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47. The time line is:
0 123456 …20
PV $3,50
0
$3,50
0
$3,50
0
$3,50
0
We want to find the value of the cash flows today, so we will find the PV of the annuity, and then
bring the lump sum PV back to today. The annuity has 15 payments, so the PV of the annuity is:
Since this is an ordinary annuity equation, this is the PV one period before the first payment, so this
is the PV at t = 5. To find the value today, we find the PV of this lump sum. The value today is:
48. The time line is:
0 1
…
180
0
0
0
0
0
0
0
0
This question is asking for the present value of an annuity, but the interest rate changes during the
life of the annuity. We need to find the present value of the cash flows for the last eight years first.
The PV of these cash flows is:
Note that this is the PV of this annuity exactly seven years from today. Now we can discount this
lump sum to today. The value of this cash flow today is:
Now we need to find the PV of the annuity for the first seven years. The value of these cash flows
today is:
The value of the cash flows today is the sum of these two cash flows, so:
49. The time line for the annuity is:
0 1
…
156
$1,10
0
$1,10
0
$1,10
0
$1,10
0
$1,10
0
$1,10
0
$1,10
0
$1,10
0
$1,100
CHAPTER 27 - 2
Here we are trying to find the dollar amount invested today that will equal the FVA with a known
CHAPTER 27 - 3
Now we have:
0 1 …13
PV $289,191
.78
So, we need to find the PV of a lump sum that will give us the same FV. So, using the FV of a lump
sum with continuous compounding, we get:
50. The time line is:
0 1
…
7
…
14 15 …∞
PV $5,200 $5,200 $5,200 $5,200
To find the value of the perpetuity at t = 7, we first need to use the PV of a perpetuity equation.
Using this equation we find:
0 1
…
7
…
14
PV $126,829
.27
Remember that the PV of a perpetuity (and annuity) equations give the PV one period before the first
payment, so, this is the value of the perpetuity at t = 14. To find the value at t = 7, we find the PV of
this lump sum as:
51. The time line is:
0 1 …12
0
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 interest rate for the cash flows of the loan is:
Again, we cannot solve this equation for r, so we need to solve this equation on a financial
calculator, using a spreadsheet, or by trial and error. Using a spreadsheet, we find:
CHAPTER 27 - 4
So the APR that would legally have to be quoted is:
And the EAR is:
52. The time line is:
0 1
…
18 19 …28
The cash flows in this problem are semiannual, so we need the effective semiannual rate. The
interest rate given is the APR, so the monthly interest rate is:
To get the semiannual interest rate, we can use the EAR equation, but instead of using 12 months as
the exponent, we will use 6 months. The effective semiannual rate is:
We can now use this rate to find the PV of the annuity. The PV of the annuity is:
Note, this is the value one period (six months) before the first payment, so it is the value at Year 9.
So, the value at the various times the questions asked for uses this value nine years from now.
Note, you can also calculate this present value (as well as the remaining present values) using the
number of years. To do this, you need the EAR. The EAR is:
So, we can find the PV at Year 5 using the following method as well:
The value of the annuity at the other times in the problem is:
CHAPTER 27 - 5
CHAPTER 27 - 6
53. a. If the payments are in the form of an ordinary annuity, the present value will be:
0 12345
$15,8
00
$15,8
00
$15,8
00
$15,8
00
$15,8
00
PVA = C({1 – [1/(1 + r)t]} / r ))
If the payments are an annuity due, the present value will be:
0 12345
$15,800 $15,8
00
$15,8
00
$15,8
00
$15,8
00
PVAdue = (1 + r) PVA
b. We can find the future value of the ordinary annuity as:
FVA = C{[(1 + r)t – 1] / r}
If the payments are an annuity due, the future value will be:
FVAdue = (1 + r) FVA
c. Assuming a positive interest rate, the present value of an annuity due will always be larger than
the present value of an ordinary annuity. Each cash flow in an annuity due is received one
54. The time line is:
0 1
…
59 60
–
We need to use the PVA due equation, that is:
CHAPTER 27 - 7
PVAdue = (1 + r) PVA
CHAPTER 27 - 8
Using this equation:
Notice, to find the payment for the PVA due we simply compound the payment for an ordinary
annuity forward one period.
