Financial Management - Time Value of Money

 

FINANCIAL MANAGEMENT

Time Value of Money

Part A:	Discussion of basic theories including different relevant formulas.
Part B:	15 Illustrations with Solutions.

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

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

 

Discounting technique

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 & present value of series of payments

  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 deferred annuity or ordinary annuity or annuity in arrears)

 

(i)	M =	A [{(1 + i/x) ^n} – 1] ÷ i/x
(ii)	M =	A × FVIFA (k, n)
(iii)	V =	A [{(1 + i/x) ^n} – 1] ÷ [(i/x)(1 + i/x)^n)]
(iv)	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)

(i)	M =	(1 + i/x) A [{(1 + i/x) ^n} – 1] ÷ i/x
(ii)	M =	A × FVIFA (k, n) × (1 + i)
(iii)	V =	(1 + i/x) A [{(1 + i/x) ^n} – 1] ÷ [(i/x)(1 + i/x)^n)]
(iv)	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

 

4. Present value of a deferred annuity to begin at the end of m years and to continue for n years –

       V = A [{(1 + i) ^n} – 1] ÷ i (1 + i) ^ (m + n)

 

5. Present value of a deferred perpetuity to begin at the end of m years –

       V = A ÷ i (1 + i) ^m

 

6. Present value of growing perpetuity –

       V = A ÷ (i – g)

       Where,   A = First payment, and

                     g =  Constant rate of growth in perpetuity

                        (Growth %-age being expressed in decimal)]

 

7. Present value of growing annuity (Assuming –

       V = A [1 – {(1 + g) ÷ (1 + i)} ^n]  ÷ (i – g)

       Where,   A = First payment, and

                     g =  Constant rate of growth in annuity

                        (Growth %-age being expressed in decimal)]

  

8. Future value of growing annuity –

       M = A [(1 + i) ^n – (1 + g) ^n] ÷ (i – g)

       Where,   A = First payment, and

                     g =  Constant rate of growth in annuity

                        (Growth %-age being expressed in decimal)]

 

   Important note:

   When i = g, Formula 8 will be:

M = A x n x (1 + i) ^ (n – 1)

 

Formulas for determining future value and present value of a single payment

 

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]

 

 

FINANCIAL MANAGEMENT

Time Value of Money

Selected Problems and Solutions

 

Illustration: 1

Mr. Rohit Roy deposits Rs 1, 00,000 to a commercial bank. The bank offers 12% p.a. interest on deposits. What is the effective rate of interest, if the compounding is done: (i) half-yearly; (ii) quarterly; and (iii) monthly?

 

Solution:

Effective Rate of Interest (ERI) = [(1 + ⁱ/_x) ^x] – 1

Where, i = ^r/₁₀₀; r = Rate of interest per cent p.a.; and x = Number of times interest is compounded per year. Therefore,

(i)    ERI = [(1 + ^0.12/₂)^2] – 1 = 0.1236 or 12.36%

(ii)  ERI = [(1 + ^0.12/₄)^4] – 1 = 0.1255 or 12.55%

(iii)    ERI = [(1 + ^0.12/₁₂)^12] – 1 = 0.1268 or 12.68%

 

Illustration: 2

Compute the future value of an initial investment of Rs 70,000 after 10 years at 10% p.a. interest rate, if compounded: (i) annually; (ii) half-yearly; and (iii) quarterly.

 

Solution:

Future value, A = P (1 + ⁱ/_x) ^n

Where, i = ^r/₁₀₀; r = Rate of interest per cent p.a.; x = Number of times interest is compounded per year; n = Total number of times interest is to be compounded; and P = Principal = Present Value.

 

(i)    A = 70,000(1 + ^0.1/₁)^10 = Rs 1,81,562

(ii)  A = 70,000(1 + ^0.1/₂)^20 = Rs 1,85,731

(iii)    A = 70,000(1 + ^0.1/₄)^40 = Rs 1,87,954

 

Illustration: 3

Mr. Samir Chatterjee has received Rs 15, 00,000 at the time of his retirement. He wants to invest this amount for 7 years into a private bank in their fixed deposit schemes. The bank offers him two schemes as follows:

Scheme I: To be compounded half-yearly at 9% p.a. rate of interest; and

Scheme II: To be compounded yearly at 9.50% p.a. rate of interest.

 

Which one of these two schemes should Mr. Chatterjee choose?

 

Solution:

Future value, A = P (1 + ⁱ/_x) ^n

Where, i = ^r/₁₀₀; r = Rate of interest per cent p.a.; x = Number of times interest is compounded per year; n = Total number of times interest is to be compounded; and P = Principal = Present Value.

