Slm Unit 03 Mbf201

Financial Management Unit 3

Sikkim Manipal University Page No. 41

Unit 3 Time Value of Money

Structure:

3.1 Introduction

Learning objectives

Rationale

3.2 Future Value

Time preference rate or required rate of return

Compounding technique

Discounting technique

Future value of a single flow

Doubling period

Increased frequency of compounding

Effective vs. Nominal rate of interest

Future value of series of cash flows

Future value of annuity

Sinking fund

3.3 Present Value

Discounting or present value of a single flow

Present values of a series of cash flows

Present values of perpetuity

Present value of an uneven periodic sum

Capital recovery factor

3.4 Summary

3.5 Solved Problems

3.6 Terminal Questions

3.7 Answers to SAQs and TQs

3.1 Introduction

In the previous unit, you have learnt that wealth maximisation is far more

superior to profit maximisation. Wealth maximisation considers time value of

money, which translates cash flows occurring at different periods into a

comparable value at zero period.

For example, a firm investing in fixed assets will reap the benefits of such

investments for a number of years. However, if such assets are procured



through bank borrowings or term loans from financial institutions, there is an

obligation to pay interest and return of principle.

Decisions, therefore, are made by comparing the cash inflows

(benefits/returns) and cash outflows (outlays). Since these two components

occur at different time periods, there should be a comparison between the

two.

In order to have a logical and a meaningful comparison between cash flows

occurring over different intervals of time, it is necessary to convert the

amounts to a common point of time. This unit is devoted to a discussion of

techniques of doing so.

3.1.1 Learning objectives

After studying this unit, you should be able to:

Explain the time value of money

Understand the valuation concepts

Calculate the present and the future values of lump sums and annuity

flows

3.1.2 Rationale

“Time value of money” is the value of a unit of money at different time

intervals. The value of the money received today is more than its value

received at a later date. In other words, the value of money changes over a

period of time. Since a rupee received today has more value, rational

investors would prefer current receipts over future receipts. That is why, this

phenomena is also referred to as “Time preference of money”. Some

important factors contributing to this are:

Investment opportunities

Preference for consumption

Risk

These factors remind us of the famous English saying, “A bird in hand is

worth two in the bush”. The question now is: why should money have time

value?

Some of the reasons are:

Production

Money can be employed productively to generate real returns. For

example, if we spend Rs. 500 on materials, Rs. 300 on labour and



Rs. 200 on other expenses and the finished product is sold for Rs. 1100,

we can say that the investment of Rs. 1000 has fetched us a return of

10%.

Inflation

During periods of inflation, a rupee has higher purchasing power than a

rupee in the future.

Risk and uncertainty

We all live under conditions of risk and uncertainty. As the future is

characterised by uncertainty, individuals prefer current consumption over

future consumption. Most people have subjective preference for present

consumption either because of their current preferences or because of

inflationary pressures.

3.2 Future Value

3.2.1 Time preference rate or required rate of return

The time preference for money is generally expressed by an interest rate,

which remains positive even in the absence of any risk. It is called the risk

free rate.

For example, if an individual‟s time preference is 8%, it implies that he is

willing to forego Rs. 100 today to receive Rs. 108 after a period of one year.

Thus he considers Rs. 100 and Rs. 108 as equivalent in value. In reality

though this is not the only factor he considers. He requires another rate for

compensating him for the amount of risk involved in such an investment.

This risk is called the risk premium.

There are two methods by which the time value of money can be calculated:

Compounding technique

Discounting technique

3.2.1.1 Compounding technique

In the compounding technique, the future values of all cash inflows at the end of the

time horizon at a particular rate of interest are calculated. The amount earned on an

initial deposit becomes part of the principal at the end of the first compounding

period.

Required rate of return = Risk free rate + Risk premium



The compounding of interest can be calculated by the following equation:

Where, A = Amount at the end of the period

P = Principle at the end of the year

i = Rate of interest

n = Number of years

Example

Mr. A invests Rs. 1,000 in a bank which offers him 5% interest compounded

annually. Substituting the actual figures for the investment or Rs. 1000 in the

formula n, we arrive at the values shown in table 3.1.

