Presentation on theme: "VALUATION AND ANALYSIS: BONDS WITH EMBEDDED OPTIONS CHAPTER 9 © 2016 CFA Institute. All rights reserved."— Presentation transcript:
1 VALUATION AND ANALYSIS: BONDS WITH EMBEDDED OPTIONS CHAPTER 9 © 2016 CFA Institute. All rights reserved.
2 TABLE OF CONTENTS 2 01INTRODUCTION 02 OVERVIEW OF EMBEDDED OPTIONS 03 VALUATION AND ANALYSIS OF CALLABLE AND PUTABLE BONDS 04 INTEREST RATE RISK OF BONDS WITH EMBEDDED OPTIONS 05 VALUATION AND ANALYSIS OF CAPPED AND FLOORED FLOATING- RATE BONDS 06 VALUATION AND ANALYSIS OF CONVERTIBLE BONDS 07 BOND ANALYTICS 08SUMMARY
3 1. INTRODUCTION The valuation of a fixed-rate option-free bond generally requires determining its future cash flows and discounting them at the appropriate rates. Valuation becomes more complicated when a bond has one or more embedded options because the values of embedded options are typically contingent on interest rates. Issuers and investors should understand how embedded options — such as call and put provisions, conversion options, caps, and floors — affect bond values and the sensitivity of these bonds to interest rate movements. 3
4 2. OVERVIEW OF EMBEDDED OPTIONS The term embedded options refers to contingency provisions found in the bond’s indenture or offering circular. These options represent rights that enable their holders to take advantage of interest rate movements. These options are not independent of the bond and thus cannot be traded separately — hence the adjective “embedded.” 4
5 CALL OPTIONS A callable bond is a bond that includes an embedded call option. The call provision allows the issuer to redeem the bond issue prior to maturity. 5 Early redemption usually happens when the issuer has the opportunity to replace a high-coupon bond with another bond that has more favorable terms. Most callable bonds include a lockout period during which the issuer cannot call the bond. Callable bonds include different types of call features: European, American, or Bermudan style.
6 PUT OPTIONS AND EXTENSION OPTIONS A putable bond is a bond that includes an embedded put option. The put provision allows the bondholders to put back the bonds to the issuer prior to maturity, usually at par. 6 Similar to callable bonds, most putable bonds include lockout periods. They can be European or, rarely, Bermudan style, but there are no American-style putable bonds. An embedded option that resembles a put option is an extension option — the right to keep the bond for a number of years after maturity, possibly with a different coupon.
7 COMPLEX EMBEDDED OPTIONS Although callable and putable bonds are the most common types of bonds with embedded options, there are bonds with other types of options or combinations of options. 7 A bond can be both callable and putable. A bond can be convertible (to stock). A bond may have an option that is contingent on some particular event. A bond may contain interrelated issuer options without any investor option, such as a sinking fund bond. A “sinker” may also include an acceleration provision or a delivery option.
8 3. VALUATION AND ANALYSIS OF CALLABLE AND PUTABLE BONDS The value of a bond with embedded options is equal to the sum of the arbitrage-free value of the straight bond and the arbitrage-free values of the embedded options. Value of callable bond = Value of straight bond – Value of issuer call option Value of issuer call option = Value of straight bond – Value of callable bond Value of putable bond = Value of straight bond + Value of investor put option Value of investor put option = Value of putable bond – Value of straight bond 8
9 VALUATION OF DEFAULT-FREE AND OPTION-FREE BONDS Refresher: The approach relying on one-period forward rates provides an appropriate framework for valuing bonds with embedded options. Example: Consider the valuation of a three-year 4.25% annual coupon bond 1) callable at par; 2) putable at par (1 and 2 years from now); and 3) equivalent non-callable bond at zero volatility. Forward rates are presented in the following table. 9 We need to know the value of the bond at different points in time in the future to determine whether the embedded option will be exercised at those points in time.
