Valuation of Bonds with Embedded Options
READING 27
VALUATION AND ANALYSIS OF BONDS WITH EMBEDDED OPTIONS
EXAM FOCUS
This topic review extends the arbitrage-free valuation framework to valuation of bonds with embedded options. Understand the risk/return dynamics of embedded options, including their impact on a bond's duration and convexity. You should also know the adjustment required for the valuation of credit-risky bonds, including the process to estimate an option-adjusted spread (OAS). Finally, understand the terminology and risk/return characteristics of convertibles.
MODULE 27.1: TYPES OF EMBEDDED OPTIONS
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LOS 27.a: Describe fixed-income securities with embedded options.
Embedded options in a bond allow an issuer to (1) manage interest-rate risk and/or (2) issue the bonds at an attractive coupon rate. The embedded options can be a simple call or put option, or more complex options such as provisions for a sinking fund or an estate put.
Simple Options
Callable bonds give the issuer the option to redeem (call) the bond prior to maturity; the investor is effectively short the embedded call option. Most callable bonds have a call protection period during which the bond cannot be called. The call option can be:
- European-style (exercisable on a single day immediately after the protection period),
- American-style (exercisable at any time after the protection period), or
- Bermudan-style (exercisable on specific fixed dates after the protection period).
Putable bonds allow the investor to sell (put) the bond back to the issuer prior to maturity; the investor is long the embedded put option. A related structure is an extendible bond, which allows the investor to extend the maturity of the bond. An extendible bond can be valued as a putable bond with the extended maturity (i.e., the maturity if the bond is extended). For example, a two-year, 3% bond extendible for an additional year at the same coupon rate would be valued the same as an otherwise identical three-year European-style putable bond with a protection period of two years.
Complex Options
More complex options include:
- Estate put - a provision that allows the heirs of an investor to put the bond back to the issuer upon the investor's death. The value of this contingent put option is inversely related to the investor's life expectancy; the shorter the life expectancy, the higher the value.
- Sinking fund bonds (sinkers) - require the issuer to set aside funds periodically to retire the bond. This provision reduces the credit risk of the bond. Sinkers typically have several related issuer options (for example, call provisions, acceleration provisions, and delivery options).
LOS 27.b: Explain the relationships between the values of a callable or putable bond, the underlying option-free (straight) bond, and the embedded option.
In essence, the holder of a callable bond owns an option-free (straight) bond and is also short a call option written on the bond. Therefore the value of the embedded call option (Vcall) is the difference between the value of a straight bond (Vstraight) and the value of the comparable callable bond (Vcallable):
Vcall = Vstraight - Vcallable
Conversely, investors pay a premium for a putable bond because its holder effectively owns an option-free bond plus a put option. The value of a putable bond is:
Vputable = Vstraight + Vput
Rearranging:
Vput = Vputable - Vstraight
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MODULE 27.2: VALUING BONDS WITH EMBEDDED OPTIONS, PART 1
LOS 27.c: Describe how the arbitrage-free framework can be used to value a bond with embedded options.
LOS 27.f: Calculate the value of a callable or putable bond from an interest rate tree.
The basic process for valuing a callable (or putable) bond is similar to valuing a straight bond. However, instead of using spot rates, one-period forward rates are used in a binomial tree framework. This methodology computes the value of the bond at different points in time; these checks are necessary to determine whether the embedded option is in-the-money (and exercised).
When valuing a callable bond, at any node where the bond is callable the value must be either the call price (the price at which the issuer will call the bond) or the computed continuation value if the bond is not called, whichever is lower. This is the call rule. Similarly, for a putable bond, at any node corresponding to a put date the value used must be either the put price (the price at which the investor will put the bond) or the computed continuation value if the bond is not put, whichever is higher. This is the put rule.
PROFESSOR'S NOTE
Call date and put date in this context vary depending on whether the option is European-, American-, or Bermudan-style.
EXAMPLE: Valuation of call and put options
Consider a two-year, 7% annual-pay, $100 par bond callable in one year at $100. Also consider a two-year, 7% annual-pay, $100 par bond putable in one year at $100.
The interest-rate tree at 15% assumed volatility is as follows:
Value the embedded call and put options.
Answer:
Value of the straight (option-free) bond:
Consider the value of the bond at the upper node for Period 1, V1,U:
Similarly, the value of the bond at the lower node for Period 1, V1,L, is:
Now calculate V0, the current value of the bond at Node 0.
The completed binomial tree is shown as follows:
Valuing a Two-Year, 7.0% Coupon, Option-Free Bond
Value of the callable bond:
The call rule (call the bond if the price exceeds $100) is reflected in the boxes in the completed binomial tree, where the second line of the boxes at the one-year node is the lower of the call price or the computed value. For example, the value of the bond in one year at the lower node is $101.594. However, in this case, the bond will be called, and the investor will only receive $100. Therefore, for valuation purposes, the value of the bond in one year at this node is $100.
