Question

A five-year bond with a yield of 11% (continuously compounded) pays an 8% coupon at the end of each year. a) What is the bond’s price? b) What is the bond’s duration? c) Use the duration to calculate the effect on the bond’s price of a 0.2% decrease in its yield. d) Recalculate the bond’s price on the basis of a 10.8% per annum yield and verify that the result is in agreement with your answer to (c).

**Can you please explain step by step on how to do this question*** and please show formulas used so I can understand how to do it on my own. thank you.

Answer 1

Q (a)

	Given:
1)	Yield on 5 year Bond	11%
2)	Coupon rate at the end of each year	8%
3)	Let Face Value of Bond	$ 100
	Bond Price at the end of 1 st year	8 * e^ -.11*1 =	7.166673
	Bond Price at the end of 2 nd year	8 * e^ -.11*2 =	6.42015
	Bond Price at the end of 3 d year	8 * e^ -.11*3 =	5.75139
	Bond Price at the end of 4 th year	8 * e^ -.11*4 =	5.152291
	Bond Price at the end of 5 th year (Face value + Interest)	(100+8) *e^ -.11*5=	62.31058
	Total		86.80108
	Therefore, Bond's price = Bond price from year 1 to 5 ( ie., Bond Price at the end of year 1 + 2+3+4+5)
=>	Bond Price	8 * e^ -.11*1 + 8 * e^ -.11*2+8 * e^ -.11*3+8 * e^ -.11*4 + (108) * e^ -.11*5
=>	Bond Price	7.166673+6.42015+5.75139+ 5.152291+62.31058
Answer	Bond Price "B"	$86.80108 or $ 86.80

Q (b)

	Bond Duration = 1 / Bond Price * (1*8 * (e^ -.11*1) + 2* 8 * (e^ -.11*2) + 3*8 * (e^ -.11*3)+ 4* 8 * (e^ -.11*4) + 5* (108) * e^( -.11*5))
i)	Bond Price calculated above	86.80108422
=>	1/ Bond Price	0.011520593
ii)	1 * 8 * e^ (-.11*1) =	7.166673082
iii)	2 * 8 * e^ (-.11*2) =	12.84030077
iv)	3* 8 * e^( -.11*3) =	17.2541696
v)	4*8 * e^ (-.11*4) =	20.60916547
vi)	5* (100+8) *e^ -.11*5=	311.5528976
	Total	369.4232065
	Bond Duration = 369.4232065/0.011520593
Answer	Bond Duration "D" =	4.255974563
	or	4.26 Years

Q (c) Formula:

Difference or B = - Bond Price "B" * Bond Duration "D" * (Decrease in yield)

We have derived in (a) and (b) above, the following:

Bond price "B" = $ 86.80

Bond Duration " D" = 4.26 years

or Yield difference = 0.2%

Therefore, impact on Bond Price or = 86.80*4.26*0.2%

Bond Price Differential = 0.738846413 or 0.74

Answer : New Bond Price will be $ 86.80 + $0.74 = $ 87.54

Q (d)

This is a cross-check for the calculation made in Q (c). With a 10.8% yield, the Bond Price will be:

	Bond Price at the end of 1 st year at 10.8%	8 * e^ -.108*1 =	7.181021
	Bond Price at the end of 2 nd year @ 10.8%	8 * e^ -.108*2 =	6.445882
	Bond Price at the end of 3 d year @10.8%	8 * e^ -.108*3 =	5.786002
	Bond Price at the end of 4 th year @10.8%	8 * e^ -.108*4 =	5.193675
	Bond Price at the end of 5 th year (Face value + Interest) @ 10.8%	(100+8) *e^ -.108*5=	62.93681
	Answer : Bond Price with yield of 10.8%		87.54339

Answer Therefore, the Bond Price calculated by this method is the same as calculated in (c) above,