined, as the relationship
between rupee one in the future and rupee one today. In other words,a rupee is hand today is worth
 Time Value of

Thie
Money
than zero,
6.2

more than a rupee to be received in

the future if rate of
interest is
greater

re-investedto get further

principle
is based on the following reasons: received
be
today canproductivity of time,
t means
retuarn,given K
becauee
() Reinvestment Opportunities
Money
zero, money has a net the same value. canalways that with the
the rate of return I8 greaterthan time does not have money be
sum of money at two pointsof opportunities, a given sum of made t
availability of profitable investment of
grow with the passage of time.
(2) Uncertainty : The present is more certain than the
future
present rupees of
which is full
future rupees which imponderabl
may eors. may
A

a resutr there is a general preference for

not be received in full because of uncertainty.
money goes decreasing or the price Jevel goes an
(3) Inflation : During inflation, the value ofprefer the present to the future. preference
As a result, people generally
inereasing.money for satisfying individuals have a strong
present needs. It may be due to urgency ofusetheiy
the to
(4) present : Mainly
existing
Personal Consu mption Preference
not being in a
positionto enjoy future consumption
due
present wants or because of the risk if
to illness or death or inflation..
FUTURE VALUE OR COMPOUNDING
is worth more than what will &
a Tuture value (FV) means that a given quantity of money today
received at some point of time in the future) Conversely, the future value of a given amount of
greata
oney Will always be greater than the present value of the sum of money invested at a rate
than zero. This is because money can be reinvested to make it groW. The process of going frOm
today's value (present values) to future values is called compounding When compound interes
rate is applied, we can write in this manner :
Future Value Present Value + Interest
(FV)
That is, FV = PV+ PV(r) = PV(1 tr) idon for one year
or FV
or
PV (1 + r) + [PV(I +t r)]r for two years
FV PV(1 + n) (1+r) = PV(1 + r for two years
Similarly , for n years
(We know that the rate of interest is .1)
the rate at which an
for exçhanging money now for
money later. The
individual or a firm willbe compensated
word compound refers to the original amount is known as the
periodic
interest is earned over one period, it is addition of interest earned on the
principal. The
is earned on the added to the amount, and then principal amount.) After
the final period orprincipal as well as the prior period's
interest. This
during the next period, /nterest
year. process continues till the end of
Example Assuming that a sum of Rs. 100
:
following table can be constructed (principal) compounded
for a period of
is at 10 per cent per
1o Year three years: annum, the
Present alue
Rs. Interest Future Value
2 100 Rs.
10 Rs.
3 110
I10
121
12 121
133
Tite Value of Moneny 6.3

Using equation (1) we can write:

FV PV(1+n
Hhere, PV Principal at the beginning
t Rate of
interest
nNumber of years
FV Future value or amount at the end of period n
Thus,
FV, PV(1 + )1 = 100(1.) Rs. 110
FV, = PV(1 +r2 =100(1.1÷ Rs. 121
FV, = PV(T +)3 = 100(1.1) Rs. 133(approx.)
Note that the interest at the end of each vear is higher than that at the previous year tor
two reasons- () the starting amount is higher at each succeeding year; and (i) the interest is
also earned on interest' Future values are also called Terminal Values (TV).
It may be stated that when n is small (as in the case of the above example), we can find out the
value of F by solving the respective equations. Atternatively, one can use a Future Value
Interest Factor (FVIF) table. One such table showing the future value factor for certain combinations
of periods and interest rates is given in (Table A-Appendix I) at the end of this book. When the table
is used the mathimatical formula
FV = PV (1 + r)" = PV(FVIFr, n)
Find the future value at the end of 4 years of Rs. 1,000 invested today at an
interest rate 10% p.a.
Solution
By using the formula bate
FV = PV (1 + r
= 1,000 (1 +0.10)
= Rs. 1,000 (1.10) = Rs. 1,464
By using FVIF (Future Value Interest Factor) Table
FV = PV (FVIFr, n)
Here: PV = Rs. 1,000
= 0.10,N=4
FV = PV (PVIF 10%, 4)(Refer to FVIF table to see the factor value)
=
Rs. 1,464
Semi-Annual /Quarterly /Monthly Compounding
When payments are made more frequently than once a year, or when interest is compounded
more than once a year, we have to convert the stated interest rate to periodic rate and number of
years to the number of periods. For example, in case of semi-annual or half yearly compoundìng
their would be two compounding periods within the year and interest at a rate of one half of the
annual rate of interest because interest is actually paid after every six months. Similarly, in quarterly
compounding, there are four compounding periods in a year and the rate of interest willbe one
1. This makes the diference between simple interest and compound interest. For example, the simple interest
be earned per annum is Rs. 10 and the total simple interest to be earned is Rs. 10 x3=Rs. 30.
 64
Time Value
of
fouth of the annual rate. The effect of compounding more than once a year can
be
Money
formala as expressed in fhe
Where, mis the number of tines per year
woold be 2, while for quarterly compounding compounding is made. For semi-annual

