Unit 4 Time Value of Money

Structure
4.1 Introduction
Objectives
4.2 Concept of Time Value of Money
4.3 Compounding Method
4.4 Discounting Method
4.5 Summary
4.6 Glossary
4.7 Terminal Questions
4.8 Answers
4.9 Case Study

Caselet

Understanding the Time Value of Money

Suppose Akansha is given a choice of receiving `100 today or `100 after a
year, she would certainly like to have the money today instead of waiting for
a year. The reason is that Akansha will be able to invest `100 today and
earn some interest over the year. This example demonstrates a simple
idea: the value of money changes with the passage of time. Even a single
rupee received today will yield more value than that received after a year
and will certainly be more worthy what is received five years from now. The
essence of time value of money is well recognized in the personal and
business areas. For instance, Akansha may have to select between receiving
a heavy amount from a pension plan and receiving a string of payments in
the coming years. Another example could be when Akansha decides to
purchase a new machinery that will most likely bring in increased revenue
in the coming years. In both of these circumstances, the time value of money
can be assessed by comparing the value of present and future money.
Source: Adapted from http://www.swoknews.com/news-top/business/item/
2927-understand-the-time-value-of-your-money (Retrieved on 4 December
2012)

4.1 Introduction

In the previous unit, you learnt about budget or budgeting, its various types and
its advantages. You were also introduced to responsibility accounting.
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Compounding and discounting are two ways in which the time value of
money can be accounted for. Compounding implies receiving the principal plus
interest at the maturity of the investment. Discounting values refers to the present
value of the money before the investment period is over.
In this unit, you will study about the Discounted Cash Flow (DCF) criteria
of investment evaluation which basically includes compounding and discounting
of cash flows over a period of time.

Objectives
After studying this unit, you should be able to:
• interpret the time value of money
• analyse the compounding method
• discuss the discounting method

4.2 Concept of Time Value of Money

If an individual behaves rationally, he or she would not value the opportunity to

receive a specific amount of money now, equally with the opportunity to have
the same amount at some future date. Most individuals value the opportunity to
receive money now higher than waiting for one or more periods to receive the
same amount. Time preference for money or Time Value of Money (TVM) is an
individual’s preference for possession of a given amount of money now, rather
than the same amount at some future time. Three reasons may be attributed to
the individual’s time preference for money:
• risk
• preference for consumption
• investment opportunities
We live under risk or uncertainty. As an individual is not certain about future
cash receipts, he or she prefers receiving cash now. Most people have subjective
preference for present consumption over future consumption of goods and
services either because of the urgency of their present wants or because of the
risk of not being in a position to enjoy future consumption that may be caused by
illness or death, or because of inflation. As money is the means by which
individuals acquire most goods and services, they may prefer to have money
now. Further, most individuals prefer present cash to future cash because of
the available investment opportunities to which they can put present cash to

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earn additional cash. For example, an individual who is offered `100 now or
`100 one year from now would prefer `100 now as he could earn on it an
interest of, say, `5 by putting it in the savings account in a bank for one year. His
total cash inflow in one year from now will be `105. Thus, if he wishes to increase
his cash resources, the opportunity to earn interest would lead him to prefer
`100 now, not `100 after one year.
In case of the firms as well, the justification for time preference for money
lies simply in the availability of investment opportunities. In financial decision-
making under certainty, the firm has to determine whether one alternative yields
more cash or the other. In case of a firm, which is owned by a large number
of individuals (shareholders), it is neither needed nor is it possible to consider
the consumption preferences of owners. The uncertainty about future cash
flows is also not a sufficient justification for time preference for money. We
are not certain even about the usefulness of the present cash held; it may
be lost or stolen. In investment and other decisions of the firm what is
needed is the search for methods of improving decision-makers knowledge
about the future.
There are basically two ways for accounting for the time value of money:
• Compounding
• Discounting

Self Assessment Questions

1. Time preference for money or ________ is an individual’s preference for

possession of a given amount of money now, rather than the same amount
at some future date.
2. _________ and __________ are two ways of accounting for the time value
of money.

