Question

A 30-year maturity bond with face value of $1,000 makes semiannual coupon payments and has a coupon rate of 9.40%. (Do not round intermediate calculations. Enter your answers as a percent rounded to 3 decimal places.)

a. What is the yield to maturity if the bond is selling for $1,100?

b. What is the yield to maturity if the bond is selling for $1,000?

c. What is the yield to maturity if the bond is selling for $1,090?

Answer 1

a. the yield to maturity if the bond is selling for $1,100

YTM is the rate at which PV of Cash Inflows are equal to Bond Price, when the bond is hold till the maturity.

Low rate 4.5 %

higher rate 5 %

Periodic coupon amount = $ 1000 * 9.4% * 1/2 = $ 47

No. of periods = 30 years * 2 = 60 periods

Maturity value = $1100

Period	Cash Flow	PVF/PVAF @ 4.5 %	PV of Cash Flows	PVF/ PVAF @5 %	PV of Cash Flows
1-60	$ 47	20.6380	$                969.99	18.9293	$                   889.68
60	$ 1,100	0.0713	$                   78.42	0.0535	$                     58.89
PV of Cash Inflows			$             1,048.40		$                   948.57
PV of Cash Oiutflows			$             1,000.00		$               1,000.00
NPV			$                   48.40		$                   -51.43

PVAF is sum of PVFs

YTM per six months = Rate at which least +ve NPV + [ NPV at that rate / Change in NPV due to Inc of 0.5% in Int Rate ] * 0.5%
= 4.5 % + [48.4 / 99.84 ] * 0.5%
= 4.5 % + [0.48 * 0.5% ]
= 4.5 % + [0.2424 % ]
= 4.74 %

YTM Per anum = YTM per six months * 12 / 6
= 4.7424 % * 2
= 9.4848 %

b. the yield to maturity if the bond is selling for $1,000

Low rate 4.5 %

higher rate 5 %

Periodic coupon amount = $ 1000 * 9.4% * 1/2 = $ 47

No. of periods = 30 years * 2 = 60 periods

Maturity value = $1000

Period	Cash Flow	PVF/PVAF @ 4.5 %	PV of Cash Flows	PVF/ PVAF @5 %	PV of Cash Flows
1-60	$             47	20.6380	$                969.99	18.9293	$                   889.68
60	$       1,000	0.0713	$                   71.29	0.0535	$                     53.54
PV of Cash Inflows			$             1,041.28		$                   943.21
PV of Cash Outflows			$             1,000.00		$               1,000.00
NPV			$                   41.28		$                   -56.79

YTM per six months = Rate at which least +ve NPV + [ NPV at that rate / Change in NPV due to Inc of 0.5% in Int Rate ] * 0.5%
= 4.5 % + [41.28 / 98.06 ] * 0.5%
= 4.5 % + [0.42 * 0.5% ]
= 4.5 % + [0.2105 % ]
= 4.71 %

YTM Per anum = YTM per six months * 12 / 6
= 4.7105 % * 2
= 9.4209 %
i.e 9.42 %

c. the yield to maturity if the bond is selling for $1,090

Low rate 4.5 %

higher rate 5 %

Periodic coupon amount = $ 1000 * 9.4% * 1/2 = $ 47

No. of periods = 30 years * 2 = 60 periods

Maturity value = $1090

Period	Cash Flow	PVF/PVAF @ 4.5 %	PV of Cash Flows	PVF/ PVAF @5 %	PV of Cash Flows
1-60	$             47	20.6380	$                969.99	18.9293	$                   889.68
60	$       1,090	0.0713	$                   77.71	0.0535	$                     58.35
PV of Cash Inflows			$             1,047.69		$                   948.03
PV of Cash Outflows			$             1,000.00		$               1,000.00
NPV			$                   47.69		$                   -51.97

YTM per six months = Rate at which least +ve NPV + [ NPV at that rate / Change in NPV due to Inc of 0.5% in Int Rate ] * 0.5%
= 4.5 % + [47.69 / 99.66 ] * 0.5%
= 4.5 % + [0.48 * 0.5% ]
= 4.5 % + [0.2393 % ]
= 4.74 %

YTM Per anum = YTM per six months * 12 / 6
= 4.7393 % * 2
= 9.4785 %
i.e 9.48 %

PVAF (4.5% , 60 periods)

PV Annuity Factor = [ 1 - [(1+r)^-n]] /r
= [ 1 - [(1+0.045)^-60]] /0.045
= [ 1 - [(1.045)^-60]] /0.045
= [ 1 - [0.07129]] /0.045
= [0.92871]] /0.045
= 20.638

PVAF (5% , 60 periods)
PV Annuity Factor = [ 1 - [(1+r)^-n]] /r
= [ 1 - [(1+0.05)^-60]] /0.05
= [ 1 - [(1.05)^-60]] /0.05
= [ 1 - [0.05354]] /0.05
= [0.94646]] /0.05
= 18.9293

please comment if any further assistance is required.