formulae are used

it
monthly, weekly or daily it would equal to 12, 52would
cocmompopuonudnidnegd
be equal to 4and if interest is
and 365 respectively for which the
() When iterest is payable half yearly/Ceoni
FV =
emnaly fol owing
(b) When interest is
payable quarterly
FV =

(c) When interest is

payable monthly.
FV =

(c) When interest is

payable daily
365n
FV

Lllustration 2 Mr.
Pv1+
Ganesh places his savings of Rs.
365
a bank which yields interest @ 1,000 in a two year time
Ganesh will receive at the end of6% p.a. compounded semi-annually. Find out
deposit scheme of
the amount that M:
What will be the amount iftwo years.
the amount is put is a
quarterly ? scheme which yields interest
Solution compounded
For semi-annual compounding
FV =

0.06 |2*204
1,000 (1 +0.03y'
For quarterly compounding 1,000 (1.03)' = Rs.
1,125.51
FV =

0.06 4x2

1,000 (1 t+ 0.015
1,000 (1.015)= Rs.
1,126.49
 Meney 6,5
Tire lae ef

The FVIE Table A(Appendix II) gíven at the end of the book can be used to simplify
calculations. When compounding occurs more than once a year, we have simply to divide the
interestrate by the number of times compounding occurs, that is (1 + m) and multiply the years
number of look
hythe compounding periods per year, that is, (m Xn), In illustration 2, we haveto
A for the sum of rupee one under fourth
at Table the 3 percent column and in the row for the
yearwhen compounding is done semi-annually, the respective rate and year figures would be 1.5
per cent
and the eighth year in
quarterly compounding.
The compounding tactor for 3 percent and 4 years is 1.126 while the factor for 1.5 per cent
and 8 year is 1.127. Multiplying each of the factors by the initial savings deposit for
Rs. I.000, we find Rs. 1,126 (Rs. 1,000 x 1,126) for semi-annual compounding and Rs. 1,127
(Rs. 1.000 x 1.127) for quarterly compounding. The corresponding values found by the long
method as per solution given are Rs. 1,125,51 and Rs. 1.126,49 respectively. The difference can
be attributed to the rounding off of values in Table A.

PRESENT VALUE OR DISCOUNTING

Present value is the value today of a sum of money to be received at a future point of time.
Present values allow us to place all the figures on a current footing so that comparisons may be
made in terms of today's rupees. For example, from the table in given on page no. 1.33. if we
want to determine the present values at the beginning of year 1 of Rs. 110, Rs. 121, and Rs. 133 to
be received at the end of year 1, year 2 and year 3 respectively at 10 per cent per annum, it would
be Rs. 100 in each case. Thus, using the compound interest formula, we can find present value of
agiven amount to be received at a point in time in future as follows:
FV = PV (1 + )"
FV
PV =
(1 + r)"
Where: FV = Amount to be received at the end of year n (future value)
= Compound rate of interest
PV = Present value
solving the equation
If the value of n is small, we can find the present value (PV) by
without the help of logarithms. For example,
Rs. 110
PV,1=( 1 +)'FV, Rs. 100
1.1

Rs. 121
FV, Rs. 100
PV, = (1+ r)' (1.1

FV, Rs. 133 =Rs. 100

PV, (1 +r (1.1)
following example
Present value is the opposite of future value. Consider the
(when r = 10% per annum) :