Table 3.1: Interest compounded annually

Year 1 2 3

Beginning amount Rs.1000 Rs.1050 Rs.1102.50

Interest rate 5% 5% 5%

Amount of interest 50 52.50 55.13

Beginning principal Rs.1000 Rs.1050 Rs.1102.50

Ending principal Rs.1050 Rs.1102.50 Rs.1157.63

As seen from table 3.1, Mr. A has Rs. 1050 in his account at the end of the first

year. The total of the interest and principal amount Rs. 1050 constitutes the

principal for the next year. He thus earns Rs. 1102.50 for the second year. This

becomes the principal for the third year. This compounding procedure will

continue for an indefinite number of years.

Let us now see how the values in table 3.1 are arrived at.

Amount at the end of year 1 = Rs. 1000 (1+0.05) == Rs. 1050

Amount at the end of year 2 = Rs. 1050 (1+0.05) == Rs. 1102.50

Amount at the end of year 3 = Rs. 1102.50 (1+0.05) == Rs. 1157.63

The amount at the end of the second year can be ascertained by substituting

Rs.1000 (1+0.05) for Rs.1050, that is,

Rs.1000 (1+0.05) (1+0.05)=Rs.1102.50

Similarly, the amount at the end of the third year can be ascertained by

substituting Rs.1000 (1+0.05) for Rs.1102.50, that is,

Rs.1000 (1+0.05) (1+0.05) (1+0.05)=Rs.1157.63



3.2.1.2 Discounting technique

In the discounting technique, the present value of the future amount is

determined. Time value of the money at time 0 on the time line is calculated.

This technique is in contrast to the compounding approach where we

convert the present amounts into future amounts.

3.2.2 Future value of a single flow (lump sum)

The process of calculating future value will become very cumbersome if they

have to be calculated over long maturity periods of 10 or 20 years. A

generalised procedure of calculating the future value of a single cash flow

compounded annually is as follows:

Solved Problem

Mr.A requires Rs.1050 at the end of the first year. Given the rate of

interest as 5%,find out how much Mr. A would invest today to earn this

amount.

Solution

If P is the unknown amount, then

P (1+0.05) =1050

P=1050/ (1+0.05)

=Rs.1000

Thus Rs. 1000 would be the required principal investment to have

Rs. 1050 at the end of the first year at 5% interest rate. The present

value of the money is the reciprocal of the compounding value.

Mathematically, we have

Where P is the present value for the future sum to be received,

A is the sum to be received in future,

i is the interest rate and

n is the number of years.



Where, FVn = future value of the initial flow in n years hence

PV = initial cash flow

i = annual rate of interest

n = life of investment

The expression n)i1( represents the future value of the initial investment

of Re. 1 at the end of n number of years. The interest rate “i” is referred to

as the Future Value Interest Factor (FVIF). To help ease the calculations,

the various combinations of “i” and “n” can be referred to in the table 3.1. To

calculate the future value of any investment, the corresponding value of

n)i1( from the table 3.1 is multiplied with the initial investment.

3.2.2.1 Doubling period

A very common question arising in the minds of an investor is “how long will

it take for the amount invested to double for a given rate of interest”. There

are 2 ways of answering this question.

Solved Problem

The fixed deposit scheme of a bank offers the interest rates, as shown in

the table 3.2:

Table 3.2: Fixed deposit scheme of a bank

Period of deposit Rate per annum

<45 days 9%

46 days to 179 days 10%

180 days to 365 days 10.5%

365 days and above 11%

What will be the status of Rs. 10, 000 after three years, if it is invested at

this point of time?

Solution

FVn = PV (1+i)n or PV*FVIF (11%, 3y)

= 10000*1.368 (from the tables)

= Rs.13, 680

The status of Rs. 10, 000 after three years, if it is invested at this point of

time, would be Rs.13, 680.



1. One is called „rule of 72‟. This rule states that the period within which the

amount doubles is obtained by dividing 72 by the rate of interest.

For instance, if the given rate of interest is 10%, the doubling period is

72/10, that is, 7.2 years.

2. A much accurate way of calculating doubling period is the „rule of 69‟,

which is expressed as 0.35+69/interest rate. Going by the same example

given above, we get the number of years as 7.25 years {0.35 + 69/10

(0.35 +6.9)}.

