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Reading Reading Title Study Session Number
32 The Term Structure and Interest Rate Dynamics 12 33 The Arbitrage-Free Valuation Framework
Valuation and Analysis of Bonds with Embedded 34 Options
35 Credit Analysis Models 13
36 Credit Default Swaps
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Valuation and Analysis of Bonds with Embedded Options
Study Session 13
Reading Number 34
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.a: Describe fixed-income securities withembedded options
Ø 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
Ø Embedded options can be (a) Simple options or (b) Complex options
a) Simple options
a.1) Callable bonds • Gives the issuer the option to call back the bond • Investor is short the call option
a.2) Putable bonds • Allow the investor to put (sell) the bond back to the issuer prior to maturity • Investor is long the put option
b) Complex options
b.1) Estate put • Allows the heirs of an investor to sell the bond upon the death of the investor
b.2) Sinking Fund bonds • Require the issuerto set aside funds periodically to retire the bond • This provision reduces the credit risk
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.b: Explain the relationships between the values of a callable or putable bond, the underlying option-free (straight) bond, and the embedded option
V = value of the callable bond Price of a callable bond is lower V = V - V than the price of a comparable V = value of a straight (option-free) bond straight bond because callable bond V = V - V incorporates a right for the issuer V = value of the embedded call option
V = value of the putable bond Price of a putable bond is higher V = V + V than the price of a comparable V = value of a straight (option-free) bond straight bond because putable bond V = V - V incorporates a right for the investor V = value of the embedded put option
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.c: Describe how the arbitrage-free framework can be used to value a bond withembedded options LOS 34.f: Calculate the value of a callable or putable bond from an interest rate tree
To value a callable or putable bond Use backward induction process and a binomial interest rate framework
Note 1
When using a callable bond, the valueat any nodewhere the bond is callable, must be either the price at which the issuer will call the bond (call price) or the computed value, whichever is lower
Note 2
When using a putablebond, the valueused at any nodewhere the bond is putable, must be either the price at which the investor will put the bond (put price) or the computed value, whichever is higher
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.c: Describe how the arbitrage-free framework can be used to value a bond withembedded options LOS 34.f: Calculate the value of a callable or putable bond from an interest rate tree
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.c: Describe how the arbitrage-free framework can be used to value a bond withembedded options LOS 34.f: Calculate the value of a callable or putable bond from an interest rate tree
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.c: Describe how the arbitrage-free framework can be used to value a bond withembedded options LOS 34.f: Calculate the value of a callable or putable bond from an interest rate tree
$100.000 CALL RULE
When using a callable bond, the value at any node where the bond is callable, must be either the price at which the issuer will call the bond (call price) or the computed value, whichever is lower
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.c: Describe how the arbitrage-free framework can be used to value a bond withembedded options LOS 34.f: Calculate the value of a callable or putable bond from an interest rate tree
$100.000 PUT RULE
$103.081 When using a putable bond, the value used at any node where the bond is putable, must be either the price at which the investor will put the bond (put price) or the computed value, whichever is higher
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.d: Explain how interest rate volatility affects the value of a callable or putable bond
Ø Option values are positively relatedto the volatility of their underlying
Ø Value of a straightbond isaffected by changesin the level of interest ratesbut is unaffected by changesin the volatility of interest rates
Value of calloption Value of callable bond Interest rate volatility Value of put option Value of putable bond
V = V - V Remember
V = V + V
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LOS 34.e: Explain how changes in the level and shape of the yield curve affect the value of a callable or putable bond
1) LEVEL OF INTEREST RATE
Value of a callable bond rises less than an Interest rate (because the maximum value of a callable bond is equivalent straight bond limitedby the exercise value of the call option)
Interest rate Value of a putable bond falls less than (because the minimum value of a putable bond is an equivalent straight bond limitedby the exercise value of the put option)
2) SHAPE OF THE YIELD CURVE
The value of a call option will be lower, as the probability Higher future interest rates of the option going in the money is low. Call option value increases as anupward sloping yield flattens Upwardsloping yield curve Value of a call option The value of a put option will be higher, as the probability of the option going in the money is higher. Put option Value of a put option value declines as an upward sloping yield flattens
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.g: Explain the calculation and use of option-adjusted spreads
OAS (option adjusted spread) Is the constant spread added to each forward rate in a bechmarrk binomial interes rate tree, such that the sum of the present values of a credit risky bond´s cash flows equals its market price
Backward induction process is based on risk-free binomial interest rate tree (the valuation assumes that the underlying bond is risk-free)
If we are valuing a credit risky bond, it is necessary to add an increment for everyforward rate to adjust the price. This increment is called OAS
§ Bonds with similar credit risk should have same OAS
§ If a bond has an OAS higher than OAS of its peers, it is considered to be undervalued good investment
§ Conversely, bonds with low OAS (relative to peers) are considered to be overvalued
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.g: Explain the calculation and use of option-adjusted spreads
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.g: Explain the calculation and use of option-adjusted spreads
The estimation of OAS is a largelyiterative process (beyond the scope of the exam)
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.h: Explain how interest rate volatility affects option-adjusted spreads
“Assumed” Value OAS OAS Levelof (CALL) (PUT) volatility Calls Puts Callable Putable High High High Low High Low High Low Low Low High Low High Low
Ø When analyst uses a higher/lower than actual level of volatility (when generating his benchmark interest rate tree), computed OAS for a callable bond will be too low/high and the bond will be erroneously classified as overpriced/underpriced
Ø When analyst uses a higher/lower than actual level of volatility (when generating his benchmark interest rate tree), computed OAS for a putable bond will be too high/low and the bond will be erroneously classified as underpriced/overpriced
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.i: Calculate and interpret effective duration of a callable or putable bond
Modified duration Measures a bond´s price sensitivity to interest rate changes(assuming that cash flows do not change)
Convexity Curvature of the relation price-returnof a bond
Comparison of convexity of two bonds with similar duration § Modified duration consider the variation of price withchanges in yield to be constant, so it is only useful forsmall changes in yield.