4.3 Compounding Method

Suppose an investor can invest `100 today in a bank at an interest of 12 per

cent for one year, how much amount would he receive after a year? He will
receive his principal as well as interest on the principal. That is, 100 + 12% ×
100 = 100 × 112% = 100 × 1.12 = `112. Notice that `112 is the compound or
future value (F) of the present amount (P) of `100 at an interest rate of (i) of 12
per cent for a period (n) of one year. Thus,

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F = P + iP = P (1 + i) = 100 (1.12) = `112.

If the investment is made for two years, the investor will receive interest
on the interest amount earned during the first year:
F = P (1 + i) (1 + i) = P (1 + i)2 = 100 (1.12)2 = 100 (1.2544) = `125.44
Similarly the present amount of `100 invested at 12 per cent for 3 years
will grow to: F = 100 (1.12) (1.12) (1.12) = 100 (1.12)3 = 100 (1.4049) = `140.49;
for 4 years: 100 (1.12) (1.12) (1.12) (1.12) = 100 (1.12)4 = 100 (1.5735) = `157.35
and so on.
Compound value of a lump sum: From the preceding discussion, we can
write the formula for calculating the future value (F) of a lump sum today (P) at
a given rate of interest (i) for given period of time (n) as follows:
F = P (1 + i)n (1)
n
The term (1 + i) is the compound (future) value factor, of `1 for a given
rate of interest, i and time period, n, i.e. CVF, i, n. It always has a value greater than
1 for positive i, indicating that CVF increases with increase in either i or n or both.
In the earlier example, the values 1.12, 1.2544, 1.4049 and 1.5735 are
CVF of `1 at 12 per cent rate of interest respectively for year 1, 2, 3 and 4. Using
Eq. (1) CVFs can be calculated for any combination of interest rate and time
period. Table A in Appendix at the end of the book provides precalculated CVFs
for a range of periods and rates of interest.
Example 1
Jacob is considering investing of `15,000 in a public sector company’s bonds at
a rate of interest of 16 per cent per year for 7 years. How much amount would
he get after 7 years? The compound value can be found as follows:
F = 15,000 (1.16)7 = 15,000 × CVF, .16, 7
The term (1.16)7 gives CVF, which can be obtained from Table A in the
Appendix. Reading through seventh row for 7 year period and 16 per cent column,
we get CVF of 2.826. Thus, the compound half-yearly, for finding out the
compound value of the lump sum of `15,000 invested today at 16 per cent per
annum for 7 years is:
F = 15,000 × 2.826 = `42,390
Multiperiod compounding: Let us assume in example1 that the company will
compound interest half-yearly (semi-annually) instead of annually. Investor will
gain as he will get interest on half-yearly interest. Since interest will be
compounded half-yearly, for finding out the compound value in example 1, the
half-yearly interest rate of 8 per cent and 14 half yearly periods will be considered:

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n×2

F = P  1 + i  (2)
 2 

7× 2
 .16 
= 15,000  1 + = 15,000 (1.08)14
 2 

From Table A in the Annexure, we find that CVF at 8 per cent interest for 14
periods is 2.937. Thus, the compound value of `15,000 is:
F = 15,000 × 2.937 = `44,055
Would the compound value of Jacob’s investment of `15,000 be different
if the company compounds interest quarterly? The quarterly rate of interest will
be 4 per cent and number of quarterly periods will be 28. Thus
n× 4 7× 4
 i  .16 
F = P 1 +  = 15,000  1 +
 4   4 

= 15,000 (1.04)28 = 15,000 (2.999) = `44.995

We can observe that the compound value increases further under quarterly
compounding.
The phenomenon of compounding interest more than once in a year is
called multiperiod compounding. The compound value increases as the
frequency of compounding in a year increases. Eq. (1) can be modified as follows
to find compound value under multiperiod compound:
n×m
 i 
F = P 1 + 
 m

where m is the number of compounding in a year.