3.2.2.2 Increased frequency of compounding

So far we have seen the calculation of the time value of money. It has been

assumed that the compounding is done annually.

Let us now see the effect on interest earned when compounding is done

more frequently - half-yearly or quarterly

Going by the calculations, we see that one gets more interest if compounding is

done on a more frequent basis. The generalised formula for shorter compounding

periods is:

Example

If we have deposited Rs.10, 000 in a bank which offers 10% interest per

annum compounded semi-annually, the interest earned is as shown in

table 3.3.

Table 3.3: Interest earned

Amount invested Rs.10,000

Interest earned for first 6 months

10000*10%*1/2 (for 6 months)

Rs.500

Amount at the end of 6 months Rs.10,500

Interest earned for second 6 months

105000*10%*1/2

Rs.525

Amount at the end of the year Rs.11,025

If in the above case, compounding is done only once in a year the

interest earned will be 10000*10% which is equal to Rs. 1000 and we

will have Rs. 11000 at the end of first year.



Where, FVn = future value after n years

PV = cash flow today

i = nominal interest rate per annum

m = number of times compounding is done during a year

n = number of years for which compounding is done

3.2.2.3 Effective vs. Nominal rate of interest

We have just learnt that interest accumulation by frequent compounding is

much more than the annual compounding. This means that the rate of

interest given to us, that is 10% is the nominal rate of interest per annum.

If the compounding is done more frequently, say semi-annually, the principal

amount grows at 10.25% per annum. 0.25% is known as the “Effective

Rate of Interest”. The general relationship between the effective and

nominal rates of interest is as follows:

Where,

r = Effective rate of interest

i = Nominal rate of interest

m= number of time compounding is done during the year

m = Frequency of compounding per year.

Solved Problem

Under the ABC Bank‟s Cash Multiplier Scheme, deposits can be made

for periods ranging from 3 months to 5 years. Every quarter, interest is

added to the principal. The applicable rate of interest is 9% for

deposits less than 23 months and 10% for periods more than 24

months. What will the amount of Rs. 1000 today be after 2 years?

Solution

. m = 12/3 = 4 (quarterly compounding)

1000 (1+0.10/4)4*2

1000 (1+0.10/4)8

Rs. 1218

The amount of Rs. 1000 after years would be Rs. 1218



3.2.3 Future value of series of cash flows

An investor may be interested in investing money in instalments and wish to

know the value of his savings after n years.

Let us understand the calculation of the same with the help of a solved

problem.

Solved Problem

Calculate the effective rate of interest if the nominal rate of interest is

12% and interest is compounded quarterly.

Solution:

m= 12/3 =4 (quarterly compounding)

r = {(1+0.12/4)4}-1

r=0.126 or 12.6% p.a.

The effective rate of interest is 12.6% p.a.

Solved Problem

Mr. Madan invests Rs. 500, Rs. 1000, Rs. 1500, Rs.2000 and Rs. 2500 at the

end of each year for 5 years. Calculate the value at the end of 5 years

compounded annually if the rate of interest is 5% p.a.

Solution

The value at the end of 5 years, compounded annually at a rate of interest of

5% per annum, is calculated in the table 3.4

Table 3.4: Future value of series of cash flows

End

of

year

Amount

invested

(Rs)

Number of years

compounded

Compounded

interest factors

from tables

FV in Rs.

1 500 4 1.216 608

2 1000 3 1.158 1158

3 1500 2 1.103 1654

4 2000 1 1.050 2100

5 2500 0 1.000 2500

Amount at the end of the fifth year Rs.8020

The value at the end of the fifth year is Rs. 8020



3.2.4 Future value of an annuity

Annuity refers to the periodic flows of equal amounts. These flows can be

either termed as receipts or payments.

Example

If you have subscribed to the Recurring Deposit Scheme of a bank

requiring you to pay Rs. 5000 annually for 10 years, this stream of pay-

outs can be called “Annuities”. Annuities require calculations based on

regular periodic contribution of a fixed sum of money.

