Question

A 12.75 year maturity zero-coupon bond selling at a yield to maturity of 8% (effective annual yield) has convexity of 150.3 and modified duration of 11.81 years. A 30-year maturity 6% coupon bond making annual coupon payments also selling at a yield to maturity of 8% has nearly identical duration of 11.79 years, but considerably higher convexity of 231.2. Answer the following:

(a)I) Calculate the prices of these two bonds.

ii) do the same but this time assume the yield to maturity on both bonds decreases to 7%.

Answer 1

a) I)

lets assume face value of all bonds are $1000

calculation of price of zero coupon bond:

a zero coupon bond don't pay any coupons during its period.instead it offers to purchase at a discounted price.

the price of a zero coupon bond is present value of its face value discounted at Yield to maturity

here duration of zero coupon bond is given as 12.75 years,it means 12 years and 9 months(0.75 x 12)

yield to maturity given in question = 8%

for one year yield to maturity is 8%

for 9 months it is 8 x (9/12) = 6%

we discount the bond @8% for 12 years and for the last year we use discount rate = 6%

PV factor at (n = 12 ; r = 8) = 0.39711375861

and for the last year we discount the above value using discount rate of 6%

so for 12.75 years = 0.3971137586 / 1.06 = 0.3746356

price of zero coupon bond = 1000 x 0.3746356 = $374.6356, lets say $374.64

calculation price of bond:

value of a bond is present value of all coupon payments plus redemption value discounted at yield to maturity

face value = $1000

coupon rate = 6%

annual coupon payments = 1000 x 6% = $60

yield to maturity = 8%

duration = 30 years

PVIFA( n = 30 ; r = 8%) = 11.25778

PV factor (n = 30 ; r = 8%) = 0.099377

so price of bond = 60 x 11.25778 + 1000 x 0.099377

= 675.4668 + 99.377

= $774.84(rounded off)

ii)

calculation of price of zero coupon bond:

yield to maturity given in question = 7%

for one year yield to maturity is 7%

for 9 months it is 7 x (9/12) = 5.25%

we discount the bond @7% for 12 years and for the last year we use discount rate = 5.25%

PV factor at (n = 12 ; r = 7) = 0.4440119592

and for the last year we discount the above value using discount rate of 5.25%

so for 12.75 years = 0.4440119592 / 1.0525 = 0.4218641

price of zero coupon bond = 1000 x 0.4218641

= $421.86(rounded of to two decimals)

calculation price of bond:

value of a bond is present value of all coupon payments plus redemption value discounted at yield to maturity

face value = $1000

coupon rate = 6%

annual coupon payments = 1000 x 6% = $60

yield to maturity = 7%

duration = 30 years

PVIFA( n = 30 ; r = 7%) = 12.409041

PV factor (n = 30 ; r = 7%) = 0.131367

so price of bond = 60 x 12.409041 + 1000 x 0.131367

= 744.54 + 131.37

= $875.91(rounded off to two decimals)