This document discusses the concept of time value of money, which means that a unit of money received today is worth more than the same amount received in the future. It explains the techniques of compounding and discounting, which allow converting cash flows received or paid at different points in time to a common point for comparison. Compounding calculates the future value of an amount invested now, growing at a specified interest rate over time. Discounting calculates the present value of a future cash flow. The document provides examples of using compounding and discounting formulas to solve time value of money problems involving single and multiple cash flows over time.

1. Time Value of Money -
In this section we will learn about basics of mathematics of finance, that is time value of
money. Recognition of time value of money in financial decision making is extremely
important. As we have seen in previous sections that wealth maximisation is superior to
profit maximisation because former incorporates timing of benefits received while later
ignores it. Thus much of the subject matter of financial management is future oriented. A
financial decision taken today has implications for number of years, that is it is spread into
future.
A firm has to pay a certain sum of money to acquire fixed assets. The benefits arising out of
these fixed assets will be spread over number of years in future, till working life of the
assets. For acquiring fixed assets funds are required to raised from different sources, like
issue of equity shares, debentures, fixed deposits, loans from banks/financial institutions,
etc. This involves cash inflow at the time of raising and an obligation to pay
interest/dividend from time to time and return the principle in future.
It is on the basis of a comparison of cash outflows (outlays) and benefits (cash inflows) the
financial decisions are made. For meaningful comparison the two variables must be strictly
comparable. One basic requirement of comparability is incorporation of time element in
the calculations. In order to have a logical and meaningful comparison between cash flows
that accrue in different time periods, it is necessary to convert sum of money to a common
point of time. The technique through which this is accomplished is time value of money. In
the ensuing paragraphs we will see how timing aspect is taken care by compounding and
discounting and important applications of this technique.
Rationale –
Conceptually time value of money means the value of unit of money received today is more
than value received after some time. Conversely sum of money received in future is less
valuable than it is today. In other words present value of rupee received after some time is
less than rupee received today. Since rupee received today is more value an investor would
prefer current receipts to future receipts. Time value of money can therefore be also
referred as time preference for money. The main reasons for such preference are
i) Risk - there is famous saying that a bird in had is worth two in bush. Similarly
money to be received in future will always have some element of uncertainty.
ii) Preference - most people and companies prefer to consume today than in future
because of urgency of needs or otherwise.
iii) Investment opportunity - is the reinvestment opportunity if funds are received
early. The money so invested will earn a rate of interest which is not possible if
money is received later.

2. The expected rate of return as also the time value of money will vary from individual
depending upon inter alia his perception/expectation. (example). What applies to
individual, in a greater measure applies to business/firm.
Compounding Technique -
Interest is compounded when the interest earned on an initial deposit becomes part of the
principle at the end of first compounding period. The term principle refers to the amount of
money on which interest is received.
The compounding of interest can be calculated by the following equation:
A = P(1+r)n
In which
A = amount at the end of the period
P = principle at the beginning of the period
r = rate of interest expressed as fraction of hundred
n = number of years
Doubling Period
It refers to a period after which an invested amount will double for given interest rate.
There are three methods of computing this period.
First method – Rule of 72.
Doubling period
Second method – Rule of 69
Doubling period + 0.35
Third method –
FVn = PV (1+r)n
Or 2 *PV = PV * FVIF(r,n) or

3. FVIF(r,n) = 2
For given value of r, by looking at the FVIF tables and using interpolation value of n can be
computed.
Illustration – ICICI Ltd offers fixed deposits @ 12.00% p.a. Mr Ashok is interested in
determining the period in which his principle would double. Find the doubling period using
above three methods.
Solution –
Method 1
As per rule of 72, n = 6 years.
Method 2
As per rule of 69, n = + 0.35 = + 0.35 = 6.1 years.
Method 3
FVIF (r,n) = 2
FVIF (12%,n) = 2
By looking at the FVIF tables for 12 % we find that for n=6 it is 1.974 and n=7 it 2.211.
Therefore the value n that is doubling period will be more than 6 but less than 7. Now let us
find it by way of interpolation.
n = 6+ = 6 + = 6 + 0.1053 = 6.1053 years.
Problem -
XYZ Company currently has 5000 employees which are expected to @ 5% per year. How
many employees company will have at the end of ten years?
Solution -
5000 * (1+0.05)10
= 5000 * 1.629 = 8145.

4. Problem -
Phoenix Limited had revenues of Rs. 100 million in 2000 which increased to 1000 million by
2010. What is the compound growth in revenues?
Solution -
Let r be the rate of growth in revenues. We can use the simple compounding equation
100 (1+r)10
= 1000
(1+r)10
= 1000/100 = 10
(1+r) = 101/10
= 1.26
r = 1.26 -1 = 0.26 or 26%
Thus we can say that the revenues of the company have grown at the compound annual
rate of 26%.
Problem -
You want to buy a house after five years when it is expected to cost Rs. 2 million. How
much you should save annually if your savings earn a compound return of 12%?
Solution -
Suppose A is the amount which is required to be saved annually. We can write the formula
using FVIFA as under –
2,000,000 = A * FVIFA(5, 12) = A * 6.353
Therefore A = 2,000,000 / 6.353 = 314,812
Thus savings of Rs. 314,812 per year will have to be undertaken.
All above calculations are done with understanding that interest is paid (compounded)
annually. But in reality interest is compounded either quarterly or monthly. Let us study the
impact of this compounding on actual payment of interest i.e. let us find out effective rate
of interest.
Future Value for Increased frequency of compounding –

