# 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 firm’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 aren’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
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) Find principle
𝐴 = 𝑃(1 + 𝑖)𝑛
1000 = 𝑃(1.08)4
1000

𝑃 = (1.08)4
= \$735.03

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?

1000 = 𝑃 (1 +

0.08 4∗4
)
4

= \$728.45

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.

I would choose \$221.92 because with quarterly payments, interest is has since interest starts
accruing earlier.

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?
𝐴 = 𝑃(1 + 𝑖)𝑛
1000 = 750(1 + 𝑖)3
1000
= (1 + 𝑖)3
750
1.3333=(1 + 𝑖)3

3

3

√1.3333= √(1 + 𝑖)3
1.1006 = 1 + 𝑖
i =0.1006 = 10.06%
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?
𝐴 = 𝑃(1 + 𝑖)𝑛
1000 = 200(1 + 𝑖)4
5 = (1 + 𝑖)4
4

√5 = 1 + 𝑖

1.4953= 1+i
i=0.4953
= 49.53%
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?
𝑖 2∗𝑛

𝐴 = 𝑃 (1 + 2)

1000 = 𝑝 (1 +

0.08 2∗2.5
)
2

1000 = 1.21665𝑃
P = 821.9271

g) What is the EAR, or EFF%, earned on the bank account in Part f? What is the APR earned on the
account?
EAR= 𝐸𝐹𝐹% = (1 +

𝐼𝑁𝑂𝑀 𝑁
)
𝑁
2

−1

𝐸𝐹𝐹% = (1 + 0.08⁄2) − 1
𝐸𝐹𝐹% = 8.16%

INOM=Periodic rate * number of payments per year

8% ∗ 2 = 16%

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 numbe...

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