Solutions Manual, The Mathematics of Money
CHAPTER 2
EXERCISES 2.1
1-6: Recall that the maturity value is the amount to be paid to the lender at the end of the note’s term to pay off the loan. The proceeds is that amount given to the borrower/buyer for the note. The proceeds is the same things as what we would call the principal if we were looking at things as simple interest.
The discount is the difference between the proceeds and the maturity value. The discount is equal to what we would call the interest if we were looking at things as simple interest.
1. a. $800
b. $750
c. $800 - $750 = $50
2. a. $20,000
b. $20,000 - $1,000 = $19,000
c. $1,000
3. a. $20,000
b. $18,000
c. $20,000 - $18,000 = $2,000
4. a. $23,467.19
b. $24,500
c. $24,500 - $23,467.19 = $1,032.81
5. a. $1,308.55
b. $1275
c. $1,308.55 - $1275 = $33.55
6. a. $25
b. $20
c. $5
7.
a. D = MdT
D = ($10,000)(0.09)(200/365)
D = $493.15
b. Recall that discount is subtracted from the maturity value to get the proceeds.
$10,000 - $493.15 = $9,506.85
8. D = MdT
D = ($1,000)(0.03)(5)
D = $150
The proceeds are found by subtracting discount from the maturity value:
$1,000 - $150 = $850
9. D = MdT
D = ($100,000)(0.35)(2/12)
D = $5,833.33
This discount would be subtracted from the $100,000 maturity value:
$100,000 - $5,833.33 = $94,166.67
10. D = MdT
D = ($20,000)(0.0319)(175/365)
D = $305.89
Discount is subtracted from the maturity value to get the proceeds:
$20,000 - $305.89 = $19,694.11
11. D = MdT
$250 = M(0.125)(6/12)
250 = M(0.0625)
Divide both sides by 0.0625
M = $4,000.00
12. The amount of the discount is $25,000 - $24,325 = $675.
D = MdT
$675 = $25,000(d)(90/365)
675 = 6164.383562d
Divide both sides by 6164.383562
d = 0.1095 = 10.95%
13. The amount of the discount is $10,000 - $9,759.16 = $240.84
D=MdT
$240.84 = ($10,000)(d)(217/365)
240.84 = 5945.205479d
Divide both sides by 5945.205479
d = 0.0405099539 = 4.05%
14. The amount of the discount is $5,000 - $4,975 = $25
D = MdT
$25 = $5,000(0.04)T
25 = 200T
Divide both sides by 200
T = 0.125 years
Multiply by 365 and round to convert to days
T = 46 days
15. The amount of the discount is $800 - $785.21 = $14.79
D = MdT
$14.79 = ($800)(0.75)T
14.79 = 600T
Divide both sides by 600
T = 0.2465 years
Multiply by 365 and round to convert to days
T = 9 days
16. April 8 is day 90+8 = 98; June 17 is day 151+17 = 168. So the term of the note is 168-98 = 70 days.
D = MdT
D = ($2,000)(0.0572)(70/365)
D = $21.94
Subtract the discount from the maturity value to find the proceeds
$2,000 - $21.94 = $1,978.06
17. The amount of the discount is $180 - $160 = $20.
D = MdT
$20 = ($180)d(1/12)
20 = 15 d
Divide both sides by 15
d = 1.3333333333 = 133.33%
18. $8,000 - $7,856.02 = $143.98
19. The amount of the discount is $5,000 - $4,876.52 = $123.48
D = MdT
$123.48 = $5,000(d)(182/365)
123.48 = 2493.150685 d
Divide both sides by 2493.150685
d = 0.495276923 = 4.95%
20. $23,507 - $1,851.03 = $21,655.97
21. D = MdT
D = $875,000(0.24)(45/365)
D = $25,890.41
To find the proceeds, subtract the discount from the maturity value.
$875,000 - $25,890.41 = $849,109.59
22. June 28 is day 151+28 = 179; December 31 is day 365. The term is therefore 365 – 179 = 186.
D = MdT
D = ($1,000)(0.0589)(186/365)
D = $30.01
23. The amount of the discount is $875.39 - $850 = $25.39
D = MdT
$25.39 = ($875.39)(d)(50/365)
25.39 = 119.9164384 d
Divide both sides by 119.9164384
d = 21.17%
24. April 1 is day 90+1 = 91; November 15 is day 304+15 = 319. The term is 319 – 91 = 228 days.
D = MdT
D = ($10,000)(0.0821)(228/365)
D = $512.84
The proceeds are $10,000 - $512.84 = $9,487.16
25. D = MdT
D = $10,000(0.0532)(28/365)
D = $40.81
The proceeds are $10,000 - $40.81 = $9,959.19
26. The amount of the discount is $1,000 - $982.56 = $17.44
D = MdT
$17.44 = $1,000(d)(91/365)
17.44 = 249.3150685 d
Divide both sides by 249.3150685
d = 0.0699516484 = 7.00%
27. The term of this loan is 15 – 1 = 14 days. (You could also calculate the number of days by finding the Julian dates for May 15 and May 1 and subtracting, but since the dates fall in the same month this is not really necessary to find the term.)
