Time Value of Money

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. ST-3 TIME VALUE OF MONEY It is now January 1, 2014; and you will need $1,000 on January 1, 2018, in 4 years. Your bank compounds interest at an 8% annual rate. a. How much must you deposit today to have a balance of $1,000 on January 1, 2018? b. If you want to make four equal payments on each January 1 from 2015 through 2018 to accumulate the $1,000, how large must each payment be? (Note that the payments begin a year from today.) c. If your father offers to make the payments calculated in Part b ($221 92) or to give you $750 on January 1, 2015 (a year from today), which would you choose? Explain. d. If you have only $750 on January 1, 2015, what interest rate, compounded annually for 3 years, must you earn to have $1,000 on January 1, 2018? e. Suppose you can deposit only $200 each January 1 from 2015 through 2018 (4 years). What interest rate, with annual compounding, must you earn to end up with $1,000 on January 1, 2018? f. Your father offers to give you $400 on January 1, 2015. You will then make six additional equal payments each 6 months from July 2015 through January 2018. If your bank pays 8% compounded semiannually, how large must each payment be for you to end up with $1,000 on January 1, 2018? g. What is the EAR, or EFF%, earned on the bank account in Part f? What is the APR earned on the account? . 5-1 What is an opportunity cost? How is this concept used in TVM analysis, and where is it shown on a time line? Is a single number used in all situations? Explain. . 5-3 If a firmโ€™s earnings per share grew from $1 to $2 over a 10-year period, the total growth would be 100%, but the annual growth rate would be less than 10%. True or false? Explain. (Hint: If you arenโ€™t sure, plug in some numbers and check it out.) . 5-5 To find the present value of an uneven series of cash flows, you must find the PVs of the individual cash flows and then sum them. Annuity procedures can never be of use, even when some of the cash flows constitute an annuity, because the entire series is not an annuity. True or false? Explain. . 5-7 Banks and other lenders are required to disclose a rate called the APR. What is this rate? Why did Congress require that it be disclosed? Is it the same as the effective annual rate? If you were comparing the costs of loans from different lenders, could you use their APRs to determine the loan with the lowest effective interest rate? Explain.
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ST-3 TIME VALUE OF MONEY
Amount

=$1000

Annual interest rate

=8%

Period

=4 years

a) Find principle
๐ด = ๐‘ƒ(1 + ๐‘Ÿ)๐‘›
1000 = ๐‘ƒ(1.08)4
1000

๐‘ƒ = (1.08)4
= $735.03

b)
Amount

=$1000

Annual interest rate

=8%

Period

=4 years

Compounding periods/yr= 4

1000 = ๐‘ƒ (1 +

0.08 4โˆ—4
)
4

= $728.45

c)

If my father offered $221.92 quarterly, or give me $750 on January 1, 2015, I would choose
$221.92 because it would earn more interest.

d)
Principal
Amount
Years
Annual interest rate =?
๐ด = ๐‘ƒ(1 + ๐‘Ÿ)๐‘›
1000 = 750(1 + ๐‘Ÿ)3

=$750
=$1,000
=3

1000
= (1 + ๐‘Ÿ)3
750
1.3333=(1 + ๐‘Ÿ)3
3

3

โˆš1.3333= โˆš(1 + ๐‘Ÿ)3
1.1006 = 1 + ๐‘Ÿ
R= 1.1006-1
=0.1006
= 10.06%

e)
principal
amount
years
annual interest rate

$200
$1,000
4
?

๐ด = ๐‘ƒ(1 + ๐‘Ÿ)๐‘›
1000 = 200(1 + ๐‘Ÿ)4
5 = (1 + ๐‘Ÿ)4
4

โˆš5 = 1 + ๐‘Ÿ

1.4953= 1+r
R=0.4953
= 49.53%
f)

Fatherโ€™s offer
Annual interest rate
Amount
Compounding periods per year
๐‘Ÿ 2๐‘›

๐ด = ๐‘ƒ (1 + 2)

1000 = ๐‘ (1 +

0.08 2โˆ—2.5
)
2

1000 = 1.21665๐‘ƒ
P = $821.9271

=$400
= 8%
=$1000
=2

g) EAR= ๐ธ๐น๐น% = (1 +

๐ผ๐‘๐‘‚๐‘€ ๐‘
)
๐‘
2

โˆ’1

0.08
) โˆ’1
2
๐ธ๐น๐น% = 8.16%
๐ธ๐น๐น% = (1 +

INOM=Periodic rate * number of periods per year
0.08*2= 16%

5-1 Opportunity cost
Opportunity cost is the rate of interest a person could earn on an alternative investment with a risk
same as the risk of investment in question. It is the value of r in the Time Value of Money equations
which is displayed at the top of a time line amid the 1st and 2nd tick marks. A single number is not used in
all cases as the opportunity cost rate varies depending on the maturity and riskiness of an investment.
5-3 Annual growt...


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