Bond price § Bond A and Bond B have similar duration but different convexities
§ When changes on interest rate are high, we can observe that change in price differs due to different convexity in bonds
Bond´s modified Bond A § duration When changes in yield are high, convexity has to be taken into account
Bond B ∆ § = - D x ∆ r + x C x (∆ r) yield
duration convexity
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.i: Calculate and interpret effective duration of a callable or putable bond
Modified duration and convexity are not useful for bonds with embedded options, because cash flows of these bonds will change if theoption is exercised
In these cases we use effective duration and effective convexity which takeinto account thechanges in cash flows due to changes in interest rates
∆ ∆ Effective duration = ED = ∆y = change in yield (in decimal form) ∆ BV ∆ = estimated price if yield decreases by ∆y
BV ∆ = estimated price if yield increases by ∆y ∆ ∆ ( ) Effective convexity = EC = BV = initial observed bond price ∆
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.i: Calculate and interpret effective duration of a callable or putable bond
Process to calculate effective duration and effective convexity
Step 1: Given assumptions about benchmark interest rates, interest rate volatility, and any calls and / or puts, calculate theOAS for the issue using the current market price and the binomial model
Step 2: Impose a small parallel shift in the benchmark yield curve by an amount equal to + ∆y
Step 3: Build a new binomial interest rate treeusing the new yield curve
Step 4: Add theOAS from step 1 to each of the one-year ratesin the interest rate treeto get a “modified” tree
Step 5: Compute BV ∆ using this modified interest rate tree
Step 6: Repeat steps 2 through 5 using a parallel rate shift of - ∆y to obtain a value of BV ∆
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.j: Compare effective durations of callable, putable, and straight bonds
Ø Effective duration callable bond ≤ effective duration straight bond Ø Effective duration putable bond ≤ effective duration straight bond
Increase in interest rate, Callable bond Decrease in interest rate, decrease effective duration Putable bond decrease effective duration
Bond Bond price Call option price price Putable bond Put option Call price price
Put price Option-free bond Option-free bond
Callable bond yield yield
Remember: effective duration measures the % of variation on the bond´s price for a 1% variation on interest rate
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.j: Compare effective durations of callable, putable, and straight bonds
Ø Effective duration (zero-coupon) ≈ maturity of the bond
Ø Effective duration of fixed-rate coupon bond < maturity of the bond
Floating bonds have a low duration, since Ø Effective duration of floater ≈ time (in years) to next reset they adapt their coupon rates to changes in the reference interest rates
Recall: ModDur = MacDur / (1 + YTM)
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.k: Describe the use of one-sided durations and key durations to evaluate the interest rate sensitivityof bonds withembedded options
One sideduration • Durations that apply only when interest ratesrise or alternativelyonly when ratesfall • Are better at capturing interest rate sensitivity for bonds with embedded options, than the more common effective duration
Ø For a callable bond, when the call option is at or near the money, the change in price fora decrease in yield (one side down duration/to lower rates) will be less than the change in price foran equal amount of increase in yield (lower one-sided-down duration than one-sided-up duration) Ø Similarly, when the put option is at or near the money, the value of a putable bond is more sensitive to downward movements in yield curve versus upward movements (lower one-sided-up duration than one-sided-down duration)
Bond Bond price price
Putable bond Callable bond Straight bond Straight bond
yield yield
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.k: Describe the use of one-sided durations and key durations to evaluate the interest rate sensitivityof bonds withembedded options
Key rate duration Capture interest rate sensitivityof a bond to changes in yields (par rates) of specific benchmark (or partialduration) maturities
1. An option free bond trading at par, the only key duration that affects the bonds value is the maturity matched
2. Foran option free bond not trading at par, the maturity matchedrate is the most improtant rate
3. Bonds witha low (or zero) coupon rate mayhave negative key rate durations for horizons other than the bond´s maturity
4. Callable bonds with low coupon rates are unlikely to be called key duration that matters most is the maturity matched
5. Callable bonds with high coupon rates are likely to be called key duration that matters most is time to excercise
6. Putable bonds with high coupon rates are unlikely to be called key duration that matters most is the maturity matched
7. Putable bonds with low coupon rates are likely to be called key duration that matters most is time to excercise
Effective duration measures the sensitivity of the value of a bond to paralell movements of the yield curve (not realistic) while key rate duration measures the sensitivity to changes in yields for specific maturities
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.k: Describe the use of one-sided durations and key durations to evaluate the interest rate sensitivityof bonds withembedded options
Key rate durations of various 15-year Option-Free Bonds with different coupon rates and YTM = 3%
Key rate durations of various 15-year Callable Bonds with different coupon rates Key rate durations of various 15-year Putable Bonds with different coupon rates (Callable in 10 years at par) (Putable in 10 years at par)
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.l: Compare effective convexities of callable, putable and straight bonds
