Amortization of Loans, Residential Mortgages, and Sinking Funds
1) A debt of $10 000.00 with interest at 8% compounded quarterly is to be repaid by equal payments at
the end of every three months for two years.
a) Calculate the size of the monthly payments.
b) Construct an amortization table.
c) Calculate the outstanding balance after three payments.
Answer:
a) PV = 10 000.00; n = 2(4) = 8; i =
= 2% = 0.02; I/Y = 8; P/Y = C/Y = 4;
10000 = PMT
10000 = PMT[7.32548144]
PMT =
PMT = $1365.10
b) Amortization Table (done using Excel):
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c) PMT = 1365.10; n = 5(number of payments left); i = 0.02
PV = 1365.10
= $6434.34
2) A contractor's price for a new building was $85 100.00. Stampede Inc., the buyers of the building, paid
down and financed the balance by making equal payments at the end of every six months for
14 years. Interest is 9.8% compounded semi-annually.
a) What is the size of the semi-annual payment?
b) How much will Stampede Inc. owe after 6 years?
c) What is the total cost of the building for Stampede Inc.?
d) What is the total interest included in the payments?
Answer:
a) 68850.00 = PMT
68850.00 = PMT(15.06140866)
$4571.29 = PMT
b) PVn = 4571.29
= 4571.29(8.913452135) = $40745.97
c)
Total paid = 16250.00 + 28(4571.29)
= 16250.00 + 127996.12
= $144246.12
d) Total interest = 144246.12 - 85100.00 = $59 146.12
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3) Vicki receives payments of $4170.00 at the beginning of each month from a pension fund of $208
500.00. Interest earned by the fund is 6% compounded monthly. What is the number of payments Vicki
will receive?
Answer: 208500 = 4170
(1.005)
49.75124378 =
0.2487562189 = 1 - 1.005-n
-n ln 1.005 = ln 0.7512437811
-n(0.0049875415) = -0.2860250712
n = 57.3479079 payments
4) A investor's price for a townhouse was $160 000.00. Sepaba Investments., the buyers of the rental unit,
paid $40 000.00 down and financed the balance by making equal payments at the end of every six months
for 25 years. Interest is 6% compounded semi-annually.
a) What is the size of the semi-annual payment?
b) How much will Sepaba Investments. owe after 20 years?
c) What is the total cost of the building for Sepaba Investments?
d) What is the total interest included in the payments?
Answer:
a) 120000.00 = PMT
120000.00 = PMT(25.72976401)
$4663.86 = PMT
b) An = 4663.86
= 4663.86(8.530202837) = $39 783.67
c)
Total paid = 40000.00 + 50(4663.86)
= 40000.00 + 233193.00
= $273 193.00
d) Total interest = 273193.00 - 160000.00 = $113 193.00
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5) A loan of $14 100.00 is amortized over 11 years by equal monthly payments at 5.4% compounded
monthly. Construct an amortization schedule showing details of the first three payments, the fortieth
payment, the last three payments, and totals.
Answer: 14100 = PMT
14100 = PMT(99.3665357)
$141.90 = PMT
Partial Amortization Table (rounded to the nearest cent):
PVn = 141.90
= 141.90(75.8552917) = 10763.87
PVn = 141.90
= 141.90(2 .97320114) = $421.90
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6) A loan of $8380.00 is repaid by equal payments made at the end of every three months for 3 years. If
interest is 7% compounded quarterly, find the size of the quarterly payments and construct an
amortization schedule showing the total paid and the total cost of the loan.
Answer: 8380.00 = PMT
8380.00 = PMT(10.7395497)
$780.29 = PMT
Amortization Schedule (done using Excel):
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7) Rola Inc. borrowed $42 000.00 at 7% compounded semi-annually. The loan is repaid by payments of
$4700.00 due at the end of every six months.
a) How many payments are needed?
b) How much of the principal will be repaid by the 6th payment?
c) Prepare a partial amortization schedule showing the details of the last three payments and totals.
