Business Mathematics - Time Value of Money

Business Mathematics

Time Value of Money

(Foundation Level)

Part A: Discussion of basic theories including different relevant formulas

Part B: 12 Illustrations with Solutions

Part A

Introduction

Conceptually, time value of money means that the value of a unit of money is different in different time periods. The value of a sum of money received today is more than the value of the same received after some time. Conversely, the sum of money received in future is less valuable than it is today. In other words, the present value of a rupee received after some time will be less than a rupee. Since a rupee received today has more value, rational investors would prefer current receipt to future receipt. The main reason for such time preference for money can be found in the investment opportunities for funds which are received early. The funds so invested will earn a return which would not be possible if the funds are received at a later time. The time preference for money is, generally, expressed in terms of a rate of return or more popularly in terms of a discount rate. The expected rate of return as also the time value of money will vary from individual to individual depending, inter alia, on his perception about the investment opportunities in particular and the overall business environment in general.

 

What applies to an individual applies equally, if not in greater measure, to a business firm. It is because business firms make decisions which have ramifications extending beyond the period in which they were taken. Therefore, time value of money is of crucial significance. This requires the development of procedures and techniques for evaluating future incomes in terms of the present.

 

Techniques

There are two techniques for determining and analysing the time value of money which are:

1.           Compounding technique, and

2.           Discounting technique.

 

Compounding technique is applied to determine the future value of an investment/cash outflow or a series of investments/cash outflows.

 

Discounting technique is applied to determine the present value of a future earning/cash inflow or a series of future earnings/cash inflows.

 

Formulas for determining future value and present value of annuities

 Let,

A =	Annuity i.e. value of each instalment [Definition of annuity: An annuity is a series of regular payments/instalments of a fixed sum at regular intervals. This interval is generally a year, but it may be a half year or quarter or a month.]
M =	Future value of an annuity of Rs A after n number of instalments
V =	Present value of an annuity of Rs A for n number of instalments
i  =	Interest of Rs 1 for one year
n  =	Total number of instalments
x  =	Number of times interest is compounded per year

 

1.  When annuities are paid at the end of each compounding period

(These types of annuities are known as immediate annuity or regular annuity or ordinary annuity or annuity in arrears) –

 

(a)	M =	A [(1 + i/x) ^n – 1] ÷ i/x
(b)	M =	A × FVIFA (k, n)
(c)	V =	A [1 – (1 + i/x) ^ (−n)] ÷ i/x
(d)	V =	A × PVIFA (k, n)

 

2. When annuities are paid at the beginning of each compounding period

(These types of annuities are known as annuity due or annuity in advance) –

 

(a)	M =	(1 + i/x) A [(1 + i/x) ^n – 1] ÷ i/x
(b)	M =	A × FVIFA (k, n) × (1 + i)
(c)	V =	(1 + i/x) A [1 – (1 + i/x) ^ (−n)] ÷ i/x
(d)	V =	A × PVIFA (k, n) × (1 + i)

 

   Important note:

   If nothing is mentioned in the given problem about whether the annuities are paid at the end or beginning of the compounding period, it should be assumed that annuities are paid at the end of each compounding period (i.e. annuities should be assumed as immediate annuities).

 

3.  Present value of perpetuity –

V   =

A / i

 

Compound interest formulas

 Let,

P  =	Principal = Present Value
A  =	Amount (Principal + Total interest) = Future Value
I  =	Total interest earned
i  =	Interest of Rs 1 for one year
n  =	Total number of times interest is to be compounded
x  =	Number of times interest is compounded per year

 

 Formulas:

4	ERI =	Effective rate of interest (ERI) = [1 + i/x] ^x – 1
5 (a)	A  =	P (1 + i/x) ^n
5 (b)	A  =	P × FVIF (k, n)
6 (a)	P  =	A ÷ (1 + i/x) ^n
6 (b)	P  =	A × PVIF (k, n)
7 (a)	I  =	A – P
7 (b)	I  =	P [(1 + i/x) ^n – 1]

 

Part B

Business Mathematics

Time Value of Money

Selected Annuity Problems and Solutions

(Foundation Level)

 

 

Illustration: 1

Mr. X will receive Rs 1,500 at the end of every year starting from today and this will continue for next 5 years. How much deposit he has to make today if the interest rate is 10% p.a.?

 

Solution:

Here, we have to calculate present value of ordinary annuities of Rs 1,500 to be received for next 5 years, if the interest rate is 10% p.a.

