3. Valuation and Analysis: Bond With Embedded Options Flashcards
Explain ‘Simple Options’ (callable bonds, putable bonds, extendible bonds).
Callable bonds give the issuer the option to call back the bond; the investor is short the call option.
Putable bonds allow the investor to put (sell) the bond back to the issuer prior to maturity. The investor is long the underlying put option.
A related bond is an extendible bond, which allows the investor to extend the maturity of the bond. An extendible bond can be evaluated as a putable bond with longer maturity. 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 putable (European style) bond with lockout period of two years.
Explain ‘Complex Options.’ (State Put and Sinking Fund Bonds).
- State Put: includes a provision that allows the heirs of an investor to put the bond back to the issuer upon the death of the investor. 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): which require the issuer to set aside funds periodically to retire the bond (a sinking fund). This provision reduces the credit risk of the bond. Sinkers typically have several related issuer options (e.g. call provisions, acceleration provisions, and delivery options)
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. The value of the embedded call option (V-call) is, therefore, simply the difference between the value of a straight (V-straight) bond and the value of the comparable callable bond (V-callable):
V-callable = V-straight - V-call
- Conversely, investors are willing to pay a premium for a putable bond, since its holder effectively owns an option-free bond plus a put option. The value of a putable bond can be expressed as:
V-putable = V-straight + V-put
Describe how the arbitrage-free framework can be used to value a bond with embedded options.
The basic process for valuing a callable (or putable) bond is similar to the process for valuing straight bond. However, instead of using spot rates, one-period forward rates are used in a binomial tree framework. This change in 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).
- Call Rule: when valuing 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 (i.e. the call price) or the computed value if the bond is not called, whichever is lower
- Put Rule: similarly, for a putable bond, the value used at any node corresponding to a put date must be either the price at which the investor will put the bond (i.e. the put price) or the computed value if the bond is not put, whichever is higher
Note, call date and put date in this context vary depending on whether the option is European, American or Bermudan-style.
Explain how interest rate volatility affects the value of a callable or putable bond.
Options values are positively related to the volatility of their underlying. 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 rate but is unaffected by changes in the volatility of interest rate.
Explain the calculation and use of option-adjusted spreads.
So far our backward induction process has relied on the risk-free binomial interest rate tree; our 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 calculated value equals the market price of the risky bond. This constant spread is called the ‘Option Adjusted Spread (OAS)’.
- Note: the OAS is added to the tree after the adjustment for the embedded option. Hence the OAS is calculated after the option risk has removed.
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 OAS of its peers, it is considered to be undervalued and hence an attractive investment.
Explain how interest rate volatility affects option-adjusted spreads.
To summarize, as the assumed level of volatility used in an interest rate tree increases
- the computed OAS (for a given maket price) for a callable bond decreases
- the computed OAS of a putable bond increases
Explain ‘One-sided Durations’.
The computation of effective duration relied on computing the value of the bond for equal parallel shifts of the yield curve up and down (by the same amount). This metric is ok 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 amount of increase in yield. In other words, callable bonds will have lower one-sided down-duration than one-sided up duration.
Similarly, the value of a putable bond is more sensitive to downward movements in yield curve versus upward movements: putable bond will have larger one-sided down-duration than one-sided up-duration.
Explain ‘Key Rate Duration’.
Key Rate Duration or ‘Partial Durations’ capture the interest rate sensitivity of a bond to changes in yields (par rates) of specific benchmark maturities. Key rate duration is used to identify the interest rate risk from changes in the shape of the yield curve (shaping risk).
Tabela com key rate durations for several 15-year option-free bonds. In these figures usually the market interest rate is not changing; rather we consider a number of different bonds that have different coupons (pag.94).
The following generalizations can be made about key rates:
1) 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 is the same as its effective duration, and all other rate durations are zero
2) For option-free bonds not trading at par, the maturity-matched rate is still the most important rate
3) A bond with a low (or zero) coupon rate may have negative key rate durations for horizons other than its maturity
4) Callable bonds with low coupon rates are unlikely to be called; hence, their maturity-matched rate is their most critical rate (i.e. the highest key rate duration corresponds to the bond’s maturity)
5) Keeping everything else (including market interest rates) constant, 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
6) Putable bonds with high coupon rates are unlikely to be put and are most sensitive to their maturity-matched rates
7) Keeping everything else (including market interest rates) constant, lower coupon bonds are more likely to be put, and therefore the time-to-exercise rate will tend to dominate the time-to-maturity rate
Compare effective convexities of callable, putable, and straight bonds.
Straight bonds have positive effective convexity: the increase in the value of an option-free bonds is higher when rates fall than the decrease in value when rates increase by an equal 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, its effective convexity turns negative; the upside potential of the bond’s price is limited due to the call (while the downside is not protected).
Putable bonds exhibit positive convexity throughout.
Explain the capped and floored floating-rate bond.
A floating-rate bond (‘floater’) pays a coupon that adjusts every period based on an underlying reference rate. The coupon is typically paid in arrears, meaning the coupon rate is determined at the beginning of a period but is paid at the end of that period.
A capped floater effectivelly contains an issuer option that prevents the coupon rate from rising above a specified maximum rate known as the cap.
V embedded cap = V straight floater - V capped floater
A related floating rate bond is the floored floater, with a embedded option belongs to investor to protect against falling interest rates.
V embedded floor = V floored floater - V straight floater
Explain the components of a convertible bond —
- conversion period
- conversion ratio
- conversion price
- conversion value
- straight value
- minimum value of a convertible bond
- market conversion price
- market onversion premium ratio
- premium over straight value
- Conversion Period: The specified timeframe during a owner of a convertible bond has the right to convert the bond into a fixed number of common shares of the issuer
- Conversion Ratio: number of common shares for which a convertible bond can be exchanged
- Conversion Price = (Issue Price of the bond)/Conversion Ratio
- Conversion Value: (market price of stock) x (Conversion Ratio)
- Straight Value: or investment value, of a convertible bond is the value of the bond if it were not convertible; the PV of the bond’s cash flows discounted at the return required on a comparable option-free issue
- Minimum Value of a Convertible Bond: is the greater of its conversion value or its straight value
- Market Conversion Price = (market price of convertible bond)/(Conversion Ratio)
- Market Conversion Premium Per Share = (Market Conversion Price) - (Stock’s Market Price)
- Market Conversion Premium Ratio = (Market Conversion Premium Per Share)/(Market Price of Commom Stock)
- Premium Over Straight Value = (Market Price of Convertible Bond)/(Straight Value) - 1
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:
Callable Convertible Bond Value = Straight Value of bond + Value of Call Option on Stock - Value of Call Option on Bond
