Time value of money

THE TIME VALUE OF MONEY
Would you prefer to
have Rs.1 million now or
Rs.1 million 10 years
from now?
2
Of course, we would all
prefer the money now!
This illustrates that there
is an inherent monetary
value attached to time.

CONCEPT
nA Rupee today is more valuable than a Rupee a
year hence.
3
Why ?
• Individuals prefer current consumption to future consumption
• Capital can be employed productively to generate positive returns i.e.,
interest.
• In an inflationary period, a rupee today represents a greater real
purchasing power than a rupee a year hence.

WHY REQUIRED TO STUDY?
• Financial problems involve cash flows occurring at
different points of time.
• They must be bought to the same point of time for
purpose of comparison & aggregation.
• Tools of compounding & discounting underlie the finance
activities.
4

THE TERMINOLOGY OF TIME VALUE
5
n Present Value - An amount of money today, or
the current value of a future cash flow
n Future Value - An amount of money at some
future time period
n Period - A length of time (often a year, but can
be a month, week, day, hour, etc.)
n Interest Rate - The compensation paid to a
lender (or saver) for the use of funds expressed
as a percentage for a period (normally expressed
as an annual rate)

TIMELINES
6
0 1 2 3 4 5
PV FV
Today
! A timeline is a graphical device used to
clarify the timing of the cash flows for an
investment
! Each tick represents one time period

TYPES OF TIME VALUE OF MONEY
CONCEPTS
8
Time
Present Future
Time Value
Single Entity
Annuity
Ordinary Annuity Due
Series Flow
Uneven

TYPES OF TVM CALCULATIONS
• There are many types of TVM calculations
• The basic types include:
• Present value of a lump sum
• Future value of a lump sum
• Present and future value of uneven cash flow streams
• Present and future value of annuities
• Keep in mind that these forms can, should, and
will be used in combination to solve more
complex TVM problems
9

COMPOUNDING TECHNIQUE
• It is used to find out the FV of a present money .
• In this the cash flows of different time periods can
be made comparable by compounding the present
money to future date i.e., by finding out the FV of
a present money.
• The compounding technique to find out the FV of
a present money can be explained with references
to:
1.The FV of a single present cash flow and
2.The FV of a series of cash flows .

CALCULATING THE FUTURE VALUE
11
n Suppose that you have an extra Rs.100 today that
you wish to invest for one year. If you can earn 10%
per year on your investment, how much will you have
in one year?
0 1 2 3 4 5
-100 ?
( )
FV1 100 1 010 110
= + =
.

CALCULATING THE FUTURE VALUE
(CONT.)
12
n Suppose that at the end of year 1 you decide to
extend the investment for a second year. How much
will you have accumulated at the end of year 2?
0 1 2 3 4 5
-110 ?
( )( )
( )
FV
or
FV
2
2
2
100 1 010 1 010 121
100 1 010 121
= + + =
= + =
. .
.

GENERALIZING THE FUTURE VALUE
13
n Recognizing the pattern that is
developing, we can generalize the
future value calculations as follows:
( )
FV PV i
N
N
= +
1
! If you extended the investment for a third year,
you would have:
( )
FV3
3
100 1 010 13310
= + =
. .

THE FV OF A SINGLE PRESENT CASH
FLOW
FV =PV(1+r)n
Where ,
FV = Future value
PV = Present value
r = % Rate of interest , and
n= Time gape after which FV is to be
ascertained
If any one of these three variables (PV , r , n)
changes, the FV will also change.

• The mathematicians have made these calculations easer by
finding out the value of (1+r)n for various combinations of
‘r ’ and ‘n’ in Compound Interest Table.
• This table gives the pre-calculated values of different
combinations of ‘r’ and ‘n’ which may be called the
Compound Value Factor (CVF). This can be denoted as
CVF(r,n).
FV=PV*CVF(r,n)
• Assumptions:
1.For a given period, higher the interest rate, the greater will be the FV
and
2.For a given rate of interest, the longer the time period, the higher will
be the FV.