Compound value of an annuity: An annuity is a fixed payment or receipt of
each period for a specified number of periods. Let us assume that an investor
decides to deposit `100 at the end of each for 4 years at 10 per cent rate of
interest. Thus, `100 deposited at the end of first year will compound for 3 years,
`100 at the end of second year for 2 years, `100 at the end of third year for one
year and `100 at the end of fourth year would remain constant. Thus, the
compound value will be as given below:
End of year 1F = 100 (1.10)3 = 100 (1.331) = 331.1
2F = 100 (1.10)2 = 100 (1.210) = 121.0

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3F = 100 (1.10)1 = 100 (1.100) = 110.0

4F = 100 (1.10)0 = 100 (1.000) = 100.0
Total compound value 100 (4.641) = 464.1
As we can observe from Table B, we can obtain the compound value of an
annuity (A) by aggregating CVFs for the given periods and then multiplying by
the amount of annuity. For example:
F = A (1 + i)3 + A (1 + i)2 + A (1 + i) + A = A [(1 + i)3 + (1 + i)2 + (1 + i) + 1]
= 100 [(1.10)3 + (1.10)2 + (1.10)1 + 1] = 100 [1.331 + 1.210 + 1.100 + 1]
= 100 (4.641) = `464.1
The factor 4.641 is the compound value factor of an annuity of `1 for 4
years at 10 per cent rate of interest. A short-cut formula for calculating the
compound value of an annuity is as follows:

 (1 + i)n − 1
F=A   (3)
 i 

The expression [(1 + i)n – 1] i gives the compound value factor for an annuity of
`1 for a given rate of interest, i and time period, n, i.e., CVAF, i, n. Table B in the
Appendix at the end of the book provides precalculated compound value factor
for an annuity, CVFA, of `1 for a range of interest rates and periods of time.
Example 2
A person deposits `10,000 at the end of each year for 5 years at 12 per cent rate
of interest. How much would the annuity accumulate to at the end of the fifth
year?
Looking up the fifth row and 12 per cent column in Table B, we obtain
CVAF of 6.353. Thus
F = A (CVAF, 12%, 5) = 10,000 × 6.353 = `63,530
If the interest is compounded quarterly, how much will be the compound
value? For quarterly compounding, interest rate of 3 per cent and time period of
20 periods will be considered. Returning to Table B we find that:
F = 2,500 × 26.870 = `67,175
Sinking fund: Suppose you want to accumulate `4,00,000 at the end of 10
years to pay for the acquisition of a flat. If the interest rate is 12 per cent, how
much amount should you invest each year so that it grows to `4,00,000 at the
end of 10 years? This is a sinking fund problem. A fund which is created out of
fixed payments each year for a specified period of time is called sinking fund.

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The desired sum of `4,00,000 is the compound value of an annuity, say,

A, at 12 per cent rate of interest for 10 years. Thus,
F = A (CVAF, 12%, 10)
4,00,000 = A (17.549)
A = 4,00,000 (1/17.549)
= 4,00,000 (0.57) = `22,800
It may be noticed that the future sum, `4,00,000, is multiplied by the
reciprocal of the compound value annuity factor (CVAF, .057 = (1/17.549), to
obtain the amount of annuity.
The reciprocal of CVAF is called the sinking fund factor (SFF).