The future value of a regular annuity for a period of n years at “i” rate of

interest can be summed up as under:

Where, FVAn = Accumulation at the end of n years

i = Rate of interest

n = Time horizon or no. of years

A = Amount invested at the end of every year for n years

The expression )i/)1)i1( n is called the Future Value Interest Factor for

Annuity (FVIFA). This represents the accumulation of Re.1 invested at the

end of every year for n number of years at “i” rate of interest. From the

tables 3.4 and 3.5, different combinations of “i” and “n” can be calculated.

We just have to multiply the relevant value with A and get the accumulation

in the formula given above.



We notice that we can get the accumulations at the end of n period using the tables.

Calculations for a long time horizon are easily done with the help of reference

tables. Annuity tables are widely used in the field of investment banking as ready

beckoners.

Solved Problem

Mr. Ram Kumar deposits Rs. 3000 at the end of every year for five years

into his account. Interest is being compounded annually at a rate of 5%.

Determine the amount of money he will have at the end of the fifth year.

Solution

The amount of money Mr. Ram Kumar will have at the end of the fifth

year is calculated from the table 3.5.

Table 3.5: Computation of future value of annuity

End of year

Amount invested

(Rs)

Number of years compounded

Compounded interest factors from tables

FV in Rs.

1 2000 4 1.216 2432

2 2000 3 1.158 2316

3 2000 2 1.103 2206

4 2000 1 1.050 2100

5 2000 0 1.000 2000

Amount at the end of the fifth year 11054

OR

Refer FVIFA table to compute the value at the end of 5th year:

= 2000 * FVIFA (5%, 5y)

= 2000 * 5.526

= Rs. 11052



3.2.5 Sinking fund

Sinking fund is a fund which is created out of fixed payments each period, to

accumulate for a future sum after a specified period.

The sinking fund factor is useful in determining the annual amount to be put in a

fund, to repay bonds or debentures or to purchase a fixed asset or a property at the

end of a specified period.

is called the Sinking Fund Factor.

Example: Manas Limited has an obligation to redeem Rs.50,00,000

debentures 6 years hence. How much should the company deposit annually

in the sinking fund account yielding 14 percent interest to cumulate

Rs.50,00,000 six years from now?

Solution: n=6 years, r= 14%, Accumulated sum = 50,00,000

Annual sinking fund deposit should be:

A = 50,00,000

FVIFA (14%, 6yrs)

Referring FVIFA table the factor is 8.536

= 50,00,000 = Rs.5,85,754

8.536

Solved Problem

Calculate the value of an annuity flow of Rs.5000 done on a yearly basis of five

years, yielding at an interest of 8% p.a.

Solution

= 5000 FVIFA (8%, 5y)

= 5000* 5.867

= Rs. 29335

The value of annuity flow is Rs. 29,335.



3.3 Present Value

Given the interest rate, compounding technique can be used to compare the

cash flows separated by more than one time period. With this technique, the

amount of present cash can be converted into an amount of cash of

equivalent value in future.

Likewise, we may be interested in converting the future cash flows into their

present values. The “Present Value” (PV) of a future cash flow is the amount

of the current cash that is equivalent to the investor. The process of

determining present value of a future payment or a series of future

payments is known as discounting.

3.3.1 Discounting or present value of a single flow

We can determine the PV of a future cash flow or a stream of future cash flows

using the formula:

Where, PV = Present Value

FVn = Amount

i = Interest rate

n = Number of years

Self Assessment Questions

Fill in the blanks

1. The important factors contributing to time value of money are

__________, ________________ and _______.

2. During periods of inflation, a rupee has a ___than a rupee in future.

3. As future is characterised by uncertainty, individuals prefer _________

consumption to __________ consumption.

4. There are two methods by which time value of money can be calculated

by _________ and _________ techniques.



3.3.2 Present value of a series of cash flows

In a business scenario, the businessman will receive periodic amounts (annuity) for

a certain number of years. An investment done today will fetch him returns spread

over a period of time. He would like to know if it is worthwhile to invest a certain sum

now in anticipation of returns he expects after a certain number of years. He should

therefore equate the anticipated future returns to the present sum he is willing to

Solved Problem

If Ms. Sapna expects to have an amount of Rs. 1000 after one year what

should be the amount she has to invest today, if the bank is offering 10%

interest rate?