Straight bond Callable bond Straight bond
Positive effective convexity Positive convexity at high rates Positive effective convexity Negative convexity at low rates
“Increase in the value is higher when “Increase in the value is lower when “Increase in the value is higher when rates fall than the decrease in value rates fall than the decrease in value rates fall than the decrease in value when rates increase by an equal when rates increase by an equal when rates increase by an equal amount” amount” amount”
Bond Bond Bond price price price
Putable bond Callable bond Straight bond Straight bond Straight bond
yield yield yield
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.m: Calculate the value of a capped or floored floating-rate bond
Floating rate bond (“floater”) Pays a coupon that adjusts every period based on an underlying reference rate
Ø Capped floater Contains an issuer option that prevents the coupon rate from rising above a specifiedmaximum rate known as the cap
Value of a capped floater = value of a “straight” floater - value of the embedded cap
Ø Floored floater Has a coupon rate that will not fall below a specifiedminimum rate known as the floor. (Here the option belongs to the investor and offer protection from falling interest rates)
Value of a floored floater = value of a “straight” floater + value of the embedded floor
Standard backward induction methodology in a binomial interest rate tree can be used to value a capped o floored floater, by adjusting the value of the floaterat each node (as with the valuation of a bond withembedded options)
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.m: Calculate the value of a capped or floored floating-rate bond
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.m: Calculate the value of a capped or floored floating-rate bond
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.m: Calculate the value of a capped or floored floating-rate bond
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.n: Describe defining features of a convertible bond
Convertible bond (1)
Ø 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(conversion period)
Ø Conversion ratio isthenumber of common shares for which a convertible bond can be exchanged
Ø Issuer of a convertible bond benefits from a lower borrowing cost, but existing shareholders may face dilution if the conversion option is exercised
Ø Convertible bonds allow investors to enjoy the upside on the issuer´s stock, although this comes at a cost of lower yield
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.o: Calculate and interpret the components of a convertible bond´s value
Convertible bond (2)
§ Conversion ratio Number of common shares for whicha convertible bond can be exchanged
§ Conversion value Value of the common stock into which the bond can be exchanged
Conversion value = market price of stock x conversion ratio
§ Straight value Value of the bond if it were not convertible (or investment value)
§ Minimum value Is the greaterof its conversion value or its straight value
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.o: Calculate and interpret the components of a convertible bond´s value
Convertible bond (3)
§ Market conversion price Is the price that the convertible bondholder would effectivelypay for the stock if she bought the (conversion parity price) bond and inmediately converted it
Market conversion price =
§ Market conversion Is the difference between market conversion price and the stock´s current market price premium per share Market conversion premium per share = market conversion price - stock´s market price
Market conversion premium ratio =
§ Premium over Convertible bond investor´s downside risk is limitedby the bond´s underlying straight value. This straight value downside risk is measured by the premium over straight value
Premium over straight value = - 1
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.o: Calculate and interpret the components of a convertible bond´s value
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.o: Calculate and interpret the components of a convertible bond´s value
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.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
Noncallable / nonputable = straight value of + value of call option on convertible bond value bond stock
Most of convertible bonds are callable, giving the issuerthe right to call the issue prior to maturity
Callable convertible straight value value of call option = + - value of call option bond value of bond on stock on bond
Callable and putable straight value value of call option value of call option = value of put option convertible bond value of bond + on stock - on bond + on bond
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.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
1. The major benefit from investing in convertible bonds is the price appreciationresulting from an increase in the value of thecommon stock
2. The main drawback of investing in a convertible bond versus investing directly in thestock is that when the stock price rises, thebond will underperform the stock because of the conversion premium of the bond
3. If the stock price remains stable, the returnon the bond may exceed thestock returns due to the coupon payments received from the bond
4. If the stock price falls, thestraight value of the bond limits downside risk (assuming bond yields remain stable)
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 36.4: Compare the risk-return characteristics of a convertible bond with the risk-return characteristics of a straight bond and of the underlying common stock
Market conversion price
Market stock price
Luis M. de Alfonso CFA® Preparation FI – Valuation and Analysis of Bonds with Embedded Options www.dbf-finance.com
LOS 34.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
Straight value of the bond
Purchase price of the bond
Actual market price of stock
Purchase price of the stock
Luis M. de Alfonso