Answer:
a)
n=
=
=
= 10.90304697
The number of payments is 11.
b)
PVn = 4700
= 4700.00(5.2509011) = $24 679.24
Interest = 24679.24(.035) = $863.77
Principal repaid = 4700.00 - 863.77 = $3836.23
c) Partial Amortization (rounded the nearest cent):
PVn = 4700
= 4700(2.71554292) = $12 763.05
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8) A debt of $12 500.00 with interest at 5.5% compounded semi-annually is amortized by making
payments of $1000.00 at the end of every six months. Calculate the outstanding balance after seven years.
Answer: PV = 12 500.00; n = 7(2) = 14; i =
% = 0.0275; I/Y = 5.5; P/Y = C/Y = 2
The accumulated value of the original principal after five years
FV = 12500(1.0275)14 = $18 274.93
The accumulated value of the first ten payments
FV = 1000
= $16 799.79
The outstanding balance after five years is $18 274.93 - $16 799.79 = $1475.14
9) Olfert Inc. is repaying a loan of $52 500.00 by making payments of $4700.00 at the end of every six
months. If interest is 7.5% compounded semi-annually, construct an amortization schedule showing the
total paid and the total cost of the loan.
Answer: Amortization Schedule (done using Excel)
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10) Barbara borrowed $12 000.00 from the bank at 9% compounded monthly. The loan is amortized with
end-of-month payments over five years.
a) Calculate the interest included in the 20th payment.
b) Calculate the principal repaid in the 36th payment.
c) Construct a partial amortization schedule showing the details of the first two payments, the 20th
payment, the 36th payment, and the last two payments.
d) Calculate the totals of amount paid, interest paid, and the principal repaid.
Answer:
a) PV = 12 000.00; n = 5(12) = 60; i =
= 0.75% = 0.0075; I/Y = 9; P/Y = C/Y = 12;
12000 = PMT
12000 = PMT[48.17337352]
PMT =
PMT = $249.10
b) The outstanding balance after Payment 58:
PV = 249.10
= $492.65
Programmed solution:
The principal repaid is $12 000.00.
The total amount paid is 249.10(59) + 249.09 = $14 945.99
The total interest paid is 14945.99 - 12000.00 = $2945.99
Partial Amortization Schedule: (rounded to the nearest cent):
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11) Mr. Lamb borrowed $8321.00 at 11.12% compounded monthly. He agreed to repay the loan in equal
monthly payments over five years.
a) What is the size of the monthly payment?
b) How much of the 24th payment is interest?
c) What is the principal repaid in the 37th payment?
d) Prepare a partial amortization schedule showing details of the first three payments, the last three
payments, and totals.
Answer:
a) 8321.00 = PMT
8321.00 = PMT(45.86668688)
$181.42 = PMT
b) PVn = 181.42
= 181.42(31.2027738) = $5660.81
5660.81(0.009266667) = $52.46
c) PVn = 181.42
= 181.42(21.42999316) = $3887.83
181.42 - 3887.83(.009266667) = 181.42 - 36.03 = $145.39
d) Partial amortization table (done by hand, rounded to the nearest cent).
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12) A loan of $19 000.00 is repaid by quarterly payments of $900.00 each at 8% compounded quarterly.
What is the principal repaid by the 21st payment?
Answer: n =
=
=
= 27.7016753
PVn = 900
= 900(7.07263077) = $6365.37
Interest = 6365.37(0.02) = $127.31
Principal repaid = 900.00 - 127.31 = $772.69
13) A loan of $12 000.00 is repaid by quarterly payments of $1200.00 each at 12% compounded quarterly.
What is the principal repaid by the 11th payment?
Answer: n =
=
=
= 12.06662371
PVn = 1200
= 1200(10) = $12000
Interest = 12000(0.03) = $360
Principal repaid = 12000 - 360 = $11 640
14) Duguid and Partners bought a property valued at $87 300.00 for $17 000.00 down and a mortgage
amortized over 17 years. The firm makes equal payments due at the end of every three months. Interest
on the mortgage is 6.85% compounded annually and the mortgage is renewable after five years.
a) What is the size of each quarterly payment?
b) What is the outstanding principal at the end of the five-year term?
c) What is the cost of the mortgage for the first five years?
d) If the mortgage is renewed for a further five years at 7.17% compounded semi-annually, what will be
the size of each quarterly payment?