 

Formula:

Present value, V = A × PVIFA _{(k, n)}

Where, A = Annuity i.e. amount of loan repayment at the end of every year; and PVIFA _{(k, n)} = Present Value Interest Factor for Annuity at k% p.a. rate of interest for n years.

 

Here, A = Rs 1,500, and PVIFA _{(10, 5)} = 3.7908

∴ V = Rs 1,500 × 3.7908 = Rs 5,686.20

∴ Mr. X should make a deposit of Rs 5,686.20 today.

 

Illustration: 2

Mr. P opened a recurring deposit account in a bank and started depositing semi-annual instalments in arrear amounting Rs 5,000 for 5 years. If the rate of compound interest is 6% p.a., what will be the maturity amount?

 

Solution:

Here, we have to calculate future value of annuities in arrear of Rs 5,000 to be deposited for next 5 years in semi-annual instalments, if the interest rate is 6% p.a.

 

Formula:

Future value, M = A × FVIFA _{(k, n)}

Where, A = Annuity i.e. amount of deposit at the end of every year; and FVIFA _{(k, n)} = Future Value Interest Factor for Annuity at k% p.a. rate of interest for n years.

 

Here, A = Rs 5,000, and FVIFA _{(3, 10)} = 11.464

∴ Maturity amount after 5 years will be (M)

= A × FVIFA _{(3, 10)}

= Rs 5,000 × 11.464

= Rs 57,320

 

Illustration: 3

AMS & Co. Borrows Rs 1, 20,000 for one year at 15% annual interest, compounded monthly. Find their monthly repayment.

 

Solution:

Here, we have to calculate the amount of ordinary annuities payable in 12 monthly instalments the present value of which is Rs 1, 20,000 at 15% interest p.a.

 

Formula:

Present value, V = A [1 – (1 + i/x) ^ (−n)] ÷ i/x

 

Here, V = Rs 1, 20,000, i = 0.15, x = 12, and n = 12

i.e. 1, 20,000 = A [1 – (1 + 0.15/12) ^ (−12)] ÷ 0.15/12

Or, 1, 20,000 = A × 11.0793

Or, A = 1, 20,000 ÷ 11.0793 = Rs 10,831

 

Therefore, required amount of monthly repayments = Rs 10,831

 

Illustration: 4

Calculate the present value of an annuity of Rs 3,000 received at the beginning of each year for 5 years at a discount factor of 6%.

 

Solution:

Here, we have to calculate present value of an annuity due of Rs 3,000 to be received for 5 years, if the discount factor is 6% p.a.

 

Formula:

Present value, V = A × PVIFA _{(k, n)} × (1 + i)

Where, A = Annuity i.e. amount of loan repayment at the end of every year; and PVIFA _{(k, n)} = Present Value Interest Factor for Annuity at k% p.a. rate of interest for n years.

 

Here, A = Rs 3,000, PVIFA _{(6, 5)} = 4.2124, and i = 0.06

∴ Required present value, V

= Rs 3,000 × 4.2124 × (1 + 0.06)

= Rs 13,395 (approx.)

 

Illustration: 5

Mr. Singh deposits Rs 2,000 at the beginning of each year for 5 years. How much do these accumulate at the end of 5^th year at an interest rate of 6%?

 

Solution:

Here, we have to calculate future value of an annuity due of Rs 2,000 to be deposited for 5 years, if the interest rate is 6% p.a.

 

Formula:

Future value, M = A × FVIFA _{(k, n)} × (1 + i)

Where, A = Annuity i.e. amount of deposit at the end of every year; and FVIFA _{(k, n)} = Future Value Interest Factor for Annuity at k% p.a. rate of interest for n years.

 

Here, A = Rs 2,000, and FVIFA _{(6, 5)} = 5.6371, and i = 0.06

∴ Mr. Singh at the end of 5 years will have (M)

= Rs 2,000 × 5.6371 × (1 + 0.06)

= Rs 11,951 (approx.)

 

Illustration: 6

A sum of Rs 4,895 borrowed from a moneylender at 5% p.a. compounded annually. Find the annual instalment, if amount borrowed is to be paid back in three equal instalments.

 

Solution:

Here, we have to calculate the amount of ordinary annuities payable in 3 annual instalments the present value of which is Rs 4,895 at 5% interest p.a.

