P A P E R 3 A : C O S T A C C O U N T I N G C H A P T E R 2

B Y : C A K A P I L E S H W A R B H A L L A

Time Value of Money 1

Learning objectives

Understand the Concept of time value of money.

Understand the relationship between present and future value of money and how interest rate is used to adjust the value of cash flows in-order to arrive at present (discounting) or future (compounding)

values.

Understand how to calculate the present or future value of an annuity?

Know how to use interest factor table’s in order to calculate the present or future values?

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Simple Interest

It may be defined as Interest that is calculated as a simple percentage of the original principal amount.

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Compound Interest

If interest is calculated on original principal amount it is simple interest. When interest is calculated on total of previously earned interest and the original principal it compound interest.

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Example

Mr. X deposited Rs. 10,000 in a bank today for a period of 5 years. If the bank pays interest @ 10% p.a. annually compounded, what is the maturity amount after a period of 5 years?

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Solution

P{1+r} for n years= 10000{1 + 0.1} compounded for 5 years= 16105

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Conclusion

16105 i.e. FUTURE VALUE of a PRESENT AMOUNT i.e. 10000

OR10000 i.e. PRESENT VALUE of a FUTURE AMOUNT i.e. 16105

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Section I (Part I)

FUTURE VALUE OF A PRESENT AMOUNT (Table A1 and A2)

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Table A1 (FVIF)

Future value interest factor

This table gives us the MATURITY AMOUNT of Re 1 deposited TODAY

At a given rate of interest i.e. r

For a given period of time i.e. n

Note: This table is based on ANNUAL Compounding.

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Extract of Table A1

Years Rate of Interest 10%

1 1.12 1.213 1.3314 1.4645 1.6105

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Issue

Can we use Table A1 in a situation where the FREQUENCY of compounding is more than once in a year.

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Answer

Yes, we can.

There are 2 approaches of doing this.

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Approach I

Calculate the effective rate of interest.

{1+ r/m} raise to power mWhere, M = frequency of compounding in a year

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Example

Let us say Interest rate is 10% p.a. with six monthly compounding.

Thus, Frequency of compounding is 2.Effective rate of interest is: [{1 + 0.10/2} raise to power 2] - 1= 10.25%

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Technique

Now we can see Table A1 with the effective rate of interest for a given no. of years

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Approach II

Change the ‘no of years’ and the ‘rate of interest’RULE:

Divide rate of interest by frequency of compounding in a year and multiply the no of years by the frequency of compounding in a year

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Example

Let us take the same example, 10% p.a. six monthly compounding for 5 years.

You can see Table A1 with 10 years @ 5% rate of interest

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Conclusion

10.25% for a period of 5 years

OR

5% for a period of 10 years.

Note: We will get same answer.

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Rule of 72

To calculate the doubling period

72/ rate of interest Ex: If rate of interest is 8%, Money gets doubled in 9 years.

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Table A2 FVIFA

Future Value Interest Factor for an Annuity

This table gives us the MATURITY AMOUNT of

Re 1 deposited EVERY Year END

At a given rate of interest i.e. r

For a given period of time i.e. n

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Extract of Table A2

Years Rate of Interest 10%

1 12 2.13 3.314 4.6415 6.105

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Note:

This table is also based on annual compounding.

But remember the table considers that deposit is made at EVERY YEAR END.

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Issue

Can we use Table A2 in a situation if the deposit is made at the BEGINNING of each year?

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Answer

Yes, we can use Table A2 but each value of the table needs to be multiplied by (1+r)

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Example

Mr. X wants Rs. 500000 at the end of 8 years from now. Find the amount to be deposited each year in an account offering 7% interest compounded per annum.

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Solution

In this case we have to make use of future value annuity of one rupee table i.e., table A2 since futue amount is given and we need to calculate series of amount which shall aggregate to Rs. 500000 at the end of 8 years.

Future value of annuity = Equal payment x (CFAF 9r, n))Rs. 500000 = Equal payment x 5.971Equal payment = 500000 / 5.971Equal payment = Rs. 83738.07

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Example

Mr. X is planning for his retirement. He is 50 years old today, and would like to have Rs. 500000 when he attains the age of 65 years. He intends to deposit a constant amount of money in a bank account offering 12 percent rate of interest per annum every year. How much should Mr. X invest at the end of each year for next 15 years to obtain Rs. 500000 at the end of that period?

