Chapter 6

THE TIME VALUE OF MONEY

OUTLINE

• Why Time Value

• Future Value of a Single Amount

• Future Value of an Annuity

• Present Value of a Single Amount

• Present Value of an Annuity

• Intra-year Compounding and Discounting

WHY TIME VALUE

A rupee today is more valuable than a rupee a year hence.

Why ?

• Productivity of capital

• Inflation

Many financial problems involve cash flows occurring at different points of time. For evaluating such cash flows, an explicit consideration of time value of money is required

TIME LINE

Part A

0 1 2 3 4 5 12% 12% 12% 12% 12%

10,000 10,000 10,000 10,000 10,000

 

Part B

  0 1 2 3 4 5 12% 12% 12% 12% 12%

10,000 10,000 10,000 10,000 10,000

NOTATION

PV : Present value

FVn : Future value n years hence

Ct : Cash flow occurring at the end of year t

A : A stream of constant periodic cash flow over a given time

r : Interest rate or discount rate

g : Expected growth rate in cash flows

n : Number of periods over which the cash flows occur.

FUTURE VALUE OF A SINGLE AMOUNT

Rs

First year: Principal at the beginning 1,000Interest for the year (Rs.1,000 x 0.10) 100Principal at the end 1,100

 

Second year: Principal at the beginning 1,100Interest for the year (Rs.1,100 x 0.10) 110Principal at the end 1,210

Third year: Principal at the beginning 1,210Interest for the year (Rs.1,210 x 0.10) 121Principal at the end 1,331

FORMULA

FUTURE VALUE = PRESENT VALUE (1+r)n

Value five years hence of a deposit of Rs.1,000 at various interest rates is as follows:

8% FV5 = 1000 x FVIF (8%, 5 years)= 1000 x 1.469 = Rs.1469

10% FV5 = 1000 x FVIF (10%, 5 years)= 1000 x 1.611 = Rs.1611

12% FV5 = 1000 x FVIF (12%, 5 years)= 1000 x 1.762 = Rs.1762

15% FV5 = 1000 x FVIF (15%, 5 years)= 1000 x 2.011 = Rs.2011

PRESENT VALUE OF A SINGLE AMOUNT

PV = FVn [1/ (1 + r) n]

Suppose an investor wants to find out present value of Rs. 50,000 to be received after 15 years. If interest rate is 9%.

PV = FVn [1/ (1 + r)n]

PV= 50,000 * PVF (15,9) = 50,000 * 0.275

= Rs. 13,750

Problem:6.8

• 10% PV= 10,000 * PVF (10,8) = 10,000 * 0.467 = 4670

• 12% PV= 10,000 * PVF (12,8) = 10,000 * 0.404 = 4040

• 15% PV= 10,000 * PVF (15,8) = 10,000 * 0.327 = 3270

FUTURE VALUE OF AN ANNUITY

An annuity is a series of periodic cash flows (payments and receipts ) of equal amounts 

1 2 3 4 5

1,000 1,000 1,000 1,000 1,000

+

1,100

+

1,210

+

1,331

+

1,464

Rs.6,105

Future value of an annuity = A [(1+r)n-1] r

WHAT LIES IN STORE FOR YOU

Suppose you have decided to deposit Rs.30,000 per year in your Public Provident Fund Account for 30 years. What will be the accumulated amount in your Public Provident Fund Account at the end of 30 years if the interest rate is 11 percent ?

The accumulated sum will be :Future value of an annuity = A [(1+r)n-1]

r = Rs.30,000 (FVIFA11%,30yrs)= Rs.30,000 (1.11)30 - 1

.11= Rs.30,000 [ 199.02]= Rs.5,970,600

Problem

Suppose a firm deposits Rs. 5000 at the end of each year for four years at 6 percent rate of interest. How much this annuity accumulate at the end of the fourth year?

Future value of an annuity = A [(1+r)n-1]

r= Rs.5000 (FVIFA 6 %,4 yrs)

  = Rs.5000 (1.06)4 - 1 .06

= Rs.5000 [ 4.375]= Rs.21,875

HOW MUCH SHOULD YOU SAVE ANNUALLY

You want to buy a house after 5 years when it is expected to cost Rs.2 million. How much should you save annually if your savings earn a compound return of 12 percent ?

The future value interest factor for a 5 year annuity, given an interest rate of 12 percent, is :

(1+0.12)5 - 1

FVIFA n=5, r =12% = = 6.353

0.12

  The annual savings should be :

Rs.2000,000 = Rs.314,812

6.353

ANNUAL DEPOSIT IN A SINKING FUND

Futura Limited has an obligation to redeem Rs.500 million bonds 6 years hence. How much should the company deposit annually in a sinking fund account wherein it earns 14 percent interest to cumulate Rs.500 million in 6 years time ?