Solution

= 1000/(1+0.10)1

= Rs. 909.09

The same can be calculated with the help of tables.

= 1000*PVIF (10%, 1y)

= 1000*0.909

= Rs. 909

The amount to be invested today to have an amount of Rs, 1000 after

one year is Rs. 909.

Solved Problem

An investor wants to find out the value of an amount of Rs. 10,000 to

be received after 15 years. The interest offered by bank is 9%.

Calculate the PV of this amount.

Solution

or 100000 PVIF (9%, 15y)

= 100000*0.275

= Rs. 27500

The PV of Rs. 10, 000 is Rs. 27,500.



forego. The PV of a series of cash flows can be represented by the following

formula:

The above formula or the equation reduces to:

The expression })i1(i/1)i1{( nn is known as Present Value Interest

Factor Annuity (PVIFA). It represents the PVIFA of Re. 1 for the given

values of i and n. The values of PVIFA (i, n) can be found out from the Table

3.6. It should be noted that these values are true only if the cash flows are

equal and the flows occur at the end of every year.

Solved Problem

Calculate the PV of an annuity of Rs. 500 received annually for four years,

when discounting factor is 10%.

Solution

The present value of annuity can be calculated from the table 3.6 as

shown under:

Table 3.6: Computation of PV of annuity

End of year Cash inflows PV factor PV in Rs.

1 Rs.500 0.909 454.5

2 Rs.500 0.827 413.5

3 Rs.500 0.751 375.5

4 Rs.500 0.683 341.5

3.170 1585.0

Present value of an annuity is Rs.1585.

OR

By directly looking at the table we can calculate:



= 500*PVIFA (10%, 4y)

= 500*3.170

= Rs. 1585

The present value of annuity is Rs. 1585.

3.3.3 Present value of perpetuity

An annuity for an infinite time period is perpetuity. It occurs indefinitely. A

person may like to find out the present value of his investment assuming he

will receive a constant return year after year. The PV of perpetuity is

calculated as:

Solved Problem

Find out the present value of an annuity of Rs. 10000 over 3 years when

discounted at 5%.

Solution

Present value of annuity

= 10000*PVIFA (5%, 3y)

= 10000*2.773

= Rs. 27730

Hence, the present value of annuity is Rs. 27,730.

Solved Problem

The principal of a college wants to institute a scholarship of Rs. 5000 for

a meritorious student every year. Find out the PV of investment which

would yield Rs. 5000 in perpetuity, discounted at 10%.

Solution

= 5000/0.10

= Rs. 50000

This means he should invest Rs. 50000 to get an annual return of

Rs. 5000.



3.3.4 Present value of an uneven periodic sum

In some investment decisions of a firm, the returns may not be constant. In

such cases, the PV is calculated as follows.

Or

PV= A1 PVIF (i, 1) + A2 PVIF (i, 2) + A3 PVIF (i, 3) + A4 PVIF (i, 4)

+……………..…. + An PVIF (i, n)

3.3.5 Capital Recovery Factor

Capital recovery factor is the annuity of an investment for a specified time at

a given rate of interest.

The reciprocal of the present value annuity factor is called capital recovery

factor.

Solved Problem

An investor will receive Rs. 10000, Rs. 15000, Rs. 8000, Rs. 11000 and

Rs. 4000 respectively at the end of each of five years. Find out the

present value of this stream of uneven cash flows, if the investors

interest rate is 8%.

Solution

PV= 10000/ (1+0.08) +15000/ (1+0.08)2+8000/ (1+0.08)3+11000/

(1+0.08)4+4000/ (1+0.08)5

=Rs.39276

Or by referring table we can compute

PV =10000 PVIF(8%,1yr)+15000 PVIF(8%,2yrs)+ 8000 PVIF(8%,3yrs)+

11000 PVIF(8%,4yrs)+4000 PVIF(8%,5yrs)

= 10000*0.926+15000*0.857+8000*0.794+11000*0.735+4000*0.681

= 9260+ 12855 + 6352 +8085 + 2724 = Rs.39,276

The present value of this stream of uneven cash flows is Rs. 39,276



is known as the Capital Recovery Factor.

