FIN358: An Investment firm Specializing in Fixed Income Has the Following Portfolio of Bonds: Fixed Income and Derivative Securities Assignment, SUSS, Singapore

Q1.  An investment firm specializing in fixed income has the following portfolio of bonds, all with similar credit ratings and maturity of 3 years. 0 < T1 < T2 < …..< Tn and Tj denotes time as the end of year j when a call and/or a put can be exercised by the issuer. Interest rate volatility in the next 3 years are estimated as (1) = 10%, (2) = 15%, (3) = 15%.

Note: Grey box indicates that no information is given.\

a. Bonds A and B are from related companies in the same sector of the same country. If there are arbitrage opportunities, what long or short positions would you take with regards to Bonds A and B?

b. Given that Treasury notes for 1, 2, and 3-year maturities have par coupon rates of 1%, 1.8%, 2.2%, calibrate the 3-period BDT binomial tree for Bonds C and D.

c. What are the arbitrage-free prices of Bonds C and D?

d. Effective duration is the (unsigned, usually quoted as absolute value) slope of the embedded option bond price curve with respect to the yield-to-maturity. What is the sign of the effective duration of D less the effective duration of C, especially when the interest rate rises?

e. Bond E has a current market price of 99.95. Bond F has a conversion price of 32. Techcraf’s stock is currently selling at 16 per share. What would be an approximate value of Bond F?

Q2. 

Papers Company issued at par two 5-year floating rate notes C and F. The additional credit spread (OAS) to add to Libor is +200 bps. The floating coupon is Libor + 2%. The coupons are as normal, paid in arrears. C is capped at 6%. F is floored at 4.5%. After 2 years, the bonds have 3 years left before maturity. The BDT binomial 1-period Libor rate tree is as follows.

a. What is the arbitrage-free price of C?

b. What is the arbitrage-free price of F?

c. What are estimates of the effective durations of C and F?

Q3. 

Company X’s convertible bond with 3 years till-maturity has the following features.

a. What is the estimated market conversion premium per share of X’s convertible bond?

b. If there is the announcement of a takeover of X by a larger firm, and a change of control price is effected, would the CB holder convert immediately before the convertible bond price and the stock price change?

c. If the announcement in (b) is canceled, but instead there is the announcement that X’s share price is expected to rise to $23 by the following year. What would be the estimated expected return of the convertible bond over the next year?

Q4.  Par curve rates for annual payment benchmark government bonds are given in Table 1. One-year transition matrix for historical A-rated bonds and credit spreads is shown in Table 2. Company beer has a single A credit rating, and issues at par a 3-year 4.5% annual coupon bond. The bond’s modified duration is 2.87. The recovery rate for an A-rated bond is 40%. The hazard rate is 1% per year. 1-period spot interest rate volatility is assumed to a constant 15%.

Table 2:

Note: There is a small 0.1% chance of default, which is rating D (not shown in Table). The entry of “88.0” indicates that over 1 year, the A-rated bond will remain in A-rating with 88% probability. Entry “2.5” indicates that over 1 year, the A-rated bond will migrate to a better rating of AA with 2.5% probability, and so on.

a. What is the credit spread on the EsBeer bond? (Hint: You have to first construct the risk-free BDT spot rate binomial tree to find the bond value under no default. Note that the spread indicated in Table 2 for A-rated bonds is just a group average and not specifically the spread for EsBeer.)

b. It is assumed that the market spreads and yields will remain stable during the initial year. Thus any changes to yields will be due to credit migration risk. Conditional on no default in the first year, what is the expected return of the EsBeer bond over the year?

Homework #1 

Q1. Given the following spot rates, find the forward rates F0(n,n+1) for n=1,2,3,4. Note the notation F0(n,n+m) denotes a forward rate over m periods contracted now at time t=0 for effect n periods from now at t=n.

Q2. Given 1-year, 2-year, 3-year par rates (or yield-to-maturity) of an institution’s bonds are 2%, 3%, 4%,

a. find the 1-year, 2-year, 3-year spot rates. Use these spot rates to price a fixed rate 3-year 5% coupon bond issued by the same institution.

b. Suppose interest rate volatility is incorrectly perceived by the market as 15% p.a. when it should be 20% p.a., can you make arbitrage profits by trading in these institution’s bonds?

c. Suppose on top of the information given above, you also know that next year’s 1-year spot rate will likely be below 4%, can you make arbitrage profits? How?

Q3. An investor’s investment time horizon is two years. Spot rates for on-the-run annual-coupon government securities and swap spreads are shown as Maturity (years)

The investor believes that interest rates will remain stable and the yield curve will not change its level or shape for the next two years. Swap spreads will also remain unchanged. The investor is considering 3 possible investments as follows. (Assume annual coupon payments throughout.)

1. Employ a “riding the yield curve” strategy by buying a high-quality four-year, zero-coupon corporate bond and then selling it after two years when yield would be lower. Assume the

swap rate proxies for this corporate bond yield. What would be the annualized return over this horizon?

2. Buy an off-the-run 1.6% government bond with two years to maturity.

3. Buy a lower-quality, two-year 3% coupon corporate bond with a Z-spread of 200 bps with inside information that the bond is most unlikely to default in the next 3 years.

What are the annualized returns of the 3 possible investments, and which will fetch the highest return?

Q4. A portfolio comprises 2-year, 5-year, and 10-year zero-coupon bonds with $100 value in each position. Factor movements affecting key rates of the yield curve are as follows.

For a parallel factor increase of 1%, year 2 bond yield increases by 1%, year 5 bond yield increases by 1%, and year 10 bond yield increases by 1%. For a steepness factor increase of 1%, Year 1 bond yield decreases by 1%, and year 10 bond yield increases by 1%. For a curvature factor increase of 1% (yield curve becoming less convex), Year 1 bond yield increases by 1%, Year 2 bond yield decreases by ½%, and year 10 bond yield increases by 1%.

a. What is the key rate duration of the portfolio to the two-year rate, 5-year rate, and 10-year rate? (Assume current 2, 5, 10 year yields are very small, 1 to 2 %).

b. If X, Y, Z denote % changes in the parallel, steepness, and curvature factors, and we define P/P = − A1 X − A2 Y − A3 Z where P is the current value of the portfolio, P= $300, then find the values of A1, A2, and A3.

c. Suppose there is a steepening of the yield curve the next moment and only the steepness factor increases by ½ %, what is the change in the dollar value of the portfolio?

Q5.  Using the background from slide 9 of Lecture 3, the benchmark spot rates are as follows.

However, the market interest rate volatility is 20%. Apply the Black-Derman-Toy model. Find the price as well as the call option of the 3-year 4.25% callable bond.