 

Formula:

Present value, V = A × PVIFA _{(k, n)}

Where, A = Annuity i.e. amount of loan repayment at the end of every year; and PVIFA _{(k, n)} = Present Value Interest Factor for Annuity at k% p.a. rate of interest for n years.

 

Here, V = Rs 4,895, and PVIFA _{(5, 3)} = 2.7232

∴ 4,895 = A × 2.7232

Or, A = 4,895 ÷ 2.7232 = Rs 1,797.52

 

Therefore, amount of equal instalments of annual repayments = Rs 1,797.52

 

Illustration: 7

AMS & Co. makes a monthly payment of Rs 11,350 for one year at 11% annual interest, compounded monthly. Find the amount borrowed by them.

 

Solution:

Here, we have to calculate the present value of ordinary annuities Rs 11,350 payable in 12 monthly instalments at 11% interest p.a.

 

Formula:

Present value, V = A [1 – (1 + i/x) ^ (−n)] ÷ i/x

 

Here, A = Rs 11,350, i = 0.11, x = 12, and n = 12

∴ V = 11,350 [1 – (1 + 0.11/12) ^ (−12)] ÷ 0.11/12

Or, V = Rs 1, 28,420 (approx.)

 

Therefore, amount borrowed by the company = Rs 1, 28,420

 

Illustration: 8

Calculate the present value of an annuity of Rs 5,000 received annually for 4 years at a discount factor of 5%.

 

Solution:

Here, we have to calculate present value of ordinary annuities of Rs 5,000 to be received annually for 4 years at a discount rate of 5% p.a.

Formula:

Present value, V = A × PVIFA _{(k, n)}

Where, A = Annuity i.e. amount of loan repayment at the end of every year; and PVIFA _{(k, n)} = Present Value Interest Factor for Annuity at k% p.a. rate of interest for n years.

 

Here, A = Rs 5000, and PVIFA _{(5, 4)} = 3.546

∴ Present value, V = Rs 5,000 × 3.546

= Rs 17,730

 

Illustration: 9

Find the present value of perpetuity of Rs 2,725 at 0.5% per month compound interest.

 

Solution:

Formula:

Present value of perpetuity, V = A ÷ i

 

Here,

A = Rs 2,725, and i = (0.5 ÷ 100) × 12 = 0.06

Therefore, required present value, V

= Rs 2,725 ÷ 0.06 = Rs 45,417 (approx.)

 

Illustration: 10

Mr. A wants to create an endowment fund of Rs 51,750 to provide for a prize to be given every year. If the fund can be invested at 11% p.a. compound interest, find the amount of prize.

 

Solution:

Here, we have to calculate the amount of annuity involved in a perpetuity present value of which is Rs 51,750 at the discount rate of 11% p.a.

 

Formula:

Present value of perpetuity, V = A ÷ i

 

Here, V = Rs 51,750, and i = 0.11

∴ 51,750 = A ÷ 0.11

Or, A = 51,750 × 0.11 = Rs 5,693 (approx.)

 

Therefore, the amount of prize that can be given every year is Rs 5,693.

 

Illustration: 11

Assume a person has the opportunity to receive an ordinary annuity that pays Rs 50,000 per year for the next 25 years, with a 6% discount rate, or to take a lump-sum payment of Rs 6, 50,000 now. Which is the better option?

 

Solution:

Present value of an ordinary annuity of Rs 50,000 per year to be received for next 25 years at a 6% discount rate,

V = A [1 – (1 + i/x) ^ (−n)] ÷ i/x

Or, V = 50,000 [1 – (1 + 0.06) ^ (−25)] ÷ 0.06

Or, V = Rs 6, 39,168 (approx.)

 

i.e. Present value of the ordinary annuity is less than the lump-sum payment. Therefore, the better option for the person is to choose the lump-sum payment.

 

Illustration: 12

Assume a person has the opportunity to receive an annuity due that pays Rs 50,000 per year for the next 25 years, with a 6% discount rate, or to take a lump-sum payment of Rs 6, 50,000 now. Which is the better option?

 

Solution:

Present value of an annuity due of Rs 50,000 per year to be received for 25 years at a 6% discount rate,

V = (1 + i/x) A [1 – (1 + i/x) ^ (−n)] ÷ i/x

Or, V = (1 + 0.06) 50,000 [1 – (1 + 0.06) ^ (−25)] ÷ 0.06

Or, V = Rs 6, 77,518 (approx.)

 

i.e. Present value of the annuity due is more than the lump-sum payment. Therefore, the better option for the person is to choose the annuity due payments.