Question

Both Bond Sam and Bond Dave have 8 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has 4 years to maturity, whereas Bond Dave has 17 years to maturity.

   

	If interest rates suddenly rise by 3 percent, what is the percentage change in the price of Bond Sam?

    

	If interest rates suddenly rise by 3 percent, what is the percentage change in the price of Bond Dave?

    

	If rates were to suddenly fall by 3 percent instead, what would the percentage change in the price of Bond Sam be then?

    

	If rates were to suddenly fall by 3 percent instead, what would the percentage change in the price of Bond Dave be then?

Answer 1

Solution:
1.	If interest rate suddenly rise by 3%
a.	% change in price of Bond Sam	= -9.50%
b.	% change in price of Bond Dave	= -22.86%
2.	If interest rate suddenly fall by 3%
a.
	% change in price of Bond Sam	=10.76%
b.
	% change in price of Bond Dave	=34.09%
Working Notes:
	AS both the bonds priced at par value, YTM of both bonds is equal to Coupon Rate that is 8%
1st case is of interest rate suddenly rise by 3% means YTM becomes 8%+3%=11%
	then price of both bond will fall
Bond Sam price =$904.9815105
	Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond
	Coupon Rate = 8%
	Annual coupon = Face value of bond x Coupon Rate = 1,000 x 8% = $80
	Semi annual coupon = Annual coupon / 2 = $80/2=$40
	YTM= 11% p.a (annual)
	Semi annual YTM= 11%/2 = 5.5%
	n= no.of coupon = No. Of years x no. Of coupon in a year
	= 4 x 2 = 8
	Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond
	= $40 x Cumulative PVF @ 5.5% for 1 to 8th + PVF @ 5.5% for 8th period x 1,000
	= 40 x   6.334565988 + 1000 x 0.651598871
	=$904.9815105
Cumulative PVF @ 5.5 % for 1 to 8th is calculated = (1 - (1/(1 + 0.055)^8) ) /0.055 = 6.334565988
PVF @ 5.5% for 8th period is calculated by = 1/(1+i)^n = 1/(1.055)^8 =0.651598871
Bond Dave price =$771.44451
	Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond
	Coupon Rate = 8%
	Annual coupon = Face value of bond x Coupon Rate = 1,000 x 8% = $80
	Semi annual coupon = Annual coupon / 2 = $80/2=$40
	YTM= 11% p.a (annual)
	Semi annual YTM= 11%/2 = 5.5%
	n= no.of coupon = No. Of years x no. Of coupon in a year
	=17 x 2=34
	Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond
	= $40 x Cumulative PVF @ 5.5% for 1 to 34th + PVF @ 5.5% for 34th period x 1,000
	= 40 x 15.23703257 + 1000 x 0.161963209
	=$771.44451
Cumulative PVF @ 5.5 % for 1 to 34th is calculated = (1 - (1/(1 + 0.055)^34) ) /0.055 = 15.23703257
PVF @ 5.5% for 34th period is calculated by = 1/(1+i)^n = 1/(1.055)^34 =0.161963209
	Percentage change in price = (New price – Original price) / Original price
	% change in price Bond Sam=($904.9815105 -1000)/1000		= -9.50%
	= -0.09501849
	% change in price Bond Dave=($771.44451 -1000)/1000		= -22.86%
	=-0.22855549
2nd case If interest rate suddenly fall by 3%
Working Notes:
	2nd case is of interest rate suddenly fall by 3% means YTM becomes 8%-3% =5%
	then price of both bond will rise
Bond Sam price =$1,107.552058
	Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond
	Coupon Rate = 8%
	Annual coupon = Face value of bond x Coupon Rate = 1,000 x 8% = $80
	Semi annual coupon = Annual coupon / 2 = $80/2=$40
	YTM= 5% p.a (annual)
	Semi annual YTM= 5%/2 = 2.5%
	n= no.of coupon = No. Of years x no. Of coupon in a year
	= 4 x 2 = 8
	Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond
	= $40 x Cumulative PVF @ 2.5% for 1 to 8th + PVF @ 2.5% for 8th period x 1,000
	= $40 x   7.170137167 + 1000 x 0.820746571
	=$1,107.552058
Cumulative PVF @ 2.5 % for 1 to 8th is calculated = (1 - (1/(1 + 0.025)^8) ) /0.025 = 7.170137167
PVF @ 2.5% for 8th period is calculated by = 1/(1+i)^n = 1/(1.025)^8 =0.820746571
Bond Dave price =$1,340.85679
	Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond
	Coupon Rate = 8%
	Annual coupon = Face value of bond x Coupon Rate = 1,000 x 8% = $80
	Semi annual coupon = Annual coupon / 2 = $80/2=$40
	YTM= 5% p.a (annual)
	Semi annual YTM= 5%/2 = 2.5%
	n= no.of coupon = No. Of years x no. Of coupon in a year
	= 17 x 2 = 34
	Bond Price = Periodic Coupon Payments x Cumulative PVF @ periodic YTM (for t= to t=n) + PVF for t=n @ periodic YTM x Face value of Bond
	= $40 x Cumulative PVF @ 2.5% for 1 to 34th + PVF @ 2.5% for 34th period x 1,000
	= 40 x 22.72378628 + 1000 x 0.431905343
	=$1,340.85679
Cumulative PVF @ 2.5% for 1 to 34th is calculated = (1 - (1/(1 + 0.025)^34) ) /0.025 = 22.72378628
PVF @ 2.5% for 34th period is calculated by = 1/(1+i)^n = 1/(1.025)^34 =0.431905343
Percentage change in price = (New price – Original price) / Original price
% change in price Bond Sam=($1,107.552058 -1000)/1000
	=0.107552058	=10.76%
% change in price Bond Dave=($1,340.85679 -1000)/1000
	=0.34085679	=34.09%
Please feel free to ask if anything about above solution in comment section of the question.