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63. This is the same question as before, with different values. So:
The time line is:
0 1
–$9,700 $11,20
0
The effective rate is not affected by the loan amount since it drops out when solving for r.
64. To find the breakeven points, we need to set the cash flows from the loan with points equal to the
interest rate for the loan without points. In general, we should make sure that the EARs of the cash
flows are equivalent. However, since both mortgages are monthly, we can make the APRs equal
since this will also make the EARs equal. So, the monthly rate with the original interest rate is:
The payments for the loan with the points is based off the original amount borrowed and the original
interest rate will be:
The amount actually received up front on the mortgage is the amount borrowed plus the points.
Letting X be the dollar amount of the points, we get:
So, the time line is:
0 1
…
360
Now we can solve for the maximum number of points that results in these cash flows having the new
interest rate of 3.75 percent. The monthly rate is:
CHAPTER 27 - 2
Solving the cash flows for the maximum points, we find:
PVA = C({1 – [1 / (1 + r) t] } / r)
Since this is the maximum dollar amount we would pay and the points are a percentage of the
amount borrowed, we find:
65. We will have the same loan payments as in the previous problem for the first 8 years, but now there
will be a balloon payment at the end of 8 years. Since there will be 22 years, or 264 months, of
payments not made, the balloon payment will be:
PVA = C({1 – [1 / (1 + r) t] } / r)
So, the time line is:
0 1
…
96
CHAPTER 27 - 3
To find the maximum number of points we would be willing to pay, we need to set the APR (and
EAR) of the loan with points equal to the APR (and EAR) of the loan without points, which is:
PVA = C({1 – [1 / (1 + r) t] } / r)
Since this is the maximum dollar amount we would pay and the points are a percentage of the
amount borrowed, we find:
66. First we will find the APR and EAR for the loan with the refundable fee. Remember, we need to use
The time line is:
0 1
…
360
0
Solving for the payment under these circumstances, we get:
We can now use this amount in the PVA equation with the original amount we wished to borrow,
0 1
…
360
Solving for r, we find:
Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives:
CHAPTER 27 - 4
With the nonrefundable fee, the APR of the loan is simply the quoted APR since the fee is not
considered part of the loan. So:
67. The time line is:
0 1
…
36
$1,000 –
–
–
–
–
–
–
–
–
Be careful of interest rate quotations. The actual interest rate of a loan is determined by the cash
Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives:
It’s called add-on interest because the interest amount of the loan is added to the principal amount of
the loan before the loan payments are calculated.
68. Here we are solving a two-step time value of money problem. Each question asks for a different
possible cash flow to fund the same retirement plan. Each savings possibility has the same FV, that
is, the PV of the retirement spending when your friend is ready to retire. The time line for the amount
needed at retirement is:
30 31 …50
The amount needed when your friend is ready to retire is:
This amount is the same for all three parts of this question.
a. If your friend makes equal annual deposits into the account, this is an annuity with the FVA equal
to the amount needed in retirement. The time line is:
0 1 …30
C C C C C C C C C
CHAPTER 27 - 5
$1,112,371
.50
The required savings each year will be:
b. Here we need to find a lump sum savings amount. The time line is:
0 1
…
30
PV $1,112,371
.50
Using the FV for a lump sum equation, we get:
c. In this problem, we have a lump sum savings in addition to an annual deposit. The time line is:
0 1
…
10
…
30
–
$175,000
$1,112,371.5
0
–
$3,500
–$3,500 –
$3,500
–
$3,500
–
$3,500
–
$3,500
–$3,500 –$3,500
C C C C C C C C
Since we already know the value needed at retirement, we can subtract the value of the lump sum
savings at retirement to find out how much your friend is short. Doing so gives us:
So, the amount your friend still needs at retirement is:
Using the FVA equation, and solving for the payment, we get:
This is the total annual contribution, but your friend’s employer will contribute $3,500 per year,
so your friend must contribute:
69. We will calculate the number of periods necessary to repay the balance with no fee first. We simply
need to use the PVA equation and solve for the number of payments.