EXAMPLE:
• Find out the FV of Rs. 5,000 invested for 10 years at 5%
rate of interest.(CVF=1.629)
FV = PV (1+r)n
FV = 5000(1+0.05)10
FV= 8,145
FV= PV*CVF(r,n)
FV=5000*1.629
FV=8,145

nNON-ANNUAL COMPOUNDING:
FV = PV(1+r/m)m.n
Where,
m = no. of compounding periods in a year.
Example:
A deposited of Rs. 1,000 is made to earn interest a
12% p.a. compounded half-yearly.
Solution:
PV=Rs. 1,000
m = 2
r = 12%
n = 1
FV = 1,000(1+12/100*2)1*2
FV = 1,123.60

Compounding Period Number of Periods(m) FV (Rs.1000)
Annual 1 I,120.00
Half-yearly 2 1,123.60
Quarterly 4 1,125.51
Monthly 12 1,126.83
Daily 365 1127.47
More frequently the compounding is made, the faster is
the growth in the FV.
EFFECT OF COMPOUNDING ON THE FV

EXAMPLE:
• A deposit of Rs. 10,000 is made in a bank for a period of
1 year. The bank offers two options: (i) to receive
interest at 12% p.a. compound monthly, or (ii) to receive
interest at 12.25% p.a. compounded half-yearly. Which
should be accepted?

SOLUTION:
nOption(1) Rate of interest 12% p.a. compounded monthly.
(1+re) = (1+r/m)m
= (1+0.12/12)12
= 1.1268
re = 0.1268 or 12.68%
nOption(2) Rate of interest 12.25% p.a. compounded half-
yearly.
(1+re) = (1+r/m)m
= (1+0.1225/2) 2
= 1.1263
re = 0.1263 or 12.63%
In this case, the normal rate of returns higher in option (2)
i.e., 12.25% but the effective rate of interest is higher in
option(1) i.e., 12.68%. Therefore, the depositor should
select the option(1).

GRAPHIC VIEW OF COMPOUNDING
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 2 4 6 8 10 12
Periods
FVIF(r,n)
0%
6%
12%
23

APPLICATIONS
• Similarly, in the same example, we have Rs
1,00,000 to invest either in option A or in option
B project.
• Here, the option A generate return on interest
rate @ 5% & option B generates on 7% interest
rate.
• So, the project whose future value will be more
gets more preference.
• These techniques are adopted by venture
capitalists during the analysis of project report
for investing money.
24

FUTURE VALUE OF A CASH FLOW STREAM
• The future value of a cash flow stream is equal to the
sum of the future values of the individual cash flows.
• The FV of a cash flow stream can also be found by
taking the PV of that same stream and finding the FV of
that lump sum using the appropriate rate of return for
the appropriate number of periods.
25

EXAMPLE OF FV OF A CASH
FLOW STREAM
• Assume Joe has the same cash flow stream from his
investment but wants to know what it will be worth at
the end of the fourth year
1. Draw a timeline:
26
0 1 2 3 4
Rs.100 Rs.300 Rs. 500 Rs.1000
i = 10%
Rs.1000
?
?
?

FLOW STREAM
2. Write out the formula using symbols
n
FV = S [CFt * (1+r)n-t]
t=0
OR
FV = [CF1*(1+r)n-1]+[CF2*(1+r)n-2]+[CF3*(1+r)n-3]+[CF4*(1+r)n-4]
3. Substitute the appropriate numbers:
FV = [Rs.100*(1+.1)4-1]+[Rs.300*(1+.1)4-2]+[Rs.500*(1+.1)4-3]
+[Rs.1000*(1+.1)4-4]
27

FLOW STREAM
4. Solve for the Future Value:
FV = Rs.133.10 + Rs. 363.00 +
Rs.550.00 + Rs.1000
FV = Rs.2046.10
28

ANNUITIES
29
n An annuity is a series of nominally equal payments
equally spaced in time
n Annuities are very common:
n Rent
n Mortgage payments
n Car payment
n Pension income
n The timeline shows an example of a 5-year, Rs.100
annuity
0 1 2 3 4 5
100 100 100 100 100

EXAMPLE OF FV OF AN ANNUITY
• Assume that Sally owns an investment that will pay
her Rs.100 each year for 20 years. The current interest
rate is 15%. What is the FV of this annuity?
1. Draw a timeline
30
0 1 2 3 …………………………. 19 20
Rs.
100
Rs.
100
Rs.
100 Rs.100 Rs.100
i = 15%
?