10 VALUATION OF DEFAULT-FREE AND OPTION-FREE BONDS 10 Today1 Year2 Year3 Year Cash Flow4.250 104.250 Discount Rate2.500%3.518%4.564% Value of the Callable Bond Value of the Putable Bond Value of Option-Free Bond Example (continued).
11 VALUATION OF DEFAULT-FREE AND OPTION-FREE BONDS Example (continued). In the exercise shown on the previous slide, the values of respective call and put options are: Value of issuer call option = 102.114 – 101.707 = $0.407 Value of investor put option = 102.397 – 102.114 = $0.283 11
12 INTEREST RATE VOLATILITY The value of any embedded option, regardless of the type of option, increases with interest rate volatility. The greater the volatility, the more opportunities exist for the embedded option to be exercised. 12 All else being equal, the call option increases in value with interest rate volatility. Thus, as interest rate volatility increases, the value of the callable bond decreases. All else being equal, the put option increases in value with interest rate volatility. Thus, as interest rate volatility increases, the value of the putable bond increases.
13 YIELD CURVE EFFECTS The value of a callable or putable bond is also affected by changes in the level and shape of the yield curve. For a callable bond: 13 All else being equal, the value of the call option increases as the yield curve flattens or inverts (effect of the shape). If the yield curve shifts down, the value of the callable bond rises less rapidly than the value of the straight bond, limiting the upside potential for the investor (level effect).
14 YIELD CURVE EFFECTS For a putable bond: 14 If the yield curve shifts up, the value of the putable bond falls slower than the value of the straight bond, limiting the downside loss for the investor (level effect). All else being equal, the value of the put option decreases as the yield curve flattens or inverts (effect of the shape).
15 VALUATION OF DEFAULT-FREE CALLABLE AND PUTABLE BONDS WITH INTEREST RATE VOLATILITY The procedure to value a bond with an embedded option in the presence of interest rate volatility is as follows: 15 Generate a tree of interest rates based on the given yield curve and interest rate volatility assumptions. At each node of the tree, determine whether the embedded options will be exercised. Apply the backward induction valuation methodology to calculate the bond’s present value. This methodology involves starting at maturity and working back from right to left to find the bond’s present value.
16 VALUATION OF DEFAULT-FREE CALLABLE AND PUTABLE BONDS WITH INTEREST RATE VOLATILITY Example. Consider a default-free three-year 4.25% annual coupon bond using the interest rate tree below (10% volatility) if in years 1 and 2 they are 1) callable and 2) putable at par: 16 2.5% 3.8695% 3.1681% 5.5258% 4.5245% 3.7041% Year 0Year 1 Year 2
17 VALUATION OF DEFAULT-FREE CALLABLE AND PUTABLE BONDS WITH INTEREST RATE VOLATILITY 17 V=101.540 C=4.25 V=99.658 C=4.25 V=100.922 Called at 100 C=4.25 V=98.791 C=4.25 V=99.738 C=4.25 V=100.526 Called at 100 C=4.25 V=100 C=4.25 V=100 C=4.25 V=100 C=4.25 V=100 2.5% 3.1681% 3.3695% 5.5258% 4.5242% 3.7041% Example (continued). 1)Callable bond [where C = cash flow (% of par) and V = value of the callable bond’s future cash flows (% of par).] Year 0Year 1Year 2 Year 3
18 Example (continued). 2) Putable bond VALUATION OF DEFAULT-FREE CALLABLE AND PUTABLE BONDS WITH INTEREST RATE VOLATILITY 18 V=102.522 C=4.25 V=100.366 C=4.25 V=101.304 C=4.25 V=98.791 Put at 100 C=4.25 V=99.738 Put at 100 C=4.25 V=100.526 C=4.25 V=100 C=4.25 V=100 C=4.25 V=100 C=4.25 V=100 2.5% 3.1681 % 3.3695 % 5.5258 % 4.5242% 3.7041% Year 0Year 1Year 2Year 3
19 VALUATION OF RISKY CALLABLE AND PUTABLE BONDS The approach for default-free (sovereign) bonds can be extended to risky (corporate) bonds. 19 The industry-standard approach is to increase the discount rates above the default-free rates to reflect default risk. The second approach to valuing risky bonds is by making the default probabilities explicit — that is, by assigning a probability to each time period going forward.