 

Future value under Scheme I:

A = 15, 00,000(1 + ^0.09/₂) ^14 = Rs 27, 77,917

 

Future value under Scheme II:

A = 15, 00,000(1 + ^0.095/₁) ^7 = Rs 28, 31,327

 

Future value under Scheme II > Future value under Scheme I

∴ Mr. Chatterjee should choose Scheme II for his investment.

 

Illustration: 4

Mr. Uttam Saha deposits Rs 10,000 at the end of every year for 4 years and the deposits earn him a compound interest @ 10% p.a. Determine the amount of money Mr. Saha will have at the end of 4 years.

 

Solution:

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 10,000, and FVIFA _{(10, 4)} = 4.6410

Therefore,

Mr. Saha at the end of 4 years will have (M)

= A × FVIFA _{(10, 4)} = Rs 10,000 × 4.6410 = Rs 46,410

 

Illustration: 5

Mr. Sundaram deposits Rs 50,000 at the beginning of each year for 5 years in a private bank and the deposits earn him a compound interest @ 6% p.a. How much money he will have at the end of 5 years?

 

Solution:

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

Where, A = Annuity i.e. amount of deposit at the end of every year; i = ^r/₁₀₀; r = Rate of interest p.c. p.a.; and FVIFA _{(k, n)} = Future Value Interest Factor for Annuity at k% p.a. rate of interest for n years.

 

Here, A = Rs 50,000; i = 0.06; and FVIFA _{(6, 5)} = 5.6371

Therefore,

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

= A × FVIFA _{(6, 5)} × (1 + 0.06)

= Rs 50,000 × 5.6371 × 1.06 = Rs 2, 98,766

 

Illustration: 6

Tansen Limited has Rs 2, 00,000, 6% Debentures outstanding today. The company has to redeem the debentures after 5 years and for this purpose it establishes a Sinking Fund to provide necessary funds for the redemption. Sinking Fund Investments earn interest @ 10% p.a. The investments are made at the end of each year. What annual payment must the company make to ensure that the required amount of Rs 2, 00,000 is available as on the date of redemption?

 

Solution:

Here, future value of an annuity (payable at the end of each year) for 5 years at interest rate 10% p.a. (M) = Rs 2,00,000 and FVIFA _{(10, 5)} = 6.1051. Amount of annuity (A) is to be found.

 

By the formula, M = A × FVIFA _{(10, 5)}

2, 00,000 = A × 6.1051

∴ A = Rs 32,760 (rounded off)

 

∴ The Company must make an annual payment of Rs 32,760 to the Sinking Fund to ensure that the required amount of Rs 2, 00,000 is available as on the date of redemption.

 

Illustration: 7

An investor makes successive investments for 5 years at an annual interest rate of 5%. His successive investments are shown below:

End of year	1	2	3	4	5
Investments (Rs)	15,000	20,000	30,000	35,000	40,000

 

Calculate the amount that he will receive after 5 years.

 

Solution:

Computation of total future value

End of year	Amount invested	Compounding Factor	Future Value
A	B (Rs)	C	D = B × C (Rs)
1	15,000	(1.05)^4 = 1.2155	18,233
2	20,000	(1.05)^3 = 1.1576	23,152
3	30,000	(1.05)^2 = 1.1025	33,075
4	35,000	(1.05)^1 = 1.05	36,750
5	40,000	(1.05)^0 = 1	40,000
Total future value at the end of Year: 5			1,51,210

 

∴ after 5 years, the investor will receive Rs 1, 51,210.

 

Illustration: 8

Mr. P. C. Sarkar is to receive Rs 20,000 after 10 years from now. If compound interest rate is 8% p.a., calculate its present value by using discount factor table.

 

Solution:

Present value, P = A × PVIF _{(k, n)}

Where, A = Amount = Future Value; and PVIF _{(k, n)} = Present Value Interest Factor at k% p.a. rate of interest for n years.

 

Here, A = Rs 20,000; and PVIF _{(8, 10)} = 0.4632

∴ Required present value, P = 20,000 × 0.4632 = Rs 9,264

 

Illustration: 9

Mr. Das expects to receive Rs 5,000 at the end of year 1, Rs 10,000 at the end of year 2, Rs 10,000 at the end of year 3, Rs 3,000 at the end of year 4, and Rs 2,000 at the end of year 5. He wants to know the present value of his cash inflows, given that his required rate of return is 10% p.a.

