CORPORATE FINANCE-I Lecture# 2 & 3: TIME VALUE OF MONEY Faculty: Prof. Kulbir Singh (Imt-nagpur)...

CORPORATE FINANCE-I

Lecture# 2 & 3:

TIME VALUE OF MONEY

Faculty: Prof. Kulbir Singh (Imt-nagpur)

12/28/2010

Faculty: Prof. Kulbir Singh (IMT-Nagpur) 2

Introduction Consider the following situations:

You expect a liability of Rs.100,000 in 5 years and you wish to know how much you should set aside today to meet the liability.

Your car dealer offers you two schemes – A & B. Under the first scheme, you can borrow the entire amount at an interest rate of 15%. Under the second scheme, the car dealer gives a discount of 10% on the sticker price but the interest charged on the amount borrowed would be 19%. You have to decide on the scheme.

You come across advertisements inserted by 2 banks in a daily. Bank A offers 13% interest per annum on deposits. Bank B offers 12% semiannual interest. You have to choose between these banks

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Basis for Time Value of Money• Inflation. • Preference for Current • Riskiness of the cash flow

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Future Value and Compound Interest

• Let us suppose you deposited Rs.1000 in Global Trust Bank for one year and the bank pays 13% interest.

Interest = 0.13 x 1000 = Rs.130

• Amount at the end of 1 year = Principal + Interest

= Rs.(1000 + 130) = Rs.1310.• In general,

Value of investment after 1 year = Initial investment x (1+r) • where ‘r’ is the interest rate in decimal form.

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• Now, if you wish to keep the balance in the bank, the principal for the second year would be Rs.1310 (and not 1000) and interest would be paid on this amount.

• Interest for the second year = 0.13 x 1310 = Rs.170.30• Amount at the end of second year = Rs.(1310 + 170.30)

= Rs.1480.30• An investment of Rs.1000 grew to Rs.1310 in one year and

Rs.1480.30 in two years. • The amount available at the end of the investment horizon is

called future value and the process of multiplying the investment is called compounding.

• Note: interest earned in the first year also earned interest in the second year.

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• In general,

• The term (1 + r)n is called Future Value Interest Factor or FVIF. It is the value of Re.1 interest at r % after n years.

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Future Value of an amount, FV = Amount (1 + r)n

Time 0 n 1 2 3 Amount A A (1+r) A(1+r)(1+r) A(1 + r)2 (1 + r) = A(1 + r)2 = A(1 + r)3 A(1 + r)n


Future Value Interest Factor (FVIF)• The Value of Re.1 for different interest rates and time periods is

readily available in the form of a table – Future Value Interest Factor Table (See Table A-1 at the end of the book).

• A section of the table is reproduced in Exhibit 2-1.

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Exhibit 2-1: Future Value Interest Factor ------------------------------------------------------------------------------------------------- Interest rate r Year (n) 8% 9% 10% 11% 12% ------------------------------------------------------------------------------------------------- 1 1.080 1.090 1.100 1.110 1.120 2 1.166 1.188 1.210 1.232 1.254 3 1.260 1.295 1.331 1.368 1.405 4 1.360 1.412 1.464 1.518 1.574 5 1.469 1.539 1.611 1.685 1.762 -------------------------------------------------------------------------------------------------


FVIF• To find the value of Re.1 at 10 % for 5 years. • Similarly try 12 % and 4 years. • Now find the future value of Rs.282 at 10% for 3 years.

• Future Value = A(1 + r)n = 282 * FVIF 10,3 = 282 x 1.331 = Rs.375.50

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Future Value of a series• You invest Rs.500 today, Rs.1000 at the end of 1st year and Rs.1500

at the end of 2nd year. • You are required to find the future value at the end of 2nd year. • Assume 8% interest. • The future value of this series is nothing but the sum of future values

of each cash flow as shown below:

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Time 0 1 2 Cash flow 500 1000 1500


Future Value of a series• Future Value = FV of Rs.500 + FV of Rs.1000 + 1500 • Note that 1500 itself is the future value of the Cash flow occurring at

the end of 2nd year.