Activity 1
Suppose you are an IT professional working in an IT company. You wish to
double `10,000 - the amount of your savings in five years. What is the
compound interest you will receive on your principal?
Hint: Apply the formula of compound interest [F = P(I + i)n]

Self Assessment Questions

3. A fund created out of fixed payments each year for a specified time is
called __________.
4. The phenomenon of compounding interest more than once in a year is
called multiperiod compounding. (True/False)

4.4 Discounting Method

Suppose a bank offers an investor to return a sum of `115 in exchange for `100
to be deposited by the investor today, should he accept the offer? His decision
will depend on the rate of interest which he can earn on his `100 from an alternative
investment of similar risk. Suppose the investor’s rate of interest is 11 per cent.
The alternative investment opportunity will provide the investor `100 (1.11) =
`111 after a year. Since the bank is offering more than this amount, the investor
should accept the offer. Let us ask a different question. Between what amount
today (P) and `115 after a year (F ), will the investor be indifferent? He will be

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indifferent to that amount of which `115 is exactly equal to 111 per cent or 1.11
times. Thus

F
P =
(1 + i)
F = P (1 + i)
115 = P (1.11)
115
P = = `103.60
1.11
Note that `103.60 invested today at 11 per cent grows to `115 after a year.
`103.60 is the present or discounted value of `115. That is:
F = P (1 + i)2
115 = P (1.11)2

F
therefore, P = (1 + i)2

115
P = = 115 × 0.812
(1.11)2

The formula for calculating the present value (P) of a lump sum in future
(F) at a given rate of interest (i) for given periods of time is as follows:
F = P (1 + i)n

 1 
P = F  (1 + i)n  (4)
 
The term 1/(1 + i)n provides the present value factor of `1 for a given rate of
interest, i and time period, n, i.e. PVF, i, n, (it may be written as PVF, i, n). It
always has a value lesser than 1 for positive i, indicating that PVF decreases
with increase in either i or n or both.
Present value of an annuity: Suppose Narsimham pays `10,000 at the end of
each year for 5 years into a public provident fund. The interest rate being 12 per
cent per year. What is the present value of the series of `10,000 paid each year
for 5 years? We can treat each payment as a lump sum and calculate the
present value as follows:

 1 
End of year 1P = 10,000  1
= 10,000 × 0.893 = `8,930
 (1.12) 

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 1 
2P = 10,000  2 
= 10,000 × 0.797 = ` 6,360
 (1.12) 

 1 
3P = 10,000  3 
= 10,000 × 0.712 = `7,120
 (1.12) 

 1 
4P = 10,000  4 
= 10,000 × 0.636 = ` 6,360
 (1.12) 

 1 
5P = 10,000  5 
= 10,000 × 0.567 = `5,670
 (1.12) 
10,000 × 3.605 = `36,050

Aggregating PVFs (of a lump sum of `1) for the given periods and then
multiplying by the amount of annuity. Thus

A A A A A
P= + + + +
(1 + i) (1 + i)2 (1 + i)3 (1 + i)4 (1 + i)5
 1 1 1 1 1 
=A + 2
+ 3
+ 4
+ 
 (1 + i) (1 + i) (1 + i) (1 + i) (1 + i)5 
 1 1 1 1 1 
= 10,000  + 2
+ 3
+ 4
+ 
 (1.12) (1.12) (1.12) (1.12) (1.12)5 
= 10,000 [0.893 + 0.797 + 0.712 + 0.636 + 0.576]
= 10,000 × 3.614 = `36,140

The factor 3.614 is the present value factor of an annuity of `1 for 5 years
at 12 per cent rate of interest. A short-cut formula for calculating the present
value of an annuity is as follows:

 1 
1−
P = A  (1 + i)n 
(5)
 
 i 

Table D in the Annexure at the end of the book provides precalculated

present value factor for an annuity of `1 for a given rate of interest, i and time
period, n, i.e. PVFA i, n, for a range of interest rates and periods of time.