20 VALUATION OF RISKY CALLABLE AND PUTABLE BONDS Use an issuer-specific curve (might be impossible due to cost and availability of data). Raise the one-year forward rates derived from the default-free benchmark yield curve by a fixed Z-spread. There are two standard approaches to construct a suitable yield curve for a risky bond: 20 A second approach can be used for risky bonds with embedded options: Option-adjusted spread (OAS) is the constant spread that, when added to all the one-period forward rates on the interest rate tree, makes the arbitrage-free value of the bond equal to its market price.
21 VALUATION OF RISKY CALLABLE AND PUTABLE BONDS The dispersion of interest rates on the tree is volatility dependent, and so is the OAS. 21 As interest rate volatility increases, the OAS for the callable bond decreases.
22 4. INTEREST RATE RISK OF BONDS WITH EMBEDDED OPTIONS 22 Effective duration indicates the sensitivity of the bond’s price to a 100 bps parallel shift of the benchmark yield curve assuming no change in the bond’s credit spread.
23 CALCULATING A BOND’S EFFECTIVE DURATION IN PRACTICE In practice, the estimation procedure is usually as follows: 23 Given a price (PV0), calculate the implied OAS to the benchmark yield curve at an appropriate interest rate volatility Shift the benchmark yield curve down, generate a new interest rate tree, and then revalue the bond using the OAS calculated in Step 1. This value is PV–. Shift the benchmark yield curve up by the same magnitude, generate a new interest rate tree, and then revalue the bond using the OAS calculated in Step 1. This value is PV+. Calculate the bond’s effective duration.
24 INTEREST RATE RISK OF BONDS WITH EMBEDDED OPTIONS The effective duration of a callable bond cannot exceed that of the straight bond. The effective duration of a putable bond cannot exceed that of the straight bond. 24 When interest rates are high relative to the bond’s coupon, the callable and straight bonds have similar effective durations. When interest rates fall, the effective duration of the callable bond is lower than that of the straight bond. When interest rates are low relative to the bond’s coupon, the putable and straight bonds have similar effective durations. When interest rates rise, the effective duration of the putable bond is lower than that of the straight bond.
25 ONE-SIDED DURATION AND KEY RATE DURATION It is better at capturing the interest rate sensitivity of a callable or putable bond than the (two-sided) effective duration, particularly when the embedded option is near the money. One-sided duration is an effective duration when interest rates go up or down. Key rate durations help portfolio managers and risk managers identify the “shaping risk” for bonds — that is, the bond’s sensitivity to changes in the shape of the yield curve (e.g., steepening and flattening). Key rate duration reflects the sensitivity of the bond’s price to changes in specific maturities on the benchmark yield curve. 25
26 EFFECTIVE CONVEXITY Effective convexity is calculated for callable, putable bonds similar to the straight bond. 26 When interest rates are high, the callable and straight bond experience very similar positive convexity. The effective convexity of the callable bond turns negative when the call option is near the money. Putable bonds always have positive convexity. It is similar to the straight bond when interest rates are low. When the put option is near the money, the convexity of a putable bond becomes larger than that of a straight bond.