The calculation for the current value of the bond at Node 0 (today), assuming the simplified call rules of this example, is:
The completed binomial tree is shown as follows:
Valuing a Two-Year, 7.0% Coupon, Callable Bond, Callable in One Year at 100
Value of the putable bond:
Similarly, for a putable bond, the put rule is to put the bond if the value falls below $100. The put option would therefore be exercised at the upper-node in Year 1 and hence the $99.830 computed value is replaced by the exercise price of $100.
Value of the embedded options:
MODULE QUIZ 27.1, 27.2
1.
Which of the following statements concerning the calculation of value at a node in a binomial interest rate tree is most accurate? The value at each node is the: A. present value of the two possible values from the next period.
B.
average of the present values of the two possible values from the next period.
C.
sum of the present values of the two possible values from the next period.
Use the following binomial interest rate tree to answer Questions 2 through 4.
2.
The value today of an option-free, 12% annual coupon bond with two years remaining until maturity is closest to:
A.
110.525.
B.
111.485.
C.
112.282.
3.
The value of the bond and the value of the embedded call option, assuming the bond in Question 2 is callable at $105 at the end of Year 1, are closest to:
4.
The value of the bond and the value of the embedded put option, assuming the bond in Question 2 is putable at $105 at the end of Year 1, are closest to:
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MODULE 27.3: VALUING BONDS WITH EMBEDDED OPTIONS, PART 2
LOS 27.d: Explain how interest rate volatility affects the value of a callable or putable bond.
Option values are positively related to the volatility of their underlying. Accordingly, when interest-rate volatility increases, the values of both call and put options increase. The value of a straight bond is affected by changes in the level of interest rates but is unaffected by changes in the volatility of interest rates.
When interest-rate volatility increases, the value of a callable bond (where the investor is short the call option) decreases and the value of a putable bond (where the investor is long the put option) increases.
LOS 27.e: Explain how changes in the level and shape of the yield curve affect the value of a callable or putable bond.
Level of Interest Rates
As interest rates decline, the short call in a callable bond limits the bond's upside, so the value of a callable bond rises less rapidly than the value of an otherwise-equivalent straight bond.
As interest rates increase, the long put in a putable bond hedges against the loss in value; the value of a putable bond falls less rapidly than the value of an otherwise-equivalent straight bond.
Call option value is inversely related to the level of interest rates, while put option value varies directly with the level of interest rates.
Shape of the Yield Curve
The value of an embedded call option increases as interest rates decline. When the yield curve is upward sloping (normal), the more distant one-period forward rates are higher than near-term forward rates. Because a higher interest-rate scenario limits the probability of the call option being in the money, the value of a call option will be lower for an upward-sloping yield curve. As an upward-sloping yield curve becomes flatter, the call option value increases.
The value of a put option increases with interest rates. When the yield curve is upward sloping, the probability of the put option going in the money is higher. Put option value therefore declines as an upward-sloping yield curve flattens.
MODULE 27.4: OPTION-ADJUSTED SPREAD
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LOS 27.g: Explain the calculation and use of option-adjusted spreads.
Our backward-induction valuation used a risk-free binomial interest-rate tree; the valuation assumed that the underlying bond was risk-free. If risk-free rates are used to discount cash flows of a credit-risky corporate bond, the calculated value will be too high. To correct for this, a constant spread must be added to all one-period rates in the tree such that the computed value equals the market price of the risky bond. This constant spread is called the option-adjusted spread (OAS).
PROFESSOR'S NOTE
The OAS is added to the tree after the adjustment for the embedded option (i.e., after applying the call/put rule at nodes). Hence the OAS is calculated after the option risk has been removed.
EXAMPLE: Computation of OAS
A $100-par, three-year, 6% annual-pay ABC, Inc., callable bond trades at $99.95. The underlying call option is a Bermudan-style option that can be exercised in one or two years at par.
The benchmark interest-rate tree assuming volatility of 20% is provided here:
Compute the OAS on the bond.
Answer:
The value of the bond using the benchmark interest-rate tree is $101.77, as shown here:
To force the computed value to equal the current market price of $99.95, a constant spread (OAS) of 100 bps is added to each interest rate in the tree, as shown here:
PROFESSOR'S NOTE
The OAS computed using the methodology just listed is the spread implied by the current market price and hence assumes that the bond is priced correctly. Note that the actual estimation of OAS is largely an iterative process and is beyond the scope of this exam.
OAS is used by analysts in relative valuation; bonds with similar credit risk should have the same OAS. If the OAS for a bond is higher than the OAS of its peers, it is considered to be undervalued and hence an attractive investment (i.e., it offers higher compensation for a given level of risk). Conversely, bonds with low OAS (relative to peers) are considered to be overvalued.
LOS 27.h: Explain how interest rate volatility affects option-adjusted spreads.
Consider the difference between the calculated value of a bond from a tree and the bond's actual market price. The greater this difference, the greater the OAS we must add to the rates in the tree to force the calculated bond value down to the market price.
Suppose that a 7%, 10-year callable bond of XYZ Corp. is trading for $958.
Analyst A assumes 15% volatility for future rates in generating her benchmark interest-rate tree and calculates the value of the bond as $1,050. She then computes the OAS (the increase in discount rate required to lower the calculated bond value to the market price of $958) for this bond to be 80 bps.