Answer:
a) p =
- 1 = 0.01670189
70300 = PMT
70300 = PMT(40.4615867)
$1737.45 = PMT
b) PVg = 1737.45
= 1737.45(32.8376079) = $57053.70
c)
Total paid = 1737.45(20)
= $34 749.00
Principal repaid = 70300.00 - 57053.70 = $13 246.30
Interest cost = 34749.00 - 13246.30
= $21 502.70
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d) p =
- 1 = 0.01776716
57053.70 = PMT
57053.70 = PMT(32.1146991)
$1776.56 = PMT
15) The owner of the Pink Flamingo Motel borrowed $19 500.00 at 8.11% compounded semi-annually and
agreed to repay the loan by making payments of $1110.00 at the end of every three months.
a) How many payments will be needed to repay the loan?
b) How much will be owed at the end of five years?
c) How much of the payments made at the end of five years will be interest?
Answer:
a) p =
- 1 = 0.02007353
n=
=
=
= 21.879938
The number of payments is 22.
b) PVg = 1110
= 1110(1.826974901) = $2027.94
c)
= $22 200
= $17 472.06
= $4727.94
Total paid = 1110(20)
Principal repaid = 19500.00 - 2027.94
Interest = 22200.00 - 17472.06
16) A debt of $12 970.00 with interest at 7.23% compounded semi-annually is repaid by payments of
$1880.00 made at the end of every three months. Construct an amortization schedule showing the total
paid and the total cost of the debt.
Answer: p =
- 1 = 0.01791454
Loan Amortization Schedule (done in Excel):
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17) A $30 000.00 mortgage is amortized by monthly payments over twenty years and is renewable after
five years.
a) If the interest rate is 8.5% compounded semi-annually, calculate the outstanding balance at the end of
the five-year term.
b) If the mortgage is renewed for a further three-year term at 8% compounded semi-annually, calculate
the size of the new monthly payment.
c) Calculate the payout figure at the end of the three-year term.
Answer:
a) PV = 30 000.00; n = 20(12) = 240; i =
= 0.0425; I/Y = 8.5; P/Y = 12; C/Y = 2; c =
p = (1 + i)c = (1 + 0.0425)1/6 - 1 = 0.006961062
30000 = PMT
30000 = PMT[116.4742048]
PMT = $257.57
The number of outstanding payments after five years is 15(12) = 180
PV = 257.57
PV = $26 386.10
b) The outstanding balance of $26 386.10 is to be amortized the remaining 15 years.
PV = 26 386.10; n = 15(12) = 180; i =
p=
=
= 0.04; I/Y = 8; P/Y = 12; C/Y = 2; c =
- 1 = 0.006558197
26386.10 = PMT
26386.10 = PMT[105.4682161]
PMT = $250.18
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=
=
c) At the end of the 3 year term the outstanding payments is
144.PV = 250.18
PV = $23 265.45
BONDS
1) A $5000 bond that pays 6% semi-annually is redeemable at par in 14 years. Calculate the purchase
price if it is sold to yield 8% compounded semi-annually.
Answer: PMT =
=
$150PP = 5000(1 + 0.04)-28 + 150
PP = 1667.39 + 2499.46
PP = $4166.85
2) A $25 000, 6% bond redeemable at par is purchased 11 years before maturity to yield 6.9%
compounded semi-annually. If the bond interest is payable semi-annually, what is the purchase price of
the bond?
Answer: FV = $5000; n = 11(2) = 22; P/Y = C/Y = 2; I/Y = 6.9; i =
= 0.0345; PMT = 25000(0.06)
=
$750.00
PP = 25000(1.0345)-22 + 750
= 25000.00(.4741646) + 750.00(15.2416064)
= 11854.11 + 11431.20
= $23 285.32
3)
A $50 000 bond bearing interest at 6.5% bond payable semi-annually matures in 10 years. If it is boughtto
yield 5.7% compounded semi-annually, what is the purchase price of the bond?
Answer: FV = $50 000; n = 10(2) = 20; P/Y = C/Y = 2; PMT = 50000(0.065)
PP = 50000(1.0285)-20 + 1625
= $1625; i =
= 0.0285
= 28502.57 + 24514.61 = $53 017.18
4) A $200 000 bond is redeemable at 108 in 14 years. If interest on the bond is 5.5% payable semi-annually,
what is the purchase price to yield 4% compounded semi-annually?
