1. Fixed Income Analysis Week 9 Bonds with Options
Commerce 4FJ3
Fixed Income Analysis
Week 9
Bonds with Options

2. Traditional Yield Spread
Traditional yield spread is 109 basis points
Ignores term structure of interest rates
The bond, if called in 10 years, should be compared to a 10 year treasury

3. Static Spread
Find the treasury spot rate term structure using the bootstrapping method
Find the present value of the cash flows for the bond using the spot rate plus a spread
Solve for the spread that gives the current price
Called the static or zero volatility spread

4. Static Spread Example

5. Value of Static Spread
Gives the return in excess of treasury over the entire term structure
Takes into account that there are expected different reinvestment rates at different points in the future
Assumes that the required spread over treasury is constant over time

6. Static vs. Traditional If yield curve is flat, no difference
If yield curve is rising, the static spread will be higher, with bigger differences for long maturities and steeper term structures
Static may be smaller if inverted yield curve
Bigger differences in spread for amortizing securities (MBS, asset backed securities)

7. Callable Bonds Two main disadvantage for buyers
Extra reinvestment risk: if the bond is called before the investors time horizon they will face extra reinvestment risk, probably at a lower rate
Price compression: if yields fall, the bond price will not rise as much as it should because the bond can be bought back at a fixed price

8. Traditional Valuation
As mentioned earlier, calculate yield to each call, yield to worst, and then decide on the price that is reasonable
Assumes: bond will be called at that date
Ignores: reinvestment rates
can be mitigated by comparing to treasuries of the maturity of the called bond

9. Price vs. Yield for Callable

10. Negative Convexity The normal price/yield curve is convex
With price compression the level of convexity can become negative (technically it is now concave)
Price change from increasing interest rates becomes larger than the change from falling interest rates

11. Price vs. Call Price
Note that the price of the bond can still be higher than the face value plus call premium if the bond has time before the call
13% bond, callable at 5% premium in one year, market rate is 5%

12. Bonds as Bundles
Bonds with embedded options can be seen as a package of bonds and options
Callable bond: package of; long an option free bond and short a call option on bond
Putable (retractable) bond: a package of; long an option free bond and long a put option on the same bond

13. Value of Options
The option value is difficult to calculate since most pricing models assume that the price volatility of the underlying asset does not change over time
The price volatility (modified duration) of the bond changes with time and also with the level of interest rates

14. Another Problem
Call options on bonds are American options while pricing models are based on European options
Argument for using Euro for models is that it is usually not worth exercizing early due to loss of time value does not hold here since there are intermediate cash flows

15. Interest Rate Volatility
The major influence on the price of a bond is interest rates
Changes in interest rates can be measured over time and the volatility can be estimated
Can be used to create an interest rate model
Textbook model is single factor, lognormal random walk, binomial interest ladder or lattice, estimating potential forward rates

16. Interest Rate Lattice
A bond can be valued by taking the present value of each cash flow, discounted by the product of all applicable forward rates
The model assumes that the forward rate will take one of two equally likely values
The higher rate = lower rate x e2s
Rates are found for each node using trial and error

17. Option-Free Value
Once the interest rate lattice has been constructed, other bonds can be analysed
Starting with the final cash flows (since the intermediate prices can not be determined in advance), fill in the nodes on the lattice
The price found should be identical to the one found using the static spread analysis

18. Valuation with Options
As with the option-free bond, add the value of the bond plus coupon to each node, but if the bond is likely to be called (greater than call price + refunding cost), replace that value with the call price
As above, but replace market values below the put price with the put price

19. Modelling Risk
If the assumptions that the model is based on is incorrect, the values derived from the model will not be useful
The volatility assumption is critical
The higher the volatility, the higher the value of an option, the lower the price of a callable bond
It is important to stress test the model

20. Option Adjusted Spread
The spread that would explain the current price of a bond with an embedded option
Can be constructed over the treasury term structure or the issuer’s term structure
Since there is disagreement between market participants, knowing which assumption they are using is critical

21. Option Value in Spread Terms
If we have the OAS in terms of the treasury forward rate structure, we can calculate the amount of the spread that is due to the embedded option
option value = static spread - OAS
Main reason for spreads is because some market participants prefer to talk about all investments in terms of rate of return

22. Effective Duration and Convexity
Found using the approximation formulas
Similar to modified if the option is deeply out of the money
P-= price if yield down
P+= price if yield up
P0= original price

23. Finding P- and P+ Five step process for binomial model
Calculate OAS for the bond
Shift the treasury yield curve down/up a few basis points
Construct the interest rate tree
Add the OAS to each node’s interest rate
Determine the value of the security

24. Convertible Bonds Another type of embedded option
A call option on a number of the issuer’s common share where the exercise price is the bond, regardless of current market value
Number of shares is conversion ratio
Can be physical or cash settle
Exchangeable bonds are similar options, but on other company’s shares

25. Conversion Price
The conversion price is simply the implied exercise price of the option on a per share basis
If the bond is issued at par the conversion price is

26. Other Features
Conversion ratio may change over time, on a schedule given in the issue
Conversion ratio is adjusted for stock splits and stock dividends
Most convertibles are also callable, which may trigger early conversion
Some are putable (hard or soft put)

27. Sample Convertible

28. Minimum Price
The bond will trade at a minimum of the greater of the conversion value or straight (debt) value
conversion value: how much the stock that the bond can be converted to is worth
straight value: the value of the convertible if it did not have the conversion option

29. Sample Minimum Price For the sample bond, conversion value
= $17 x 50 = $850
Given a 14% yield on non-convertible otherwise similar bonds, straight value
= PVcoupons+ PVface = $788
This bond should trade for a minimum of $850 since that is the higher value

30. Market Conversion Prices
Since the exercise price is the bond, the effective price of the common stock changes over time

31. Sample Conversion Market conversion price = $950/50 = $19
Market conversion premium per share = $19 - $17 = $2
Market conversion premium ratio = $2/$17 = 11.8%

32. Current Income
One reason for not converting a convertible bond before maturity, are the coupon payments
FIDPS = Coupon/(conversion ratio) - dividend
Premium payback period (break-even time) = Market premium per share ÷ Favourable income differential per share

33. Sample Income
Coupon interest from bond = $100
Dividend per share = $1
Conversion ratio = 50
Favourable income differential per share = $100/50 - $1 = $1
Premium Payback Period = $2/$1 = 2 years

34. = (Market value/Straight value) - 1
Downside Risk
Often measured as the premium over straight value
= (Market value/Straight value) - 1
Sample bond = $950/$ = 21%
Note: the investor has more than 21% downside risk since the YTM could increase, decreasing the straight value

35. Jargon
A convertible where the option is well out of the money is called a bond equivalent or busted convertible
A convertible with a conversion value much higher than its straight value is called an equity equivalent
Between those it is a hybrid security

36. Payoff
Share price goes up to $34 Shareholder return = 100% Convertible holder return = 79%
Share price goes down to $7 Shareholder return = -59% Convertible holder return = -17%
The convertible is less risky

37. Call Risk
One reason for issuing convertible bonds is that the company would prefer to issue equity, but considers the current price to be too low to be worth issuing common shares
Conversion ratio is set to reflect “reasonable pricing”
Call options can be used to force conversion

38. Takeover Risk
If the issuer gets taken over before the price of the shares make conversion reasonable, the bond holders may be left with a bond that pays a lower coupon than similar corporate bonds

39. Options Approach
Similar to callable bonds, convertibles can be viewed as a bond and an option
An additional problem here is that the exercise price on the share changes over time as the bond’s market price is affected by changes in interest rates
To make matters worse, most convertible bonds are also callable