D = MdT
D = $17,500,000(0.0398)(14/365)
D = $26,715.07
The proceeds are $17,500,000 - $26,715.07 = $17,473,284.93
28. a. The amount of the discount is $5,000 - $4,905.75 = $94.25
D = MdT
$94.25 = ($5,000)(0.04)T
94.25 = 200T
Divide both sides by 200
T = 0.47125 years
Multiply by 365 and round to convert years to days
T = 172 days
b. March 12 is day 59+12 = 71. 172 days after that is day 71+172 = 243. This date is actually one of the numbers listed in the abbreviated table. The maturity date is August 31.
29. The amount of the discount is $100,000 - $99,353.29 = $646.71
D = MdT
$646.71 = $100,000(0.0444)T
646.71 = 4440T
Divide both sides by 4440
T = 0.1456554054 years
Multiply by 365 and round to convert years to days
T = 53 days
April 3 is day 90+3 = 93. 53 days after that is day 93+53=146.
The largest number in the abbreviated table less than 146 is 120, the end of April. So the maturity date is 146-120 = 26 days after that: May 26, 2007.
30. The amount of the discount is $150,000 - $117,300 = $32,700
D = MdT
$32,700 = $150,000(d)(9/12)
32,700 = 112,500 d
Divide both sides by 112,500
d = 29.07%
Note that this rate is based on an assumed remaining life expectancy. Because the insured’s actual time left can not be known for certain, the actual simple discount rate cannot be known in advance.
EXERCISES 2.2
1-4. Recall the definition of these terms.
Both the principal (a) and proceeds (e) refer to the amount borrowed.
The maturity value (b) and (f) refers to the amount to be paid by the borrower to the lender at the end of the loan’s term, whether simple interest or simple discount.
The amount of interest (c) is the same as the amount of discount (g).
The face value (d) when using simple interest is the same as the principal. When using simple discount, the face value (h) is the same as the maturity value.
1. a) $875
b) $900
c) $900 - $875 = $25
d) $875 (same as (a))
e) same as a)
f) same as b)
g) same as c)
h) same as b)
2. a) $4,753,259
b) $5,000,000
c) $5,000,000 - $4,753,259 = $246,741
d) same as a)
e) same as a)
f) same as b)
g) same as c)
h) same as b)
3. a) $352.45
b) $363.79
c) $363.79 - $352.45 = $11.34
d) same as a)
e) same as a)
f) same as b)
g) same as c)
h) same as b)
4. a) $18,355.17
b) $18,759.15
c) $18,759.15 - $18,355.17 = $403.98
d) same as a)
e) same as a)
f) same as b)
g) same as c)
h) same as b)
5. The amount of interest is $10,000 - $9,393.93 = $606.07. This is also the amount of discount.
a) I = PRT
$606.07 = $9,393.93(R)(300/365)
606.07 = 7721.038356 R
R = 7.85%
b) D = MdT
$606.07 = $10,000(d)(300/365)
606.07 = 8219.178082 d
d = 7.37%
6. The amount of interest/amount of discount is $70,000 - $62,500 = $7,500.
a) I = PRT
$7,500 = $62,500(R)(9/12)
7,500 = 46,875 R
R = 16.00%
b) D = MdT
$7,500 = $70,000(d)(9/12)
