Time-Value-of-Money chapter 3 finance

finance

1. 6-1
CHAPTER 3
Time Value of Money
Future value
Present value
Annuities
Rates of return
Amortization

2. 6-2
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. FV
n=PV(1+i)
n
PV=FV
n/(1+i)
n

3. 6-3
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 = FV
n
/ ( 1 + i )
n

PV = FV
3 / ( 1 + i )
3
= Tk.100 / ( 1.10 )
3
= Tk.75.13

4. 6-4
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

5. 6-5
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

6. 6-6
Classifications of interest
rates

EFF% for 10% semiannual investment
EFF%= ( 1 + i
NOM
/ 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.

7. 6-7
Effective Annual Rate
EFF% = ( 1 + i
NOM / 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%

8. 6-8
Why is it important to consider
effective rates of return?
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.
EAR
ANNUAL
10.00%
EAR
QUARTERLY
10.38%
EAR
MONTHLY
10.47%
EAR
DAILY (365)10.52%

9. 6-9
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.

10. 6-10
Annuity
100 100100
0 1 2 3
i%
3 year $100 ordinary
annuity.
PV?

11. 6-11
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. 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 FV
1 of Tk.100
or) Tk.90.91 and (FV
2 of Tk.100) or Tk.82.64.

12. 6-12
PVIFA=
1-
1  
(1+i)
n
 
i
Formulae for Present Value
Interest Factor of Annuity
(PVIFA)

13. 6-13
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

14. 6-14
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?

15. 6-15
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

16. 6-16
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

17. 6-17
Amortization Schedule
n OPENG BALANCEINSTALLMENTINTEREST PAIDPRINCIPAL PAIDCLOSING BALANCE
150001281.4501231.43768.6
23768.61281.437.6861243.72524.9
32524.91281.425.2491256.21268.7
41268.71281.412.6871268.70.0

18. 6-18
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?

19. 6-19
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

20. 6-20
Problem 9a

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?

21. 6-21
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.
4734583

22. 6-22
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

23. 6-23
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.

24. 6-24
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

25. 6-25
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=?

26. 6-26
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)

27. 6-27
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).

28. 6-28
What is the difference between an
ordinary annuity and an annuity
due?
Ordinary Annuity
PMT PMTPMT
0 1 2 3
i%
PMT PMT
0 1 2 3
i%
PMT
Annuity Due

29. 6-29
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?

30. 6-30
Solution: Problem 12(a)

(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

31. 6-31
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