Transcript Chapter 6 Time Value of Money

CHAPTER 3
Time Value of Money
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Future value
Present value
Annuities
Amortization
6-1
Time Value of Money
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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
6-2
Solving for PV:
The arithmetic method
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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:
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PV = FVn / ( 1 + i )n
PV = FV3 / ( 1 + i )3
= Tk.100 / ( 1.10 )3
= Tk.75.13
6-3
Finding the interest rate
and time period
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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
6-4
Compounding more than
once in a year
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For round year case:
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Step 1: “i” (interest rate) should be
divided by “m” (how many times to be
compounded in a year)
Step 2: “n” (number of years) should
be multiplied by “m” (how many times
to be compounded in a year)
6-5
Compounding more than
once in a year (Contd.)
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For broken year case:
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Step 1: “i” (interest rate) should be divided
by “m” (how many times to be
compounded in a year)
Step 2: Look at the interest rate in step 1.
Now the power would be determined on
the basis of how many times such interest
rate gets compounded throughout the
whole life.
6-6
Compounding more than
once in year (Example)
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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
6-7
Classifications of interest rates
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EFF% for 10% semiannual investment
EFF% = ( 1 + iNOM / m )m - 1
= ( 1 + 0.10 / 2 )2 – 1 = 10.25%
An investor would be indifferent between
an investment offering a 10.25% annual
return and one offering a 10% annual
return, compounded semiannually.
6-8
Effective Annual Rate
EFF% = ( 1 + iNOM / m )m - 1
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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%
6-9
Why is it important to consider
effective rates of return?
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An investment with monthly payments is
different from one with quarterly payments.
Must put each return on an EFF% basis to
compare rates of return. Must use EFF% for
comparisons. See following values of EFF% rates
at various compounding levels.
EARANNUAL
EARQUARTERLY
EARMONTHLY
EARDAILY (365)
10.00%
10.38%
10.47%
10.52%
6-10
Annuity
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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.
6-11
Annuity
3 year $100 ordinary annuity.
0
1
2
3
100
100
100
i%
PV?
6-12
Present Value Annuity
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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. For example, at
10% interest rate, you can borrow Tk.173.55 in
a 2 year annuity of Tk.100 installment. The
amount of Tk.173.55 is composed of (the PV of
FV1 of Tk.100 or) Tk.90.91 and (FV2 of Tk.100)
or Tk.82.64.
6-13
Formulae for Present Value
Interest Factor of Annuity (PVIFA)
PVIFA=
1
1(1+i)n
i
6-14
Present Value Annuity
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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
6-15
Present Value Annuity
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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?
6-16
Solution of Problem 5
PVIFA={1-1/(1+.085/12)12*20]}/(.085/12)
=115.2308
PV Annuity= C*PVIFA
=1000*115.2308=1,15,230.80
6-17
Present Value Annuity
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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
6-18
Amortization Schedule
n
OPENG BALANCEINSTALLMENT INTEREST PAIDPRINCIPAL PAID
CLOSING BALANCE
1
2
3
4
5000
3768.6
2524.9
1268.7
1281.4
50 1231.4 3768.6
1281.4 37.686 1243.7 2524.9
1281.4 25.249 1256.2 1268.7
1281.4 12.687 1268.7
0.0
6-19
Present Value Annuity
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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?
6-20
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
6-21
Problem 9a
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In 1992, a 60 year old nurse bought a
$12 dollar lottery ticket and won the
biggest jackpot to that date of $9.3
million. Later it turned up that she
would be paid in 20 annual installments
of $465,000 each. If the interest rate
was 8%, then what was the amount
she was deprived of in present value?
6-22
Answer to problem 9a
PV = $465,000*PVIFA i=.08, n=20
= $ 465,000 * $ 9.818147
= $4,565,417
So, she was paid less than $9.3 million
by an amount of $4,734,583.
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4734583
6-23
Future Value Annuity
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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
6-24
Composition of
Future Value of Annuity
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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.
6-25
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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6-26
Future Value Annuity (Contd.)
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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=?
6-27
Solution: Problem 11
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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)
6-28
Ordinary Annuity versus
Annuity Due
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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).
6-29
What is the difference between an
ordinary annuity and an annuity due?
Ordinary Annuity
0
i%
1
2
3
PMT
PMT
PMT
1
2
3
PMT
PMT
Annuity Due
0
i%
PMT
6-30
Annuity Due
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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?
6-31
Solution: Problem 12(a)
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(a) FV Annuity=C*FVIFA
=10000*[(1+.15/12)24-1]/(.15/12)
=10000*27.78808=Tk.2,77,880.80
Since 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
6-32
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
6-33