• FV = 500 (1.08) 2 + 1000(1.08) + 1500• = 500 x FVIF8,2 + 1000 x FVIV8,1 + 1500

• = Rs.3163• What would happen to the future value if the cash flow were to

occur at the beginning of the year? • Beginning of year 1 is same as year 0 (that is today) and beginning of

year 2 is same as end of year 1. The cash flow would be:

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Future Value of a series

• FV = (500 + 1000)(1.08) 2 + (1500 x1.08) = Rs.3369

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Time 0 1 2 Cash flow 500 + 1000 1500


Types of Cash Flows• Cash flows can exhibit different patterns. • Different periodic amounts & the series had a finite life. • A series of periodic flows of equal amounts is called an Annuity and a series of periodic flows of equal amounts but with an infinite life is called a perpetuity.

• Thus investing or receiving Rs.500 as dividends forever is a perpetuity.

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Future Value of an Annuity• You require Rs.250,000 at the end of 5 years to buy a Car. Assume

that this is the price that would prevail 5 years from now for the model you have in mind. You wish to invest a fixed (constant) sum every year to accumulate Rs.250,000 at the end of 5 years. The series of cash flows is shown below:

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Time 0 1 2 3 4 5 Cash flow A A A A A


Future Value of an Annuity• The future value of an annuity for a period of n years at a rate of of

interest ‘r’ is given by

Where• A - amount invested at the end of every year for n years.• r - rate of interest in decimal form• n - time period in years.

• The term is called Future Value Interest Factor of an Annuity or FVIFA. . It is the future value of an annuity of Re.1 for various values of r and n.

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Future Value of an Annuity• To make our life simple, the value of FVIFA for various r & n are presented

in the form of a Table (See Table A-2 at the end). A section of the Table is shown in Exhibit 2-3.

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Exhibit 2-3: Future Value Interest Factor of an Annuity ------------------------------------------------------------------------------------------------- Interest rate r % Year (n) 5 6 7 8 9 ------------------------------------------------------------------------------------------------- 1 1.000 1.000 1.000 1.000 1.000 2 2.050 2.060 2.070 2.080 2.090 3 3.152 3.184 3.215 3.246 3.278 4 4.310 4.375 4.440 4.506 4.573 5 5.526 5.637 5.751 5.867 5.985 -------------------------------------------------------------------------------------------------


Present Values• We saw that Rs.100 grows to Rs.108 at an interest rate of

8% in one year. Let’s do the reverse now. How much do we need to set aside today in order to produce Rs.108 a year from now?

• The amount is referred to as present value of Rs.108.• Future Value FV = A(1 + r)n

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Present Values• The process of bringing a cash flow to the present is called

discounting and the rate of interest is called discount rate.• The discount rate is a rate at which present and future values are

traded off. • The term [1/1 + r) t ] is called Present Value Interest Factor or PVIF.

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Present Value

Amount Rs X


Present Values• The value of [1/1 + r)n ] for various combinations of r and n is given in

the present value • interest factor table (see Table A-3) at the end of the book. A section

of the Table is reproduced in Exhibit 2-2.

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Exhibit 2-2: Present Value Interest Factor PV of Re.1 to be received after n years = [ 1/1 + r)n ] ------------------------------------------------------------------------------------------------- Interest rate % Year (n) 8 9 10 11 12 ------------------------------------------------------------------------------------------------- 1 0.926 0.917 0.909 0.901 0.893 2 0.857 0.842 0.826 0.812 0.797 3 0.794 0.772 0.751 0.731 0.712 4 0.735 0.708 0.683 0.659 0.636 5 0.681 0.650 0.621 0.593 0.567


Present Value of a Series• Consider the same cash flows again. The present value of this series

is the sum of present values of each of these cash flows.

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Time 0 1 2 Cash flow 500 1000 1500

•


Present Value of an Annuity • The present value of an annuity ‘A’ receivable at the end of each year

for a period of n years at an interest rate of r is given by:

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• The term in the bracket is called Present Value Interest Factor of an Annuity or PVIFA.

• It is the present value of an annuity of Re.1 for the given values of r and n.

• The values of PVIFA (K, n) for various combination of r and n are given in Table A-4 at the end of the book. A section is shown in Exhibit 2-4.

𝑭𝑽𝑨𝒏=𝑨 [ (𝟏+𝒓 )𝒏−𝟏𝒓 (𝟏+𝒓 )𝒏 ]


Present Value of an Annuity

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• The term in the bracket is called Present Value Interest Factor of an Annuity or PVIFA.

• It is the present value of an annuity of Re.1 for the given values of r and n. • The values of PVIFA (K, n) for various combination of r and n are given in

Table A-4 at the end of the book. A section is shown in Exhibit 2-4.

