Question

Both Bond Sam and Bond Dave have 8 percent coupons, make semiannual payments, and are priced at par value. Bond Sam has 2 years to maturity, whereas Bond Dave has 16 years to maturity.

   

	If interest rates suddenly rise by 4 percent, what is the percentage change in the price of Bond Sam?

    

	If interest rates suddenly rise by 4 percent, what is the percentage change in the price of Bond Dave?

    

	If rates were to suddenly fall by 4 percent instead, what would the percentage change in the price of Bond Sam be then?

    

	If rates were to suddenly fall by 4 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 4%
a.	% change in price of Bond Sam	= -6.93%
b.	% change in price of Bond Dave	= -28.17%
2.	If interest rate suddenly fall by 4%
a.	% change in price of Bond Sam	=7.62%
b.	% change in price of Bond Dave	=46.94%
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 4% means YTM becomes 8%+4%=12%
	then price of both bond will fall as YTM of Bond is more than its coupon rate
Bond Sam price =$930.69424
	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= 12% p.a (annual)
	Semi annual YTM= 12%/2 = 6%
	n= no.of coupon = No. Of years x no. Of coupon in a year
	= 2 x 2 = 4
	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 @ 6% for 1 to 4th + PVF @ 6% for 4th period x 1,000
	= 40 x   3.465106 + 1000 x 0.79209
	=$930.69424
Cumulative PVF @ 6% for 1 to 4th is calculated = (1 - (1/(1 + 0.06)^4) ) /0.06 = 3.465106
PVF @ 6% for 4th period is calculated by = 1/(1+i)^n = 1/(1.06)^4 =0.79209
Bond Dave price =$718.31900
	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= 12% p.a (annual)
	Semi annual YTM= 12%/2 = 6%
	n= no.of coupon = No. Of years x no. Of coupon in a year
	=16 x 2=32
	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 @ 6% for 1 to 32th + PVF @ 6% for 32th period x 1,000
	= 40 x 14.08404 + 1000 x 0.1549574
	=$718.31900
Cumulative PVF @ 6 % for 1 to 32th is calculated = (1 - (1/(1 + 0.06)^32) ) /0.06 = 14.08404
PVF @ 6% for 32th period is calculated by = 1/(1+i)^n = 1/(1.06)^32 =0.1549574
	Percentage change in price = (New price – Original price) / Original price
	% change in price Bond Sam=($930.69424 -1000)/1000		= -6.93%
	= -0.069305
	% change in price Bond Dave=($718.31900 -1000)/1000		= -28.17%
	= -0.281681
2nd case If interest rate suddenly fall by 4%
Working Notes:
	2nd case is of interest rate suddenly fall by 4% means YTM becomes 8%-4% =4%
	then price of both bond will rise
Bond Sam price =$1,076.154148
	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= 4% p.a (annual)
	Semi annual YTM= 4%/2 = 2%
	n= no.of coupon = No. Of years x no. Of coupon in a year
	= 2 x 2 = 4
	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% for 1 to 4th + PVF @ 2% for 4th period x 1,000
	= $40 x   3.8077287 + 1000 x 0.923845
	=$1,076.154148
Cumulative PVF @ 2% for 1 to 4th is calculated = (1 - (1/(1 + 0.02)^4) ) /0.02 = 3.8077287
PVF @ 2% for 4th period is calculated by = 1/(1+i)^n = 1/(1.02)^4 =0.923845
Bond Dave price =$1,469.36640
	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= 4% p.a (annual)
	Semi annual YTM= 4%/2 = 2%
	n= no.of coupon = No. Of years x no. Of coupon in a year
	= 16 x 2 = 32
	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% for 1 to 32th + PVF @ 2% for 32th period x 1,000
	= 40 x 23.468335+ 1000 x 0.530633
	=$1,469.36640
Cumulative PVF @ 2% for 1 to 32th is calculated = (1 - (1/(1 + 0.02)^32) ) /0.02 = 23.468335
PVF @ 2% for 32th period is calculated by = 1/(1+i)^n = 1/(1.02)^32 =0.530633
Percentage change in price = (New price – Original price) / Original price
% change in price Bond Sam=($1076.154148 -1000)/1000
	=0.07615415	=7.62%
% change in price Bond Dave=($1469.36640 -1000)/1000
	=0.46936640	=46.94%
Please feel free to ask if anything about above solution in comment section of the question.