Analyst B assumes 20% volatility, and because of higher assumed volatility, the computed value of the bond turns out to be $992. (This is lower than the $1,050 calculated by A-the higher the volatility, the lower a callable bond's value.) The OAS calculated by B is accordingly lower at 54 bps.
Observe that for the same bond, the OAS calculated varied depending on the volatility assumption used.
When we use a higher estimate of volatility to value a callable bond, the calculated value of the call option increases, the calculated value of the straight bond is unaffected, and the computed value (not the market price) of the callable bond decreases (since the bondholder is short the option). Hence when the assumed volatility of benchmark rates used in a binomial tree is higher, the computed value of a callable bond will be lower-and therefore closer to its market price. The constant spread that needs to be added to the benchmark rates to correctly price the bond (the OAS) is therefore lower.
Figure 27.1: Relationship Between Volatility and OAS
To summarize, as the assumed level of volatility used in an interest-rate tree increases, the computed OAS (for a given market price) for a callable bond decreases. Similarly, the computed OAS of a putable bond increases as the assumed level of volatility in the binomial tree increases.
PROFESSOR'S NOTE
Notice that the columns for the value of callable and putable bonds in the preceding figure match the corresponding OAS columns.
MODULE QUIZ 27.3, 27.4
1.
The option adjusted spread (OAS) on a callable corporate bond is 73 basis points using on-the-run Treasuries as the benchmark rates in the construction of the binomial tree. The best interpretation of this OAS is the:
A.
cost of the embedded option is 73 basis points.
B.
cost of the option is 73 basis points over Treasury.
C.
spread that reflects the credit risk is 73 basis points over Treasury.
2.
An increase in interest-rate volatility increases the value of:
A.
bonds with embedded call options.
B.
bonds with embedded put options.
C.
low-coupon bonds with embedded options, but decreases the value of high-coupon bonds with embedded options.
MODULE 27.5: DURATION
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LOS 27.i: Calculate and interpret effective duration of a callable or putable bond.
Recall that modified duration measures a bond's price sensitivity to interest-rate changes, assuming the bond's cash flows do not change as interest rates change. The standard measure of convexity can be used to improve price-change estimates from modified duration. Modified duration and convexity are not useful for bonds with embedded options because cash flows from those bonds will change if the option is exercised. To overcome this problem, effective duration and effective convexity should be used, because these measures take into account how changes in interest rates may alter cash flows.
The following expressions can be used to compute effective duration and effective convexity for any bond:
Effective duration = (BV(-Δy) - BV(+Δy)) / (2 × BV0 × Δy)
Effective convexity = (BV(-Δy) + BV(+Δy) - 2 × BV0) / (2 × BV0 × (Δy)^2)
Calculating effective duration and effective convexity for bonds with embedded options requires calculation of BV(+Δy) and BV(-Δy). The process:
- Given assumptions about benchmark interest rates, interest-rate volatility, and any calls and/or puts, calculate the OAS for the issue using the current market price and the binomial model.
- Impose a small parallel shift in the benchmark yield curve by +Δy.
- Build a new binomial interest-rate tree using the new yield curve.
- Add the OAS from step 1 to each of the one-year rates in the interest-rate tree to get a "modified" tree.
- Compute BV(+Δy) using this modified interest-rate tree.
- Repeat steps 2 through 5 using a parallel shift of -Δy to obtain BV(-Δy).
LOS 27.j: Compare effective durations of callable, putable, and straight bonds.
Both call and put options can reduce the effective life of a bond, so the durations of callable and putable bonds will be less than or equal to their straight counterparts.
Effective duration (callable) ≤ Effective duration (straight).
Effective duration (putable) ≤ Effective duration (straight).
Effective duration (zero-coupon) ≈ maturity of the bond.
Effective duration of fixed-rate coupon bond < maturity of the bond.
Effective duration of a floater ≈ time (in years) to next reset.
While effective duration of straight bonds is relatively unaffected by changes in interest rates, an increase in rates would increase the effective duration of a callable bond, and decrease the effective duration of a putable bond.
MODULE QUIZ 27.5
1. Ron Hyatt has been asked to do a presentation on how effective duration (ED) and effective convexity (EC) are calculated with a binomial model. His presentation includes the following formulas:
Are Hyatt's formulas for effective duration and effective convexity correctly presented?
A.
The formulas are both correct.
B.
One formula is correct, the other incorrect.
C.
Both formulas are incorrect.
MODULE 27.6: KEY RATE DURATION
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LOS 27.k: Describe the use of one-sided durations and key rate durations to evaluate the interest rate sensitivity of bonds with embedded options.
One-Sided Durations
The effective-duration computation discussed earlier relies on equal parallel shifts of the yield curve up and down. This metric captures interest-rate risk reasonably well for small changes in the yield curve and for option-free bonds.
For a callable bond, when the call option is at or near the money, the change in price for a decrease in yield will be less than the change in price for an equal increase in yield. The value of a callable bond is capped by its call price: the bond's value will not increase beyond the call price regardless of how low interest rates fall. Similarly, the value of a putable bond is more sensitive to downward movements in yield than upward movements.
