1. Copyright 2006, Jeffrey M. Mercer, Ph.D.1 Chapter 9: 3/23 Lecture Valuing Bonds with Embedded Options

2. Copyright 2006, Jeffrey M. Mercer, Ph.D.2 Relative Value Analysis Relative value analysis is used to help identify bonds that are underpriced (“cheap”), overpriced (“rich”), or fairly priced. We use yield spread measures to accomplish this. Yield spreads have to measured relative to interest rates from some benchmark. Our interpretation of relative value depends upon the benchmark choice.

3. Copyright 2006, Jeffrey M. Mercer, Ph.D.3 Benchmarks Benchmark interest rates come from: Treasury securities, or a specific bond sector with similar credit risk, liquidity, and maturity characteristics, or the issuer’s own (other) bonds.

4. Copyright 2006, Jeffrey M. Mercer, Ph.D.4 Interpretation of Spread Measures the nominal spread is a spread measured relative to the yield curve. YTM GM – YTM Tbond = nominal spread. the zero-volatility spread (or static spread) is a spread relative to the spot rate curve. Spot rates on GM bond = spot rates on Tbond + 500 bps. the option adjusted spread is a spread relative to the spot rate curve. The OAS adjusts GM’s z-spread for the “option value” (in bps). If option value = 100 bps, OAS = 500-100 = 400 bps. In this case, the 400 bps would compensate for default and liquidity risks.

5. Copyright 2006, Jeffrey M. Mercer, Ph.D.5 Interpretation of Spread Measures: Treasury Market Benchmark If we compare the rates on a callable corporate bond with U.S. Treasury rates, we can summarize as follows: Spread measureBenchmark Spread reflects compensation for risks: Nominalyield curvecredit, option, liquidity Zero-volatilityspot rate curvecredit, option, liquidity OASspot rate curvecredit, liquidity

6. Copyright 2006, Jeffrey M. Mercer, Ph.D.6 Interpretation of Spread Measures: Specific Bond Sector Benchmark If we compare the rates on a callable corporate bond with rates from a specific sector, we can summarize as follows: Spread measureBenchmark Reflects compensation for risks (see next slide): Nominalsector yield curvecredit, option, liquidity Zero-volatilitysector spot rate curvecredit, option, liquidity OASsector spot rate curvecredit, liquidity

7. Copyright 2006, Jeffrey M. Mercer, Ph.D.7 Risks from prior slide Where credit risk in this case means: Credit risk of a security under consideration relative to the credit risk of the sector used as the benchmark. Where liquidity risk in this case means: liquidity risk of a security under consideration relative to the liquidity risk of the sector used as the benchmark.

8. Copyright 2006, Jeffrey M. Mercer, Ph.D.8 Interpretation of Spread Measures: Issuer-Specific Benchmark If we compare a callable bond’s rates with rates on other bonds from the same issuer, we can summarize as follows: Spread measureBenchmark Reflects compensation for risks: Nominalissuer yield curveoption, liquidity Zero-volatilityissuer spot rate curveoption, liquidity OASissuer spot rate curveliquidity

9. Copyright 2006, Jeffrey M. Mercer, Ph.D.9 Relative Value Analysis and Spreads: Example When a bond has an embedded option, we must use the option-adjusted spread in relative value analysis. But the OAS has to be measured relative to a benchmark, and the choice of benchmark determines how we interpret “relative value.” Example 1: (next slide)

10. Copyright 2006, Jeffrey M. Mercer, Ph.D.10 Example 1 Benchmark: Treasury market rates The bond under consideration is a triple B rated corporate bond with an embedded option. Nominal spread between our bond and the benchmark = 170 bps Nominal spread between option-free BBB’s and the benchmark = 145 bps Z-spread = 160 bps OAS = 125 bps Is the bond cheap, rich, or fair?

11. Copyright 2006, Jeffrey M. Mercer, Ph.D.11 Example 1: continued The bond has an embedded option, so the OAS should be used. The only comparison that can be made is the OAS of 125 bps to the nominal spread for option-free BBB’s of 145. Based on this comparison, the bond is rich (i.e., overvalued). Recall that the z-spread and the nominal spread are typically close, given a reasonably sloped yield curve.

