Question

Bond A has a 4% coupon. Bond B has a 10 percent coupon. Both bonds have 8 years to maturity, make annual payments, and have a YTM of 9 percent. If interest rates suddenly rise by 3 percent, what is the percentage price change in these bonds? What does this say about the interest rate risk of lower-coupon bonds?

Please show the formulas you used.

Answer 1

Percentage change in the price of Bond A

Price of the Bond at Current Yield to Maturity of 9%

Face Value of the bond = $1,000

Annual Coupon Amount = $40 [$1,000 x 4%]

Yield to Maturity = 9%

Maturity Period = 8 Years

Price of the Bond = Present Value of the Coupon Payments + Present Value of the face Value

= $40[PVIFA 9%, 8 Years] + $1,000[PVIF 9%, 8 Years]

= [$40 x 5.53482] + [$1,000 x 0.50187]

= $221.39 + $501.87

= $723.16

Price of the Bond at Current Yield to Maturity increased by 3%

Face Value of the bond = $1,000

Annual Coupon Amount = $40 [$1,000 x 4%]

Yield to Maturity = 12%

Maturity Period = 8 Years

Price of the Bond = Present Value of the Coupon Payments + Present Value of the face Value

= $40[PVIFA 12%, 8 Years] + $1,000[PVIF 12%, 8 Years]

= [$40 x 4.96764] + [$1,000 x 0.40388]

= $198.71 + $403.88

= $602.59

Change in the bond’s price in dollars = -$120.67 [$602.59 - $723.26]

Therefore, The Percentage Change in the price of Bond A

= [Change in the bond’s price in dollars / Bond Price at 9% YTM]

= [-$120.67 / $723.26] x 100

= -16.68% (Negative)

“Hence, the Percentage change in the price of Bond A = -16.68% (Decrease)”

Percentage change in the price of Bond B

Price of the Bond at Current Yield to Maturity of 9%

Face Value of the bond = $1,000

Annual Coupon Amount = $100 [$1,000 x 10%]

Yield to Maturity = 9% [9% + 3%]

Maturity Period = 8 Years

Price of the Bond = Present Value of the Coupon Payments + Present Value of the face Value

= $100[PVIFA 9%, 8 Years] + $1,000[PVIF 9%, 8 Years]

= [$100 x 5.53482] + [$1,000 x 0.50187]

= $553.48 + $501.87

= $1,055.35

Price of the Bond at Current Yield to Maturity increased by 3%

Face Value of the bond = $1,000

Annual Coupon Amount = $100 [$1,000 x 4%]

Yield to Maturity = 12% [9% + 3%]

Maturity Period = 8 Years

Price of the Bond = Present Value of the Coupon Payments + Present Value of the face Value

= $100[PVIFA 12%, 8 Years] + $1,000[PVIF 12%, 8 Years]

= [$100 x 4.96764] + [$1,000 x 0.40388]

= $496.76 + $403.88

= $900.65

Change in the bond’s price in dollars = -$154.70 [$900.65 - $1,055.35]

Therefore, The Percentage Change in the price of Bond B

= [Change in the bond’s price in dollars / Bond Price at 9% YTM]

= [-$154.70 / $1,055.35] x 100

= -14.66% (Negative)

“Hence, the Percentage change in the price of Bond B= -14.66% (Decrease)”

NOTE

-The formula for calculating the Present Value Annuity Inflow Factor (PVIFA) is [{1 - (1 / (1 + r)ⁿ} / r], where “r” is the Yield to Maturity of the Bond and “n” is the number of maturity periods of the Bond.

--The formula for calculating the Present Value Inflow Factor (PVIF) is [1 / (1 + r)ⁿ], where “r” is the Yield to Maturity of the Bond and “n” is the number of maturity periods of the Bond.

CONCLUSION

If the interest rate increases, the price of the Price will decrease and if the interest rate decreases, the price of the bond will increase. There is always a inverse relationship between the interest rate and the price of the Bond.