Question

Bond J has a coupon rate of 3 percent and Bond K has a coupon rate of 9 percent. Both bonds have 13 years to maturity, make semiannual payments, and have a YTM of 6 percent

If rates suddenly fall by 2%

What is the percentage change in price of bond J and what is the percentage change in price of bond K ??

Answer 1

Bond J:
Let us take face value=$1000. We normally take face value as $1000 for better calculations.
Coupon rate=3%
Time to maturity=13 years
Yield to maturity=6%

As the bond makes semiannual payments,
Semiannual coupon rate=3%/2=1.5%
Semiannual coupon payments=1.5%*$1000=$15
Semiannual yield to maturity (YTM)=6%/2=3%
Number of periods=(13 years)*2 (because of semiannual payments)
So, number of periods=26

Bond K:
Let us take face value= $1000. We normally take face value as $1000 for better calculations.
Coupon rate=9%
Time to maturity=13 years
Yield to maturity=6%

As the bond makes semiannual payments,
Semiannual coupon rate=9%/2=4.5%
Semiannual coupon payments=4.5%*$1000=$45
Semiannual yield to maturity (YTM)=6%/2=3%
Number of periods=(13 years)*2 (because of semiannual payments)
So, number of periods=26

Now, if rates fall suddenly by 2%. It is referring to coupon rate.

Bond J:
New coupon rate=(3%-2%)=1%
Time to maturity=13 years
Yield to maturity=6%

As the bond makes semiannual payments,
Semiannual coupon rate=1%/2=0.5%
Semiannual coupon payments=0.5%*$1000=$5
Semiannual yield to maturity (YTM)=6%/2=3%
Number of periods=(13 years)*2 (because of semiannual payments)
So, number of periods=26

Bond K:
New coupon rate=(9%-2%)=7%
Time to maturity=13 years
Yield to maturity=6%

As the bond makes semiannual payments,
Semiannual coupon rate=7%/2=3.5%
Semiannual coupon payments=3.5%*$1000=$35
Semiannual yield to maturity (YTM)=6%/2=3%
Number of periods=(13 years)*2 (because of semiannual payments)
So, number of periods=26

Now, we have calculated the bond prices using excel.
Calculating the percentage change:

Bond J price before fall in rate=$731.85
Bond J price after fall in rate=$553.08
Percentage change=(Final value -Initial value)/Initial value
=($553.08-$731.85)/$731.85=-$178.77/$731.85=-0.244271367 or -24.43% (Rounded up to two decimal places)
So, there is a decrease in price of bond J by -24.43%

Bond K price before fall in rate=$1268.15
Bond K price after fall in rate=$1089.38
Percentage change=(Final value -Initial value)/Initial value
=($1089.38-$1268.15)/$1268.15=-$178.77/$1268.15=-0.140969128=-14.10% (Rounded up to two decimal places)
So, there is a decrease in price of bond K by -14.10%