The future value interest factor for a 5 year annuity, given an interest rate of 14 percent is :

FVIFAn=6, r=14% = (1+0.14)6 – 1 = 8.536

0.14  

The annual sinking fund deposit should be :

Rs.500 million = Rs.58.575 million

8.536

FINDING THE INTEREST RATE

A finance company advertises that it will pay a lump sum of Rs.8,000 at the end of 6 years to investors who deposit annually Rs.1,000 for 6 years. What interest rate is implicit in this offer?

The interest rate may be calculated in two steps :

1.  Find the FVIFAr,6 for this contract as follows :

Rs.8,000 = Rs.1,000 x FVIFAr,6

FVIFAr,6 = Rs.8,000 = 8.000

Rs.1,000

2.  Look at the FVIFAr,n table and read the row corresponding to 6 years

until you find a value close to 8.000. Doing so, we find that

FVIFA12%,6 is 8.115 . So, we conclude that the interest rate is slightly below

12 percent.

HOW LONG SHOULD YOU WAIT

You want to take up a trip to the moon which costs

Rs.1,000,000 the cost is expected to remain unchanged

in nominal terms. You can save annually Rs.50,000 to fulfill

your desire. How long will you have to wait if your savings

earn an interest of 12 percent ? The future value of an

annuity of Rs.50,000 that earns 12 percent is equated to

Rs.1,000,000.

50,000 x FVIFAn=?,12% = 1,000,00050,000 x 1.12n – 1 = 1,000,000

0.12  1.12n - 1 = 1,000,000 x 0.12 = 2.4 50,000  1.12n = 2.4 + 1 = 3.4

n log 1.12 = log 3.4  n x 0.0492 = 0.5315  n = 0.5315 = 10.8 years

0.0492You will have to wait for about 11 years.

PRESENT VALUE OF AN ANNUITY

1 (1+r)n

r

1 -Present value of an annuity = A

LOAN AMORTISATION SCHEDULE

Loan : 1,000,000 r = 15%, n = 5 years

1,000,000 = A x PVAn =5, r =15%

= A x 3.3522

A = 298,312

LOAN AMORTISATION SCHEDULE

Year Beginning Annual Interest Principal Remaining

Amount Instalment Repayment Balance

(1) (2) (3) (2)-(3) = (4) (1)-(4) = (5)

1 1,000,000 298,312 150,000 148,312 851,688

2 851,688 298,312 127,753 170,559 681,129

3 681,129 298,312 102,169 196,143 484,986

4 484,986 298,312 727,482 225,564 259,422

5 259,422 298,312 38,913 259,399 23*

  a     Interest is calculated by multiplying the beginning loan balance by the interest rate.

b.   Principal repayment is equal to annual instalment minus interest.

* Due to rounding off error a small balance is shown

PRESENT VALUE OF AN UNEVEN SERIES

A1 A2 An

PVn = + + …… + (1 + r) (1 + r)2 (1 + r)n

n At

= t =1 (1 + r)t

Year Cash Flow PVIF12%,n Present Value of Rs. Individual Cash Flow

1 1,000 0.893 893 2 2,000 0.797 1,594 3 2,000 0.712 1,424 4 3,000 0.636 1,908 5 3,000 0.567 1,701 6 4,000 0.507 2,028 7 4,000 0.452 1,808 8 5,000 0.404 2,020

Present Value of the Cash Flow Stream 13,376

DOUBLING PERIOD

Thumb Rule : Rule of 72

72

Interest rate

Interest rate : 15 percent

72

15

A more accurate thumb rule : Rule of 69

69 Interest rate

Interest rate : 15 percent

69

15

Doubling period =

= 4.8 yearsDoubling period =

Doubling period = 0.35 +

Doubling period = 0.35 + = 4.95 years

• If you deposit Rs.2,000 today at 6 percent rate of interest in how many years (roughly) will this amount grow to Rs.32,000 ? Work this problem using the rule of 72–do not use tables.

Rs.32,000 / Rs. 2,000 = 16= 24

According to the Rule of 72 at 6 percent interest rate doubling takes place approximately in 72 / 6 = 12 years

So Rs.2,000 will grow to Rs.32,000 in approximately 4 x 12 years = 48 years

• A finance company offers to give Rs.20,000 after 14 years in return for Rs.5,000 deposited today. Using the rule of 69, figure out the approximate interest rate offered.