12. Copyright 2006, Jeffrey M. Mercer, Ph.D.12 Example 2 Benchmark: AA-rated callable corporate bonds The bond under consideration is a triple B rated corporate bond with an embedded option. Nominal spread between our bond and the benchmark = 110 bps Nominal spread between the benchmark and option- free AA’s = 90 bps Z-spread = 100 bps OAS = 80 bps Is the bond cheap, rich, or fair?

13. Copyright 2006, Jeffrey M. Mercer, Ph.D.13 Example 2: continued The bond has an embedded option, so the OAS should be used. The only comparison that can be made is the OAS of 80 bps to the nominal spread for option-free AA’s of 90. Based on this comparison, the bond is rich (i.e., overvalued). Recall that the z-spread and the nominal spread are typically close, given a reasonably sloped yield curve.

14. Copyright 2006, Jeffrey M. Mercer, Ph.D.14 Example 3 Benchmark: Rates on other issues from the specific issuer of our bond. The bond under consideration is a triple B rated corporate bond with an embedded option. Nominal spread between our bond and the benchmark = 30 bps Nominal spread between the benchmark and option- free AA’s = 90 bps Z-spread = 20 bps OAS = -25 bps Is the bond cheap, rich, or fair?

15. Copyright 2006, Jeffrey M. Mercer, Ph.D.15 Example 3: continued The bond has an embedded option, so the OAS should be used. The only measure we need to look at is the OAS. Sine OAS is negative, the bond is rich (i.e., overvalued).

16. Copyright 2006, Jeffrey M. Mercer, Ph.D.16 “Required OAS” vs. “Security OAS” Define required OAS as the OAS available on comparable securities (i.e., same credit risk, liquidity risk, and maturity). Define security OAS as the OAS on the security under consideration. Then, “Cheap” is security OAS > required OAS. “”Rich” is security OAS < required OAS. “Fair” is security OAS = required OAS.

17. Copyright 2006, Jeffrey M. Mercer, Ph.D.17 Exhibit 11 and pp. 312 - 313 Exhibit 8 demonstrates that the theoretical price of the four year, 6.5% callable bond, with volatility = 10%, is 102.899. No suppose the market price is 102.218. The bond is underpriced (relative to our model) by $0.681. The OAS equals the constant spread that, when added to the rates in the binomial tree, will make the model value equal to the market value.

18. Copyright 2006, Jeffrey M. Mercer, Ph.D.18 Exhibit 11 and pp. 312 - 313 Exhibit 11 demonstrates that the OAS is 35 bps. Thus, a positive OAS (in this case) is consistent with the bond being undervalued. Note: “In this case” because the benchmark rates are from this specific issuer, so credit risk and liquidity risk differences are controlled for (see top of page 302 where this issuer’s bonds are first introduced).

19. Copyright 2006, Jeffrey M. Mercer, Ph.D.19 Exhibit 11 and pp. 312 - 313 Again assume that the market price is 102.218. But now, assume that our volatility forecast is 20% (rather than 10%). At 20% volatility, the model value of the bond would decrease, and the option value would increase. But since the market price didn’t change, the OAS must decrease.

20. Copyright 2006, Jeffrey M. Mercer, Ph.D.20 Exhibit 11 and pp. 312 - 313 Thus, our OAS estimate, and our relative value analysis for bonds with embedded options, depend heavily on our volatility estimate!

21. Copyright 2006, Jeffrey M. Mercer, Ph.D.21 Exhibit 11 Problem Can you solve the following problem? Calculate the bond’s price in Exhibit 11 if the OAS is 50 bps instead of 35 bps. Draw the binomial tree diagram, as in Exhibit 11, showing the computed value and call price if exercised, and the adjusted interest rates.

