copyright 2006, jeffrey m. mercer, ph.d.1 chapter 9: 3/23 lecture valuing bonds with embedded...

Copyright 2006, Jeffrey M. Mercer, Ph.D. 1

Chapter 9: 3/23 Lecture

Valuing Bonds with Embedded Options


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.


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.


Interpretation of Spread Measures

the nominal spread is a spread measured relative to the yield curve.

YTMGM – YTMTbond = 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.


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 measure BenchmarkSpread reflects

compensation for risks:

Nominal yield curve credit, option, liquidity

Zero-volatility spot rate curve credit, option, liquidity

OAS spot rate curve credit, liquidity


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 measure BenchmarkReflects compensation

for risks (see next slide):

Nominal sector yield curve credit, option, liquidity

Zero-volatility sector spot rate curve credit, option, liquidity

OAS sector spot rate curve credit, liquidity


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.


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 measure BenchmarkReflects compensation

for risks:

Nominal issuer yield curve option, liquidity

Zero-volatility issuer spot rate curve option, liquidity

OAS issuer spot rate curve liquidity


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)


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?


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.


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?




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.


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?




The only measure we need to look at is the OAS.

Sine OAS is negative, the bond is rich (i.e., overvalued).


“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.


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.



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).



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.



Thus, our OAS estimate, and our relative value analysis for bonds with embedded options, depend heavily on our volatility estimate!


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.


Original Exhibit 11


Exhibit 11 Problem

Node NHHH will be:

9.6987% = 9.5487% + 0.15% 97.084 = 106.5/(1.096987) Will not be called.

97.084

97.084

6.5

9.6987%


Exhibit 11 Problem

Node NHHL will be:


98.327

98.327

6.5

8.0312%


Exhibit 11 Problem

Node NHLL will be:


99.844

99.844

6.5

5.5483%


Exhibit 11 Problem

Node NLLL 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%


Exhibit 11 Problem

Node NHH will be:

7.5053% = 7.3553% + 0.15%

Will not be called.

96.931

96.931

6.5

7.5053%

075053.1

5.6327.98

075053.1

5.6084.97

2

1931.96


Exhibit 11 Problem

Node NHL will be:

6.2354% = 6.0854% + 0.15%

Will not be called.

99.388

99.388

6.5

6.2354%

062354.1

5.6844.99

062354.1

5.6327.98

2

1388.99


Exhibit 11 Problem

Node NLL will be:

5.1958% = 5.0458% + 0.15%

Will be called at 100.

101.166

100.000

6.5

5.1958%

051958.1

5.6100

051958.1

5.6844.99

2

1166.101


Exhibit 11 Problem

Node NH will be:

5.9289% = 5.7789% + 0.15%

Will not be called.

98.874

98.874

6.5

5.9289%

059289.1

5.6388.99

059289.1

5.6084.97

2

1874.98


Exhibit 11 Problem

Node NL will be:

4.9448% = 4.7948% + 0.15%


101.190

100.000

6.5

4.9448%

049448.1

5.6100

049448.1

5.6388.99

2

1190.101


Exhibit 11 Problem

Node N will be:

4.0000% = 3.850% + 0.15%


101.863

4.0000%

04000.1

5.6100

04000.1

5.6874.98

2

1863.101


Effective Duration and Convexity

You will not have to calculate effective duration or effective convexity on the exam (pp. 314-315).


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.


Exhibit 14: Putable Bond


Exhibit 15

Computed price is below 100.

It will be put.

Computed price is below 100.

It will be put.


Exhibit 15

From Node NHH:

070053.1

5.6100

070053.1

5.6100

2

1528.99


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

bond free-option of Valuebond putable of ValueOptionPut of Value


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).

copyright 2006, jeffrey m. mercer, ph.d.1 chapter 9: 3/23 lecture valuing bonds with embedded...

Documents