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66. First, we will find the APR and EAR for the loan with the refundable fee. Remember, we need to use
the actual cash flows of the loan to find the interest rate. With the $2,400 application fee, you will
need to borrow $227,400 to have $225,000 after deducting the fee. The time line is:
0
1
…
360
$227,400
C
C
C
C
C
C
C
C
C
–$2,500
$112.66
$112.66
$112.66
$112.66
$112.66
$112.66
$112.66
$112.66
$112.66
EAR = (1 + .0289)12 – 1 = .4078, or 40.78%
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. The payments are a growing annuity, so we use the equation for the present value of a growing annuity.
The payment growth rate is 2.5 percent and the EAR 11 percent. Since the payments are quarterly, we
need the APR, which is:
EAR = (1 + APR/m)m – 1
.11 = (1 + APR/4)4 – 1
PV = C{[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t}
PV = $220,000{[1/(.0264 – .025)] – [1/(.0264 – .025)] × [(1 + .025)/(1 + .0264)]25}
Value of offer = $5,269,524.67 + 500,000
Value of offer = $5,769,524.67
69. We will calculate the number of periods necessary to repay the balance with no fee first. We need to
use the PVA equation and solve for the number of payments.
Without fee and annual rate = 18.20%:
t = ln{1/[1 – ($10,000/$175)(.0152)]}/ln(1.0152)
t = ln 7.5/ln 1.0152
t = 133.86 months
Without fee and annual rate = 7.90%:
PVA = $10,000 = $175[(1 – 1/1.00658t)/.00658] where .00658 = .079/12
Solving for t, we get:
t = ln{1/[1 – ($10,000/$175)(.00658)]}/ln(1.00658)
t = ln 1.60305/ln 1.00658
t = 71.92 months
So, you will pay the card off:
133.86 – 74.70 = 59.16 months sooner
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 = $750(1.10)5 = $1,207.88
FV5 = $950(1.10)1 = $1,045
Value at year six = $1,207.88 + 1,098.08 + 1,131.35 + 1,028.50 + 1,045 + 950
Value at year six = $6,460.81
PV = ($500,000/1.0859)/1.106
PV = $3,010.32
The premiums still have the higher cash flow. At time zero, the difference is $636.64. Whenever you
are comparing two or more cash flow streams, the cash flow with the highest value at one time will
have the highest value at any other time.
Here is a question for you: Suppose you invest $636.64, the difference in the cash flows at time zero,
for six years at a 10 percent interest rate, and then for 59 years at an 8 percent interest rate. How much
will it be worth? Without doing calculations, you know it will be worth $105,742.96, the difference in
the cash flows at Time 65!
71. Since the payments occur at six month intervals, we need to get the effective six-month interest rate.
We are assuming 365 days per year, in other words, we are ignoring leap year. We can calculate the
daily interest rate since we have an APR compounded daily, so the effective six-month interest rate is:
Effective six-month rate = (1 + Daily rate)182.5 – 1
Effective six-month rate = (1 + .08/365)182.5 – 1
Effective six-month rate = .0408 or 4.08%
Now, we can use the PVA equation to find the present value of the semi-annual payments. Doing so,
This is the value six months from today, which is one period (six months) prior to the first payment.
So, the value today is:
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:
PVA = $32,000[{1 – [1/(1 + r)4]}/r]
And the FV of the savings is:
Setting these two equations equal to each other, we get:
$32,000[{1 – [1/(1 + r)4]}/r] = $15,000{[(1 + r)6 – 1]/r}
Reducing the equation gives us:
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:
PV = $15,000/r
And the PV of the annuity is:
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
number of days in two years as the exponent. (We use the number of days in two years since it is daily
compounding; if monthly compounding was assumed, we would use the number of months in two
years.) So, the effective two-year interest rate is:
Effective 2-year rate = [1 + (.11/365)]365(2) – 1
Effective 2-year rate = .2460, or 24.60%
We can use this interest rate to find the PV of the perpetuity. Doing so, we find:
75. To solve for the PVA due:
PVA =
) (1
....
) (1
) (1 2t
r
C
r
C
r
C
+
++
+
+
+
PVAdue =
) (1
....
) (1
1 - t
r
C
r
C
C+
++
+
+
C
C
C
And the FVA due is:
76. The time line is:
0
6
9
…
351
357
360
$600,000
$900,000
$900,000
$600,000
$1,500,000
Since the cash flows are every six months and every nine months, we need the six- and nine-month
interest rates, which are:
Nine-month rate = .0624, or 6.24%
Now we can find the present value of the cash flows. Even though there is a cash flow every 18 months
that is the combined value of the two cash flows, we can still treat each cash flow as an individual
PVA2 = $900,000{[1 – 1/(1 + .0624)]60}/.0624}
PVA2 = $13,139,463.25
Present value = $26,413,314.09
77. a. The APR is the interest rate per week times 52 weeks in a year, so:
APR = 52(5.5%)
APR = 286%
EAR = (1 + .055)52 – 1
EAR = 15.1856, or 1,518.56%
b. In a discount loan, the amount you receive is lowered by the discount, and you repay the full
the APR and the EAR:
EAR = 17.9472, or 1,794.72%
c. Using the cash flows from the loan, we have the PVA and the annuity payments and need to find
the interest rate, so:
PVA = $76.12 = $25[{1 – [1/(1 + r)4]}/ r]
Using a spreadsheet, trial and error, or a financial calculator, we find:
78. To answer this, we can diagram the perpetuity cash flows, which are: (Note, the subscripts are only to
differentiate when the cash flows begin. The cash flows are all the same amount.)
C3 ….
C2 C2
C1 C1 C1
Thus, each of the increased cash flows is a perpetuity in itself. So, we can write the cash flows stream
The present value of this perpetuity is:
PV = (C/R)/R = C/R2
PV = C/R + C/R2
79. Since it is only an approximation, we know the Rule of 72 is exact for only one interest rate. Using
the basic future value equation for an amount that doubles in value and solving for t, we find:
0 = (.72/R)/[ln(2)/ln(1 + R)]