Answer: PP = 200000.00(1.08)(1.02)-28 + 5500.00
= 200000.00(1.08)(.5743746) + 5500.00(21.2812724)
= 124064.90 + 117047.00 = $241 111.90
Diff: 2
Type: SA
Page Ref: 604-618
Topic: 15.2 Purchase Price of a Bond when Market Rate Does Not Equal Bond Rate
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Objective: 15-2: Determine the purchase price of a bond when the market rate does not equal the bond rate.
5) What is the purchase price of a $10 000, 3.5% bond with semi-annual coupons redeemable at 104 in
seven years if the bond is bought to yield 2.5% compounded semi-annually?
Answer: FV = $10000(1.04) = $10400; P/Y = C/Y = 2; PMT = 10000(0.035/2) = $175.00; n = 7(2) = 14.
PP = 10400.00(1.0125)-14 + 175.00
= 10400.00(.8403681) + 175.00(12.7705528)
= 87398.28143 + 2234.84673
= $89 633.13
Diff: 1
Type: SA
Page Ref: 604-618
Topic: 15.2 Purchase Price of a Bond when Market Rate Does Not Equal Bond Rate
Objective: 15-2: Determine the purchase price of a bond when the market rate does not equal the bond rate.
6) A $100 000 bond is redeemable at 105 in 20 years. If interest on the bond is 5% payable semi-annually,
what is the purchase price to yield 6% compounded semi-annually?
Answer: FV = 100000(1.05) = $105000; PMT = 100000(0.05)
= $2500; n = 20(2) = 40; P/Y = C/Y = 2; i =
0.03
PP = 105000(1.03)-40 + 2500
= 105000(.3065568408) + 2500(23.11477197)
= 32188.47 + 57786.93 = $89 975.40
7) A $100 000, 4% bond with semi-annual coupons is redeemable at 108. What is the purchase price to
yield 5.5% compounded semi-annually seven years before maturity?
Answer: FV = $100000(1.08) = $108 000; PMT = 100000(0.04/2) = $2000; P/Y = C/Y = 2; i =
= 0.0275; n
=7(2) = 14.
PP = 108000(1.0275)-14 + 2000
= 73871.70584 + 22982.01627
= $96 853.72
8) A $5000, 6.5% bond with semi-annual coupons redeemable at 108 is bought two years before maturity
to yield 6% compounded semi-annually. What is the purchase price?
Answer: FV = $5000(1.08) = $5400; P/Y = C/Y = 2; PMT = $5000
= $162.50; i =
= 0.03
PP = 5400(1.03)-4 + 162.50
= 5400.00(.888487) + 162.50(3.7170984)
= 4797.83 + 604.03 = $5401.86
9) A $200 000.00, 6% bond with semi-annual coupons is redeemable at par. What is the purchase price of
the bond six years before maturity to yield 8% compounded semi-annually?
Answer: FV = $200 000; P/Y = C/Y = 2; PMT = 200000(0.06/2) = $6000; i =
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= 0.04; n = 6(2) =
=
12.PP = 200000(1.04)-12 + 6000
= 124919.4099 + 56310.44256
= $181 229.85
10) A $100 000, 7.5% bond with semi-annual coupons redeemable at 106 is bought three years before
maturity to yield 8% compounded semi-annually. What is the purchase price?
Answer: FV = $100000(1.06) = $106000; P/Y = C/Y = 2; PMT = $100000
PP = 106000.00(1.04)-6 + 3750.00
= 106000.00(.7903145257) + 3750.00(5.242136857)
= 83773.34 + 19658.01 = $103 431.35
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= $3 750; n = 3(2) = 6
11) A $150 000 bond bearing interest at 6% payable semi-annually is bought eight years before maturity to
yield 4.5% compounded annually. If the bond is redeemable at par, what is the purchase price?
Answer: FV = $150 000; PMT = 150000(0.06)0.5 = $4500; n = 8(2) = 16; P/Y = 2; C/Y = 1; c =
= 0.5; i =
= 0.045
p = (1 + 0.054)1/2 - 1 = 0.0222524
PP
= 150000(1.0222524)-16 + 4500
= 150000(.7031853) + 4500(13.3385481)
= 105477.77 + 60023.46 = $165 501.23
12) A $100 000 bond bearing interest at 8% payable semi-annually is bought five years before maturity to
yield 6% compounded annually. If the bond is redeemable at par, what is the purchase price?