For bonds with embedded options, one-sided durations-durations that apply only when interest rates rise (or only when rates fall)-are better at capturing interest-rate sensitivity than simple effective duration. When the underlying option is at-the-money (or near-the-money), callable bonds will have lower one-sided down-duration than one-sided up-duration: the price change of a callable when rates fall is smaller than the price change for an equal increase in rates. Conversely, a near-the-money putable bond will have larger one-sided down-duration than one-sided up-duration.
Key Rate Duration
Key rate durations (partial durations) capture the interest-rate sensitivity of a bond to changes in yields (par rates) of specific benchmark maturities. Key rate duration identifies shaping risk-the sensitivity to changes in the shape of the yield curve.
The process is similar to computing effective duration except that only one specific par rate (the key rate) is shifted before measuring the price impact.
Figure 27.2 shows key rate durations for several 15-year option-free bonds with different coupon rates and YTM = 3%.
Figure 27.3 shows key rate durations for several 15-year European-style callable bonds (callable in 10 years at par) with different coupon rates.
Figure 27.4 shows key rate durations for several 15-year European-style putable bonds (putable in 10 years at par) with different coupon rates.
The following generalizations can be made about key rates:
- If an option-free bond is trading at par, the bond's maturity-matched rate is the only rate that affects the bond's value. Its maturity key-rate duration equals its effective duration, and all other key-rate durations are zero.
- For an option-free bond not trading at par, the maturity-matched rate is still the most important rate; its maturity key-rate duration is usually the highest.
- A bond with a low (or zero) coupon may have negative key-rate durations for horizons other than the bond's maturity.
- A callable bond with a low coupon rate is unlikely to be called; hence the bond's maturity-matched rate is the most critical rate (the highest key-rate duration).
- All else equal, higher-coupon bonds are more likely to be called, and therefore the time-to-exercise rate will tend to dominate the time-to-maturity rate.
- Putable bonds with high coupon rates are unlikely to be put, and thus are most sensitive to their maturity-matched rates.
- All else equal, lower-coupon putable bonds are more likely to be put, and therefore the time-to-exercise rate will tend to dominate the time-to-maturity rate.
LOS 27.l: Compare effective convexities of callable, putable, and straight bonds.
Straight bonds have positive effective convexity: the increase in value when rates fall is greater than the decrease in value when rates rise by the same amount. When rates are high, callable bonds are unlikely to be called and will exhibit positive convexity. When the underlying call option is near the money, the effective convexity of a callable bond can turn negative because the upside potential of price increases is limited by the call while the downside is not protected. Putable bonds exhibit positive convexity throughout.
MODULE 27.7: CAPPED AND FLOORED FLOATERS
LOS 27.m: Calculate the value of a capped or floored floating-rate bond.
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A floating-rate bond (floater) pays a coupon that adjusts every period based on an underlying reference rate. Coupons are typically paid in arrears, meaning the coupon rate is determined at the beginning of a period and paid at the end of that period.
A capped floater effectively contains an issuer option that prevents the coupon rate from rising above a specified maximum rate known as the cap. A floored floater contains an investor option that ensures the coupon rate will not fall below a specified minimum rate known as the floor.
Valuation uses backward induction in a binomial interest-rate tree. At each node where a cap or floor becomes relevant, adjust the node cash flows to reflect exercise of the option.
EXAMPLE: Value of a capped and floored floating-rate bond
Susane Albright works as a fixed-income analyst with Zedone Banks, NA. She has been asked to value a $100 par, two-year, floating-rate note that pays MRR (set in arrears). The underlying bond has the same credit quality as reflected in the swap curve. Albright has constructed the following two-year binomial MRR tree:
The tasks:
- The value of the floater, assuming it is an option-free bond.
- The value of the floater, assuming it is capped at a rate of 6%. Also compute the value of the embedded cap.
- The value of the floater, assuming it is floored at a rate of 5%. Also compute the value of the embedded floor.
Answer:
1. An option-free floater whose coupon equals the required rate of return is valued at par. Hence, the straight value of the floater is $100.
2. The value of the capped floater is $99.47, as shown here:
The upper node in Year 2 shows exercise of the cap (the coupon is capped at $6.00 instead of rising to $7.18).
Note that when the option is not in the money, the floater is valued at par.
The Year 0 value is the average of the Year 1 values (including their adjusted coupons) discounted for one period. In this case, the Year 1 coupons require no adjustment, as the coupon rate is below the cap rate.
Thus the value of the embedded cap = $100 - $99.47 = $0.53.
3. The value of the floored floater is $100.41, as shown here:
The nodes for Year 2 show the coupons for that period (none of the rates are below the floor, and hence the floor is not exercised). Strikethroughs for both nodes in Year 1 indicate that the floor was in the money; for valuation we replace the MRR-based coupon with the coupon based on the floor strike rate of 5%.
The Year 0 value is the average of the Year 1 values (including their adjusted coupons) discounted for one period.
Thus the value of the embedded floor = $100.41 - $100 = $0.41.
MODULE QUIZ 27.6, 27.7
Use the following information to answer Questions 1 through 8.