In 14 years Rs.5,000 grows to Rs.20,000 or 4 times. This is 22 times the initial deposit. Hence doubling takes place in 14 / 2 = 7 years.

According to the Rule of 69, the doubling period is 0.35 + 69 / Interest rate

We therefore have 0.35 + 69 / Interest rate = 7Interest rate = 69/(7-0.35) = 10.38 %

Growth Rate

Phoenix Ltd. had revenues of 100 million

in 2000 which increased to Rs. 1000

million in 2010. What was the compound

growth rate in revenues?

Growth rate in revenues = 100 (1+g)10 = 1000(1+g)10 = 1000 =10

100(1+g) = 10 1/10 -1 = 1.26 -1 = 0.26 or 26%

PRESENT VALUE OF A GROWING ANNUITYThe present value of a growing annuity can be determined using the following formula :

(1 + g)n

1- (1 + r)n

PV of a Growing Annuity=A(1+g)

r – g

The above formula can be used when the growth rate is less than the discount rate (g < r) as well as when the growth rate is more than the discount rate (g > r). However, it does not work when the growth rate is equal to the discount rate (g = r) – in this case, the present value is simply equal to n A.

1 –

PRESENT VALUE OF A GROWING ANNUITY

For example, suppose you have the right to harvest a teak plantation

for the next 20 years over which you expect to get 100,000 cubic feet

of teak per year. The current price per cubic foot of teak is Rs 500,

but it is expected to increase at a rate of 8 percent per year. The

discount rate is 15 percent. The present value of the teak that you

can harvest from the teak forest can be determined as follows:

PV of teak = Rs 500 x 100,000 (1.08)

= Rs.551,736,683

1- (1.08)20

(1.15)20

0.15 – 0.08

ANNUITY DUE

A A … A A

0 1 2 n – 1 n A A A … A

0 1 2 n – 1 n

Thus,

Annuity due value = Ordinary annuity value (1 + r) This applies to both present and future values

Ordinary annuity

Annuitydue

PRESENT VALUE OF PERPETUITY

A Present value of perpetuity =

r

GROWING PERPETUITY

PV = A

r - g

SHORTER COMPOUNDING PERIOD

Future value = Present value 1+ r mxn

m

Where r = nominal annual interest rate

m = number of times compounding is done in a

year

n = number of years over which compounding is

done

Example : Rs.5000, 12 percent, 4 times a year, 6 years

5000(1+ 0.12/4)4x6 = 5000 (1.03)24

= Rs.10,164

EFFECTIVE VERSUS NOMINAL RATE

r = (1+k/m)m –1

r = effective rate of interest

k = nominal rate of interest

m = frequency of compounding per year

Example : k = 8 percent, m=4

r = (1+.08/4)4 – 1 = 0.0824

= 8.24 percent Nominal and Effective Rates of InterestNominal and Effective Rates of Interest

Effective Rate %

  Nominal Annual Semi-annual Quarterly Monthly

Rate % Compounding Compounding Compounding Compounding

8 8.00 8.16 8.24 8.30

12 12.00 12.36 12.55 12.68

Problem:

• You can save Rs.5,000 a year for 3 years, and Rs.7,000 a year for 7 years thereafter. What will these savings cumulate to at the end of 10 years, if the rate of interest is 8 percent?

Solution:

Saving Rs.5000 a year for 3 years and Rs.6000 a year for 7 years thereafter is equivalent to saving Rs.5000 a year for 10 years and Rs.2000 a year for the years 4 through 10.

Hence the savings will cumulate to:5000 x FVIFA (8%, 10 years) + 2000 x FVIFA (8%, 7 years)

= 5000 x 14.487 + 2000 x 8.923= Rs.90281

Problem:

• At the time of his retirement, Rahul is given a choice between two alternatives: (a) an annual pension of Rs120,000 as long as he lives, and (b) a lump sum amount of Rs.1,000,000. If Rahul expects to live for 20 years and the interest rate is expected to be 10 percent throughout , which option appears more attractive

Solution:

The present value of an annual pension of Rs.120,000 for 20 years when r = 10% is:120,000 x PVIFA (10%, 20 years)= 120,000 x 8.514 = Rs.1,021,680

The alternative is to receive a lump sum of Rs 1,000,000

Rahul will be better off with the annual pension

amount of Rs.120,000.

Problem:

• What is the present value of an income stream which provides Rs.30,000 at the end of year one, Rs.50,000 at the end of year three , and Rs.100,000 during each of the years 4 through 10, if the discount rate is 9 percent ?