CHAPTER 27 - 6
Without fee and annual rate = 18.60%:
Solving for t, we get:
Without fee and annual rate = 9.20%:
Solving for t, we get:
1/1.007667t = 1 – ($12,000/$225)(.007667)
So, you will pay off your new card:
Note that we do not need to calculate the time necessary to repay your current credit card with a fee
since no fee will be incurred. It will still take 113.94 months to pay off your current card. The time to
repay the new card with a transfer fee is:
With fee and annual rate = 9.20%:
Solving for t, we get:
1/1.007667t = 1 – ($12,240/$225)(.007667)
So, you will pay off your new card:
70. We need to find the FV of the premiums to compare with the cash payment promised at age 65. We
have to find the value of the premiums at year 6 first since the interest rate changes at that time. So:
FV1 = $700(1.09)5 = $1,077.04
CHAPTER 27 - 7
Finding the FV of this lump sum at the child’s 65th birthday:
The policy is not worth buying; the future value of the deposits is $184,626.72, but the policy
contract will pay off $150,000. The premiums are worth $34,626.72 more than the policy payoff.
Note, we could also compare the PV of the two cash flows. The PV of the premiums is:
And the value today of the $150,000 at age 65 is:
The premiums still have the higher cash flow. At time zero, the difference is $663.45. Whenever you
Here is a question for you: Suppose you invest $663.45, the difference in the cash flows at time zero,
71. The monthly payments with a balloon payment loan are calculated assuming a longer amortization
schedule, in this case, 30 years. The payments based on a 30-year repayment schedule would be:
Now, at Time 8, we need to find the PV of the payments which have not been made. The balloon
payment will be:
CHAPTER 27 - 8
72. Here we need to find the interest rate that makes the PVA, the college costs, equal to the FVA, the
savings. The PV of the college costs are:
And the FV of the savings is:
Setting these two equations equal to each other, we get:
Reducing the equation gives us:
Now we need to find the roots of this equation. We can solve using trial and error, a root-solving
calculator routine, or a spreadsheet. Using a spreadsheet, we find:
73. Here we need to find the interest rate that makes us indifferent between an annuity and a perpetuity.
To solve this problem, we need to find the PV of the two options and set them equal to each other.
The PV of the perpetuity is:
And the PV of the annuity is:
Setting them equal and solving for r, we get:
$30,000 / r = $35,000[{1 – [1 / (1 + r)15]} / r ]
74. The cash flows in this problem occur every two years, so we need to find the effective two-year rate.
One way to find the effective two-year rate is to use an equation similar to the EAR, except use the
We can use this interest rate to find the PV of the perpetuity. Doing so, we find:
CHAPTER 27 - 9
This is an important point: Remember that the PV equation for a perpetuity (and an ordinary
annuity) tells you the PV one period before the first cash flow. In this problem, since the cash flows
The second part of the question assumes the perpetuity cash flows begin in four years. In this case,
when we use the PV of a perpetuity equation, we find the value of the perpetuity two years from
today. So, the value of these cash flows today is:
75. To solve for the PVA due:
PVA =
C
(1 +r)+C
(1 +r)2+. .. .+C
(1 +r)t
C+C
(1 +r)+. . ..+C
(1 +r)t - 1
FVA = C + C(1 + r) + C(1 + r)2 + …. + C(1 + r)t – 1
76. We need to find the lump sum payment into the retirement account. The present value of the desired
amount at retirement is:
PV = FV / (1 + r)t
This is the value today. Since the savings are in the form of a growing annuity, we can use the
growing annuity equation and solve for the payment. Doing so, we get:
PV = C {[1 – ((1 + g) / (1 + r))t ] / (r – g)}
CHAPTER 27 - 10
This is the amount you need to save next year. So, the percentage of your salary is:
Note that this is the percentage of your salary you must save each year. Since your salary is
increasing at 3 percent, and the savings are increasing at 3 percent, the percentage of salary will
remain constant.
