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58. To answer this question, we should find the PV of both options, and compare them. Since we are
purchasing the car, the lowest PV is the best option. The PV of the leasing option is the PV of the
lease payments, plus the $2,400. The interest rate we would use for the leasing option is the same as
the interest rate of the loan. The PV of leasing is:
0 1
…
36
$2,400 $380 $380 $380 $380 $380 $380 $380 $380 $380
The PV of purchasing the car is the current price of the car minus the PV of the resale price. The PV
of the resale price is:
0 1
…
36
$28,00
0
–
$17,00
0
In this case, it is cheaper to buy the car than lease it since the PV of the leasing cash flows is lower.
To find the break-even resale price, we need to find the resale price that makes the PV of the two
options the same. In other words, the PV of the decision to buy should be:
59. To find the quarterly salary for the player, we first need to find the PV of the current contract. The
cash flows for the contract are annual, and we are given a daily interest rate. We need to find the
EAR so the interest compounding is the same as the timing of the cash flows. The EAR is:
The player wants the contract increased in value by $2,700,000, so the PV of the new contract will
be:
PV = $35,371,651.95 + 2,700,000
PV = $38,071,651.95
The player has also requested a signing bonus payable today in the amount of $10 million. We can
subtract this amount from the PV of the new contract. The remaining amount will be the PV of the
future quarterly paychecks.
60. The time line for the cash flows is:
0 1
–
$16,86
0
$20,0
00
To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the
interest rate quoted in the problem is only relevant to determine the total interest under the terms
given. The cash flows of the loan are the $20,000 you must repay in one year, and the $16,860 you
borrow today. The interest rate of the loan is:
61. The time line is:
–24 –23
…
–12 –11
…
0 1
…
60
$3,083.
33
$3,083.
33
$3,250 $3,250 $3,583.
33
$3,583.
33
$3,583.
33
$150,0
00
$25,00
0
Here, we have cash flows that would have occurred in the past and cash flows that would occur in
the future. We need to bring both cash flows to today. Before we calculate the value of the cash flows
today, we must adjust the interest rate, so we have the effective monthly interest rate. Finding the
APR with monthly compounding and dividing by 12 will give us the effective monthly rate. The
APR with monthly compounding is:
Notice we found the FV of the annuity with the effective monthly rate, and then found the FV of the
lump sum with the EAR. Alternatively, we could have found the FV of the lump sum with the
effective monthly rate as long as we used 12 periods. The answer would be the same either way.
Now, we need to find the value today of last year’s back pay:
FVA = ($39,000 / 12) [{[ 1 + (.0753 / 12)]12 – 1} / (.0753 / 12)]
FVA = $40,375.34
Next, we find the value today of the five year’s future salary:
62. To find the interest rate of a loan, we need to look at the cash flows of the loan. Since this loan is in
the form of a lump sum, the amount you will repay is the FV of the principal amount, which will be:
So, the time line is:
0 9
–
$9,700
$10,8
00
Now, we find the interest rate for this PV and FV.
With a quoted interest rate of 11 percent and two points, the EAR is:
Loan repayment amount = $10,000(1.11)
Loan repayment amount = $11,100
The effective rate is not affected by the loan amount, since it drops out when solving for r.
63. 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 $2,900 application fee, you will
need to borrow $227,900 to have $225,000 after deducting the fee. The time line is:
0 1
…
360
$227,90
0
CCCC CCCCC
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,
$225,000.
0 1
…
360
–
$225,00
0
$1,308.3
3
$1,308.3
3
$1,308.3
3
$1,308.3
3
$1,308.3
3
$1,308.3
3
$1,308.3
3
$1,308.3
3
$1,308.
33
Solving for r, we find:
PVA = $225,000 = $1,308.33[{1 – [1 / (1 + r)]360} / r]
Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives:
r = .4764% per month
64. The time line is:
0 1
…
36
–$1,000 $44.95 $44.95 $44.95 $44.95 $44.95 $44.95 $44.95 $44.95 $44.95
Be careful of interest rate quotations. The actual interest rate of a loan is determined by the cash
flows. Here, we are told that the PV of the loan is $1,000, and the payments are $44.95 per month for
three years, so the interest rate on the loan is:
65. 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 = 18.6%:
PVA = $10,000 = $225{[1 – (1 / 1.0155)t ] / .0155 } where .0155 = .186 / 12
Solving for t, we get:
Without fee and annual rate = 8.2%:
You will pay off your account:
75.91 – 53.15 = 22.76 months quicker
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. The time to repay the new card with a transfer fee is:
With fee and annual rate = 8.20%:
66. 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:
Value at Year 6 = $674.02 + 759.04 + 820.58 + 862.47 + 888 + 900
Value at Year 6 = $4,904.11
Finding the FV of this lump sum at the child’s 65th birthday:
FV = $4,904.11(1.07)59
FV = $265,584.56
The policy is not worth buying; the future value of the policy is $265,584.56, but the policy contract
will pay off $250,000. The premiums are worth $15,584.56 more than the policy payoff.
Note, we could also compare the PV of the two cash flows. The PV of the premiums is:
The premiums still have the higher cash flow. At time zero, the difference is $153.86. 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 $153.86, the difference in the cash flows at time zero,
67. Since the payments occur at six month intervals, we need to get the effective six-month interest rate.
We can calculate the daily interest rate since we have an APR compounded daily, so the effective
six-month interest rate is:
Now, we can use the PVA equation to find the present value of the semi-annual payments. Doing so,
we find:
This is the value six months from today, which is one period (six months) prior to the first payment.
So, the value today is:
PV = $29,017,924.95 / (1 + .0460)
PV = $27,741,219.20
This means the total value of the lottery winnings today is:
68. 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 is:
PVA = $25,000[{1 – [1 / (1 + r)]4 } / r ]
And the FV of the savings is:
69. The time line is:
0 1 …10 …∞
–
$20,00
0
–
$20,00
0
–
$20,00
0
–
$20,00
0
–
$20,00
0
–
$20,00
0
–
$20,00
0
–
$20,00
0
$34,0
00
$34,00
0
$34,00
0
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:
Setting them equal and solving for r, we get:
70. The time line is:
0 1 3
…
∞
$50,00
0
$50,00
0
$50,00
0
$50,00
0
$50,00
0
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:
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,
71. 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
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:
r = 15.02% per week
