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35. The time line is:
0
1
…
15
PV
$5,250
$5,250
$5,250
$5,250
$5,250
$5,250
$5,250
$5,250
$5,250
The relationship between the PVA and the interest rate is:
PVA@15% = $5,250{[1 – (1/1.15)15]/.15} = $30,698.69
36. The time line is:
0
1
…
?
–$25,000
$190
$190
$190
$190
$190
$190
$190
$190
$190
t = ln 1.95943/ln 1.00729
t = 92.59 payments
37. The time line is:
0
1
…
60
–$105,000
$2,025
$2,025
$2,025
$2,025
$2,025
$2,025
$2,025
$2,025
$2,025
The APR is the periodic interest rate times the number of periods in the year, so:
APR = 12(.492%) = 5.90%
38. The time line is:
0
1
…
360
PV
$875
$875
$875
$875
$875
$875
$875
$875
$875
39. The time line is:
0
1
2
3
4
–$5,800
$1,300
?
$1,900
$2,450
We are given the total PV of all four cash flows. If we find the PV of the three cash flows we know, and
subtract them from the total PV, the amount left over must be the PV of the missing cash flow. So, the
PV of the cash flows we know are:
PV of Year 1 CF: $1,300/1.08 = $1,203.70
PV of Year 3 CF: $1,900/1.083 = $1,508.28
PV of Year 4 CF: $2,450/1.084 = $1,800.82
So, the PV of the missing CF is:
$5,800 – 1,203.70 – 1,508.28 – 1,800.82 = $1,287.19
The question asks for the value of the cash flow in Year 2, so we must find the future value of this
amount. The value of the missing CF is:
$1,287.19(1.08)2 = $1,501.38
Amount borrowed = .80($3,900,000) = $3,120,000
The time line is:
0
1
…
360
–$3,120,000
$18,250
$18,250
$18,250
$18,250
$18,250
$18,250
$18,250
$18,250
$18,250
Using the PVA equation:
PVA = $3,120,000 = $18,250[{1 – [1/(1 + r)360]}/r]
Unfortunately, this equation cannot be solved to find the interest rate using algebra. To find the interest
rate, we need to solve this equation on a financial calculator, using a spreadsheet, or by trial and error.
If you use trial and error, remember that increasing the interest rate decreases the PVA, and decreasing
the interest rate increases the PVA. Using a spreadsheet, we find:
r = .481%
The APR is the monthly interest rate times the number of months in the year, so:
APR = 12(.481%)
APR = 5.77%
And the EAR is:
EAR = (1 + .00481)12 – 1
EAR = .0593, or 5.93%
42. The time line is:
And the firm’s profit is:
Profit = $109,678.71 – 102,000
Profit = $7,678.71
To find the interest rate at which the firm will break even, we need to find the interest rate using the
PV (or FV) of a lump sum. Using the PV equation for a lump sum, we get:
0
3
–$102,000
$150,000
$102,000 = $150,000/(1 + r)3
r = ($150,000/$102,000)1/3 – 1
r = .1372, or 13.72%
43. The time line is:
0
1
…
6
7
…
30
$3,500
$3,500
$3,500
$3,500
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 24 payments, so the PV of the annuity is:
PVA = $3,500{[1 – (1/1.076)24]/.076}
PVA = $38,113.74
Since this is an ordinary annuity equation, this is the PV one period before the first payment, so it is
the PV at t = 6. To find the value today, we find the PV of this lump sum. The value today 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:
PV = $121,161.48/[1 + (.114/12)]84
PV = $54,756.22
45. The time line for the annuity is:
0
1
…
180
$1,250
$1,250
$1,250
$1,250
$1,250
$1,250
$1,250
$1,250
$1,250
FV
Here, we are trying to find the dollar amount invested today that will equal the FVA with a known
46. The time line is:
0
1
…
7
…
14
15
…
∞
PV
$3,250
$3,250
$3,250
$3,250
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:
PV = $3,250/.064
PV = $50,781.25
0
1
…
7
…
14
PV
$50,781.25
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:
PV = $50,781.25/1.0647
PV = $32,893.64
47. The time line is:
0
1
…
12
–$23,000
$2,234.83
$2,234.83
$2,234.83
$2,234.83
$2,234.83
$2,234.83
$2,234.83
$2,234.83
$2,234.83
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:
PVA = $23,000 = $2,234.83{(1 – [1/(1 + r)]12 )/r }
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:
0
1
…
…
18
19
…
28
$6,500
$6,500
$6,500
$6,500
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:
Semiannual rate = (1.075)6 – 1 = 4.59%
We can now use this rate to find the PV of the annuity. The PV of the annuity is:
PVA @ T = 9: $6,500[(1 – 1/1.045910)/.0459] = $51,217.83
Note, that this is the value one period (six months) before the first payment, so it is the value at t = 9.
So, the value at the various times the questions asked for uses this value 9 years from now.
PV @ T = 5: $51,217.83/1.04598 = $35,781.50
Note, that 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:
PV @ T = 0: $51,217.83/1.045918 = $22,853.63
PV @ T = 0: $51,217.83/1.09389 = $22,853.63
49. a. The time line for the ordinary annuity is:
0
1
2
3
4
5
PV
$17,500
$17,500
$17,500
$17,500
$17,500
If the payments are in the form of an ordinary annuity, the present value will be:
PVA = C({1 – [1/(1 + r)t]}/r ))
PVA = $17,500[{1 – [1/(1 + .074)5]}/ .074]
PVA = $70,991.47
The time line for the annuity due is:
0
1
2
3
4
5
PV
$17,500
$17,500
$17,500
$17,500
$17,500
PVAdue = $76,244.84
b. The time line for the ordinary annuity is:
0
1
2
3
4
5
FV
$17,500
$17,500
$17,500
$17,500
$17,500
We can find the future value of the ordinary annuity as:
FVA = C{[(1 + r)t – 1]/r}
FVA = $17,500{[(1 + .074)5 – 1]/.074}
FVA = $101,444.28
The time line for the annuity due is:
0
1
2
3
4
5
$20,000
$20,000
$20,000
$20,000
$20,000
FV
If the payments are an annuity due, the future value will be:
FVAdue = (1 + r)FVA
FVAdue = (1 + .074)$101,444.28
FVAdue = $108,951.16
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 period
earlier, which means there is one period less to discount each cash flow. Assuming a positive
interest rate, the future value of an ordinary due will always be higher than the future value of an
ordinary annuity. Since each cash flow is made one period sooner, each cash flow receives one
extra period of compounding.
50. The time line is:
0
1
…
59
60
–$83,000
C
C
C
C
C
C
C
C
C
We need to use the PVA due equation, that is:
PVAdue = (1 + r)PVA
Using this equation:
PVAdue = $83,000 = [1 + (.0489/12)] × C[{1 – 1/[1 + (.0489/12)]60}/(.0489/12)
C = $1,555.79
Notice, to find the payment for the PVA due we compound the payment for an ordinary annuity
forward one period.