Answer: FV = $100000; PMT = $100000(0.08)0.5 = $4000; P/Y = 2; C/Y = 1; c =
i=
= 0.06; p =
; n = 5(2) = 10;
- 1 = .029563014
PP = 100000(1.029563014)-10 + 4000
= 100000(.7472581729) + 4000(8.549257745)
= 74725.82 + 34197.03 = $108 922.85
13) Six $1500 bonds with 4.5% coupons payable semi-annually are bought to yield 5% compounded
monthly. If the bonds are redeemable at par in eight years, what is the purchase price?
Answer: FV = $1500(6) = $9 000.00; PMT = 9000(0.05)0.5 = $225.00; n = 8(2) = 16; P/Y = 2; C/Y = 12;
c=
PP
= 6; p =
- 1 = 0.02526187
= 9000 (1.02526187)-16 + 225
= 6037.90 + 2931.40
= $8969.30
14) A $150 000 bond redeemable at par on October 1, 2026, is purchased on January 15, 2014. Interest is
6% payable semi-annually and the yield is 7.5% compounded semi-annually.
a) What is the market price of the bond?
b) How much interest has accrued?
c) What is the cash price?
Answer: FV = $150 000; PMT = $4500; P/Y = C/Y = 2;
The interest date preceding the date of purchase is October 1, 2013. The time period from October 1,
2013to the date of maturity is 13 years. n = 13(2) = 26; i =
= 0.0375
a) PP on October 1, 2013 = 150000(1.0375)-26 + 4500
= 57597.09 + 73922.33
= $131 519.42
Market price = $131 519.42
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b) Days in the interest period Oct 1, 2013-Apr 1, 2014: 182
days;Days between Oct 1, 2013 and Jan 15, 2014: 106
Accrued interest = 150000(.03)
= $2620.88
c) Cash price = 131519.42 + 2620.88 = $134
140.30
15) A $50 000 bond bears interest at 4.5% payable semi-annually and is redeemable at par on November 4,
2022. The bond is sold on March 20, 2013, to yield 5.5% compounded semi-annually. What is the cash
price?
Answer: FV = 50 000; PMT = 5000
= $1125.00; P/Y = C/Y = 2; i =
= 0.0275
Since the maturity date is Nov.4, the semi-annual interest dates are May 4 and Nov. 4. The interest date
preceding the date of purchase is November 4, 2012. The time period from Nov. 4, 2012 to November 4,
2022 is 10 years.
n = 10(2) = 20
The purchase price on the date preceding the date of purchase
PP(Nov. 4, 2012) = 50000(1.0275)-20 + 1125
= 29062.52832 + 17130.65865 = $46 193.19
The purchase price is $46 193.19
The number of days from Nov.4, 2012 to May 4, 2013 is 181 days.
The number of days from Nov.4, 2012 to March 20, 2013 is 136 days.
PV = $46 193.19; r = i = 0.0225; t =
FV = $46193.19
= $46 974.14
The cash price on March 20, 2013 is $46 974.14
16) A $250 000, 7% bond redeemable at par with interest payable semi-annually is bought 4 years before
maturity. Determine the premium or discount and the purchase price if the bond is purchased to yield:
a) 6.5% compounded semi-annually
b) 8% compounded semi-annually
Answer: FV = $250 000; P/Y = 2; C/Y = 2; b =
$8750.00a) i =
= 0.035; PMT = 250 000(0.035) =
= 0.0325; n = 4(2) = 8.
Since b > i, the bond will sell at a premium.
PP = 250 000(1.0325)-8 + 8750
=
= 193 561.744 + 60 779.661
= $254 341.41
The premium is $254 341.41-$250 000 =
$4341.41b) i =
= 0.04; n = 4(2) = 8.
Since b < i, the bond will sell at a
discount.PP = 250 000(1.04)-8 + 8750
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=
= 182 675.551 + 58 911.518
= $241 587.07
The discount is $250 000 - $241 587.07 = $8412.93
17) A $125 000 bond, redeemable at par in three years with 7.5% coupons payable quarterly, is bought to
yield 6% compounded quarterly.
(i) Compute the premium or discount and the purchase price.
