77. a. The APR is the interest rate per week times 52 weeks in a year, so:
b. In a discount loan, the amount you receive is lowered by the discount, and you repay the full
principal. With a discount of 7.5 percent, you would receive $9.25 for every $10 in principal, so
the weekly interest rate would be:
Note the dollar amount we use is irrelevant. In other words, we could use $.925 and $1, $92.50
and $100, or any other combination and we would get the same interest rate. Now we can find the
APR and the EAR:
c. Using the cash flows from the loan, we have the PVA and the annuity payments and need to find
the interest rate, so:
Using a spreadsheet, trial and error, or a financial calculator, we find:
78. To answer this, we need to 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
C2C2
C1C1C1
Thus, each of the increased cash flows is a perpetuity in itself. So, we can write the cash flows
stream as:
C1/r C2/r C3/r C4/r….
CHAPTER 27 – 2
So, we can write the cash flows as the present value of a perpetuity, and a perpetuity of:
C2/r C3/r C4/r….
The present value of this perpetuity is:
PV = (C/r) / r = C/r2
So, the present value equation of a perpetuity that increases by C each period is:
PV = C/r + C/r2
79. We are only concerned with the time it takes money to double, so the dollar amounts are irrelevant.
So, we can write the future value of a lump sum as:
FV = PV(1 + r)t
Solving for t, we find:
ln(2) = t[ln(1 + r)]
Since r is expressed as a percentage in this case, we can write the expression as:
To simplify the equation, we can make use of a Taylor Series expansion:
Since r is small, we can truncate the series after the first term:
Combine this with the solution for the doubling expression:
t = ln(2) / (r/100)
This is the exact (approximate) expression, Since 69.3147 is not easily divisible, and we are only
concerned with an approximation, 72 is substituted.
For a 10 percent interest rate, the time to double your money is:
CHAPTER 27 – 3
$2 = PV(1 + .10)t
So, for a 10 percent interest rate, it takes 7.27 periods to double, which is closer to 73 than to 72.
80. We are only concerned with the time it takes money to double, so the dollar amounts are irrelevant.
So, we can write the future value of a lump sum with continuously compounded interest as:
$2 = $1ert
Since we are using interest rates while the equation uses decimal form, to make the equation correct
with percentages, we can multiply by 100:
Calculator Solutions
1.
CFo $0 CFo $0 CFo $0
C01 $680 C01 $680 C01 $680
F01 1F01 1F01 1
2.
Enter 8 5% $4,700
N I/Y PV PMT FV
Enter 5 5% $6,700
N I/Y PV PMT FV
Enter 8 15% $4,700
N I/Y PV PMT FV
CHAPTER 27 – 4
Enter 5 15% $6,700
N I/Y PV PMT FV
3.
Enter 3 8% $1,225
N I/Y PV PMT FV
Enter 2 8% $1,345
N I/Y PV PMT FV
Enter 1 8% $1,460
N I/Y PV PMT FV
Enter 3 11% $1,225
N I/Y PV PMT FV
Enter 2 11% $1,345
N I/Y PV PMT FV
Enter 1 11% $1,460
N I/Y PV PMT FV
Enter 3 24% $1,225
N I/Y PV PMT FV
Enter 2 24% $1,345
N I/Y PV PMT FV
Enter 1 24% $1,460
N I/Y PV PMT FV
CHAPTER 27 – 5
Solve for $1,810.40
4.
Enter 15 6% $5,500
N I/Y PV PMT FV
Enter 40 6% $5,500
N I/Y PV PMT FV
Enter 75 6% $5,500
N I/Y PV PMT FV
5.
Enter 15 5.8% $38,000
N I/Y PV PMT FV
6.
Enter 7 7.8% $57,000
N I/Y PV PMT FV
7.
Enter 20 9.7% $4,000
N I/Y PV PMT FV
Enter 40 9.7% $4,000
N I/Y PV PMT FV
8.
Enter 12 6.2% $50,000
N I/Y PV PMT FV
9.
Enter 5 7.5% $50,000
N I/Y PV PMT FV
12.
