42 Berk/DeMarzo/Harford ? Fundamentals of Corporate Finance, Third Edition, Global Edition

28. Plan: This problem is asking us to solve for the rate of return (r). Because there are no

recurring payments, we can use Eq. 4.1 to represent the problem and then just solve algebraically for r. We have FV = 200, PV = 100, n = 10. Execute:

FV ? PV(1 ? r)n

200 ? 100(1 ? r)10, so r = (200/100) 1/10 – 1 = 0.072 or 7.2% Evaluate:

The implicit return we earned on the savings bond was 7.2%. Our money doubled in 10 years, which by the rule of 72 meant that we earned about 72/10 = 7.2% and our calculation confirmed that.

29. Plan: This problem is again asking us to solve for r. We will represent the investment with Eq.

4.1 and solve for r. We have PV = 2,000, FV = 10,000, n = 10. The second part of the problem asks us to change the rate of return going forward and calculate the FV in another 10 years. Execute:

a. FV= PV(1 + r)n

10,000 ? 2,000(1 ? r)10, so r = (10,000/2,000) 1/10 – 1 = 0.1746, or 17.46% b. FV ? 10,000(1.12)10 ? $31,058.48

30. Plan: Draw a timeline and determine the IRR of the investment. Execute:

0 –15,000 1 20,000

IRR is the r that solves:

15,000 ? 20,000/(1 ? r), so r = (20,000/15,000) – 1 = 33.33% Evaluate: You are making a 33.33% IRR on this investment.

31. Plan: Draw a timeline to demonstrate when the cash flows will occur. Then solve the problem

to determine the payments you will receive. Execute:

0 1 2 –500 C C P = C/r, So, C = P × r = 500 × 0.08 = $40

3 C

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Chapter 4 Time Value of Money: Valuing Cash Flow Streams 43

Evaluate: You will receive $40 per year into perpetuity.

32. Plan: Draw a timeline to determine when the cash flows occur. Solve the problem to determine

the annual payments. Timeline (from the perspective of the bank): Execute:

0 –3,000,000 1 C C C 2 3

C

30 C?3,000,000?$240,727.76

1?1??1??0.05?1.0520? which is the annual payment.

N

I/Y

5.00%

PV

?3000000

PMT

$240727.76

FV

0

Excel Formula

?PMT(0.05,20,?3000000,0)

Given: 20

Solve for PMT:

Evaluate: You will have to pay the bank $240,727.76 per year for 20 years in mortgage payments.

*33. Plan: Draw a timeline to demonstrate when the cash flows will occur. Determine the annual

payments.

Execute:

0 0 –500,00 2 1 C 4 2 C 6 3 C

20 10 C

This cash flow stream is an annuity. First, calculate the two-year interest rate: The one-year rate is 4%, and $1 today will be worth (1.04)2 ? 1.0816 in two years, so the two-year interest rate is 8.16%. Using the equation for an annuity payment:

C?50,0001?1?1??0.0816??(1.0816)10??$7,505.34

which is the payment you must make every two years.

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44 Berk/DeMarzo/Harford ? Fundamentals of Corporate Finance, Third Edition, Global Edition

N I/Y PV PMT FV 0

Excel Formula

?PMT(0.0816,10,?50000,0)

Given: 10 8.16% ?50,000.00

Solve for PMT: $7,505.34

Evaluate: You must pay the art dealer $7505.34 every two years for 20 years.

*34. Plan: Draw a timeline to determine when the cash flows occur. Timeline (where X is the balloon payment):

0 –300,000 1 23,500 23,500 2 3 23,500

23,500 ? X

30

Note that the PV of the loan payments must be equal to the amount borrowed. Execute:

300,000?23,500?1?1?0.07?1.0730??X ?30(1.07)? Solving for X:

23,500?1??X??300,000?1?(1.07)30?30??? 0.07?1.07????$63,848

N

Given: 30 Solve for PV:

I/Y 7.00%

PV

291,612.47

PMT ?23,500

FV 0

Excel Formula

?PV(0.07,30,?23500,0)

The present value of the annuity is $291,612.47, which is $8,387.53 less than the $300,000.00. To make up for this shortfall with a balloon payment in year 30 would require a payment of $63,848.02.

Given: Solve for FV:

N 30

I/Y

PV

PMT 0

FV

Excel Formula

7.00% 8,387.53

(63,848.02) ?FV(0.07,30,0,8387.53)

Evaluate: At the end of 30 years you would have to make a $63,848 single (balloon) payment to the bank.

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Chapter 4 Time Value of Money: Valuing Cash Flow Streams 45

*35. Plan: Draw a timeline to demonstrate when the cash flows occur. We know that you intend to

fund your retirement with a series of annuity payments and the future value of that annuity is $2 million.

22 0 C C 23 1 C C 24 2 25 3

C

65 43

Execute: FV ? $2 million.

The PV of the cash flows must equal the PV of $2 million in 43 years. The cash flows consist of a 43-year annuity, plus the contribution today, so the PV is:

PV?

C?1??1???C 0.05??1.05?43? The PV of $2 million in 43 years is:

2,000,000?$245,408.80

(1.05)43 N I/Y PV PMT 0

FV 2,000,000

Excel Formula

?PV(0.05,43,0,2000000)

Given: 43 5.00%

Solve for PV: (245,408.80) Setting these equal gives

C?1??C?245,408.80?1?43?0.05?(1.05)? 245,408.80?C??$13,232.501?1?1????10.05?(1.05)43?

We need $245,408.80 today to have $2,000,000 in 43 years. If we do not have $245,408.80 today, but wish to make 44 equal payments (the first payment is today, making the payments an annuity due) then the relevant Excel command is:

?PMT(rate,nper,pv,(fv),type ?PMT(.05,44,245,408.80,0,1) ? 13,232.50

Type is set equal to 1 for an annuity due as opposed to an ordinary annuity.

Evaluate: You would have to put aside $13,232.50 annually to have the $2 million you wish to have in retirement.

36. Plan: This problem is asking you to solve for n. You can do this mathematically using logs, or

with a financial calculator or Excel. Because the problem happens to be asking how long it will

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