 

Solution:

Computation of total present value

End of year	Amount receivable	Discounting Factor	Present Value
A	B (Rs)	C	D = B × C (Rs)
1	5,000	(1 ÷ 1.1)^1 = 0.9091	4,545.50
2	10,000	(1 ÷ 1.1)^2 = 0.8264	8,264.00
3	10,000	(1 ÷ 1.1)^3 = 0.7513	7,513.00
4	3,000	(1 ÷ 1.1)^4 = 0.6830	2,049.00
5	2,000	(1 ÷ 1.1)^5 = 0.6209	1,241.80
Total present value of the cash inflows			23,613.30

 

∴ Total present value of the cash inflows receivable by Mr. Das = Rs 23,613.30.

 

Illustration: 10

Mr. Mitra wants to withdraw Rs 20,000 every year from his bank for 5 years. He wants to know how much he should deposit in his bank today given that interest rate is 6% p.a.

 

Solution:

Here, we have to find the present value of annuity of Rs 20,000 for 5 years, because that is the amount which is required to be deposited in bank now to be able to withdraw every year Rs 20,000 for 5 years.

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

Where, A = Annuity i.e. amount to be withdrawn from bank 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.

 

∴The amount to be deposited in bank now, V

= Rs 20,000 × PVIFA _{(6, 5)}

= Rs 20,000 × 4.2124

= Rs 84,248

 

Illustration: 11

Mr. Lahiri takes a loan of Rs 8, 00,000 for 7 years. The bank charges an interest of 10% p.a. on loans. How much should Mr. Lahiri pay every year to settle the loan?

 

Solution:

Here, Rs 8, 00,000 is present value of the annual loan repayments. We know,

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 8, 00,000, and PVIFA _{(10, 7)} = 4.8684

∴ 8, 00,000 = A × 4.8684

⇒ A = 8, 00,000 ÷ 4.8684 = Rs 1, 64,325

∴ Mr. Lahiri should pay Rs 1, 64,325 to bank every year to settle the loan.

 

Illustration: 12

The compound rate of interest is 8% p.a. Which option would you prefer:-

(i)    Rs 1, 00,000 today, or (ii) Rs 2, 00,000 after 10 years?

 

Solution:

We know, Present value, P = A × PVIF _{(k, n)}

Where, A = Amount = Future Value; and PVIF _{(k, n)} = Present Value Interest Factor at k% p.a. rate of interest for n years.

 

∴ Present value of option (ii), P

    = Rs 2, 00,000 × PVIF _{(8, 10)}

    = Rs 2, 00,000 × 0.4632

    = Rs 92,640

 

Whereas, present value of option (i) = Rs 1, 00,000

 

∴ Option (i) should be preferred.

 

Illustration: 13

Mr. Kirlosker retires at the age of 60 years and his employer gives him a pension of Rs 2, 00,000 per year for the rest of his life. His company gives him another offer to receive a lump sum of Rs 15, 00,000 at the time of his retirement. Reckoning his expectation of life to be 12 years and considering the rate of compound interest as 8% p.a., advice Mr. Kirlosker for his best alternative.

 

Solution:

We know, 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.

 

∴ Present value of annual pension of Rs 2, 00,000:-

V = 2, 00,000 × PVIFA _{(8, 12)}

⇒ V = 2, 00,000 × 7.5361

⇒ V = Rs 15, 07,220

 

Now we find present value of annual pension of Rs 2, 00,000 is more than the lump sum Rs 15, 00,000. ∴ Mr. Kirlosker should opt for the first option of annual pension.

 

Illustration: 14

Ramprasad Ganguly has taken a 20 month car loan of Rs 6, 00,000. The rate of compound interest is 12% p.a. What will be the amount of monthly loan amortisation?

 

Solution:

Here, loan amount of Rs 6, 00,000 is the present value of all the equal loan amortisation amounts.

 

∴By the formula, Present value, V = A × PVIFA _{(k, n)}

Where, A = Annuity i.e. amount of loan amortisation 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,

6, 00,000 = A × PVIFA _{(1, 20)}

⇒ 6, 00,000 = A × 18.046

⇒ A = 6, 00,000 ÷ 18.046 = 33,248

 

∴The amount of monthly loan amortisation will be Rs 33,248.

 

Illustration: 15

Mathurmohan Biswas bought a TV costing Rs 13,000 by making a down payment of Rs 3,000 and agreeing to make equal annual payment for 4 years. How much would be each payment if the interest on unpaid amount be 14% compounded annually?

 

Solution:

Here, present value of the 4 equal annual instalments = 13,000 – 3,000 = Rs 10,000.

 

∴By the formula, Present value, V = A × PVIFA _{(k, n)}

Where, A = Annuity i.e. amount of equal annual instalment 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,

 

10,000 = A × PVIFA _{(14, 4)}

⇒ 10,000 = A × 2.9137

⇒ A = 10,000 ÷ 2.9137

⇒ A = 3,432 (rounded off)

 

∴ Amount of each of 4 equal instalments will be Rs 3,432.