27 5. VALUATION AND ANALYSIS OF CAPPED AND FLOORED FLOATING-RATE BONDS Capped and floored floaters can be valued by using the arbitrage-free framework. A capped floater protects the issuer against rising interest rates and is thus an issuer option. The investor is long in the bond but short in the embedded option. Example: Consider a three-year LIBOR floater capped at 4.5% at 10% interest rate volatility. The interest rate tree is the same as in earlier examples. 27 Value of embedded cap Value of straight bond Value of capped floater
28 Example (continued). Capped Floater: Value of embedded cap = 100 – 99.761 = 0.239. VALUATION OF A CAPPED FLOATER 28 V=99.761 C=2.5 V=99.521 C=2.5 V=99.989 C=3.8695 V=99.028 C H =3.3695 V=99.977 C L =3.1681 C=3.1681 V=100.000 C=5.5258 4.5 V=100 2.5% 3.1681% 3.3695% 5.5258% 4.5242% 3.7041% C=5.5258 4.5 V=100 C=4.5242 4.5 V=100 C=4.5242 4.5 V=100 C=3.7041 V=100 C=3.7041 V=100 Year 1Year 0 Year 2Year 3
29 VALUATION OF A FLOORED FLOATER The floor provision in a floater prevents the coupon rate from decreasing below a specified minimum rate. A floored floater protects the investor against declining interest rates and is thus an investor option. The investor is long both in the bond and in the embedded option. Example: Consider a three-year LIBOR floater floored at 3.5% at 10% interest rate volatility. The interest rate tree is the same as in earlier examples. 29 Value of embedded floor Value of straight bond Value of floored floater
30 Example (continued). Floored Floater: Value of embedded floor = 101.133 – 100 = 1.133. VALUATION OF A FLOORED FLOATER 30 V = 101.133 C = 2.500 3.500 V = 100 C = 2.500 3.500 V = 100.322 C = 3.8695 V = 100 C H = 3.8695 V = 100 C L = 3.1681 3.500 C = 3.1681 3.500 V = 100 C = 5.5258 V = 100 2.5% 3.1681% 3.3695% 5.5258% 4.5242% 3.7041% C = 5.5258 V = 100 C = 4.5242 V = 100 C = 4.5242 V = 100 C = 3.7041 V = 100 C = 3.7041 V = 100 Year 0Year 1Year 2Year 3
31 6. VALUATION AND ANALYSIS OF CONVERTIBLE BONDS A convertible bond is a hybrid security that presents the characteristics of an option-free bond and an embedded conversion option. 31 The conversion option is a call option on the issuer’s common stock, which gives bondholders the right to convert their debt into equity during a pre-determined period (known as the conversion period) at a pre-determined price (known as the conversion price). The number of shares of common stock that the bondholder receives from converting the bonds into shares is called the “conversion ratio.”
32 CONVERSION VALUE The conversion value or parity value of a convertible bond indicates the value of the bond if it is converted at the market price of the shares. The minimum value of a convertible bond is equal to the greater of the following: 32 Conversion ratio Underlying share price Conversion value Value of the underlying option- free bond and
33 MARKET CONVERSION PRICE AND PREMIUM 33
34 DOWNSIDE RISK AND UPSIDE POTENTIAL OF CONVERTIBLE BONDS 34 Thus, convertible bond investors should be familiar with the techniques used to value and analyze common stocks.
35 VALUATION OF A CONVERTIBLE BOND The most commonly used model to value convertible bonds is the arbitrage-free framework. = stock 35 Value of convertible bond Value of straight bond Value of call option on the issuer’s stock Value of callable convertible bond Value of straight bond Value of call option on the issuer’s stock Value of issuer call option Value of callable putable convertible bond Value of straight bond Value of call option on the issuer’s stock Value of issuer call option Value of investor put option
36 RISK–RETURN CHARACERISTICS OF A CONVERTIBLE BOND, A STRAIGHT BOND, AND THE UNDERLYING STOCK In between the bond and the stock extremes, the convertible bond trades like a hybrid instrument. 36 In contrast, when the underlying share price is above the conversion price, a convertible bond exhibits mostly stock risk–return characteristics. When the underlying share price is well below the conversion price, the convertible bond is described as “busted convertible” and exhibits mostly bond risk–return characteristics...
37 7. BOND ANALYTICS Some market participants, in particular financial institutions, develop bond analysis system in-house. How can a practitioner tell if such a system is adequate? -The system should be able to report the correct cash flows, discount rates, and present value of the cash flows. The discount rates can be verified by hand or on a spreadsheet. -Even if it is difficult to verify that a result is correct, it may be possible to establish that it is wrong by doing the following checks: 37 Check that the put–call parity holds. Check that the value of the underlying option-free bond does not depend on interest rate volatility. Check that the volatility term structure slopes downward.