(ii) Construct a schedule for amortization of premium.
Answer: FV = $125 000; P/Y = C/Y = 4, b =
= 0.01875; i =
= 0.015; n = 3(4) = 12;
PMT = 125 000(0.01875) = $2 343.75. Since b > i, bond sells at a
premium.(i) Premium = [125000(0.01875) - 125000(0.015)]
= 468.75(10.907505) = $5 112.89
PP = 125 000.00 + 5112.89 = $130 112.89
(ii)
18) A $100 000 bond, redeemable at 110 in seven years with 6.75% coupons payable annually, is bought to
yield 7.25% compounded annually.
(i) Determine the discount and the purchase price.
(ii) Construct a schedule of accumulation of discount.
Answer: FV = $100 000; P/Y = C/Y = 1; n = 7(1) = 7; b =
= 0.0675; i =
PMT = 100 000(0.0675) = $6750.00. Since b < i, the bond sells at a
discount.(i) Discount = [100000(0.0675) -110000(0.0725)]
= -1225(5.342633) = -$6544.73
PP = 110000.00 - 6544.73 = $103 455.27
(ii)
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= 0.0725;
19) A $5000.00 6.5% bond redeemable at 109 with interest payable annually is purchased six years before
maturity to yield 7% compounded annually. Construct the appropriate bond schedule.
Answer: FV = $5 000(1.09) = $5450.00; P/Y = C/Y = 1; b =
= 0.065; i =
0.07;PMT = $5000(0.065) = $325
Discount = [5000(0.065) - 5450(0.07)]
= -56.50(4.7665397) = -$269.31
PP = 5450.00 - 269.31 = $5180.69
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=
20) A $5000, 6.25% bond with interest payable annually redeemable at par in eight years is purchased to
yield 7.5% compounded annually. Find the premium or discount and the purchase price and construct
the appropriate bond schedule.
Answer: FV = $5000; P/Y = C/Y = 1; b = 0.0625; i = 0.075; n = 8(1) = 8
Discount = [5000.00 * .0625 - 5000.00 * .075]
= -$62.50(5.8573036) = -$366.08
PP = 5000.00 - 366.08 = $4633.92
1) A $100 000, 6.0% bond with semi-annual coupons redeemable at par is bought 17.5 years before
maturity at 74.25. What was the approximate yield rate?
Answer: Quoted price: 100000.00 ∗ .7425 = $74 250.00
Average book value: .5(100000.00 + 74250.00) = $87 125.00
Semi-annual interest: 100000.00 * .03 = $3000.00
Total interest: 35(3000.00) = $105000.00
Discount: 100000.00 - 74250.00 = $25750.00
Average income:
(3000.00 + 105000.00) = $3085.71
Approximate value of i is
= .0354170445
Annual rate: 2(.0354170445) = 7.0834%
2) A $10 000.00, 5% bond with semi-annual coupons redeemable at 105 in 23 years is purchased at 102.5.
What is the approximate yield rate?
Answer: Initial book value: 100000.00 * 1.025 = $102 500.00
Redemption price: 100000.00 * 1.05 = $105 000.00
Average book value: .5(102500.00 + 105000.00) = $103 750.00
Total interest: 46(100000(0.025) = $115 000.00
Discount = 102500.00 - 105000.00 = $2500.00
Average income per interest payment interval =
Approximate value of i:
= $2554.35
= .0246202
Annual rate: 2(.0246202) = 4.92405%
3) A $50 000 bond that pays 5% semi-annually is redeemable at par on July 15, 2025.It is quoted at 97.5
on December 2, 2013. Determine the yield rate.
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Answer: The interest dates are January 15 and July 15. The nearest interest date is January 15, 2004 (11.5
years to maturity)
The quoted price is 50 000(0.975) = $48 750
The redemption value is 50 000
The average book value is
(48750 + 50000) = $49375.00
The semi-annual interest is 50000
= $1250
The number of interest payments to maturity is 11.5(2) = 23
The total interest payments are 23(1250) = $28 750.00
Discount is 50 000 - 48 750 = $1250
Average income per interest interval is
Approximate value of i is
= $1304.35
= 0.026417215 = 2.64%
The yield rate is 2(2.64) = 5.28%
4) A $250 000, 6.5% bond with semi-annual coupons redeemable at par is bought 17.5 years before
maturity at 78.25. What was the approximate yield rate?
