Q1 (Essential to cover) The following bonds are trading in the market:

Bond   Time-to-Maturity   Face value   Coupon rate       Price

In addition to the bonds above, you also observe some other bond (bond E) trading in the market at $138. Bond E has a time-to-maturity of two years, a face value of $100 and pays a coupon rate of 25%. Show that there is an arbitrage opportunity and how to exploit it.

Q2 (Essential to cover) Suppose the following bonds are trading in the market.

In addition to the bonds above, you also observe the 1-year forward rate in 2 year’s time 2f3  is 5.40%. 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 0.50% 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)?

Q3 (Essential to cover) Consider a four-year bond with a face value of $100 and a coupon rate of 15%. The term structure of interest rates is flat at 6%, i.e.   = 6% for all t.

a. Please calculate the duration of this bond, and use the duration rule to estimate the change in price (in dollars) if the term structure of interest shifts to 7%?

b. What would be the actual price change?

c. Could you please explain the approximation error of using duration rule by the price-yield curve and thus the relationship between yield and duration?

d. Now let’s assume that the convexity of this bond is 13.47. Please estimate the price change by using both duration and convexity.

e. Would the dollar error using the duration approximation be larger or smaller if the term structure would shift from 15% to 16% (instead of from 6% to 7%)? Why?

Q4 (Essential to cover) In this problem the term structure of interest rates is flat at 5%. The following bonds and liabilities are given:

• Bond A: A zero-coupon bond with a face value of $100 and a time to maturity of 3 years.

• Bond B: A zero-coupon bond with a face value of $100 and a time to maturity of 6 years.

• Bond C: A zero-coupon bond with a face value of $100 and a time to maturity of 10 years.

• Liability X: A one-time liability of $100 maturing in 4 years.

• Liability Y: A one-time liability of $100 maturing in 8 years.

a. Suppose you have liability X and want to immunize it using bonds A and B. How would you invest in each bond?

b. Suppose you have liability X and want to immunize it using bonds B and C. How would you invest in each bond?

c. Suppose you have both liabilities X and Y and want to immunize your position using bonds B and C. How would you invest in each bond?

Q5 (Essential to cover) Consider the following bonds:

• Bond A: A 2-year zero-coupon bond with a face value of $100 and 6% YTM.

• Bond B: A 2-year par-value bond with a face value of $100 and 6% coupon.

• Bond C: A 2-year par-value bond with a face value of $100 and 7% coupon.

• Bond D: A 3-year par-value bond with a face value of $100 and 7% coupon.

• Bond E: A 4-year par-value bond with a face value of $100 and 7% coupon.

• Bond F: A 4-year discount bond with a face value of $100 and 7% coupon.

If the yield curve shifts upwards by one percent,

a. Which bond among bonds A, B and C will experience the largest percentage price change? Which will have the lowest percentage price change?

b. Which bond of bonds C and D will experience a larger percentage price change?

c. Would you expect the difference in percentage price change to be bigger between bonds C and D or between bonds D and E?

d. Which bond of bonds E and F will experience a larger percentage price change?