Vincent Osagae, CFA, is the fixed-income portfolio manager for Alpha Specialists, an institutional money manager. Vanessa Alwan, an intern, has suggested a list of bonds for Osagae's consideration as shown in Figure 1. The benchmark yield curve is currently upward sloping.
Osagae then turns his attention to a newly issued 4%, 15-year bond issued by Suni Corp. The bond has a Bermudan-style call option exercisable annually at any time after Year 4. Osagae computes an OAS of 145 bps for this bond using a binomial interest-rate tree. The tree was generated with U.S. Treasury rates and an assumed volatility of 15%, consistent with historical data. Option markets are currently pricing interest-rate options at an implied volatility of 19%.
Every week, Alwan meets Osagae to discuss what she had learned over the week. At one of their weekly meetings, Alwan makes the following statements about key rate duration:
Statement 1: The highest key rate duration of a low-coupon callable bond corresponds to the bond's time-to-exercise.
Statement 2: The highest key rate duration of a high-coupon putable bond corresponds to the bond's time-to-maturity.
Beth Grange, a senior analyst reporting to Osagae, has generated the following two-year MRR tree using a 15% volatility assumption:
Figure 2: MRR Tree
Grange has also compiled OAS for two very similar callable corporate bonds as shown in Figure 3.
Figure 3: OAS at 10% Volatility
1.
In Figure 1, which bond's embedded option is most likely to increase in value if the yield curve flattens?
A.
Bond Y.
B.
Bond Z.
C.
Neither Bond Y nor Bond Z.
2.
If Osagae had used implied volatility from interest-rate options to compute the OAS for Suni Corp bond, the estimated OAS would most likely have been:
A.
lower than 145 bps.
B.
higher than 145 bps.
C.
equal to 145 bps.
3.
Which bond in Figure 1 is most likely to have the highest effective duration?
A.
Bond X.
B.
Bond Y.
C.
Bond Z.
4.
Regarding Alwan's statements about key rate durations:
A.
only one statement is correct.
B.
both statements are correct.
C.
neither statement is correct.
5.
Which bond in Figure 1 is least likely to experience the highest increase in value, given a parallel downward shift of 150 bps in the yield curve?
A.
Bond X.
B.
Bond Y.
C.
Bond Z.
6.
Which bond in Figure 1 is most likely to experience an increase in effective duration due to an increase in interest rates?
A.
Bond X.
B.
Bond Y.
C.
Bond Z.
7.
In Figure 3, relative to bond A, bond B is most likely:
A.
overpriced.
B.
underpriced.
C.
fairly priced.
8.
Using the data in Figure 2, the value of a $100 par, two-year, 3% capped floater is closest to:
A.
$0.31.
B.
$98.67.
C.
$99.78.
MODULE 27.8: CONVERTIBLE BONDS
LOS 27.n: Describe defining features of a convertible bond.
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The owner of a convertible bond has the right to convert the bond into a fixed number of common shares of the issuer during a specified timeframe (the conversion period) and at a fixed amount of money (the conversion price). Convertibles allow investors to enjoy upside on the issuer's stock, though at the cost of a lower yield. The issuer benefits from a lower borrowing cost, but existing shareholders may face dilution if conversion occurs.
The conversion ratio is the number of common shares for which a convertible bond can be exchanged. For example, a convertible bond issued at par with an initial conversion ratio of 10 allows its holder to convert one $1,000 par bond into 10 shares of common stock. Equivalently, the conversion price is $1,000 / 10 shares = $100. For bonds not issued at par, conversion price equals the issue price divided by the conversion ratio. Offer documents indicate how the conversion ratio is adjusted for corporate actions such as stock splits or stock dividends.
Offer documents may also provide a contingent put option in the event of change-of-control events such as mergers. Such a contingent put option can be exercised for a specific period after the change of control. Alternatively, a lower conversion price may be specified following a change of control. The conversion ratio may be adjusted upward if the company pays a dividend in excess of a specified threshold dividend, protecting bondholders from dilution when unusually large dividends are paid. Other put options exercisable during specific periods may also be embedded; these put options can be hard puts (redeemable for cash) or soft puts (issuer chooses whether to redeem for cash, stock, subordinated debentures, or a combination).
LOS 27.o: Calculate and interpret the components of a convertible bond's value.
The conversion value of a convertible bond is the value of the common stock into which the bond can be converted. Conversion value equals:
Conversion value = Market price of stock × Conversion ratio
The straight value (investment value) of a convertible bond is the value of the bond if it were not convertible-the present value of the bond's cash flows, discounted at the return required on a comparable option-free issue.
The minimum value of a convertible bond is the greater of its conversion value or its straight value:
Minimum value = max(straight value, conversion value)
EXAMPLE: Calculating the minimum value of a convertible bond
Business Supply Company, Inc. (BSC) has a convertible bond with a 7% coupon currently selling at $985 with a conversion ratio of 25 and a straight value of $950. Suppose the value of BSC common stock is $35 per share and it pays $1 per share in dividends annually. What is the bond's minimum value?