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Example 3
Anant Rao is considering paying `5,000 half-yearly into his public provident fund
for 10 years. Suppose the interest rate is 12 per cent per annum. How much is
the present value of his payment? Since the annuity is in terms of half-yearly
payments, the number of periods to be considered is 20 and half-yearly interest
to be 6 per cent. Referring to Table D of the Annexure, the present value may be
calculated as follows:
P = 5,000 × PVAF, 6%, 20 = 5,000 × 11.470 = `57,350
Capital recovery: The reciprocal of the present value annuity factor is called
the Capital Recovery Factor (CRF). It is useful in determining the income to be
earned to recover an investment at a given rate of interest.
Suppose Priyan is considering investing `20,000 today for a period of 3
years. If he expects a return of 16 per cent per year, how much annual income
should he earn? The amount of `20,000 is the present value of a 3-year annuity,
A, given the rate of interest of 15 per cent. Thus
P = A (PVAF, 16%, 3)
20,000 = A (2.246)
 1 
A = 20,000   = 20,000 × 0.445 = `8,900
 2.246 
It may be observed that the present sum, `20,000, is multiplied by the
reciprocal of the present value annuity factor, PVAF, 0.445 = (1/2.246) to obtain
the amount of annuity.

Activity 2
Suppose Akhilesh an engineer would like to invest `15,000 for ten years in
his public provident fund. Assume the interest rate to be 10 per cent per
year. Calculate the present value of the series of `15,000 paid each year for
ten years?

Self Assessment Questions

5. The reciprocal of the present value annuity factor is called the __________.
6. CRF is useful in determining the income to be earned to recover an
investment at a given rate of interest. (True/False)

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4.5 Summary

Let us recapitulate the important concepts discussed in this unit:

• The time value for money can be assessed as the discounted value – the
current actual before the money is invested ; the compounded value is
the principal plus interested accumulated at the end of the investment
period.
• The DCF criteria of investment evaluation are based on the concept of
time value of money.
• The phenomenon of compounding interest more than once in a year is
called multiperiod compounding.
• An annuity is a fixed payment or receipt of each period for a specified
number of periods.
• A fund which is created out of fixed payments each year for a specified
period of time is called sinking fund.
• The reciprocal of CVAF is called the sinking fund factor.

4.6 Glossary

• Annuity: An annuity is a fixed payment or receipt of each period for a

specified number of periods.
• Capital recovery: The reciprocal of the present value annuity factor is
called the capital recovery factor (CRF).
• Sinking fund: A fund which is created out of fixed payments each year
for a specified period of time is called sinking fund.

4.7 Terminal Questions

1. What do you understand by the concept time value of money?

2. Illustrate, with an example, how compounded value of money is calculated.
3. Explain, with an example, how the compounded value of an annuity is
calculated.
4. Explain with an example, the concept of sinking fund.

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5. Give an example of capital recovery.

6. Interpret the concept of discounting, by giving an example.

4.8 Answers

Self Assessment Questions

1. Time value of money

2. Compounding, discounting
3. Sinking fund
4. True
5. CRF
6. True

Terminal Questions

1. Time preference for money or Time Value of Money (TVM) is an individual’s

preference for possession of a given amount of money now, rather than
the same amount at some future time. For more details, refer section 4.2.
2. Here is an example of calculating compound value of money. For more
details, refer section 4.3.
3. An annuity is a fixed payment or receipt of each period for a specified
number of periods. For more details, refer section 4.3.
4. A fund which is created out of fixed payments each year for a specified
period of time is called sinking fund. For more details, refer section 4.3.
5. The reciprocal of the present value annuity factor is called the Capital
Recovery Factor (CRF). It is useful in determining the income to be earned
to recover an investment at a given rate of interest. For more details, refer
section 4.3.
6. To illustrate discounting method, let us suppose a bank offers an investor
to return `115 in exchange of `100 to be deposited by the investor today.
Should the investor accept the offer? For more details, refer section 4.4.