FUTURE VALUE OF A SERIES OF EQUAL
CASH FLOWS OR ANNUITY OF CASH
FLOWS:
• Quite often a decision may result in the occurrence of
cash flows of the same amount every year for a number
of years consecutively, instead of a single cash flow.
• Formula:
FV= Annuity Amount*CVAF(r,n)
where, CVAF= Compound value of Annuity Factor

EXAMPLE OF FV OF AN ANNUITY
2. Write out the formula using symbols:
FVAt = PMT * {[(1+r)t –1]/r}
FVA20 = Rs.100 * {[(1+.15)20 –1]/.15
4. Solve for the FV:
FVA20 = Rs.100 * 102.4436
FVA20 = Rs.10,244.36
32

EXAMPLE:
• A firm decides to make a deposit of Rs. 10,000 at the end of
each of the next 10 years at 10% rate of return. What will be
the total cumulative deposit at the end of 10th year from today?
The firm may also be interested to know the total deposit if the
rate of interest is 9% or 11%?

SOLUTION:
• Annuity Amount = Rs. 10,000
n = 10 years
r = 10%/9%/11%
CVAF(10%,10y) = 15.937
CVAF(9%,10y) = 15.193
CVAF(11%,10y) = 16.722
Therefore,
at 10% is 15.937*10,000 = Rs. 1,59,370
at 9% is 15.193*10,000 = Rs. 1,51,930
at 11% is 16.722*10,000 = Rs. 1,67,220

DISCOUNTING TECHNIQUE
nThis process is reverse of compounding technique .
nIn this the cash flows of different time periods can
be made comparable by discounting the future
money to present date i.e., by finding out the PV of
a future money.
nThe discounting technique to find out the PV of a
future money can be explained with references to:
1. The PV of a future sum,
2. The PV of a future series.

THE PV OF A FUTURE SUM:
• Present Value of a given future amount is equal to an
amount which if invested today would accumulate to
the future amount at a given rate of interest after the
specific period.
• Persent value of a future sum will be worth less than
the future sum because of interest during that period.
• Formula:
PV = FV/(1+r)n
OR
PV = FV *PVF(r,n)
where,
PVF= Present value of a future sum

THE PV OF A FUTURE SERIES:
• A decision taken today may result in a series of future
cash flows of the same amount over a period of number
of years.
• Formula:
PV = Annuity Amount * PVAF(r,n)
Where,
PVAF = Present value annuity factor.

CALCULATING THE PRESENT VALUE
38
n So far, we have seen how to calculate
the future value of an investment
n But we can turn this around to find the
amount that needs to be invested to
achieve some desired future value:
FVt
PV = -----------
(1+r) t

PRESENT VALUE: AN EXAMPLE
39
n Suppose that a five-year old daughter has just
announced her desire to attend college. After some
research, the father determine that he will need
about Rs.100,000 on her 18th birthday to pay for
four years of college. If you can earn 8% per year
on your investments, how much do you need to
invest today to achieve your goal?
100,000
PV = -------------- = Rs. 36,769.79
(1.08)13

EXAMPLE OF PV OF A LUMP SUM
• How much would Rs.100 received five years from
now be worth today if the current interest rate is
10%?
1. Draw a timeline
The arrow represents the flow of money and the
numbers under the timeline represent the time period.
Note that time period zero is today.
41
0 1 2 3 4 5
Rs.100
?
i = 10%