Answer: Quoted price: 250000(0.7825) = $195 625.00
Average book value: .5(250 000 + 195 625) = $222 812.50
Semi-annual interest: 250 000(0.0325) = $8125.00
Total interest: 35(8125.00) = $284 375.00
Discount: 250000.00 - 195625.00 = $54 375.00
Average income:
(8125.00 + 284375.00) = $8357.14
Approximate value:
= .0375075
Annual rate: 2(.0375075) = 7.5015% ..
INVESTMENT DECISION
1) An obligation can be settled by making a payment of $7500.00 now and a final payment of $10 000.00 in
five years. Alternatively the obligation can be settled by payments of $750.00 at the end of every three
months for five years. If interest rate is 10% compounded quarterly, determine the preferred alternative.
Answer: PV of 10 000: n = 5(4) = 20; i =
= 0.025; FV = 10 000; I/Y = 10;P/Y = C/Y = 4
PV = 10000(1.025)-20 = 6102.71
ALT. 1 = 7500 + 6102.71 = 13 602.71 = $13 603
For ALT. 2:
PMT = 750; n = 5(4) = 20; i = 0.025; I/Y = 10; P/Y = C/Y = 4
PV = 750
= 11 691.87 = $11 692
ALT. 2 is better (less to pay)
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2) You win a lottery and have a choice of taking $200 000.00 immediately or taking payments of $7311.15
at the end of every three months for ten years. Which offer is preferable if interest is 8% compounded
quarterly?
Answer: PMT = 7311.15; n = 10(4) = 40; i = 0.02; I/Y = 8; P/Y = C/Y = 4
PV = 7311.15
= $200 000.01 = $200 000
Either option is preferable.
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3) An obligation can be settled by making a payment of $12 000 now and a final payment of $32 000 in 4
years. Alternatively, the obligation can be settled by payments of $2700 at the end of every three months
for five years. Interest is 10% compounded quarterly.
Compute the present value of each alternative and determine the preferred alternative according to the
discounted cash flow criterion.
Answer:
ALT. 1 = 12000.00 + 32000.00(1.025)-16
= 12000.00 + 32000.00(.6736249)
= 12000.00 + 21556.00 = $33 556
ALT. 2 = 2700.00
= 2700.00(15.5891623) = $42090.74 = $42 091
Since the present value of Alternative 1 is less than the present value of Alternative 2, Alternative 1 is
preferable.
4) An obligation can be settled by making a payment of $16 000 now and a final payment of $30 000 in 3
years. Alternatively, the obligation can be settled by payments of $2500 at the end of every three months
for four years. Interest is 12% compounded quarterly.
Compute the present value of each alternative and determine the preferred alternative according to the
discounted cash flow criterion.
Answer:
ALT. 1 = 16000.00 + 30000.00(1.03)-12
= 16000.00 + 30000.00(.701378802)
= 16000.00 + 21041.40 = $37041.40 = $37 041.
ALT. 2 = 2500.00
= 2500.00(12.56110203) = $31402.76 = $31 403
At 12% compounded quarterly Alternative 2 is preferable.
5) An expenditure may be met by outlays of $1700 now and $2210 at the end of every six months for 6
years or by making monthly payments of $500 in advance for seven years. Interest is 11% compounded
annually.
Compute the present value of each alternative and determine the preferred alternative according to the
discounted cash flow criterion.
Answer:
ALT. 1 p =
- 1 = .05356538
= 1700.00 + 2210.00
= 1700.00 + 2210.00(8.6876858) = 1700.00 + 19199.79 = $20899.79 = $20 900. BEST
ALT. 2 p =
= 500.00
- 1 = .0087346
(1.0087346)
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= 500.00(59.3435117)(1.0087346) = $29930.93 = $29 931
6) An expenditure may be met by outlays of $3000 now and $1000 at the end of every six months for 5
years or by making monthly payments of $250 in advance for three years. Interest is 12% compounded
annually.
Compute the present value of each alternative and determine the preferred alternative according to the
discounted cash flow criterion.