7,500 = 52,500 d
d = 14.29%
7. The amount of interest/amount of discount is $20 - $15 = $5.
a) I = PRT
$5 = $15(R)(14/365)
5 = 0.5753424658 R
R = 869.05%
b) D = MdT
$5 = $20(d)(14/365)
5 = 0.767123877 d
d = 651.79%
8. The amount of interest/amount of discount is $725,000 - $576,300 = $148,700
a) I = PRT
$148,700 = $576,300(R)(2)
148,700 = 1,152,600 R
R = 12.90%
b) D = MdT
$148,700 = $725,000(d)(2)
148,700 = 1,450,000 d
d = 10.26%
9. D = MdT
D = ($7,500)(0.12)(1)
D = $900
The proceeds are $7,500 - $900 = $6,600
I = PRT
$900 = ($6,600)R(1)
900 = 6,600R
R = 13.64%
10. D = MdT
D = ($5,000)(0.0835)(150/365)
D = $171.58
The proceeds are $5,000 - $171.58 = $4,828.42
I = PRT
$171.58 = ($4,828.42)R(150/365)
171.58 = 1,984.282192 R
R = 8.65%
11. D = MdT
D = ($4,250)(0.0955)(37/365)
D = $41.14
The proceeds are $4,250 - $41.14 = $4,208.86
I = PRT
$41.14 = ($4,208.86)(R)(37/365)
41.14 = 426.6515616 R
R = 9.64%
12. D = MdT
D = ($10,000)(0.0548)(73/365)
D = $109.60
The proceeds are $10,000 - $109.60 = $9,890.40
I = PRT
$109.60 = $9,890.40(R)(73/365)
109.60 = 1978 R
R = 5.54%
13. D = MdT
D = ($120,000)(0.40)(1)
D = $48,000
The proceeds are $120,000 - $48,000 = $72,000
I = PRT
$48,000 = ($72,000)(R)(1)
48,000 = 72,000(R)
R = 66.67%
14. a) (1.5%)($1,043.59) = (0.015)($1,043.59) = $15.65
b) $1,043.59 - $15.65 = $1,027.94
c) D = MdT
$15.65 = ($1,043.59)(d)(7/365)
15.65 = 21.01405479 d
d = 78.20%
d) I = PRT
$15.65 = ($1,027.94)(R)(7/365)
15.65 = 19.71391781 R
R = 79.39%
15. a) (0.0175)($789.95)+$5=$18.82
b) D = MdT
$18.82 = ($789.95)d(10/365)
18.82 = 21.64246575 d
d = 86.96%
c) The proceeds (=principal) is $789.95 - $18.82 = $771.13
I = PRT
$18.82 = ($771.13)(R)(10/365)
18.82 = 21.12684932 R
R = 89.08%
16. The amount of the discount is (0.04)($3,279.46) = $131.18
Discount rate: D = MdT
$131.18 = ($3,279.46)d(30/365)
131.18 = 269.5446575d
d = 48.67%
Interest rate: The proceeds (= principal) is $3,279.46 - $131.18 = $3,148.28
I = PRT
$131.18 = ($3,148.28)(R)(30/365)
131.18 = 258.7627397 R
R = 50.70%
17. The amount of the discount is (0.01)($750,000) = $7,500
Discount rate: D = MdT
$7,500 = ($750,000)d(16/365)
7,500 = 32,876.71233 d
d = 22.81%
Interest rate: The proceeds (= principal) is $750,000 - $7,500 = $742,500.
I = PRT
$7,500 = ($742,500)R(16/365)
7,500 = 32,547.94521 R
R = 23.04%
18. The amount of interest/discount is $475,000 - $471,500 = $3,500.
Discount rate:
D = MdT
$3,500 = ($475,000)(d)(27/365)
3,500 = 35,136.9863 d
d = 9.96%
Interest rate:
I = PRT
$3,500 = $471,500(R)(27/365)
3,500 = 34,878.08219 R
R = 10.03%
19. The amount of the discount is:
D = MdT
D = ($10,000)(0.0444)(135/365)
D = $164.22
The proceeds were $10,000 - $164.22 = $9,835.78
I = PRT
$164.22 = ($9,835.78)R(135/365)
164.22 = 3,637.891233 R
R = 4.51%
20. a) $482.36
b) $500
c) $500 - $482.36 = $17.64
d) same as a)
e) same as a)
f) same as b)
g) same as c)
h) same as b)
21. The discount is (0.10)($79.95) = $8.00. The amount paid would be $79.95 - $8.00 = $71.95.
I = PRT
$8.00 = $71.95(R)(3/12)
8 = 17.9875 R
R = 44.48%
22. D = MdT
D = ($25,000)(0.0509)(18/365)
D = $62.75
The proceeds would have been $25,000 - $62.75 = $24,937.25
I = PRT
$62.75 = ($24,937.25)(R)(18/365)
62.75 = 1,229.782192 R
R = 5.10%
23. D = MdT
D = ($1,800,000)(0.095)(3/12)
D = $42,750
So the company would receive proceeds of $1,800,000 - $42,750 = $1,757,250
To compare the two offers, we need to know the equivalent simple interest rate, to be able to compare it to the other offer’s simple interest rate.
I = PRT
$42,750 = ($1,757,250)(R)(3/12)
42,750 = 439,312.5 R
R = 9.73%
Even though 9.63% sounds like a higher rate compared to 9.5%, in actuality 9.63% is a better deal since the 9.5% discount rate is equivalent to a 9.73% interest rate.
24. $1,295 - $1,249.35 = $45.65
25. The amount of discount/amount of interest is $45.65. We can now calculate the simple discount rate.
I = PRT
$45.65 = $1249.35(R)(45/365)
45.65 = 154.0294521 R
R = 29.64%
This is actually higher than the rate charged by the credit card.
26. I = PRT
I = ($14,357)(0.15)(100/365)
I = $590.01
The maturity value is $14,357 + $590.01 = $14,947.01
D = MdT
$590.01 = ($14,947.01)d(100/365)
590.01 = 4,095.071233 d
d = 14.41%
27. The simple interest rate should always be higher than the simple discount rate, because the principal is always less than the maturity value. 7/31/07 must be the misprint.
28. D = MdT
D = ($1,000)(0.25)(2)
D = $500
The proceeds are $1,000 - $500 = $500
I = PRT
$500 = ($500)R(2)
500 = 1000R
R = 50%
29. D = MdT
D = ($1,000)(0.18)(5)
D = $900
The proceeds are $1,000 - $900 = $100
I = PRT
$900 = ($100)R(5)
900 = 5000R