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Chapter 9 Valuing Bonds with Embedded Options • Major learning outcomes: – Understand the different types of embedded options (cal s, puts, convertible features). – Be able to calculate the nominal, static (z) spread, and understand the option adjusted spread (OAS). – Understand the comparative statics of bonds with embedded options. – Be able to understand how cal able bonds are valued under Fabozzi’s binominal tree method.

Key Learning Outcomes • Explain the importance of the benchmark interest rates in interpreting spread measures. • Explain the binomial interest rate modeling strategy. • Explain the backward induction valuation methodology within the binomial interest rate tree framework. • Compute the value of a cal able bond from an interest rate tree given the cal schedule and the rule for cal ing a bond. • Explain how the value of an embedded option is determined.

Key Learning Outcomes • Explain the relationship among the values of a bond with an embedded option, the corresponding option-free bond, and the embedded option. • Explain the effect of volatility on the arbitrage-free value of a bond with an embedded option. • Calculate an option-adjusted spread using the binomial model and interpret an option-adjusted spread with respect to the benchmark interest rates. • Calculate effective duration and effective convexity using the binomial model. • Compute the value of a putable bond using the binomial model.

Key Learning Outcomes • Describe the basic features of a convertible bond. • Compute and explain the meaning of the fol owing for a convertible bond: conversion value, straight value, market conversion price, market conversion premium per share, market conversion premium ratio, premium payback period, and premium over straight value. • Discuss the components of a convertible bond’s value that must be included in an option-based valuation approach. • Compare the risk/return characteristics of a convertible bond’s value to the risk/return characteristics of the underlying common stock.

Elements of a Bond Valuation Model • The potential benchmark interest rates that can be used in bond valuation are those in the Treasury market, a specific bond sector with a given credit rating, or a specific issuer. • Benchmark interest rates can be based on either an estimated yield curve or an estimated spot rate curve.

Relative Valuation Measures • Yield spread measures are used in assessing the relative value of securities. • Relative value analysis is used to identify securities as being overpriced (‘‘rich’ ), underpriced (‘ cheap’ ), or fairly priced relative to benchmark interest rates. • The interpretation of a spread measure depends on the benchmark used.

Traditional Yield Measures • Nominal spread• Static (z) spread• Yield to worst, i.e. smallest of: – Yield to maturity– Al yields to cal s and puts • These measures do not consider the effect of embedded options (reinvestment or call features).

Traditional Yield Measures

Nominal Yield Spread • The nominal spread is the difference between a non-Treasury bond’s yield and the YTM for a benchmark Treasury coupon security. – The nominal yield spread measures the compensation for the additional credit risk, option risk, and liquidity risk an investor is exposed to by investing in a non- Treasury security with the same maturity.

Zero-Volatility Spread • The zero-volatility or Z- spread is a measure of the spread the investor would realize over the entire Treasury spot rate curve if the bond is held to maturity. – It is not the spread off of one point on the Treasury yield curve (nominal spread), it is an average over al spot rates. • The Z-spread is also cal ed a static spread – and is calculated as the spread which wil make the present value of the cash flows from the non-Treasury bond, when discounted at the Treasury spot rate plus the spread, equal to the non-Treasury bond’s price. – Trial and error is used to determine the Z-spread.

What is the best spread? Option Adjusted Spread (OAS) • The Z-spread, which looks at measuring the spread over a spot rate curve, has a problem in that it fails to take future interest rate volatility into consideration – which could change the cash flows for bonds with embedded options. • The option-adjusted spread (OAS) was developed to take the dol ar difference between the fair valuation and the market price and convert it to a yield spread measure. – The OAS is used to reconcile the fair price (value) and the market price by finding a return (spread) that will equate the two. – The spread is measured in basis points.

Option Adjusted Spread (OAS) • The option-adjusted spread is a spread after adjusting for the option risk. • Depending on the benchmark interest rates used to generate the interest rate tree, the option-adjusted spread may or may not capture credit risk. • The option-adjusted spread is not a spread off of one maturity of the benchmark interest rates; rather, it is a spread over the forward rates in the interest rate tree that were constructed from the benchmark interest rates.

Relationship Between the Benchmark, OAS, and Relative Value

Bond Valuation Model • A valuation model must produce arbitrage-free values; that is, a valuation model must produce a value for each on-the-run issue that is equal to its observed market price. • There are several arbitrage-free models that can be used to value bonds with embedded options but they al follow the same principle— they generate a tree of interest rates based on some interest rate volatility assumption, they require rules for determining when any of the embedded options will be exercised, and they employ the backward induction methodology. • A valuation model involves generating an interest rate tree based on (1) benchmark interest rates, (2) an assumed interest rate model, and (3) an assumed interest rate volatility.