Answer:
ALT. 1 p =
- 1 = .05833005
= 3000.00 + 1000.00
= 3000.00 + 1000.00(7.419712748) = 3000.00 + 7419.71 = $10419.71 = $10 420
ALT. 2 p =
= 250.00
- 1 = .00948879
(1.00948879)
= 250.00(30.3747539)(1.009488793) = $7665.74 = $7666 BEST
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7) Sean and Jessica want to sell their interest in a small business. They have received two offers. If they
accept Offer A they will receive $40 000 immediately and $30 000 in two years. If the accept Offer B they
will receive $50 000 now and $2000 at the end of every six months for 5 years. If interest is 8%, which offer
is preferable?
Answer: p =
-1 = .0392305
ALT. A = 40000.00 + 30000.00(1.08)-2
= 40000.00 + 30000.00(.857339) = 40000.00 + 25720.16 = $65720.16 = $65 720
ALT. B = 50000.00 + 2000.00
= 50000.00 + 2000.00(8.142056025) = 50000.00 + 16284.11 = $66284.11 = $66 284 BEST
8) A piece of property may be acquired by making an immediate payment of $125 000 and payments of
$37 500 and $50 000 three and five years from now respectively. Alternatively, the property may be
purchased by making quarterly payments of $11 150 in advance for five years. Which alternative is
preferable if money is worth 12.2% compounded semi-annually?
Answer:
ALT. 1 = 125000.00 + 37500.00(1.061)-6 + 50000.00(1.061)-10
= 125000.00 + 37500.00(.7009833) + 50000.00(.5531541)
= 125000.00 + 26286.87 + 27657.71 = $178944.58 = $178 945.
ALT. 2: I/Y = 12.2%; P/Y = 4; C/Y = 2; c =
= 0.5; n = 4(5) = 20; p =
PV of ALT.2 = 11 150
(1.03004854)
-1 = .03004854
= 11150.00(14.87080004)(1.03004854) = $170791.75 = $170 792
Since the present value of the Alternative 2 is smaller than the present value of the Alternative 1,
Alternative 2 is preferable.
9) A contract is estimated to yield net returns of $7000.00 quarterly for seven years. To secure the
contract, an immediate outlay of $80 000.00 and a further outlay of $60 000.00 three years from now are
required. If interest is 6% compounded quarterly, determine if the investment should be accepted or
rejected.
Answer: PV of 60 000: n = 3(4) = 12; i =
= 0.015; FV = 60 000; I/Y = 6; P/Y = C/Y = 4
PV = 60000(1.015)-12 = 50183.25
PVout = 80 000 + 50 183.25 = 130 183.25 = $130 183
For PVIN:
PMT = 7000; n = 7(4) = 28; i = 0.015; I/Y = 6; P/Y = C/Y = 4
PVIN = 7000
= 159087.02 = $159 087
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10) A restaurant may be purchased for $250 000. Annual net income from the operation of the
restaurantis expected to be $61 000 for each of the first 4 years and $30 000 for each of the next
three years. After 7 years, the restaurant can be sold for $315 000. Determine the rate of return on
the investment correct to the nearest tenth of a percent.
a) Use linear interpolation to find the approximate value of the rate of return.
b) Find the answer using Cash Flow
and IRR.Answer: PVOUT = 250 000 315 000(1 + i)-7;
PVIN = 61 000
+ 30 000
(1 +
i)-4;NPV = PVIN - PVOUT
Index =
× 100%; Increase/Decrease in Rate = (Index -
100%)/4Calculations below show the attempts in approximating
the ROI
Attempts
PVOUT
PVIN
NPV
Index
Inc/Decr Rate
i = 16%
i = 28%
i = 24%
i = 22%
138 544
194 045
180 119
171 695
207 901
69 357
150
12
ROI > 16%
157 580
-364 644
81
-5
ROI < 28%
171 802
-8317
95
-1
ROI < 24%
179 768
8073
105
1
ROI > 22%
Hence, 22% < ROI < 24%. Use linear approximation to find the approximate value of ROI.
d=
= 0.985%
The rate of return, correct to the nearest tenth of a percent, is 22 + 0.985
= 23.0%b) CF0 = 250 000±
C01 = 61 000
F01 = 4
C02 = 30 000
F02 = 2
C03 = 345 000 F03 = 1
Compute IRR = 22.958989 = 23.0%
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