978-0077861704 Chapter 6 Solutions Manual Part 2

Unlock access to all the studying documents.

View Full Document

23. The time line is:

0 1

∞

$215,00

Here we need to find the interest rate that equates the perpetuity cash flows with the PV of the cash

flows. Using the PV of a perpetuity equation:

PV = C / r

We can now solve for the interest rate as follows:

The interest rate is .65% per month. To find the APR, we multiply this rate by the number of months

in a year, so:

And using the equation to find an EAR:

EAR = [1 + (APR / m)]m – 1

24. The time line is:

0 1

…

360

This problem requires us to find the FVA. The equation to find the FVA is:

FVA = C{[(1 + r)t – 1] / r}

25. The time line is:

0 1

…

In the previous problem, the cash flows are monthly and the compounding period is monthly. This

compounding periods are still monthly, but since the cash flows are annual, we need to use the EAR

to calculate the future value of annual cash flows. It is important to remember that you have to make

CHAPTER 27 – 2

EAR = [1 + (APR / m)]m – 1

CHAPTER 27 – 3

Using the FVA equation, we get:

FVA = C{[(1 + r)t – 1] / r}

26. The time line is:

0 1

…

The cash flows are simply an annuity with four payments per year for four years, or 16 payments.

We can use the PVA equation:

PVA = C({1 – [1 / (1 + r)t] } / r)

27. The time line is:

01234

The cash flows are annual and the compounding period is quarterly, so we need to calculate the EAR

EAR = [1 + (APR / m)]m – 1

And now we use the EAR to find the PV of each cash flow as a lump sum and add them together:

28. The time line is:

01234

Here the cash flows are annual and the given interest rate is annual, so we can use the interest rate

given. We can find the PV of each cash flow and add them together.

CHAPTER 27 – 4

Intermediate

29. The total interest paid by First Simple Bank is the interest rate per period times the number of

periods. In other words, the interest by First Simple Bank paid over 10 years will be:

First Complex Bank pays compound interest, so the interest paid by this bank will be the FV factor

of $1 minus the initial investment of $1, or:

Setting the two equal, we get:

30. Here we need to convert an EAR into interest rates for different compounding periods. Using the

equation for the EAR, we get:

EAR = [1 + (APR / m)]m – 1

EAR = .125 = (1 + r)2 – 1; r = (1.125)1/2 – 1 = .0607, or 6.07% per six months

Notice that the effective six-month rate is not twice the effective quarterly rate because of the effect

of compounding.

31. Here we need to find the FV of a lump sum, with a changing interest rate. We must do this problem

in two parts. After the first six months, the balance will be:

This is the balance in six months. The FV in another six months will be:

The problem asks for the interest accrued, so, to find the interest, we subtract the beginning balance

from the FV. The interest accrued is:

CHAPTER 27 – 5

32. Although the stock and bond accounts have different interest rates, we can draw one time line, but

we need to remember to apply different interest rates. The time line is:

0 1

…

360

…

660

We need to find the annuity payment in retirement. Our retirement savings ends and the retirement

withdrawals begin, so the PV of the retirement withdrawals will be the FV of the retirement savings.

So, we find the FV of the stock account and the FV of the bond account and add the two FVs.

So, the total amount saved at retirement is:

Solving for the withdrawal amount in retirement using the PVA equation gives us:

33. We need to find the FV of a lump sum in one year and two years. It is important that we use the

number of months in compounding since interest is compounded monthly in this case. So:

…

There is also another common alternative solution. We could find the EAR, and use the number of

years as our compounding periods. So we will find the EAR first:

Using the EAR and the number of years to find the FV, we get:

0 1

$1 FV

CHAPTER 27 – 6

0 1 2

$1 FV

34. Here we are finding the annuity payment necessary to achieve the same FV. The interest rate given is

Starting today:

0 1 …480

$1,000,0

CCCC CCCC C

Starting in 10 years:

0 1

…

120 121

…

480

$1,000,0

CCCCCC C

Starting in 20 years:

0 1 …240 241 …480

$1,000,0

C C C C C

Notice that a deposit for half the length of time, i.e., 20 years versus 40 years, does not mean that the

annuity payment is doubled. In this example, by reducing the savings period by one-half, the deposit

necessary to achieve the same ending value is about 10 times as large.