Solution:

The present value of the income stream is:

30,000 x PVIF (9%, 1 year) + 50,000 x PVIF (9%, 3 years) + 100,000 x PVIFA (9 %, 7 years) x PVIF(9%, 3 years)

= 30,000 x 0.917 + 50,000 x 0.772 + 100,000 x 5.033 x 0.0.772 = Rs.454,658.

Problem:

• What is the present value of an income stream which provides Rs.1,000 a year for the first three years and Rs.5,000 a year forever thereafter, if the discount rate is 12 percent?

Solution:

The present value of the income stream is:

1,000 x PVIFA (12%, 3 years) + (5,000/ 0.12) x PVIF (12%, 3 years)

= 1,000 x 2.402 + (5000/0.12) x 0.712= Rs.32,069

Problem:

• Mr. Ganapathi will retire from service in five years .How much should he deposit now to earn an annual income of Rs.240,000 forever beginning from the end of 6 years from now ? The deposit earns 12 percent per year.

Solution:

To earn an annual income of Rs.240,000 forever , beginning from the end of 6 years from now, if the deposit earns 12% per year a sum of

Rs.240,000 / 0.12 = Rs.2,000,000 is required at the end of 5 years.

The amount that must be deposited to get this sum is:Rs.2,000,000 PVIF (12%, 5 years) = Rs.2,000,000 x 0.567 = Rs. 1,134,000

Problem:

• Ravikiran deposits Rs.500,000 in a bank now. The interest rate is 9 percent and compounding is done quarterly. What will the deposit grow to after 5 years? If the inflation rate is 3 percent per year, what will be the value of the deposit after 5 years in terms of the current rupee?

Solution:

FV5 = Rs.500,000 [1 + (0.09 / 4)]5x4

= Rs.500,000 (1.0225)20= Rs.500,000 x 2.653= Rs.780,255

If the inflation rate is 3 % per year, the value of Rs.780,255 5 years from now, in terms of the current rupees is:Rs.780,255 x PVIF (3%, 5 years)= Rs.780,255 x 0. 863 = Rs.673,360

Problem:

• A person requires Rs.100,000 at the beginning of each year from 2015 to 2019. Towards this, how much should he deposit ( in equal amounts) at the end of each year from 2007 to 2011, if the interest rate is 10 percent.

Solution:

The discounted value of Rs.100,000 receivable at the beginning of each year from 2015 to 2019, evaluated as at the beginning of 2014 (or end of 2013) is:

Rs.100,000 x PVIFA (10%, 5 years)= Rs.100,000 x 3.791= Rs.379,100

The discounted value of Rs.379,100 evaluated at the end of 2011 is

Rs.379,100 x PVIF (10 %, 2 years)= Rs.379,100 x 0.826= Rs.313,137If A is the amount deposited at the end of each year from 2007 to 2011 thenA x FVIFA (10%, 5 years) = Rs.313,137A x 6.105 = Rs.313,137A = Rs.313,137/ 6.105 = Rs.51,292

Problem:

• After eight years Mr.Tiwari will receive a pension of Rs.10,000 per month for 20 years. How much can Mr. Tiwari borrow now at 12 percent interest so that the borrowed amount can be paid with 40 percent of the pension amount? The interest will be accumulated till the first pension amount becomes receivable.

Solution:

40 per cent of the pension amount is 0.40 x Rs.10,000 = Rs.4,000

Assuming that the monthly interest rate corresponding to an annual interest rate of 12% is 1%, the discounted value of an annuity of Rs.4,000 receivable at the end of each month for 240 months (20 years) is:

Rs.4,000 x PVIFA (1%, 240) 1- 1

(1.01)240Rs.4,000 x ---------------- = Rs.363,278

.01 If Mr. Tiwari borrows Rs.P today on which the monthly interest rate is 1%

P x (1.01)96 = Rs. 363,278P x 2.60 = Rs. 363,278

Rs. 363,278P = ------------ = Rs.139,722

2.60

Problem:

• Metro Corporation has to retire Rs.20 million of debentures each at the end of 6, 7, and 8 years from now. How much should the firm deposit in a sinking fund account annually for 5 years, in order to meet the debenture retirement need? The net interest rate earned is 10 percent.

Solution:

The discounted value of the debentures to be redeemed between 6 to 8 years evaluated at the end of the 5th year is:Rs.20 million x PVIFA (10%, 3 years) = Rs.20 million x 2.487= Rs.49.74millionIf A is the annual deposit to be made in the sinking fund for the years 1 to 5, thenA x FVIFA (10%, 5 years) = Rs.49.74 millionA x 6.105 = Rs.49.74 millionA = Rs.8,147,420

End