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CHAPTER 4 and 5The Time Value of Money and Discounted Cash Flow Valuation

Future value Present value Annuities Rates of return Amortization

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Time Value of Money Definition: Value of money changes as time

changes. This is because of the positive rate of interest in all the markets. If the market interest rate is 10%, then Tk.100 today has the same value as Tk.110 after 1 year from now and Tk.121 after 2 years from now. So the present value of Tk.110 of the next year is Tk.100, or the future value of Tk.100 now is Tk.110 in the next year. FVn=PV(1+i)n

PV=FVn/(1+i)n

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Solving for PV:The arithmetic method Problem 1: How much should you set

aside now to get Tk.100 after 3 years from now?Solve the general FV equation for PV: PV = FVn / ( 1 + i )n

PV = FV3 / ( 1 + i )3

= Tk.100 / ( 1.10 )3

= Tk.75.13

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Finding the interest rate and time period Problem 2. What is the rate of interest by

what Tk.100 becomes Tk.200 in 4 years? 200=100(1+i)4

(1+i)4=2, 1+i=2 1/4=2.25 =1.1892, i=18.92% Problem 3. How long time it takes to double

an amount if the interest rate is 15% per annum?

200=100(1+.15)n

(1.15)n=2, n log(1.15)=log(2)n=log(2)/log(1.15)=4.96 years

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Compounding more than once in year

Problem 4: You like to set aside an amount of money so that you get Tk.50,000 after 5 years from now. Bank One offers you 10% annual interest rate and Bank Two offers you 9.5% interest rate compounded monthly. Where should you put the money?

Bank One: PV=50,000/(1.1)5=Tk.31046.07 Bank Two:

PV=50,000/(1+.095/12))60=Tk.31152.46 Bank One is a better choice

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Effective Annual RateEAR% = ( 1 + iNOM / m )m - 1

Problem 5: A Credit card charges 2% interest rate per month. What is the effective interest rate?

EAR=(1+.24/12)12-1=(1.02)12-1=26.82%

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Annuity Definition: A series of equal payments

is made against what an accumulated sum can be received either at the beginning or at the end of the period of annuity. If the accumulated sum takes place at the beginning then it is a Present Value Annuity, and if the accumulated sum takes place at the end then it is a Future Value Annuity.

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Annuity

100 100100

0 1 2 3i%

3 year $100 ordinary annuity.

PV?

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Present Value Annuity All kinds of consumers’ credit schemes

follow present value annuity. A lump sum amount is borrowed now against what payments would be made in equal installments at a regular interval for a definite period of time.

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PVIFA=1-

1  (1+i)n  

i

Formulae for Present Value Interest Factor of Annuity (PVIFA)

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Present Value Annuity Problem 6: At 10% interest rate, How

much can you borrow now against the repayment 3 equal annual installments of Tk.1000?

PV Annuity=C*(PVIFA)=C{[1-(1/(1+i)n)]/i}=1000{[1-(1/(1.1)3]/.1}=1000*2.4869=2486.90

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Present Value Annuity Problem 7: You have a plan to deposit

Tk.1,000 per month in a bank for next 20 years. If the interest rate is 8.5% per annum then how much can you borrow from the bank against that?

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Solution of Problem 7PVIFA={1-1/(1+.085/12)12*20]}/

(.085/12)=115.2308

PV Annuity= C*PVIFA

=1000*115.2308=1,15,230.80

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Present Value Annuity Problem 8: Find the amount of

installment of a loan of Tk.5,000 to be repaid in 4 equal monthly installment at 12% interest. Make an amortization schedule.