22. Copyright 2006, Jeffrey M. Mercer, Ph.D.22 Original Exhibit 11

23. Copyright 2006, Jeffrey M. Mercer, Ph.D.23 Exhibit 11 Problem Node N HHH will be: 9.6987% = 9.5487% + 0.15% 97.084 = 106.5/(1.096987) Will not be called. 97.084 6.5 9.6987%

24. Copyright 2006, Jeffrey M. Mercer, Ph.D.24 Exhibit 11 Problem Node N HHL will be: 8.0312% = 7.8812% + 0.15% 98.327 = 106.5/(1.08312) Will not be called. 98.327 6.5 8.0312%

25. Copyright 2006, Jeffrey M. Mercer, Ph.D.25 Exhibit 11 Problem Node N HLL will be: 6.6660% = 6.5160% + 0.15% 99.844 = 106.5/(1.06666) Will not be called. 99.844 6.5 5.5483%

26. Copyright 2006, Jeffrey M. Mercer, Ph.D.26 Exhibit 11 Problem Node N LLL will be: 5.5483% = 5.3983% + 0.15% 100.902 = 106.5/(1.055483) Will be called at 100. 100.902 100.000 6.5 5.5483%

27. Copyright 2006, Jeffrey M. Mercer, Ph.D.27 Exhibit 11 Problem Node N HH will be: 7.5053% = 7.3553% + 0.15% Will not be called. 96.931 6.5 7.5053%

28. Copyright 2006, Jeffrey M. Mercer, Ph.D.28 Exhibit 11 Problem Node N HL will be: 6.2354% = 6.0854% + 0.15% Will not be called. 99.388 6.5 6.2354%

29. Copyright 2006, Jeffrey M. Mercer, Ph.D.29 Exhibit 11 Problem Node N LL will be: 5.1958% = 5.0458% + 0.15% Will be called at 100. 101.166 100.000 6.5 5.1958%

30. Copyright 2006, Jeffrey M. Mercer, Ph.D.30 Exhibit 11 Problem Node N H will be: 5.9289% = 5.7789% + 0.15% Will not be called. 98.874 6.5 5.9289%

31. Copyright 2006, Jeffrey M. Mercer, Ph.D.31 Exhibit 11 Problem Node N L will be: 4.9448% = 4.7948% + 0.15% Will be called at 100. 101.190 100.000 6.5 4.9448%

32. Copyright 2006, Jeffrey M. Mercer, Ph.D.32 Exhibit 11 Problem Node N will be: 4.0000% = 3.850% + 0.15% Will be called at 100. 101.863 4.0000%

33. Copyright 2006, Jeffrey M. Mercer, Ph.D.33 Effective Duration and Convexity You will not have to calculate effective duration or effective convexity on the exam (pp. 314-315).

34. Copyright 2006, Jeffrey M. Mercer, Ph.D.34 Valuing Putable Bonds Recall that with a putable bond the owner has the right to force the issuer to redeem the bond and pay it off. Using the binomial model, the process we follow is the same as with callable bonds, except that we assume the bond will be “put” if the price falls below some level. Let’s use the same interest rate tree as in Exhibit 5, and consider a 4 year 6.5% bond that is putable in one year at a price of 100.

35. Copyright 2006, Jeffrey M. Mercer, Ph.D.35 Exhibit 14: Putable Bond

36. Copyright 2006, Jeffrey M. Mercer, Ph.D.36 Exhibit 15 Computed price is below 100. It will be put. Computed price is below 100. It will be put.

37. Copyright 2006, Jeffrey M. Mercer, Ph.D.37 Exhibit 15 From Node N HH :

38. Copyright 2006, Jeffrey M. Mercer, Ph.D.38 Value of the Put Option We know from earlier coverage that this bond, if it is option-free, is priced at $104.643. Exhibit 14 shows the putable bond is priced at $105.327. Since: the option value is $105.327 - $104.643 = $0.684

39. Copyright 2006, Jeffrey M. Mercer, Ph.D.39 Value of the Put Option We know that the option’s value increases with increases in volatility. Therefore, if we increase our volatility estimate, our model will produce a higher price estimate for the putable bond. This makes sense since the bondholder owns the option (which is increasing in value).