Question

X P6-10 (similar to) Coupon rates. What are the coupon rates for the following bonds, Hint: Make sure to round all intermediate calculations to at least six decimal places. a. What is the coupon rate for the following bond? (Round to two decimal places.) Coupon Rate % Years to Maturity Yield to Maturity Price Coupon Frequency Par Value $1,000.00 0 Data Table (Click on the following icon in order to copy its contents into a spreadsheet.) Coupon Rate Years to Maturity Yield to Maturity 8% 30 Par Value $1,000.00 $1,000.00 $5,000.00 $1.000.00 79 Price $660.65 $1,125.65 $5.210.62 $697.27 Coupon Frequency semiannual quarterly annual monthly 12% Print Done

Answer 1

Par Value	Coupon Rate	Years to Maturity	Yield to Maturity	Price	Coupon Frequency
$1,000	5%	30	8%	$660.65	Semi-annual
$1,000	10%	5	7%	$1,125.65	Quarterly
$5,000	7%	5	6%	$5,210.62	Annual
$1,000	8%	20	12%	$697.27	Monthly

Explanation:

Bond price = C x PVIFA (n, i) + F x PVIF (n, i)

C = Annual coupon amount

n = Number of periods to maturity

i = Periodic rate

F = Face value

1)

n = 30 x 2 = 60

i = 8 % p.a. or 0.08/2 = 0.04 semi-annually

F = $ 1,000

Bond price = C x PVIFA (60, 4%) + F x PVIF (60, 4%)

                    = C x [{1-(1+0.04)^-60}/0.04] + $ 1,000 x (1+0.04)^-60

                    = C x [{1-(1.04)^-60}/0.04] + $ 1,000 x (1.04)^-60

                    = C x [(1-0.0950604010209042)/0.04] + $ 1,000 x 0.0950604010209042

                    = C x (0.9049395989790958/0.04) + $ 95.0604010209042

                    = C x 22.6234899744774 + $ 95.0604010209042

$ 660.65 = C x 22.6234899744774 + $ 95.0604010209042

C x 22.6234899744774 = $ 660.65 - $ 95.0604010209042

                                       = $ 565.5895989790958

C = $ 565.5895989790958/22.6234899744774 = $ 25.0001038574139

C = Face value x coupon rate/annual coupon frequency

$ 25.0001038574139 = $ 1,000 x coupon rate/2

Coupon rate/2 = $ 25.0001038574139/$ 1,000

                        = 0.02500010385741

Coupon rate = 2 x 0.02500010385741 = 0.05000020771482 or 5 %

2)

i = 0.07/4 = 0.0175 p. q.

n = 5 x 4 = 20, F = $ 1,000

Bond price = = C x PVIFA (20, 0.0175) + F x PVIF (60, 0.0175)

                    = C x [{1-(1+0.0175)^-20}/0.0175] + $ 1,000 x (1+0.0175)^-20

                    = C x [{1-(1.0175)^-20}/0.0175] + $ 1,000 x (1.0175)^-20

                    = C x [(1-0.70682457716687)/0.0175] + $ 1,000 x 0.70682457716687

                    = C x (0.29317542283313/0.0175) + $ 706.82457716687

                    = C x 16.7528813047503 + $ 706.82457716687

$ 1,125.65 = C x 16.7528813047503 + $ 706.82457716687

C x 16.7528813047503 = $ 1,125.65 - $ 706.82457716687

                                      = $ 418.82542283313

C = $ 418.82542283313/16.7528813047503 = $ 25.0002023660474

C = Face value x coupon rate/annual coupon frequency

$ 25.0002023660474 = $ 1,000 x Coupon rate/4

Coupon rate/4 = $ 25.0002023660474/$ 1,000

                        = 0.0250002023660474

Coupon rate = 4 x 0.0250002023660474 = 0.10000080946419 or 10 %

3)

i = 6 % p. a., n = 5, F = $ 5,000

Bond price = = C x PVIFA (5, 6 %) + F x PVIF (5, 6%)

                    = C x [{1-(1+0.06)^-5}/0.06 + $ 5,000 x (1+0.06)^-5

                    = C x [{1-(1.06)^-5}/0.06] + $ 5,000 x (1.06)^-5

                    = C x [(1-0.747258172866057)/0.06] + $ 5,000 x 0.747258172866057

                    = C x (0.252741827133943/0.06) + $ 3,736.29086433028

                    = C x 4.21236378556572 + $ 3,736.29086433028

$ 5,210.62 = C x 4.21236378556572 + $ 3,736.29086433028

C x 4.21236378556572 = $ 5,210.62 - $ 3,736.29086433028

                                       = $ 1,474.32913566972

C = $ 1,474.32913566972/4.21236378556572 = $ 350.000429858817

C = Face value x coupon rate/annual coupon frequency

$ 350.000429858817 = $ 5,000 x Coupon rate

Coupon rate = $ 350.000429858817 /$ 5,000

                            = 0.0700000859717634 or 7 %

4)

i = 12 % p. a. or 0.12/12 = 0.01 p.m.

n = 20 x 12 = 240, F = $ 1,000

Bond price = = C x PVIFA (240, 1 %) + F x PVIF (240, 1%)

                    = C x [{1-(1+0.01)^-240}/0.01 + $ 1,000 x (1+0.01)^-240

                    = C x [{1-(1.1)^-240}/0.01] + $ 1,000 x (1.01)^-240

                    = C x [(1-0.0918058365169841)/0.01] + $ 1,000 x 0.0918058365169841

                    = C x (0.9081941634830159/0.01) + $ 91.8058365169841

                    = C x 90.8194163483016 + $ 91.8058365169841

$ 697.27 = C x 90.8194163483016 + $ 91.8058365169841

C x 90.8194163483016 = $ 697.27 - $ 91.8058365169841

                                       = $ 605.4641634830159

C = $ 605.4641634830159/90.8194163483016 = $ 6.66668194784472

C = Face value x coupon rate/annual coupon frequency

$ 6.66668194784472 = $ 1,000 x Coupon rate/12

Coupon rate/12 = $ 6.66668194784472/$ 1,000

                          = 0.00666668194784472

Coupon rate = 0.00666668194784472 x 12 = 0.0800001833741367 or 8 %