3.4 Summary

Money has time preference. A rupee in hand today is more valuable than a

rupee a year later. Individuals prefer possession of cash now rather than at

a future point of time. Therefore cash flows occurring at different points in

time cannot be compared. Interest rate gives money its value and facilitates

comparison of cash flows occurring at different periods of time.

Compounding and discounting are two methods used to calculate the time

value of money.

Self Assessment Questions

Fill in the blanks

5. _________________ is created out of fixed payments each period to

accumulate for a future sum after a specified period.

6. The ________________ of a future cash flow is the amount of the

current cash that is equivalent to the investor.

7. An annuity for an infinite time period is called ______________.

8. The reciprocal of the present value annuity factor is called __________.

Solved Problem

A loan of Rs. 100000 is to be repaid in 5 equal annual instalments. If the

loan carries a rate of 14% p.a, what is the amount of each instalment?

Solution

Instalment*PVIFA (14%, 5) = 100000

Instalment=100000/3.433 = Rs. 29128.

The amount of each instalment has been calculated.



3.5 Solved Problems

1. What is the future value of a regular annuity of Re. 1.00 earning a rate of

12% interest p.a. for 5 years?

2. If a borrower promises to pay Rs.20000 eight years from now in return for

a loan of Rs.12550 today, what is the annual interest being offered?

3. A loan of Rs. 500000 is to be repaid in 10 equal instalments. If the loan

carries 12% interest p.a. what is the value of one instalment?

Solution: FVAn = A * FVIFA (12%,5yrs)

= 1*FVIFA (12%, 5y) = 1*6.353 = Rs. 6.353

Solution:

PV = A*PVIFA (12%, 10y)

500000 = A *5.650

500000/5.650 = A

Rs. 88492 = A (instalment amount)

Solution: PV = A * PVIF(r%,8 yrs)

12550 = 20000*PVIF (r%, 8yrs)

12550/20000 = PVIF (r%,8yrs)

0.627 = PVIF (r%,8yrs)

Refer PVIF table and search for 0.627 in the 8th year row. You can

identify this number under 6% column.

Hence the annual interest (r) = 6%



4. A person deposits Rs. 25000 in a bank that pays 6% interest half-yearly.

Calculate the amount at the end of 3 years

5. Find the present value of Rs. 100000 receivable after 10 years if 10% is

the time preference for money

3.6 Terminal Questions

1. If you deposit Rs.10000 today in a bank that offers 8% interest, how

many years will the amount take to double?

2. An employee of a bank deposits Rs. 30000 into his PF A/c at the end of

each year for 20 years. What is the amount he will accumulate in his PF

at the end of 20 years, if the rate of interest given by PF authorities is

9%?

3. A person can save _____________ annually to accumulate Rs. 400000

by the end of 10 years, if the saving earns 12%

4. Mr. Vinod has to receive Rs. 20000 per year for 5 years. Calculate the

present value of the annuity assuming he can earn interest on his

investment at 10% per annum

Solution

:

I /m= 6% ; m= 2 ; n=3 yrs

25000*(1+0.06)3*2 = 25000*1.194 = Rs. 29850

Solution:

Refer PVIF table (10%,10yrs)

PV =100000*(0.386)

= Rs. 38600



5. Aparna invests Rs. 5000 at the end of each year at 10% interest p.a.

What is the amount she will receive after 4 years?

3.7 Answers to SAQs and TQs

Answers to Self Assessment Questions

1. Investment opportunities, preference for consumption, risk

2. Higher purchasing power

3. Current and future

4. Compounding and discounting

5. Sinking fund

6. Present Value

7. Perpetuity

8. Capital Recovery Factor

Answers to Terminal Questions

1. 9 years (using rule of 72); 8.975 years (using rule of 69)

2. 30000*FVIFA (9%, 20Y) = 30000*51.160 = Rs. 1534800

3. A*FVIFA (12%, 10y) = 400000 which is 400000/17.549 = Rs. 22795

4. 20000*PVIFA (105, 5y)=20000*3.791 = Rs. 75820

5. 5000*FVIFA (10%, 4y) = 5000*6.105 = Rs. 23205