55. The payment for a loan repaid with equal payments is the annuity payment with the loan value as the
PV of the annuity. So, the loan payment will be:
The interest payment is the beginning balance times the interest rate for the period, and the principal
payment is the total payment minus the interest payment. The ending balance is the beginning
balance minus the principal payment. The ending balance for a period is the beginning balance for
the next period. The amortization table for an equal payment is:
Year
Beginning
Balance
Total
Payment
Interest
Payment
Principal
Payment
Ending
Balance
1 $67,500.00 $16,462.62 $4,725.00 $11,737.62 $55,762.38
In the third year, $3,024.22 of interest is paid.
56. This amortization table calls for equal principal payments of $13,500 per year. The interest payment
is the beginning balance times the interest rate for the period, and the total payment is the principal
payment plus the interest payment. The ending balance for a period is the beginning balance for the
next period. The amortization table for an equal principal reduction is:
Year
Beginning
Balance
Total
Payment
Interest
Payment
Principal
Payment
Ending
Balance
1 $67,500.00 $18,225.00 $4,725.00 $13,500.00 $54,000.00
CHAPTER 27 - 9
In the third year, $2,835 of interest is paid.
Notice that the total payments for the equal principal reduction loan are lower. This is because more
principal is repaid early in the loan, which reduces the total interest expense over the life of the loan.
Challenge
57. The time line is:
0 1
…
120 …360 361 …660
0
The cash flows for this problem occur monthly, and the interest rate given is the EAR. Since the cash
flows occur monthly, we must get the effective monthly rate. One way to do this is to find the APR
based on monthly compounding, and then divide by 12. So, the preretirement APR is:
And the post-retirement APR is:
First, we will calculate how much he needs at retirement. The amount needed at retirement is the PV
of the monthly spending plus the PV of the inheritance. The PV of these two cash flows is:
So, at retirement, he needs:
He will be saving $2,500 per month for the next 10 years until he purchases the cabin. The value of
his savings after 10 years will be:
After he purchases the cabin, the amount he will have left is:
CHAPTER 27 - 10
He still has 20 years until retirement. When he is ready to retire, this amount will have grown to:
So, when he is ready to retire, based on his current savings, he will be short:
This amount is the FV of the monthly savings he must make between years 10 and 30. So, finding
the annuity payment using the FVA equation, we find his monthly savings will need to be:
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 leasing is the PV of the lease
payments, plus the $1,200. 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
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
The PV of the decision to purchase is:
In this case, it is cheaper to buy the car than leasing it since the PV of the purchase cash flows is
The break-even resale price is the FV of this value, so:
CHAPTER 27 - 11
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 PV of the current contract offer is the sum of the PV of the cash flows. So, the PV is:
PV = $6,500,000 + $5,100,000 / 1.0492 + $5,600,000 / 1.04922 + $6,100,000 / 1.04923
The player wants the contract increased in value by $2,000,000, so the PV of the new contract will
be:
The player has also requested a signing bonus payable today in the amount of $10 million. We can
To find the quarterly payments, first realize that the interest rate we need is the effective quarterly
rate. Using the daily interest rate, we can find the quarterly interest rate using the EAR equation,
with the number of days being 91.25, the number of days in a quarter (= 365 / 4). The effective
quarterly rate is:
Now we have the interest rate, the length of the annuity, and the PV. Using the PVA equation and
solving for the payment, we get:
60. The time line is:
0 1
$21,650 –
$25,00
0
To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the
CHAPTER 27 - 12
Because of the discount, you only get the use of $21,650, and the interest you pay on that amount is
15.47%, not 13.4%.
61. The time line is:
–24 –23
…
–12 –11
…
0 1 …60
Here we have cash flows that would have occurred in the past and cash flows that would occur in the
To find the value today of the back pay from two years ago, we will find the FV of the annuity, and
then find the FV of the lump sum. Doing so gives us:
Notice we found the FV of the annuity with the effective monthly rate, and then found the FV of the
Now, we need to find the value today of last year’s back pay:
Next, we find the value today of the five year’s future salary:
The value today of the jury award is the sum of salaries, plus the compensation for pain and
suffering, and court costs. The award should be for the amount of:
As the plaintiff, you would prefer a lower interest rate. In this problem, we are calculating both the
PV and FV of annuities. A lower interest rate will decrease the FVA, but increase the PVA. So, by a
CHAPTER 27 - 13
62. Again, 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:
The amount you will receive today is the principal amount of the loan times one minus the points.
The time line is:
0 1
$9,800 –
$10,80
0
Now, we find the interest rate for this PV and FV.