5000=C(PVIFA, i=.12, m=12, n=4)=C(3.901966)

C=5000/3.901966=1281.405

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Amortization Schedule

n OPENG BALANCEINSTALLMENT INTEREST PAIDPRINCIPAL PAIDCLOSING BALANCE

1 5000 1281.4 50 1231.4 3768.62 3768.6 1281.4 37.686 1243.7 2524.93 2524.9 1281.4 25.249 1256.2 1268.74 1268.7 1281.4 12.687 1268.7 0.0

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Present Value Annuity Problem 9: You need Tk.12 lakh now

to buy a car, under the terms and condition of monthly installments for 10 year. Interest rate is 15% per annum. (a) What would be the amount of installments? (b) How much would be the accumulated liability of interest?

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Solution: Problem 9(a) Installment =PV Annuity/PVIFA

=12,00,000/61.98285=Tk.19,360.19(b) Accumulated Interest=Total payments

– Present value of annuity=(19,360.19*120)-12,00,000=23,23,223-12,00,000=11,23,223

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Future Value Annuity Definition: FV Annuity is different from

PV Annuity in that the accumulated sum takes place at the end of the period of the annuity. In a savings scheme if you deposit equal installment regularly and at the maturity of the annuity receive the accumulated sum then it is an example of future value annuity. It is composed of the principal amounts and the interest thereof.

FVIFA=[(1+i)n-1]/i FV of Annuity=C*FVIFA

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Composition of Future Value of Annuity Suppose, there is a 2 year annuity

of $100 installments at 10% interest. The future value is

FV Annuity= C*FVIFA==100*[(1.1)2-1]/0.1=$210

This is composed of $110 and $100.

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Future Value Annuity (Contd.) Problem 10: You like to deposit

Tk.1000 per month for a period of 15 years. Assuming an interest of 10% how much would you get at the end?

FV Annuity=C*(FVIFA)=1000*{[(1+.1/12)15*12]-1}/(.1/12)

=1000*414.4703=Tk.4,14,470.30

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Future Value Annuity (Contd.)

Problem 11: You need to have Tk.1 million after 20 years from now. Assuming the market interest rate of 13% per annum if you like to deposit equal quarterly installments during the period in a bank then how much would be the amount of each installment? What is the interest accumulation in the annuity?

Given, FV=Tk.1,000,000, i=.13/4, n=20*4, C=?

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Solution: Problem 11 C=FV/FVIFA.

C=1,000,000/366.7164=Tk.2,726.90 Interest accumulation=FV Annuity-

Total payments=1,000,000-(C*n)=1,000,000-(2726.90*80)=Tk.781,847.80 (This is 78.18% of face value)

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Ordinary Annuity and Annuity Due The installments of an annuity can be

paid either at the beginning or at the end of the period. If it is paid at the end of the period then it is called ordinary annuity. If it is paid at the beginning of the period then it is called annuity due. Both present value annuity and future value annuity can be an ordinary annuity or annuity due. To convert ordinary annuity into annuity due multiply the value by (1+i).

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What is the difference between an ordinary annuity and an annuity due?

Ordinary Annuity

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3i%

PMT

Annuity Due

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Annuity Due Problem 12: You need to receive Tk.10,000

monthly for a period of 2 years to pursue your MBA program. You make an arrangement with a Bank that says the interest rate is 15%.

(a) How much will you have to return back to the bank at the end?

(b) How much should you deposit to the bank now to get the same monthly installments throughout the MBA program?

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Solution: Problem 12(a) (a) FV Annuity=C*FVIFA

=10000*[(1+.15/12)24-1]/(.15/12) =10000*27.78808=Tk.2,77,880.80Since you need the money at the beginning of the month so it is an annuity due. In that case, FV Annuity Due=2,77,880.80*(1+.15/12)=Tk.2,81,354.40

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Solution: Problem 12(b)(b) This is the present value annuity due. PV Annuity due=C*PVIFA*(1+i)

=10,000*20.62423*(1+.15/12)=2,08,820.4

Also notice: you can get answer to (b) by dividing answer to (a) by (1+i)n or [(1+.15/12)2*12]

Or, you can get (a) through multiplying (b) by (1+i)n factor

For example, 208820.4[(1+.15/12)2*12]=208820.4*[(1.0125)24]=281354.40