Question

Question 4 (25 marks / Bond Market and Term Structure of Interest Rates)

1. a) You are considering investing in bonds and have collected the following information about the prices of a 1-year zero-coupon bond and a 2-year coupon bond.

   • - The 1-year discount bond pays $1,000 in one year and sells for a current price of $950.

   • - The 2-year coupon bond has a face value of $1,000 and an annual coupon of $60. The bond currently sells for a price of $1,050.

   1. i) What are the implied yields to maturity on one- and two-year discount bonds?

   2. ii) What is the implied forward rate between years 1 and 2?

   3. iii) Consider a 2-year annuity with annual coupon payments of $800. What is the most that you would be willing to pay for this annuity?

Answer 1

i)Computation of the YTM of the bonds

We know that at YTM , discounted value of the future cash inflows is equal to the investment amount

Year	Cash flow	Disc @ 5%	Discounted cash flows	Disc @ 6%	Discounted cash flows
0	($950)	1	($950)	1	($950)
1	$1,000	0.952380952	$952	0.943396226	$943
	NPV		$2		($7)

Discount rate lies between 5% and 6%. To findout the exact figure, we have to use interpolation method

For 1 % Chage in interest rate, NPV turns from $ 2 to ( $ 7)

Change in interest rate	Change in NPV
1%	$9	( $ 2+$ 7)
X	$2

By doing criss cross multiplication we get

$ 2 = $ 9X

X = 0.2222%

Hence YTM = 5.2222%

Calculation of YTM on 2 year bond

Year	Cash flow	Disc @6%	Discounted cash flows	Disc @ 7%	Discounted cash flows
0	($1,050)	1.0000	($1,050)	1.0000	($1,050.00)
1	$60	0.9434	$57	0.9346	$56.07
2	$60	0.8900	$53	0.8734	$52.41
2	$1,000	0.9434	$943	0.8734	$873.44
	NPV		$3		($68.08)

Discount rate lies between 6% and 7%. To findout the exact figure, we have to use interpolation method

For 1 % Chage in interest rate, NPV turns from $ 3 to ( $ 68)

Change in interest rate	Change in NPV
1%	$71	( $ 68.08+$ 3)
X	$3

$ 3= $ 71X

X = 0.04225

Hence the YTM is 6.04225 for two yearbond

ii)Computation of the implied forward rate in year 2:

Year	YTM
1	5.2222%.
2	6.04225%

Let interest rate in the second year be x%

Total 2 years interest rate= Interest rate in year 1+x %

(1+6.04225% ) ^2= ( 1+5.2222%) ( 1+x%)

( 1.0604225)^2=( 1.052222)(1+x%)

1.1244=( 1.052222)( 1+x%)

1.1244/1.052222=1+x%

By solving the above equation , we get X = 6.8597%

Hence the implied forward rate between year 1 & Year 2 is 6.8597%.

Case III:Computation of present value of annuity

Year	Cash inflow	Disc @ 6.04225%	Discoounted cash flows
1	$800	0.9430	$754.42
2	$800	0.8893	$711.43
	Present value		$1,465.85

The most relavent Discount rate is 2 years YTM

Annuity value = $ 1465.85,