Valuing a Bond with an Embedded Option Using the Binomial Model

Valuing a Bond with an Embedded Option Using the Binomial Model

Valuing a Bond with an Embedded Option Using the Binomial Model • A valuation model involves generating an interest rate tree based on (1) benchmark interest rates, (2) an assumed interest rate model, and (3) an assumed interest rate volatility. • The assumed volatility of interest rates incorporates the uncertainty about future interest rates into the analysis. • The interest rate tree is constructed using a process that is similar to bootstrapping but requires an iterative procedure to determine the interest rates that wil produce a value for the on-the-run issues equal to their market value. • At each node of the tree there are interest rates and these rates are effectively forward rates; thus, there is a set of forward rates for each year.

Valuing a Bond with an Embedded Option Using the Binomial Model

Valuing a Bond with an Embedded Option Using the Binomial Model • Using the interest rate tree the arbitrage-free value of any bond can be determined. • In valuing a cal able bond using the interest rate tree, the cash flows at a node are modified to take into account the cal option. • The value of the embedded cal option is the difference between the value of an option-free bond and the value of the cal able bond.

Valuing a Bond with an Embedded Option Using the Binomial Model

Valuing a Bond with an Embedded Option Using the Binomial Model

Valuing a Bond with an Embedded Option Using the Binomial Model • The volatility assumption has an important impact on the arbitrage-free value. • The option-adjusted spread is the constant spread that when added to the short rates in the binomial interest rate tree wil produce a valuation for the bond (i.e. arbitrage- free value) equal to the market price of the bond. • The interpretation of the OAS, or equivalently, what the OAS is compensating an investor for, depends on what benchmark interest rates are used.

Valuing a Bond with an Embedded Option Using the Binomial Model

Valuing a Bond with an Embedded Option Using the Binomial Model

Valuing a Bond with an Embedded Option Using the Binomial Model

Valuing a Bond with an Embedded Option Using the Binomial Model

Valuing a Bond with an Embedded Option Using the Binomial Model

Effective Duration and Effective Convexity • The required values for calculating effective duration and effective convexity are found by shifting the on-the-run yield curve, calculating a new binomial interest rate tree, and then determining the required values after adjusting the tree by adding the OAS to each short rate. • For a bond with any embedded option or options, application of the binomial model requires that the value at each node of the tree be adjusted based on whether or not the option wil be exercised; the binomial model can be used to value bonds with multiple or interrelated embedded options by determining at each node of the tree whether or not one of the options wil be exercised.

Effective Duration and Effective Convexity

Effective Duration and Effective Convexity

Valuing a Putable Bond • With a putable bond, the option wil be exercised if the value at a node is less than the price at which the bondholder can put the bond to the issuer. • The value of a putable bond is greater than the value of an otherwise option-free bond. • The binomial model can be used to value a single step-up cal able note or a multiple step-up cal able note.

Valuing a Putable Bond

Valuing a Putable Bond

Valuing a Putable Bond

Valuing a Floating-Rate Note • To value a floating-rate note that has a cap, the coupon at each node of the tree is adjusted by determining whether or not the cap is reached at a node; if the rate at a node does exceed the cap, the rate at the node is the capped rate rather than the rate determined by the floater’s coupon formula. • For a floating-rate note, the binomial method must be adjusted to account for the fact that a floater pays in arrears; that is, the coupon payment is determined in a period but not paid until the next period.

Convertible Securities • Convertible and exchangeable securities can be converted into shares of common stock. • The conversion ratio is the number of common stock shares for which a convertible security may be converted. • Almost al convertible securities are cal able and some are putable.

Convertible Securities • The conversion value is the value of the convertible bond if it is immediately converted into the common stock. • The market conversion price is the price that an investor effectively pays for the common stock if the convertible security is purchased and then converted into the common stock. • The premium paid for the common stock is measured by the market conversion premium per share and market conversion premium ratio. • The straight value or investment value of a convertible security is its value if there was no conversion feature. • The minimum value of a convertible security is the greater of the conversion value and the straight value.

Convertible Securities • A fixed income equivalent (or a busted convertible) refers to the situation where the straight value is considerably higher than the conversion value so that the security wil trade much like a straight security. • A common stock equivalent refers to the situation where the conversion value is considerably higher than the straight value so that the convertible security trades as if it were an equity instrument. • A hybrid equivalent refers to the situation where the convertible security trades with characteristics of both a fixed income security and a common stock instrument.

Convertible Securities • While the downside risk of a convertible security usual y is estimated by calculating the premium over straight value, the limitation of this measure is that the straight value (the floor) changes as interest rates change. • An advantage of buying the convertible rather than the common stock is the reduction in downside risk. • The disadvantage of a convertible relative to the straight purchase of the common stock is the upside potential give-up because a premium per share must be paid.

Convertible Securities • An option-based valuation model is a more appropriate approach to value convertible securities than the traditional approach because it can handle multiple embedded options. • There are various option-based valuation models: one-factor and multiple-factor models. • The most common convertible bond valuation model is the one-factor model in which the one factor is the stock price movement.