EXAMPLE OF PV OF A LUMP SUM
PV = CFt / (1+r)t
3. Insert the appropriate numbers:
PV = 100 / (1 + .1)5
4. Solve the formula:
PV = Rs.62.09
42

PRESENT VALUE OF A SINGLE
AMOUNT
• The present value of a sum is the amount that would
need to be invested today in order to be worth that sum
in the future.
• Computing the present value of a sum is known as
discounting.
Where, FV = Future value
r = discounted rate
n= number of periods
43
Formula : PV = FVⁿ
(1+ r)ⁿ

GRAPHIC VIEW OF DISCOUNTING
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12
Periods
PVIF(r,n)
0%
6%
10%
14%
44

APPLICATION
• In order to ascertain the value of 2 projects in terms of
their cost involvement as against their profit generation.
• As, 2 projects A & B both can generate Rs 10,00,000
after 5 years. But, the discounted rate of option A is
10% & option B is 15%.
• So, in this case the co. will invest into that project whose
present value is less.
45

PRESENT VALUE OF A CASH FLOW STREAM
nA cash flow stream is a finite set of
payments that an investor will receive or
invest over time.
nThe PV of the cash flow stream is equal to
the sum of the present value of each of the
individual cash flows in the stream.
nThe PV of a cash flow stream can also be
found by taking the FV of the cash flow
stream and discounting the lump sum at the
appropriate discount rate for the
appropriate number of periods.
46

EXAMPLE OF PV OF A CASH
FLOW STREAM
n Joe made an investment that will pay Rs. 100
the first year, Rs. 300 the second year, Rs. 500
the third year and Rs. 1000 the fourth year. If
the interest rate is 10%, what is the present
value of this cash flow stream?
1. Draw a timeline:
47
0 1 2 3 4
?
Rs.100 Rs.300 Rs.500 Rs.1000
?
?
?
i = 10%

EXAMPLE OF PV OF A CASH FLOW
STREAM
n
PV = S [CFt / (1+r)t]
t=0
OR
PV = [CF1/(1+r)1]+[CF2/(1+r)2]+[CF3/(1+r)3]+[CF4/(1+r)4]
PV = 100/(1+.1)1]+[300/(1+.1)2]+[500/(1+.1)3]+[1000/(1.1)4]
48

EXAMPLE OF PV OF A CASH FLOW
STREAM
Solve for the present value:
PV = Rs. 90.91 + Rs. 247.93 +
Rs. 375.66 + Rs. 683.01
PV = Rs. 1397.51
49

RULE OF THUMB/ RULE OF 72
• According to this rule, the doubling period is obtained by
dividing 72 by the interest rate.
• For example: if interest rate (r) is 8%, then the doubling
period is 9 years as (72/8)
50
Doubling Period = 72/r

RULE OF THUMB/ RULE OF 69
51
n According to this rule, the doubling period is obtained
by dividing 69 by the interest rate and then adding
0.35.
n For example: if interest rate (r) is 8%, then the
doubling period is 8.975 i.e. 9 years as [0.35+(69/8)]
Doubling Period = 0.35+69/r

PRESENT VALUE OF A PERPETUITY
• A perpetuity may be defined as an infinite series of equal
cash flows occurring at regular interval.
• It has indefinitely long life.
• Formula:
PVp = Annual Cash Flow/r

ANNUITIES
• An annuity is a cash flow stream in which the
cash flows are all equal and occur at regular
intervals.
• Note that annuities can be a fixed amount, an
amount that grows at a constant rate over time,
or an amount that grows at various rates of
growth over time. We will focus on fixed
amounts.
53

ANNUITY
nAn annuity is a stream of constant cash flows
(payments or receipts) occurring at regular
intervals. For ex:- life insurance policy premium.
nOrdinary Annuity :- cash flow occur at the end of
each period.
Annuity Due :- cash flow occur at the beginning of
each period.
54
0 1
0 2 3 n-1 n
A A A
A A
0 1 2 3 n-1 n
A
A A A
A A

