Question

The following table gives information about several bonds.

Bond Principal	Time to Maturity (years)	Semi-Annual Coupon ($)	Bond Price ($)
100	0.5	0	97.53
100	1.0	0	94.65
100	1.5	4	102.74
100	2.0	5	105.46

1. Calculate the zero rates for maturities 0.5, 1.0, 1.5 and 2.0 years. (4 points) (Hint: to compute the zero rates for maturities 1.5 and 2, you need to “Bootstrap” the zero curve – please refer to the Addendum on Bootstrapping on BB for details.) (Pg 29 manual)
2. What are the forward rates for the periods: 6 months to 12 months, 12 months to 18 months, 18 months to 24 months? (4 points)
3. Compute the price and yield of a 2-year bond providing a semi-annual coupon of $300 and a principal of $5,000. (4 points)

Answer 1

(a) Let the 0.5 year zero rate be 2R1

0.5 year maturity bond has zero coupon, face value of $ 100 and Price of $ 97.53

Therefore, 97.53 = 100 / (1+R1)

R1 = [(100/97.53) - 1] = 0.02533 or 2.533 %

0.5 year Zero Rate = 2.533 x 2 = 5.066 %

Let the 1 year zero rate be 2R2

1 year maturity bond has zero coupon, face value of $ 100 and Price of $ 94.65

Therefore, 94.65 = 100 / (1+R2)^(2)

R2 = [(100/94.65)^(1/2) - 1] = 0.02787 or 2.787 %

1 year Zero Rate = 2 x 2.787 = 5.574 %

Let the 1.5 year zero rate be 2R3

1.5 year bond has $100 face value, $ as semi-annual coupon, 102.74 $ as price

Therefore, 102.74 = 4 / (1.02533) + 4 / (1.02787)^(2) + 104 / (1+R3)^(3)

102.74 = 3.901 + 3.786 + 104/(1+R3)^(3)

95.053 = 104/(1+R3)^(3)

R3 = [(104/95.053)^(1/3) - 1] = 0.03044 or 3.044 %

1.5 Year zero rate = 2 x 3.044 = 6.088 %

Let the 2 year zero rate be 2R4

2 year bond has $ 100 face value, $ 5 as semi-annual coupons and $ 105.46 as price

Therefore, 105.46 = 5 / (1.02533) + 5 / (1.02787)^(2) + 5 / (1.03044)^(3) + 105 / (1+R4)^(4)

105.46 = 4.8765 + 4.7325 + 4.5698 + 105 / (1+R4)^(4)

91.2812 = 105 / (1+R4)^(4)

R4 = [(105/91.2812)^(1/4) - 1] = 0.03562 or 3.562 %

2 Year Zero Rate = 2 x 3.562 = 7.124 %

(b) Forward Rate between 0.5 year and 1 year = f(0.5,1) = 2 x {[1.02787]^(2) / [1.02533]} - 1 = 2 x  0.03042 or 6.083 %

Similarly f(1,1.5) = 2 x  {[(1.03044)^(3)/(1.02787)^(2)] - 1} = 2 x 0.03559 or 7.118 %

f(1.5,2) = 2 x {[(1.03562)^(4)/(1.03044)^(3)] - 1} = 2 x 0.05132 or 10.264 %

(c) Semi-Annual Coupon = $ 300, Principal = $ 5000 and tenure = 2 years

Therefore, Bond Price = 300 / (1.02533) + 300 / (1.02787)^(2) + 300 / (1.03044)^(3) + 5300 / (1.03562)^(4) = $ 5458.33

Let the bond's yield to maturity be 2R

Therefore, 5458.33 = 300 x (1/R) x [1-{1/(1+R)^(4)] + 5000/(1+R)^(4)

Using EXCEL's Goal Seek Function/ Hit and Trial Method to solve the above equation, we get:

R = 0.03504 or 3.504 %

Therefore, Yield to Maturity = 2 x 3.504 = 7.008 % ~ 7.01 %