Question

1) Find the price for a 7.5% coupon bond under the following conditions.

1a. 30 years to maturity, required return is 9%

1b. 30 years to maturity, required return is 7.5%

1c. 30 years to maturity, required return is 6%

1d. 10 years to maturity, required return is 9%

1e. 10 years to maturity, required return is 7.5%

1f. 10 years to maturity, required return is 6%

1g. 2 years to maturity, required return is 9%

1h. 2 years to maturity, required return is 7.5%

1i. 2 years to maturity, required return is 6%

2) The current price of a 9.75% coupon bond with 20 years to maturity is $1318, what is the YTM? If the bond contains a call provision that allows the company to call the bond for $1050 7-years from now, what is the YTC? Based on the available information, is this bond likely to be called?

3) Find the price of a 20-year zero coupon bond if the required return on such a bond was 12%? What if the required return was 10%?

Answer 1

1) Find the price for a 7.5% coupon bond under the following conditions.

..

Assume bond par value is $1000

The formula is:

Value of Bond = ( Coupon interest * PVIFA ) + ( Par * PVIF )

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1a. 30 years to maturity, required return is 9%

.

Coupon interest = 1000 * 7.5% = 75

PVIFA = Present value of interest factor annuity, 9%, 30 years = 10.2736

Par value = 1000

PVIF = Present value interest factor, 9%, 30 years = 0.07537

.

Value of Bond = ( 75 * 10.2736 ) + ( 1000 * 0.07537 )

Value of Bond = 770.52 + 75.37 = $845.89

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1b. 30 years to maturity, required return is 7.5%

,

Coupon interest = 1000 * 7.5% = 75

PVIFA = Present value of interest factor annuity, 7.5%, 30 years = 11.81038

Par value = 1000

PVIF = Present value interest factor, 7.5%, 30 years = 0.1142

.

Value of Bond = ( 75 * 11.81038 ) + ( 1000 * 0.1142 )

Value of Bond = 885.8 + 114.2 = $1000

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1c. 30 years to maturity, required return is 6%

,

Coupon interest = 1000 * 7.5% = 75

PVIFA = Present value of interest factor annuity, 6%, 30 years = 13.76483

Par value = 1000

PVIF = Present value interest factor, 7.5%, 30 years = 0.17411

.

Value of Bond = ( 75 * 13.76483 ) + ( 1000 * 0.17411 )

Value of Bond = 1032.36 + 174.11 = $1206.47

.

.

1d. 10 years to maturity, required return is 9%

.

Coupon interest = 1000 * 7.5% = 75

PVIFA = Present value of interest factor annuity, 9%, 10 years = 6.41766

Par value = 1000

PVIF = Present value interest factor, 7.5%, 30 years = 0.42241

.

Value of Bond = ( 75 * 6.41766 ) + ( 1000 * 0.42241 )

Value of Bond = 481.32 + 422.41 = $903.73

..

1e. 10 years to maturity, required return is 7.5%

.

Coupon interest = 1000 * 7.5% = 75

PVIFA = Present value of interest factor annuity, 7.5%, 10 years = 6.86408

Par value = 1000

PVIF = Present value interest factor, 7.5%, 10 years = 0.4852

.

Value of Bond = ( 75 * 6.86408 ) + ( 1000 * 0.4852 )

Value of Bond = 514.8 + 485.2 = $1000

.

1f. 10 years to maturity, required return is 6%

.

Coupon interest = 1000 * 7.5% = 75

PVIFA = Present value of interest factor annuity, 6%, 10 years = 7.36

Par value = 1000

PVIF = Present value interest factor, 6%, 10 years = 0.5584

.

Value of Bond = ( 75 * 7.36 ) + ( 1000 * 0.5584 )

Value of Bond = 552 + 558.40 = $1110.40

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1g. 2 years to maturity, required return is 9%

.

Coupon interest = 1000 * 7.5% = 75

PVIFA = Present value of interest factor annuity, 9%, 2 years = 1.7591

Par value = 1000

PVIF = Present value interest factor, 9%, 2years = 0.84168

.

Value of Bond = ( 75 * 1.7591 ) + ( 1000 * 0.84168 )

Value of Bond = 131.93 + 841.68 = $973.61

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1h. 2 years to maturity, required return is 7.5%

.

Coupon interest = 1000 * 7.5% = 75

PVIFA = Present value of interest factor annuity, 7.5%, 2 years = 1.79556

Par value = 1000

PVIF = Present value interest factor, 7.5%, 2 years = 0.8653

.

Value of Bond = ( 75 * 1.79556 ) + ( 1000 * 0.8653)

Value of Bond = 134.70 + 865.30   = $1000

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1i. 2 years to maturity, required return is 6%

.

Coupon interest = 1000 * 7.5% = 75

PVIFA = Present value of interest factor annuity, 6%, 2 years = 1.83339

Par value = 1000

PVIF = Present value interest factor, 6%, 2 years = 0.88999

.

Value of Bond = ( 75 * 1.83339 ) + ( 1000 * 0.88999 )

Value of Bond = 137.50 + 889.90 = $1027.40

.

2) The current price of a 9.75% coupon bond with 20 years to maturity is $1318, what is the YTM? If the bond contains a call provision that allows the company to call the bond for $1050 7-years from now, what is the YTC? Based on the available information, is this bond likely to be called?

.

YTM = (( c * ( P- M ) / N )) / ( ( M + P) / 2 )

Where,

C = 9.75% coupon * 1000 = 97.5

M = 1318

P = 1000

N = 20

.

YTM = (( 97.5 + ( 1000 - 1318 ) / 20 ) ) / ( ( 1318 + 1000 ) / 2 )

YTM = ( 97.5 + -15.9 ) / 1159

YTM = 81.6 / 1159 = 0.0704 or 7.04%

.

.

Yield to call

YTC = (C + (CP - M ) / N) / ((CP + M ) / 2)

Where:

YTC = yield to call

C = annual coupon = 97.5

CP = call price of the bond = 1050

M = price of the bond = 1318

N = time in years remaining until the call date = 7

.

YTC = ( 97.5 + ( 1050 - 1318 ) / 7 ) / ( ( 1050 + 1318 ) / 2 )

YTC = ( 97.5 + -38.28 ) / 1184

YTC = 59.22 / 1184 = 0.05 or 5%

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3) Find the price of a 20-year zero coupon bond if the required return on such a bond was 12%? What if the required return was 10%?

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Value = Maturity value / (1 + r )^{^n}

^.

required return on such a bond was 12%

Value = 1000 / ( 1 + 12%)^{^20} = 1000 / 9.6462 = $103.66

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What if the required return was 10%

Value = 1000 / ( 1 + 10%)^{^20} = 1000 / 2.5937 = $385.54