Question

Below is information on several bonds of different maturities and coupon rates that trade in the same market. If the bonds pay a coupon, you may assume that the coupon is paid semi-annually, with the next coupon payment six months away. All bonds have a face value of $1,000.

For Bonds 1-4, assuming that they are fairly valued, calculate the spot interest rates for maturities of 0.5, 1.0, 1.5 and 2.0 years.

What is the Yield to Maturity (YTM) for each of the four (4) bonds?

Bond 5 trades in this market place. It has a maturity of 1 year, a coupon rate of 3.5% per year (paid semi-annually) and a face value of $1,000. What is the value of the bond?

	Coupon	Maturity	Price
Bond 1	0%	6 months	995.02
Bond 2	0%	1 year	989.09
Bond 3	2%	1.5 years	1012.86
Bond 4	4%	2 years	1056.22

Answer 1

Spot Rates

	Coupon	Maturity	Price	Face Value	semi annual rate
Bond 1	0%	0.5	995.02	1000	0.005
Bond 2	0%	1	989.09	1000	0.0055
Bond 3	2%	1.5	1012.86	1000	.0057
Bond 4	4%	2	1056.22	1000	.00575
duration	Spot Rate(semi annual rate*2)
6months	1.001%
1year	1.10%
1.5years	1.13%
2years	1.15%

Bond (6 months)

995.02=1000/(1+r1)

r1=.5% Spot 6months =1.001%

Bond (1 year)

989.09=1000/(1+r2)2

r2= .55% Spot 1year=1.1%

Bond (1.5 year)

1012.86=20/(1+.005) +20/(1+.0055)2 +1010/(1+r3)3

r3=.57% Spot 1.5years=1.13%

Bond 4 (2 year)

1056.22=20/(1+.005) +20/(1+.0055)2 +20/(1+.0057)3 +1020/(1+r4)4

r4=.00575% Spot 2years=1.15%

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Yield To Maturity:

Bond1: Same as spot rate; YTM= 1.001% (zero coupon bond)

Bond2: Same as spot rate; YTM=1.1%

Bond 3: YTM=1.133% ( IRR of the cash flows. Can be calculated in excel including the price paid in present year)

Bond 4: YTM=1.148% ( IRR of the cash flows. Can be calculated in excel including the price paid in present year)

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Price of Bond 5;

Price= semi coupon/(1+r1) + (face value+semi coupon)/(1+r2)^2

=17.5/(1.005) + 1017.5/(1.0055)^2

Price=1023.81$