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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:
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
are two years apart, we have found the value of the perpetuity one period (two years) before the first
payment, which is one year ago. We need to compound this value for one year to find the value
today. The value of the cash flows today is:
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
PVAdue =
C+C
(1 +r)+. . ..+C
(1 +r)t - 1
PVAdue =
(1 +r)
(
C
(1 +r)+C
(1 +r)2+.. . .+C
(1 +r)t
)
PVAdue = (1 + r)PVA
And the FVA due is:
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:
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.
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
Note the dollar amount we use is irrelevant. In other words, we could use $.932 and $1, $93.20
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….
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:
Since r is expressed as a percentage in this case, we can write the expression as:
t = ln(2)/ln(1 + r/100)
ln(1 + r) = r – r2/2 + r3/3 – ...
ln(1 + r) = r
Combine this with the solution for the doubling expression:
For a 10 percent interest rate, the time to double your money is:
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 $530 C01 $530 C01 $530
F01 1F01 1F01 1
C02 $690 C02 $690 C02 $690
F02 1F02 1F02 1
C03 $875 C03 $875 C03 $875
F03 1F03 1F03 1
C04 $1,090 C04 $1,090 C04 $1,090
F04 1F04 1F04 1
2.
N I/Y PV PMT FV
N I/Y PV PMT FV
N I/Y PV PMT FV
N I/Y PV PMT FV
3.
N I/Y PV PMT FV
N I/Y PV PMT FV
N I/Y PV PMT FV
Solve for $1,447.20
N I/Y PV PMT FV
N I/Y PV PMT FV
N I/Y PV PMT FV
Solve for $1,487.40
N I/Y PV PMT FV
N I/Y PV PMT FV
N I/Y PV PMT FV
Solve for $1,661.60
4.
N I/Y PV PMT FV
N I/Y PV PMT FV
N I/Y PV PMT FV
5.
N I/Y PV PMT FV
6.
N I/Y PV PMT FV
7.
N I/Y PV PMT FV
N I/Y PV PMT FV
8.
N I/Y PV PMT FV
9.
N I/Y PV PMT FV
12.
NOM EFF C/Y
NOM EFF C/Y
NOM EFF C/Y
13.
NOM EFF C/Y
NOM EFF C/Y
NOM EFF C/Y
14.
NOM EFF C/Y
NOM EFF C/Y
15.
NOM EFF C/Y
16.
N I/Y PV PMT FV
17.
N I/Y PV PMT FV
N I/Y PV PMT FV
N I/Y PV PMT FV
18.
N I/Y PV PMT FV
