Question #12

of 98 Using the following tree of semiannual interest rates what is the value of a 5% callable bond that has one year remaining to maturity, a call price of 99 and pays coupons semiannually? 7.76% 6.20% 5.45% A) 97.17. B) 98.29. C) 99.01. Explanation The callable bond price tree is as follows: 100.00 A → 98.67 98.29 100.00 99.00 100.00 As an example, the price at node A is obtained as follows: PriceA = min[(prob × (Pup + (coupon / 2)) + prob × (Pdown + (coupon/2)) / (1 + (rate / 2)), call price] = min[(0.5 × (100 + 2.5) + 0.5 × (100 + 2.5)) / (1 + (0.0776 / 2)), 99} = 98.67. The bond values at the other nodes are obtained in the same way. (Module 27.2, LOS 27.f) Kate Inka is a new hire for Maya Incorporated, a fixed income fund manager. On her first week on the job, she is asked to prepare a presentation on valuation and analysis of bonds with embedded options. Inka starts her presentation with the following three statements: Statement 1: "In times of increased expectations of interest rate volatility the value of callable bonds will fall." Statement 2: "When trying to analyze the return for credit and liquidity risk on a corporate callable bond relative to a government bond, the Z-spread must be calculated. The Z-spread can be viewed as the constant spread added to treasury spot rates such that the present value of the callable bonds coupons and principal equate to its market price." Statement 3: "When analyzing the interest rate risk of a callable bond it is worth keeping in mind that its effective convexity will be less than or equal to the equivalent option free bond." Inka is analyzing a three-year, 6% annual coupon, $100 par callable bond. The bond has a European call feature allowing it to be called at 101% of par in two years' time. Inka uses a binomial tree assuming interest rate volatility of 20% as shown in Exhibit 1. Exhibit 1: Binomial Lattice T0 T1 T2 6.34% 5.45% 3% ????? 3.65% 2.85% Inka makes the following three comments about her binomial tree exercise: Comment 1: "If the spot and expected future 1-period rates in the binomial tree have been derived from treasury securities we should be aware that the backwardly induced value of a corporate bond would be too high relative to its market price." Comment 2: "For a corporate callable bond, the option adjusted spread must be added as a fixed margin to all the treasury spot and expected future 1-period rates so that the backwardly induced price converges with market price." Comment 3: "If we were to increase our assumption of interest rate volatility used to create the binomial tree, the estimated option adjusted spread would be smaller." Finally, Inka makes three comments on her use of effective duration: Comment 1: "Given that a corporate callable bond will exhibit negative convexity when yields are low, care must be taken when interpreting effective duration, as essentially the computation averages the impact of the up and down shock on bond price. Perhaps the non-symmetrical price reaction to yield increases and decreases would be better captured by looking at one- sided durations." Comment 2: "Effective duration is an incomplete measure of interest rate risk as it fails to adequately capture option risk. For example, callable bonds are more sensitive to interest rate risk due to embedded options and as such have a higher effective duration." Comment 3: "One method of capturing shaping risk is to compute one-sided durations. A 20-year bond callable after 10 years with a low coupon is likely to have the highest one-sided duration corresponding to the call date. If the coupon is increased the one- sided duration corresponding to the call date declines but the maturity matched 20-year one-sided duration increases."