Answer:
Conversion value = 25 × $35 = $875. Since straight value $950 > conversion value $875, the bond's minimum value is $950.
The market conversion price (conversion parity price) is the effective price per share the convertible bondholder pays for the stock if she converts:
Market conversion price = Market price of convertible bond / Conversion ratio
EXAMPLE: Calculating market conversion price
Answer:
The market conversion price for the BSC bond is $985 / 25 = $39.40.
The market conversion premium per share is:
Market conversion premium per share = Market conversion price - Stock's market price
EXAMPLE: Calculating market conversion premium per share
Answer:
Since BSC is selling for $35 per share, market conversion premium per share = $39.40 - $35 = $4.40.
The market conversion premium ratio expresses the premium per share as a ratio:
Market conversion premium ratio = (Market conversion price - Market price of stock) / Market price of stock
EXAMPLE: Calculating market conversion premium ratio
Answer:
The BSC bond market conversion premium ratio is:
The convertible bond investor's downside risk is limited by the bond's straight value because the convertible will not fall below its straight value, so the premium over straight value measures downside cushion:
Premium over straight value = Market price of convertible bond - Straight value
EXAMPLE: Calculating premium over straight value
Answer:
The premium over straight value for the BSC bond is:
All else equal, the greater the premium over straight value, the less attractive the convertible bond.
Note: the premium-over-straight metric is flawed because the straight value is not constant; it varies with interest rates and credit spread.
LOS 27.p: Describe how a convertible bond is valued in an arbitrage-free framework.
Investing in a noncallable/nonputable convertible bond is equivalent to buying:
- an option-free bond, and
- a call option on an amount of the common stock equal to the conversion ratio.
The value of a noncallable/nonputable convertible bond can be expressed as:
Value = Value of straight bond + Value of call option on stock (with underlying = conversion ratio × stock)
Most convertible bonds are callable. Incorporating a callable feature (issuer right to call prior to maturity) changes the expression:
Value = Value of straight bond + Value of call on stock - Value of call on bond (issuer's call)
For a convertible that is both callable and putable, value becomes more complex and includes the put value adjustments as well.
LOS 27.q: Compare the risk-return characteristics of a convertible bond with the risk-return characteristics of a straight bond and of the underlying common stock.
Buying convertible bonds rather than stocks limits downside risk because the straight bond value provides a floor. The cost of this downside protection is reduced upside potential due to the conversion premium. Convertible investors must also be concerned with credit risk, call risk, interest-rate risk, and liquidity risk.
EXAMPLE: Risk and return of a convertible bond, part 1
Calculate returns on the convertible bond and the common stock if the market price of BSC common stock increases to $45 per share.
Answer:
The return from investing in the convertible bond is:
The return from investing directly in the stock is:
The lower return from the convertible bond investment is attributable to the effective purchase of the stock at the market conversion price ($39.40).
EXAMPLE: Risk and return of a convertible bond, part 2
Calculate returns if the market price of BSC common stock falls to $30 per share.
Answer:
The conversion value in this scenario is 25 × $30 = $750. Assuming the straight value remains $950, the bond will trade at $950. Thus the return from the convertible bond is:
The return from investing directly in the stock is:
The loss is less for the convertible bond investment because the straight value serves as a floor. Even if the straight value changed, the loss would likely be less than on a direct stock investment.
Comparisons between owning the stock and the convertible bond:
- When the stock price falls, returns on convertibles exceed stock returns because the convertible has a floor equal to the straight value.
- When the stock price rises, the convertible will underperform due to the conversion premium.
- If the stock price is stable, convertible returns may exceed stock returns because of coupon payments (assuming no change in interest rates or credit risk).
- A convertible whose associated stock price is so low that conversion is irrelevant trades like a straight bond-called a fixed-income equivalent or busted convertible.
- A convertible whose stock price is high enough to behave like equity is called a common-stock equivalent. Most convertibles are hybrids between bond and equity characteristics.
MODULE QUIZ 27.8
1.
An analyst has gathered the following information on a convertible bond and the common equity of the issuer.
Market price of bond: $925.00
Annual coupon: 7.5%
Conversion ratio: 30
Market price of stock: $28.50
Annual stock dividend: $2.15 per share
The market conversion premium ratio for the convertible bond is closest to:
A.
7.56%.
B.
7.77%.
C.
8.18%.
2.
Which of the following statements concerning a comparison between the risk and return of convertible bond investing versus common stock investing is least accurate, assuming interest rates are stable?
A.
When stock prices fall, the returns on convertible bonds are likely to exceed those of the stock because the convertible bond's price has a floor equal to the straight bond value.
B.
The main drawback of investing in convertible bonds versus direct stock purchases is that when stock prices rise, the convertible bond will likely underperform the stock due to the conversion premium.
C.
Buying convertible bonds instead of direct stock investing limits upside potential to that of buying a straight bond, at the cost of increased downside risk due to the conversion premium.
3.
Data on two convertible bonds are shown in the following table.
Which factors are most likely to influence the market prices of ABC and XYZ: factors that affect equity prices, or factors that affect option-free bond prices?
A.
Both will be more affected by equity factors.
B.