EXAMPLE OF PV OF AN ANNUITY
• Assume that Sally owns an investment that will pay
her Rs.100 each year for 20 years. The current
interest rate is 15%. What is the PV of this
annuity?
1. Draw a timeline
55
0 1 2 3
………………………….
19 20
Rs.
100
Rs.
100
Rs.
100
Rs.
100
Rs.
100
?
i = 15%

EXAMPLE OF PV OF AN ANNUITY
Write out the formula using symbols:
PVA = PMT {1/(1+r)t}
Substitute appropriate numbers:
PVA = Rs.100 * [1/(1+.15)20]
Solve for the PV
PVA = Rs.100 * 6.2593
PVA = Rs.625.93 56
t=0
n
Formula:-
PVAⁿ = A[1- (1/1+r)ⁿ]
r

PRESENT VALUE OF AN ANNUITY DUE:
nThe discussion on FV or the PV of an annuity
was based on the presumption that cash
flows occur at the end of each of the
periods starting from now.
nFV of an annuity due:
FV = Annuity Amount*CVAF(r,n)*(1+r)
nPV of an annuity due:
PV = Annuity Amount*PVAF(r,n)*(1+r)

APPLICATION
• Period of Loan Amortization, while taking loan for
starting the new business or, for its seed capital etc.,
• Ex: we borrow Rs 1,80,000 for business, approaches the
bank which charges 12.5% interest. We pay Rs 1,08,000
per year toward loan amortization. So, what should be
the maturity period?
58

SOLUTION
• PVA ⁿ = Rs 1,08,000
• A = Rs 1,80,000
• r = 12.5%
• n= ?
• By applying the formula
PVAⁿ = A[1- (1/1+r)ⁿ]
r
n = 11.76 years or 12 years.
59

USES OF TIME VALUE OF MONEY
• Time Value of Money, or TVM, is a concept that
is used in all aspects of finance including:
• Bond valuation
• Stock valuation
• Accept/reject decisions for project management
• Financial analysis of firms
• And many others!
62

NOTE:
nIn both of the examples, note that if you
were to perform the opposite operation on
the answers (i.e., find the future value of Rs.
62.09 or the present value of Rs.161.05) you
will end up with your original investment of
Rs. 100.
nThis illustrates how present value and future
value concepts are intertwined. In fact,
they are the same equation . . .
n Take PV = FVt / (1+r)t and solve for FVt. You will get FVt = PV
* (1+r)t.
63

EQUATED MONTHLY INSTALMENT
• Equal Monthly instalment payable at specific interval of
time (EMIs)
• Regular periodic payment whose present value,
discounted at specified rate of interest, adds to the loan
value.
64

A LOAN OF 1000 REPAYABLE IN 3 YEARS
INSTALMENT WITH AN INTEREST RATE OF
10%.
• For Re 1
PVA(10%,3) = Rs. 2.4869
For Rs. 1000
1000/2.4869
= Rs 402.11 per annum
65

CALCULATION
nMr. X has taken a loan of Rs. 20,000 against a
television from Consumer Bank. The loan amount is
repayable in 36 monthly instalments. The bank
charges the interest at 10% per annum and the
instalments are payable in advance.
Solution
66

FORMULA
67
PVFA:
EMI = LOAN AMOUNT/PVFA

PRACTICAL APPLICATION
nPresent value of an infinite life Annuity
nA financial manager is often interested in
determining the size of annual payment to
accumulate a future sum to repay an existing liability
at some future date or to provide funds for
replacement of an existing machine/asset after its
useful life
nWhen the amount of loan taken from a commercial
bank or financial institution is to be repaid in a
specific number of equal monthly installment, the
finance manager will be interested in determining
the amount of annual installment 68

nAn investor may often be interested in finding
the rate of growth in dividend paid by a
company over a period of time. It is because
growth in dividends has a significant bearing on
the price of the share.
nTo determine the current value of debenture
nEffective/Nominal Rate of Return
nPresent value of an Annuity payable PTHLY
69

Time value of money

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