Enter 7% 4
CHAPTER 27 – 6
NOM EFF C/Y
Enter 17% 12
NOM EFF C/Y
Enter 13% 365
NOM EFF C/Y
13.
Enter 12.4% 2
NOM EFF C/Y
Enter 11.7% 12
NOM EFF C/Y
Enter 9.5% 52
NOM EFF C/Y
14.
Enter 12.4% 12
NOM EFF C/Y
Enter 12.7% 2
NOM EFF C/Y
15.
Enter 16.5% 365
NOM EFF C/Y
16.
Enter 17 × 2 7.9% / 2 $2,400
N I/Y PV PMT FV
17.
Enter 5 365 6.7% / 365 $7,000
N I/Y PV PMT FV
CHAPTER 27 – 7
Enter 10 365 6.7% / 365 $7,000
N I/Y PV PMT FV
Enter 20 365 6.7% / 365 $7,000
N I/Y PV PMT FV
18.
Enter 10 365 7% / 365 $65,000
N I/Y PV PMT FV
19.
Enter 384% 12
NOM EFF C/Y
20.
Enter 60 5.8% / 12 $79,500
N I/Y PV PMT FV
Enter 5.8% 12
NOM EFF C/Y
21.
Enter 1.5% $18,000 $500
N I/Y PV PMT FV
22.
Enter 1,733.33% 52
NOM EFF C/Y
23.
Enter 7.81% 12
NOM EFF C/Y
24.
Enter 30 12 10% / 12 $450
N I/Y PV PMT FV
CHAPTER 27 – 8
25.
Enter 10% 12
NOM EFF C/Y
Enter 30 10.47% $5,400
N I/Y PV PMT FV
26.
Enter 4 4 .43% $2,200
N I/Y PV PMT FV
27.
Enter 9% 4
NOM EFF C/Y
CFo$0
C01 $790
F01 1
C02 $860
F02 1
28.
CFo$0
C01 $2,480
F01 1
C02 $0
30.
Enter 12.5% 2
CHAPTER 27 – 9
NOM EFF C/Y
Solve for 12.13%
Enter 12.5% 4
NOM EFF C/Y
Enter 12.5% 12
NOM EFF C/Y
31.
Enter 6 .50% / 12 $7,000
N I/Y PV PMT FV
Enter 6 18.5% / 12 $7,017.52
N I/Y PV PMT FV
32. Stock account:
Enter 360 10% / 12 $850
N I/Y PV PMT FV
Enter 360 6% / 12 $350
N I/Y PV PMT FV
Enter 300 5% / 12 $2,272,995.00
N I/Y PV PMT FV
33.
Enter 12 .74% $1
N I/Y PV PMT FV
CHAPTER 27 – 10
Enter 24 .74% $1
N I/Y PV PMT FV
34.
Enter 480 10.5% / 12 $1,000,000
N I/Y PV PMT FV
Enter 360 10.5% / 12 $1,000,000
N I/Y PV PMT FV
Enter 240 10.5% / 12 $1,000,000
N I/Y PV PMT FV
35.
Enter 12 / 3 $1 $3
N I/Y PV PMT FV
36.
Enter 2 × 12 7% /12 $75,000 / 12
N I/Y PV PMT FV
Enter 2 × 12 7% /12 $64,000 / 12
N I/Y PV PMT FV
39.
Enter 13 10% $7,500
N I/Y PV PMT FV
Enter 13 5% $7,500
N I/Y PV PMT FV
Enter 13 15% $7,500
N I/Y PV PMT FV
CHAPTER 27 – 11
Solve for $41,873.60
40.
41.
Enter 60 $89,000 $1,850
N I/Y PV PMT FV
42.
Enter 360 5.25% / 12 $975
N I/Y PV PMT FV
Enter 360 5.25% / 12 $63,434.72
N I/Y PV PMT FV
Solve for $305,385.86
43.
CFo$0
C01 $1,300
F01 1
Value of missing CF:
Enter 2 9% $1,631.34
N I/Y PV PMT FV