Question

Q2 (Essential to cover) Suppose the following bonds are trading in the market. Bond Time-to-Maturity Face value Coupon rate Price E 1 $ 100 0% $ 94.79 F 2 $ 100 2% $ 92.25 G 4 $ 100 0% $ 74.88 In addition to the bonds above, you also observe the 1-year forward rate in 2 year’s time 2f3 is 8.50%. You wish to price Bond H, which is 4-year 10% coupon bond with a face value of $100. Assume all bonds (and the forward rate) are risk-free and that Bond F and Bond H are annual coupon bonds. a. Infer the term structure of interest rates: y1, y2, y3 and y4 (i.e. derive the pure yield curve for years 1-4). b. Price Bond H of the pure yield curve. c. Based on the pure yield curve, infer the 2-year forward rate commencing in 2 year’s time 2f4. d. Assume the Liquidity Preference Hypothesis holds and the annual liquidity premium is flat at 1.00% for all t. What is the expected future 1-year spot rate (i.e. the short rate) in 3 year’s time E(3y4)? e. Assume the Expectations Hypothesis holds. What is the expected 1 year future spot rate (i.e. the short rate) in 1 year’s time E(1y2)?

Answer 1

The information is summarised as under :

Bond	Time to maturity	Face Value	Coupon rate	Price
E	1	100	0%	94.79
F	2	100	2%	92.25
G	4	100	0%	74.88

a) y1 = 100/94.79 -1 = 0.0549636 or 5.50%

y2 is given by price of bond F

2/1.0549636 + 102/(1+y2)^2 = 92.25

=> (1+y2)^2 = 1.12889

y2 = 0.06249or 6.25%

(1+y3)^3 = (1+y2)^2*(1+2f3) = 1.06249^2*1.085 = 1.224846

=> y3 = 1.224846^(1/3)-1 = 0.0699427 or 6.99%

y4 is given by price of Bond G

y4 = (100/74.88)^(1/4)-1 = 0.07500019 or 7.50%

b) Price of Bond H

10/1.055+10/1.0625^2+10/1.0699^3+110/1.075^4 = $108.87

c) (1+y2)^2* (1+2f4)^2 = (1+y4)^4

=> 1.12889*(1+2f4)^2 = 1.33547

=> (1+2f4)^2 =1.18299

2f4 = 0.087655 or 8.77%

d) E(3y4) as per expectations theory

= (1+y4)^4/(1+y3)^3-1

=1.33547/1.224846 -1

= 0.0903 or 9.03%

So, E(3y4) as per liquidity preference hypothesis = 9.03% -1% = 8.03%

e)

E(1y2) as per expectations theory

= (1+y2)^2/(1+y1)-1

=1.12889/1.0549636 -1

= 0.070075 or 7.01%