One will be more affected by equity factors, the other by bond factors.
C.
Both will be more affected by bond factors.
4.
The difference between the value of a callable convertible bond and the value of an otherwise comparable option-free bond is closest to the value of the:
A.
call option on the stock minus value of the call option on the bond.
B.
put option on the stock plus value of the call option on the bond.
C.
call option on the stock plus value of call option on the bond.
5.
With respect to the value of a callable convertible bond, the most likely effects of a decrease in interest-rate volatility or a decrease in the underlying stock price volatility are:
A.
both will result in an increase in value.
B.
one will result in an increase in value, the other in a decrease.
C.
both will result in a decrease in value.
KEY CONCEPTS
LOS 27.a
Bonds with embedded options allow issuers to manage interest-rate risk or issue bonds at attractive coupon rates. Embedded options can be simple call or put options or more complex options such as sinking funds, estate puts, etc.
LOS 27.b
Value of option embedded in a callable or putable bond:
Vcall = Vstraight - Vcallable
Vput = Vputable - Vstraight
LOS 27.c
To value a callable or putable bond, employ backward induction and a binomial interest-rate tree calibrated to benchmark bonds so the tree reproduces benchmark prices.
LOS 27.d
When interest-rate volatility increases, the value of both call and put options on bonds increases. As volatility increases, the value of a callable bond (investor short the call) decreases and the value of a putable bond (investor long the put) increases.
LOS 27.e
The short call in a callable bond limits the investor's upside when rates decrease; the long put in a putable bond hedges the investor against rate increases. A call option has lower value with an upward-sloping yield curve because the probability of the option going in the money is lower; the call gains value when the curve flattens. Conversely, a put option has a higher probability of going in the money with an upward-sloping curve and loses value as the curve flattens.
LOS 27.f
Use backward induction in a binomial interest-rate tree using one-period forward rates. At any node corresponding to a call (put) date, the node value is the lower (higher) of the computed continuation value or the call (put) price.
LOS 27.g
The option-adjusted spread (OAS) is the constant spread added to each forward rate in a benchmark binomial tree such that the discounted cash flows of a credit-risky bond equal its market price, after removing option effects.
LOS 27.h
High-volatility binomial trees give higher call option values and lower computed values for callable bonds, thus reducing the OAS needed to match a given market price. Using too-low (too-high) assumed volatility leads to computed OAS that are too high (too low) for callable bonds and too low (too high) for putable bonds, respectively, potentially misclassifying bonds as underpriced or overpriced.
LOS 27.i / LOS 27.j
Effective duration (callable) ≤ Effective duration (straight)
Effective duration (putable) ≤ Effective duration (straight)
Effective duration (zero-coupon) ≈ maturity
Effective duration (fixed-rate coupon) < maturity
Effective duration (floater) ≈ time to next reset
LOS 27.k
For bonds with embedded options, one-sided durations (up-only or down-only) better capture sensitivity than symmetric effective duration. Callable bonds near-the-money have lower down-duration than up-duration; putable bonds near-the-money have higher down-duration than up-duration. Low-coupon callable bonds are unlikely to be called, so maturity-matched key-rate duration dominates; as coupon rises, time-to-exercise key-rate duration becomes more important. For putable bonds, the reverse applies.
LOS 27.l
Straight and putable bonds have positive convexity at all rate levels. Callable bonds have positive convexity when rates are high (unlikely to be called), but can show negative convexity at lower rates when the call option limits price appreciation.
LOS 27.m
A capped floater contains an issuer option that prevents the coupon from rising above a specified cap rate; a floored floater contains an investor option that prevents the coupon falling below a specified floor rate.
LOS 27.n
The owner of a convertible bond can exchange the bond for common shares of the issuer; the convertible includes an embedded call option giving the bondholder the right to buy the stock.
LOS 27.o
The conversion ratio is the number of shares the bond can be exchanged for.
Conversion value = Market price of stock × Conversion ratio.
Market conversion price = Market price of convertible bond / Conversion ratio.
Market conversion premium per share = Market conversion price - Market price of stock.
The minimum trading value is the higher of the straight value or conversion value.
LOS 27.p
The value of a bond with embedded options equals the value of the straight bond plus (or minus) the value of options that the investor is long (or short).
LOS 27.q
The major benefit of convertibles is potential price appreciation from the underlying stock. The main drawback is limited upside versus direct equity due to the conversion premium. If the stock price remains stable, coupon income can make the convertible's total return exceed stock return. If the stock price falls, the straight value limits downside risk.
ANSWER KEY FOR MODULE QUIZZES
Module Quiz 27.1, 27.2
1. B - The value at any node in a binomial tree is the average of the present values of the two possible values in the next period, discounted at the one-period rate at the node.
2. C - The tree should look like this:
Consider the value of the bond at the upper node for Period 1, V1,U:
Similarly, the value of the bond at the lower node for Period 1, V1,L is:
Now calculate V0, the current value of the bond at Node 0.