5. FVn = PV (1+r/m)mn
Where FVn = future value after n years PV = present value of cash flow today
r = nominal rate of interest per annum m = number of compounding per annum
n = number of years
therefore FVn = PV * FVIF (r/m, mn)
Problem - An investor deposits Rs. 100/- in bank account for one year at 10% interest per
annum. Find out the amount he will have in his account if interest is compounded yearly,
half yearly, quarterly and monthly.
Solution –
a – 100*0.10 = Rs. 10/- interest for one year
b – 100*0.05 = 5, 100 + 5 = 105*0.05 = 5.25 Total intt = 5+5.25 = 10.25
so the effective interest will be 10.25 percent.
C – 100*0.025 = 2.50, 102.5*0.025 = 2.5625, 105.0625*0.025 = 2.6266,
107.689*0.025 = 2.692 total intt 110.381 – 100 = 10.381
Effective rate of interest would be 10.38%
D – we do calculations exactly in the similar manner total interest would work out to
Rs.10.59, thus effective rate of interest would be 10.59%
COMPOUNDING FREQUENCY ANNUAL HALF YRLY QUARTERLY MONTHLY
EFFECTIVE RATE OF INTT 10.00 10.25 10.38 10.59
From the above example we can conclude that the actual or effective rate of interest will
depend on the compounding frequency. If the compounding frequency is annual then the
effective rate would be same as indicated rate also called as nominal rate. From the above
example we also observe that as the compounding frequency increases, the effective rate
of interest goes up.
Symbolically effective rate of interest can be calculated as
r = (1+k/m)m
- 1
k = nominal rate of interest, m = compounding frequency, r = effective rate of interest.

6. Problem - Mahesh is working as finance manager in Mint Ltd. Company has a fixed deposit
scheme in which depositor for 10000/- deposit gets 12625 after two years, on half yearly
compounding basis. Some depositors have demanded that compounding may be done on
quarterly basis as in case of banks. Boss has called Mahesh and asked him to find out how
much additional amount company will have to pay. What will Mahesh do?
FVn = PV * FVIF(r/m, mn)
12,625 = 10000 * FVIF(r/2, 4)
FVIF(r/2, 4) = 12625/10000 = 1.2625
Solving this equation we get r = 12%
Hence the revised maturity for quarterly compounding would be
10000*FVIF (3, 8) = 10000 * 1.2668 = 12,668.
Future Value of Multiple Cash Flows –
Suppose Milind deposits Rs. 5000/- every year for four years in a bank yielding 6% p.a. He
wants to the maturity value at the end of four years.
Solution –
Maturity value at the end of four years can be calculated as under
M.V. = [5000 * (1.06)4
] + [5000 * (1.06)3
] + [5000 * (1.06)2
] + [5000 * (1.06)1
]
= 28,185.50
Instead of doing such laborious calculation the maturity value is given by formula
M.V. = A * [(1+r)n
-1] / r
FVAn = A * FVIFA(r,n) (called as future value interest factor for annuity)
Where A = amount deposited at the end of every year for n years
r = rate of interest n = number of years
FVAn = amount accumulated at the end of n years including interest.
For ready reference tables giving value of FVIFA for change in interest rate & year are
available.

7. Problem –
Roshan deposits Rs. 4,000 every year in bank deposit offering interest at 9% pa.
Compounded annually. Compute the amount of money he will receive at the end of five
years.
Solution -
FVAn = 4000 * FVIFA(9, 5) = 4000 * 5.6371 = 23,185.50
TECHNIQUES OF DISCOUNTING
Present value of single cash flow
Up till now we have learned about compounding which gives us future value of sum
invested now. But practical financial situation we come across cash flows to be received in
future are required to be compared with cash flows going out today. To make this
comparison we have to find out present value of cash flows to be received after some time.
The procedure to find out present value of future cash flows is called as technique of
discounting. The procedure is exactly opposite of compounding.
For r = rate of intt in fraction and n = number of years
We know, FV = PV *(1 + r)n
Therefore PV = FV/(1 + R)n
= FV * [1/(1+r)n
] = FV * PVIF(r, n)
[1/(1+r)n
]is also referred to as PVIF(r,n) i.e., present value interest factor.
In order to simplify the present value calculations tables are readily available for different
values of n and r i.e., for various discount rates and years for an amount of one rupee. By
using these tables one can find out present value of any amount for given r and n.
In terms of formula it will be
PV = FV * PVIF
Problem - For making an investment of rs 650 today company has offered to pay me
rs.1000 after five years. The current going rate of bank fixed deposit for five year period is
8.5%. Please advise whether I should undertake the investment ignoring risk factor.
Solution -
PV = FV * PVIF(r, n) let us enter values in this equation
650 = 1000 * PVIF (r, 5)

