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Corporate Finance Chapter 4 Solutions

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Solutions to Chapter 4 The Time Value of Money

1. a. b. c. d. $100/(1.08)10 = $46.32 $100/(1.08)20 = $21.45 $100/(1.04)10 = $67.56 $100/(1.04)20 = $45.64 $100 × (1.08)10 = $215.89 $100 × (1.08)20 = $466.10 $100 × (1.04)10 = $148.02 $100 × (1.04)20 = $219.11

2.

a. b. c. d.

3.

$100 × (1.04)113 = $8,409.45 $100 × (1.08)113 = $598,252.29

4.

With simple interest, you earn 4% of $1,000 or $40 each year. There is no interest on interest. After 10 years, you earn total interest of $400, and your account accumulates to $1,400. With compound interest, your account grows to: $1,000 × (1.04)10 = $1480.24 Therefore $80.24 is interest on interest. PV = $700/(1.05)5 = $548.47

5.

4-1

6. Present Value a. $400 Years 11 Future Value $684 Interest Rate
⎡ 684 ⎤ ⎢ 400 ⎥ ⎣ ⎦ ⎡ 249 ⎤ ⎢ 183 ⎥ ⎦ ⎣
(1 / 11)

− 1 = 5.00%
(1 / 4 )

b.

$183

4

$249

− 1 = 8.00%
(1 / 7 )

c.

$300

7

$300

⎡ 300 ⎤ ⎢ 300 ⎥ ⎣ ⎦

− 1 = 0%

To find the interest rate, we rearrange the basic future value equation as follows:
⎡ FV ⎤ FV = PV × (1 + r) ⇒ r = ⎢ ⎣ PV ⎥ ⎦ t (1 / t )

−1

7.

You should compare the present values of the two annuities. a.

⎡ 1 ⎤ 1 − PV = $1,000 × ⎢ = $7,721.73 10 ⎥ ⎣ 0.05 0.05 × (1.05) ⎦ ⎡ 1 ⎤ 1 − PV = $800 × ⎢ = $8,303.73 15 ⎥ ⎣ 0.05 0.05 × (1.05) ⎦

b.

⎡ 1 ⎤ 1 − = $4,192.47 PV = $1,000 × ⎢ 10 ⎥ ⎣ 0.20 0.20 × (1.20) ⎦ ⎡ 1 ⎤ 1 − PV = $800 × ⎢ = $3,740.38 15 ⎥ ⎣ 0.20 0.20 × (1.20) ⎦

c.

When the interest rate is low, as in part (a), the longer (i.e., 15-year) but smaller annuity is more valuable because the impact of discounting on the present value of future payments is less significant.

8.

$100 × (1 + r)3 = $115.76 ⇒ r = 5.00% $200 × (1 + r)4 = $262.16 ⇒ r = 7.00% $100 × (1 + r)5 = $110.41 ⇒ r = 2.00%

4-2

9.

PV = ($200/1.06) + ($400/1.062) + ($300/1.063) = $188.68 + $356.00 + $251.89 = $796.57

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