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Solution

GivenRequired sum in future Rs. 500000Period of investment (years) 15Interest rate 12%

Future value factor of annuity at 12% for 15 years37.28 (using table A2)

Let ‘R’ be the amount deposited every year for the given period, therefore, we have

R x (37.28) = 500000R = 500000 / 37.28 = Rs. 13412.01

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Section II (Part I)

Present Value of a Future Amount (Table A3 and A4)

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Part I (PVIF)

Present Value Interest factor

This table gives us the discounted or present value

Of an amount which is to be received after ‘n’ no of years

If received TODAY

Discounted at a given rate of interest i.e. r

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Table A3 Extract

Years Rate of Interest 10%

1 0.9092 0.8263 0.7514 0.6835 0.621

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Notes

This table is basically INVERSE of Table A1. FV = PV (1+r) Thus,PV = FV x 1/(1+r)

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Example

If we have to receive Rs. 10000 after 5 years from Mr. Y, if it is received today:

Taking a discount/interest rate 10%, We will receive 10000 x 0.621= 6210

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Example II

Find the present value of Rs. 8000, in following cases :

Received today

Received three year from now.

Received five years from now

Received nine years from now.

Received twelve years from now.

Given required rate of 12%.

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Solution

In the above cases, we shall make use of table A3 i.e., present value of one rupee.

Rs. 8000 received is equivalent to Rs. 8000.

Present value of Rs. 8000 received at the end of three years from now

= Rs. 8000 x PVF (12%, 3 years)

= Rs. 8000 x 0.712 = 5696

Present value of Rs. 8000 received at the end of five years from now

= Rs. 8000 x 0.567 = 4536

Present value of Rs. 8000 received at the end of nine years from now

= Rs. 8000 x 0.361 = 2888

Present value of Rs. 8000 received at the end of twelve years from now

= Rs. 8000x 0.257 = 2056

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Part II (PVIFA)

Present Value Interest Factor for an Annuity

This table gives us PRESENT VALUE

Of amount to be received at the end of every year

If received TODAY

Discounted at a rate of interest ‘r’

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Table A4 Extract

Years Rate of interest 10%

1 0.9092 1.7353 2.4864 3.1695 3.791

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Example

If we receive Rs. 10,000 every year end for the next 5 years,If the entire money is received today, using a discount rate of 10%,

We will receive 10,000 x 3.791= 37910

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Example

Mr. X is planning to retire this year. He is given two choices. His company can either pay him a lump sum retirement payment of Rs. 400000 or Rs. 60000 life time annuity. Mr. X is in good health and expects to live for at least 20 more years. If he has opportunity to earn interest at the rate of 12% p.a., which alternative should be choose? Would his decision change, if he has opportunity to earn interest rate of 14% p.a.

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Solution

Payment (presently) = 400000Annuity = Rs. 60000Period of annuity 20 yearsIf he has opportunity to earn interest rate of 12%,Present value of annuity = 60000 x PVAF (12%, 20 years) (using table A4)60000 x 7.469 = 448140If he has opportunity to earn interest rate of 14%,Present value of annuity = 60000 x PVAF (14%, 20 years) (using table A4)60000 x 6.623 = 397380

Mr. X should choose annuity payment of R. 60000 if he has opportunity to earn return of 12% p.a. However, he shall opt for lump sum payment if he has opportunity to earn return of 14% p.a.

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Example

A company is extending a loan facility of Rs. 5,00,000 for five years at the rate of 12% p.a., on compounding basis which is to be paid back in the form of five equal installments. Find the size of each installment.

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Solution

Annual amount = Total amount of loan to be repaid / PVF (r, n)Annual Installment = 5,00,000 / 3.605 (Use Table A4)Annual Installment = Rs. 1,38,696.26.

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Present Value of Annuity till Perpetuity

Without growth

A/r

Where, A= Annuity R= rate of interest

With growth

A/r-g

Where, A= Annuity R= rate of interestG= growth rate

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Lesson Summary

Concept of Future Value

Concept of Present Value

Concept of Annuity

Practical application

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Thank you

All the best ………

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