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

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

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

Static Spread Example

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

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)

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

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

Price vs. Yield for Callable

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

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%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

Sample Convertible

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

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

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

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

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

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

= (Market value/Straight value) - 1 Downside Risk Often measured as the premium over straight value = (Market value/Straight value) - 1 Sample bond = $950/$788 - 1 = 21% Note: the investor has more than 21% downside risk since the YTM could increase, decreasing the straight value

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

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

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

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

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