CHAPTER 27 – 7

CHAPTER 27 – 8

35. The time line is:

0 4

$1 $3

Since we are looking to quadruple our money, the PV and FV are irrelevant as long as the FV is four

36. Here we need to compare two cash flows, so we will find the value today of both sets of cash flows.

We need to make sure to use the monthly cash flows since the salary is paid monthly. Doing so, we

find:

0 1

…

0 1

…

37. We can use the present value of a growing annuity equation to find the value of your winnings today.

Doing so, we find:

PV = C {[1 / (r – g)] – [1 / (r – g)] × [(1 + g) / (1 + r)]t}

38. Since your salary grows at 3 percent per year, your salary next year will be:

This means your deposit next year will be:

Since your salary grows at 3 percent, your deposit will also grow at 3 percent. We can use the present

value of a growing perpetuity equation to find the value of your deposits today. Doing so, we find:

CHAPTER 27 – 9

PV = C {[1 / (r – g)] – [1 / (r – g)] × [(1 + g) / (1 + r)]t}

Now, we can find the future value of this lump sum in 40 years. We find:

FV = PV(1 + r)t

This is the value of your savings in 40 years.

39. The time line is:

0 1

…

The relationships between the value of an annuity and the interest rate are:

PVA falls as r increases, and PVA rises as r decreases

FVA rises as r increases, and FVA falls as r decreases

The present values of $7,500 per year for 13 years at the various interest rates given are:

PVA@10% = $7,500{[1 – (1 / 1.10)13] / .10} = $53,275.17

40. The time line is:

0 1

…

–

$20,00

Here we are given the FVA, the interest rate, and the amount of the annuity. We need to solve for the

number of payments. Using the FVA equation:

Solving for t, we get:

CHAPTER 27 – 10

CHAPTER 27 – 11

41. The time line is:

0 1

…

$89,000

Here we are given the PVA, number of periods, and the amount of the annuity. We need to solve for

the interest rate. Using the PVA equation:

To find the interest rate, we need to solve this equation on a financial calculator, using a spreadsheet,

The APR is the periodic interest rate times the number of periods in the year, so:

42. The time line is:

0 1

…

360

The amount of principal paid on the loan is the PV of the monthly payments you make. So, the

present value of the $975 monthly payments is:

The monthly payments of $975 will amount to a principal payment of $176,565.28. The amount of

principal you will still owe is:

0 1

…

360

This remaining principal amount will increase at the interest rate on the loan until the end of the loan

period. So the balloon payment in 30 years, which is the FV of the remaining principal will be:

CHAPTER 27 – 12

43. The time line is:

0 1 2 3 4

We are given the total PV of all four cash flows. If we find the PV of the three cash flows we know, and

subtract them from the total PV, the amount left over must be the PV of the missing cash flow. So, the

PV of the cash flows we know are:

PV of Year 1 CF: $1,300 / 1.09 = $1,192.66

So, the PV of the missing CF is:

0 1 2 3 4

–

The question asks for the value of the cash flow in Year 2, so we must find the future value of this

amount. The value of the missing CF is:

44. To solve this problem, we need to find the PV of each lump sum and add them together. It is

important to note that the first cash flow of $1 million occurs today, so we do not need to discount

that cash flow. The PV of the lottery winnings is:

PV = $1,000,000 + $1,450,000 / 1.07 + $1,900,000 / 1.072 + $2,350,000 / 1.073 + $2,800,000 / 1.074

45. Here we are finding the interest rate for an annuity cash flow. We are given the PVA, number of

periods, and the amount of the annuity. We should also note that the PV of the annuity is the amount

borrowed, not the purchase price, since we are making a down payment on the warehouse. The

amount borrowed is:

The time line is:

0 1 …360

CHAPTER 27 – 13

Using the PVA equation:

Unfortunately this equation cannot be solved to find the interest rate using algebra. To find the

interest rate, we need to solve this equation on a financial calculator, using a spreadsheet, or by trial

and error. If you use trial and error, remember that increasing the interest rate lowers the PVA, and

decreasing the interest rate increases the PVA. Using a spreadsheet, we find:

The APR is the monthly interest rate times the number of months in the year, so:

And the EAR is:

46. The time line is:

0 4

PV $157,0

The profit the firm earns is just the PV of the sales price minus the cost to produce the asset. We find

the PV of the sales price as the PV of a lump sum:

And the firm’s profit is:

To find the interest rate at which the firm will break even, we need to find the interest rate using the

PV (or FV) of a lump sum. Using the PV equation for a lump sum, we get:

0 4