8. Therefore PVIF (r, 5) = 650/1000 = 0.650
By looking into PVIF table in five year row, we find r = 9%. As this rate is greater than 8.5%
the investment should be undertaken.
Problem - a company has offered to pay Rs. 2000/- after five years for an investment of
Rs.1200/-. Ignoring the risk factor and considering bank rate for five year is 10%, please
advise whether investment should be undertaken.
Solution -
we have to look into 10% column for five years. The relative PVIF is 0.621.
Therefore the present value of the maturity value would be
PV = 2000 * 0.621 = 1242.
As the present value is more than our initial investment of Rs. 1100/-, the investment
should be undertaken.
(Also try to do the reverse calculation of compounding of Rs. 1242 for five years at 10%)
The concept of present value is exact opposite of that of future value also called as
compound value. In future value money invested now appreciates in value because
compound interest is added. In case of present value approach, money is received at some
future date and will be worth less today because corresponding interest is lost during the
period. In other words, present value of rupee that will be received at some future date will
be less than the value of rupee in hand today. Thus in contrast to the compounding
approach where we convert present sums into future sums, in present value approach
future sums are converted into present sums. Given the positive rates of interests, the
present value of future rupee, will always be lower. It is for this reason, the procedure for
finding present value is called as discounting. It is concerned with determining the present
value of future amount which is going to help in decision making.
Present Value of Multiple and Equal Cash Flows –
Annuity is defined as same amount of cash flows each year. This is particularly useful in
case of project which is going to earn same amount of profit each year.
Problem –
Mahesh deposits Rs. 5000/- each year for four years in a bank that pays 6% interest per
annum. He wants to know the maturity value at the end of four years.
Solution - the value at the end of five years can be calculated as follows

9. MV = [5000 * (1+0.06)4
] + [5000 * (1+0.06)3
] + [5000 * (1+0.06)2
] + [5000 * (1+0.06)1
]
= 5000 * [(1.06)4
+ (1.06)3
+ (1.06)2
+ (1.06)1
]
= 28185.50
However this is long way of calculating maturity value. This is given by the mathematical
formula as under –
FVAn = A * [(1+r)n
-1] / r
= A * FVIFA (r, n)
Where A = amount deposited every year for n years,
r = rate of interest, n = number of years amount is deposited
FVAn = amount accumulated at the end of n years.
[(1+r)n
-1] / r is also referred to as FVIFA(r,n) i.e., Future Value Interest Factor for Annuity.
Thus in the above example maturity value at the end of four years would be
FVAn = A * FVIFA (r, n) = 5000* FVIFA (r, n) = 5000 * 5.6731 = 28185.50
Thus for calculating future value of annuity for different rates and period we can use the
tables.
Problem –
Future Ltd., has an obligation to redeem Rs 25,000,000/- worth of debentures six years
from now. Find out how much company should deposit in a sinking fund account wherein it
earns 14% interest, so as to accumulate full amount for debenture repayment.
Solution –
Let us say the company should save Rs. X per year. Then we can put compounding value
equation 25,000,000 = X * FVIFA(14, 6) = X * 8.536
Therefore X = 25,000,000 / 8.536 = 2,928,772.26
Thus company should save an amount of Rs. 2,928,772.26 per year.
Problem -

10. An executive is about to retire at the age of sixty. His employer has offered him post
retirement benefits under two option – i) Rs. 3,00,000/- p.a. for ten years or ii) Rs.
20,00,000/- lump sum. Rate of interest on bank FD for ten years is 10%. You are his
colleague and he has sought your advice. How will you find out which option is better.
Solution - for this purpose first we will calculate present value of annuity of 300000 for ten
years for 10%.
PV = 300,000 * PVIFA(10, 10) = 300,000 * 6.1446 = 1,843,380.00
As this amount is less than the lump sum amount of Rs. 2,000,000/-, the concerned
executive should opt for lump sum payment from the company.
Problem -
Compute present value of perpetuity of Rs. 100/- if the discount rate is 10%.
Solution – PV of perpetuity = A/i = 100/0.10 = 1000.
Problem –
After reviewing your budget, you feel that you can afford to pay Rs. 12000/- per month for
three years towards a new car. You call a finance company and learn that the going rate of
interest on car finance is 1.5% per month for 36 months. How much you can borrow?
Solution -
To determine how much one can borrow, we have to calculate the present value of
Rs.12000/- per month for 36 months at 1.5% per month.
PV = 12000 * PVIFA(1.5, 36) = 12000 * 27.70 = 332,400
Thus one can borrow Rs. 332,400 for purchase of car.
Problem –
Growmoney company offers to pay you Rs. 2500/- to you for six years if you deposit Rs.
10,000/- with them. Bank rate of FD for 6 years is 11.5% . Ignoring the risk factor whether
you should undertake the investment?
Solution -
10000 = 2500 * PVIFA(r, 6) on solving this equation we get,
PVIFA(r, 6) = 4.000. looking into PVIFA table find out the rates which give value
close to 4. We find that for 13% value is 3.998. So the rate on this investment would be
13% which is higher than 11.5%. therefore the investment should be undertaken.