38 SUMMARY An embedded option represents a right that can be exercised by the issuer, by the bondholder, or automatically depending on the course of interest rates. Simple embedded option structures include call options, put options, and extension options. Complex embedded option structures include bonds with other types of options or combinations of options, including a conversion option, an estate put, or an acceleration provision for a sinking fund. Fixed-income securities with embedded options 38
39 SUMMARY The value of a bond with an embedded option is equal to the arbitrage-free values of its parts—that is, the arbitrage-free value of the straight bond and the arbitrage-free values of each of the embedded options. Relationships between the values of a callable or putable bond, the underlying option-free bond, and the embedded option The value of a callable or putable bond can be calculated by discounting the bond’s future cash flows at the appropriate one-period forward rates, taking into consideration the decision to exercise the option. Arbitrage-free framework can be used to value a bond with embedded options 39
40 SUMMARY Interest rate volatility is modeled using a binomial interest rate tree. The higher the volatility, the lower the value of the callable bond and the higher the value of the putable bond. Changes in the level and shape of the yield curve affect the values of bonds with embedded options. Interest rate volatility and value of a callable/putable bond Valuing a bond with embedded options assuming an interest rate volatility requires three steps: (1) Generate a tree of interest rates based on the given yield curve and volatility assumptions; (2) at each node of the tree, determine whether the embedded options will be exercised; and (3) apply the backward induction valuation methodology to calculate the present value of the bond. Valuing a callable/putable bond from an interest rate tree 40
41 SUMMARY The option-adjusted spread is the single spread added uniformly to the one-period forward rates on the tree to produce a value or price for a bond. The OAS is sensitive to interest rate volatility: The higher the volatility, the lower the OAS for a callable bond. Option-adjusted spreads For bonds with embedded options, the best measure to assess the sensitivity of the bond’s price to a parallel shift of the benchmark yield curve is effective duration. The effective duration of a callable or putable bond cannot exceed that of the straight bond. Effective duration of a callable/putable bond 41
42 SUMMARY Because the prices of callable and putable bonds respond asymmetrically to upward and downward interest rate changes of the same magnitude, one-sided durations provide a better indication regarding the interest rate sensitivity of bonds with embedded options than (two-sided) effective duration. Key rate durations show the effect of shifting only key points, one at a time, rather than the entire yield curve. One-sided and key rate durations When the option is near the money, the convexity of a callable bond is negative, indicating that the up side for a callable bond is much smaller than the down side, whereas the convexity of a putable bond is positive, indicating that the up side for a putable bond is much larger than the downside. Effective convexities of callable, putable, and straight bonds 42
43 SUMMARY The value of a capped floater is equal to or less than the value of the straight bond. The value of a floored floater is equal to or higher than the value of the straight bond. Capped or floored floating-rate bond The characteristics of a convertible bond include the conversion price, which is the applicable share price at which the bondholders can convert their bonds into common shares, and the conversion ratio, which reflects the number of shares of common stock that the bondholders receive from converting their bonds into shares. Defining features of a convertible bond 43
44 SUMMARY There are a number of investment metrics and ratios that help analyze and value convertible bonds. The conversion value indicates the value of the bond if it is converted at the market price of the shares. The minimum value of a convertible bond sets a floor value for the convertible bond at the greater of the conversion value or the straight value. The arbitrage-free framework can be used to value convertible bonds, including callable and putable ones. Each component (straight bond, call option of the stock, and call and/or put option on the bond) can be valued separately. Valuation of a convertible bond 44
45 8. SUMMARY The risk–return characteristics of a convertible bond depend on the underlying share price relative to the conversion price. When the underlying share price is well below the conversion price, the convertible bond is “busted” and exhibits mostly bond risk–return characteristics. In contrast, when the underlying share price is well above the conversion price, the convertible bond exhibits mostly stock risk–return characteristics. Risk–return characteristics of a convertible bond 45
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