Exhibit 2- 4: Present Value Interest Factor of an Annuity ------------------------------------------------------------------------------------------------- Interest rate r % Year (n) 6 7 8 9 ------------------------------------------------------------------------------------------------- 1 0.943 0.935 0.926 0.909 2 1.833 0.873 0.857 0.826 3 2.673 0.816 0.794 0.751 4 3.465 0.763 0.735 0.688 -------------------------------------------------------------------------------------------------



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• On December 31, you buy a car for Rs.200,000 paying 20% up front and agreeing to pay the balance in 5 equal annual installments. The rate of interest is 15%.

• Amount payable = 0.8 x 200,000 = Rs.160,000

• PVAn = A [ PVIFAr,n ]

• A = [ 160,000 / 3,352 ] = Rs.47,732.70



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• Sometimes annuities grow at a constant rate. Retirement and pension benefits, for example, increase every year with a cost of living adjustment.

• The present value of a series of growing annuity can be estimated using the following equation:

• For eg., if A = Rs 6600 , r=5% , n=30 periods , g=3.5% , • PV = Rs 154,251

𝑷𝑽= 𝑨𝒓 −𝒈 [− {𝟏+𝒈

𝟏+𝒓 }𝒏 ]


Loan Amortization

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• Assume that you have borrowed Rs.10 lakh from a financial institution at an interest rate of 14% per annum. The loan is to be cleared in equal annual installments.

• Annual installment = Rs 10,00,000 / PVIFA(14,5)• = Rs.10,00,000 / 3,433 = Rs.291,290

• This installment contains interest and principal components. • The principal is retired partially every year and the interest is paid on

the outstanding balance. • The loan amortization schedule is presented in Exhibit 2-5.


Loan Amortization

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Exhibit 2-5: Loan Amortization schedule Annual Interest Principal Loan Year Installment(Rs.) Component(Rs.) Component(Rs.) Outstanding(Rs.) (1) (2) (3) (4) = (2) – (3) (5) 1 291,290 140,000 151,290 1,000,000 2 291,290 118,819 172,470 848,710 3 291,290 94,673 196,616 676,239 4 291,290 67,147 224,142 479,622 5 291,290 35,767 255,500* 255,480* * These two figures do not coincide due to rounding off errors.


Effective and Nominal Rates

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• How should we compare interest rates that are quoted for different periods?

• A car dealer offers 2 schemes – A & B. Under the first scheme you have to pay an interest of 12% compounded monthly and under the second scheme you have to pay an interest of 12% compounded annually. You are required to choose between the two.

• The trick is to convert both of them into same basis. There is a simple formula to do this :(1 + annual rate) = (1 + monthly rate) 12

• For the first scheme, the annual rate is 12%. So the rate per month is 12/12 = 1%.

• Substitute in the above equation • (1 + annual rate) = (1 + 0.01) 12 • Annual rate = 12.68%


Effective and Nominal Rates

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• From the numerical it must be clear that the shorter the compounding period, the higher will be the interest rate.

• The rate quoted on an annual basis (12%) is the simple annual rate and the rate that considers more frequent compounding is called effective rate (12.68% in the above example).

In general, Effective rate = [1 + (r/m) ]m – 1 Where r – nominal annual rate m – number of times compounding is done in a year


Perpetuity

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• Some investments make a regular income forever. • Let’s suppose the dividend per share of Company X is Rs.2. • It is expected to remain at this level forever (the firm is a going

concern). • If the appropriate discount rate is 13%, ……..the present value of this

perpetual stream = (Rs.2/0.13) =Rs.15.40.

• In general, • PV of a Perpetuity = Per Period Amount/Discount Rate

𝑷𝑽=𝑨𝒓


Perpetuity

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• If an amount to grow at a constant rate forever it is called a growing perpetuity .

• The present value of a growing perpetuity can be estimated using the following formula.

• If an amount to grow at a constant rate forever it is called a growing perpetuity . The present value of a growing perpetuity can be estimated using the following formula.

𝑷𝑽=𝑨

𝒌−𝒈

𝑷𝑽=𝑨𝒌

CORPORATE FINANCE-I Lecture# 2 & 3: TIME VALUE OF MONEY Faculty: Prof. Kulbir Singh (Imt-nagpur)...

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