(Module 27.2, LOS 27.f)
3. C - The tree should look like this:
Consider the value of the bond at the upper node for Period 1, V1,U:
Similarly, the value of the bond at the lower node for Period 1, V1,L is:
Now calculate V0, the current value of the bond at Node 0:
The value of the embedded call option is $112.282 - $111.640 = $0.642. (Module 27.2, LOS 27.f)
4. A - The tree should look like this:
Consider the value of the bond at the upper node for Period 1, V1,U:
Similarly, the value of the bond at the lower node for Period 1, V1,L, is:
Now calculate V0, the current value of the bond at Node 0:
The value of the embedded put option is $112.523 - $112.282 = $0.241. (Module 27.2, LOS 27.f)
Module Quiz 27.3, 27.4
1. C - Construct a table of the risk differences between the issuer's callable bond and on-the-run Treasuries to answer. The OAS reflects the corporate callable bond's credit risk over Treasuries, since option risk has been removed. (Module 27.4, LOS 27.g)
2. B - Like ordinary options, the value of an embedded option increases as volatility increases. The arbitrage-free value of an option-free bond is independent of assumed volatility. Thus, as volatility increases, the value of the embedded call increases, reducing the value of the callable bond; Vcallable decreases as volatility rises. The value of the putable bond Vputable increases as assumed volatility increases because the embedded put increases in value. (Module 27.3, LOS 27.d)
Module Quiz 27.5
1. B - The duration formula is correct. The convexity formula presented earlier is incorrect in that the "2" should not appear in the denominator of the convexity formula. (LOS 27.i)
Module Quiz 27.6, 27.7
1. A - When an upward-sloping yield curve flattens, call options increase in value while put options decrease in value. (Module 27.3, LOS 27.e)
2. A - When assumed volatility in a binomial tree increases, the computed OAS will decrease. Using a lower-than-actual volatility (15% here) produces a computed OAS that is too high; using the 19% implied volatility instead would have produced an estimated OAS lower than 145 bps. (Module 27.4, LOS 27.h)
3. A - Straight bonds generally have higher effective durations than bonds with embedded options. Both call and put options can reduce the bond life and hence duration of callable and putable bonds will be less than or equal to that of their straight counterparts. (Module 27.5, LOS 27.j)
4. A - Statement 1 is incorrect. Low-coupon callable bonds are unlikely to be called, so their highest key-rate duration corresponds to time-to-maturity. Statement 2 is correct. High-coupon putable bonds are unlikely to be put; their highest key-rate duration corresponds to time-to-maturity. (Module 27.6, LOS 27.k)
5. B - Straight and putable bonds exhibit positive convexity at all interest-rate levels. A callable bond's price appreciation due to a decline in interest rates is limited by the call feature; callables exhibit negative convexity at low rates. Hence a decline in rates is least likely to result in best price performance for a callable bond. (Module 27.6, LOS 27.l)
6. B - When interest rates increase, a callable bond becomes less likely to be called and its duration will increase. The put option in a putable bond would be more likely to be exercised in a rising-rate scenario and hence the duration of a putable bond would decrease. Duration of an option-free bond would also decrease as interest rates increase but not as significantly. (Module 27.6, LOS 27.k)
7. A - Relative to bond A, bond B has lower OAS. Given similar credit risk, bond B offers a lower OAS for the same level of risk as bond A and thus would be considered overpriced; bond A is more attractive (underpriced). (Module 27.4, LOS 27.g)
8. C - The value of the capped floater is $99.78. The upper node at Year 2 is subject to the 3% cap, and the coupon is adjusted accordingly. (Module 27.7, LOS 27.m)
Module Quiz 27.8
1. C - The market conversion premium per share is the market conversion price per share minus the market price per share. The market conversion price per share is
so the conversion premium per share is $30.833 - $28.50 = $2.333. (LOS 27.o)
2. C - Buying convertible bonds in lieu of direct stock investing limits downside risk to that of straight-bond investing, at the cost of reduced upside potential due to the conversion premium. This analysis assumes interest rates remain stable. When stock prices fall, convertible returns likely exceed stock returns because of the straight-bond floor. The main drawback is underperformance when stock prices rise, due to the conversion premium. If the stock price remains stable, returns on the bond may exceed stock returns if the bond's coupon payment exceeds the dividend income of the stock. (LOS 27.q)
3. B - ABC has a conversion price much less than the current stock price, so the conversion option is deep in the money. Bond ABC effectively trades like equity and is more influenced by equity factors. A busted convertible like XYZ, with a stock price significantly below conversion price, trades like a bond and is influenced more by bond factors. (LOS 27.q)
4. A - A bond that is both callable and convertible contains two embedded options: a call option on the stock (investor long) and a call option on the bond (investor short, issuer's call). Therefore, the difference in value between the callable convertible and the comparable option-free bond is equal to the value of the call on the stock minus the value of the call on the bond. (LOS 27.p)
5. B - A decrease in interest-rate volatility will decrease the value of the embedded short call on the bond (raising the convertible's value) and have no effect on the embedded call on the stock. A decrease in stock-price volatility will decrease the value of the embedded call on the stock (lowering the convertible's value) and have no effect on the embedded call on the bond. Overall, one effect increases value and the other decreases value. (LOS 27.p)