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4.9 Case Study

Value of Money handled by Divya Handtools Pvt Ltd (DHPL)

DHPL is a small-sized firm manufacturing hand tools. Its manufacturing
plant is situated in Faridabad. The company’s sales in the year ending on
31 March 2009 were `1,000 million (`100 crore) on an asset base of `650
million. The net profit of the company was `76 million. The management of
the company wants to improve profitability further. The required rate of return
of the company is 14 per cent. The company is currently considering two
investment proposals. One is to expand its manufacturing capacity. The
estimated cost of the new equipment is `250 million. It is expected to have
an economic life of 10 years. The accountant forecasts that net cash inflows
would be `45 million per annum for the first three years, `68 million par
annum from year four to year eight and for the remaining two years `30
million per annum. The plant can be sold for `55 million at the end of its
economic life.
The second proposal before the management is to replace one of the old
machines in the Faridabad plant to reduce the cost of operations. The new
machine will involve a net cash outlay of `50 million. The life of the machine
is expected to be 10 years without any salvage value. The company will
go for the replacement only if it generates sufficient cost savings to justify
the investment. If the company accepts both projects, it would need to raise
external funds of `200 million, as about `100 million internal funds are
available. The company has the following options of borrowing `200 million:
The company can borrow funds from the State Bank of India (SBI) at an
interest rate of 14 per cent per annum for 10 years. It will be required to pay
equal annual instalments of interest and repayment of principal. The
managing director of the company was wondering if it were possible to
negotiate with SBI to make one single payment of interest and principal at
the end of 10 years (instead of annual instalments). A large financial institution
has offered to lend money to DHPL at a lower rate of interest. The institution
will charge 13.5 per cent per annum. The company will have to pay equal
quarterly instalments of interest and repayment of principal.
Discussion Questions
1. What is the annual instalment of the SBI loan?
2. Would you recommend borrowing from the financial institution?

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References
• Brigham, F. E. & Houston, F.J. (2013). Fundamentals of Financial
Management (13th ed.). USA: South-West Cengage Learning.
• Ross, S., Westerfield, R. & Jaffe J. (2012). Corporate Finance. New Delhi:
McGraw-Hill.
• Brigham, F.E. & Ehrhardt, C.M., Financial Management: Theory & Practice
(2010). USA: South-West Cengage Learning.
• Berk, J., DeMarzo. P. & Thampy A. (2010). Financial Management. New
Delhi: Pearson Education.
• Paramasivan, C. & Subramanian, T. (2009) Financial Management. New
Delhi: New Age International Publishers.
• James C. Vanhorne. (2000). Fundamentals of Financial Management.
New Delhi: Prentice Hall Books.
E-References
• http://vcmdrp.tums.ac.ir/files/financial/istgahe_mali/moton_english/
financial_management_%5Bwww.accfile.com%5D.pdf (Retrieved on 28
May 2013)

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Unit 5 Cost of Capital
Structure
5.1 Introduction
Objectives
5.2 Cost of Capital
5.3 Cost of Debt
5.4 Cost of Preference Capital
5.5 Cost of Equity Capital
5.6 Approaches to Derive Cost of Equity
5.7 Weighted Average Cost of Capital and Weighted Marginal Cost of Capital
5.8 Summary
5.9 Glossary
5.10 Terminal Questions
5.11 Answers
5.12 Case Study

Caselet

Essar Steel Raising Funds through Various Debt Instruments

Essar Steel, which is currently under debt burden of `23,500 crore, has
sought the intervention of its senior management for raising `3,000 crore
through various debt instruments from both international and domestic
markets. In order to overcome its high-cost debts, the company is looking
at various options for raising long-term funds through equity, bond or any
other form of hybrid securities. The management is considering this
proposition. The company will be approaching several financial institutions,
agencies and investors for the purpose of raising funds. In the financial year
2011-12, the company completed its extension of the Hazira steel plant to
10 million tonne per annum. However, the extension has resulted in severe
loss for the company amounting to `1,251 crore. The company generated
revenue of `16,056 crore. Since 1989, Essar Steel has invested `37,500
crore at its Hazira plant. This huge investment will enable the plant to produce
10 million tonnes of steel, and as a result, the company’s revenue will
increase to `40,000 crore. Earlier this year in 2012, the company got an
authorization from the Reserve Bank of India to raise $430 million
(approximately `2,338 crore) through external commercial borrowings.