Learning Module 3: Valuation and Analysis of Bonds with Embedded Options

Learning Module 3 Valuation and Analysis of Bonds with Embedded Options SINKING FUND BONDS The underlying bond has an amortizing structure, for example, a 30-year maturity with level annual principal repayments beginning at the end of the 11th year. In this case, each payment is 5% of the original principal amount. A typical sinking fund bond may include the following options: . A standard call option above par, with declining premiums, starting at the end of Year 10. Thus, the entire bond issue could be called from Year 10 onward. An acceleration provision, such as a "triple up." Such a provision allows the issuer to repurchase at par three times the mandatory amount, or in this case 15% of the original principal amount, on any scheduled sinking fund date. Assume that the issuer wants to retire the bonds at the end of Year 11. Instead of calling the entire outstanding amount at a premium, it would be more cost-effective to "sink" 15% at par and call the rest at a premium. Thus, the acceleration provision provides an additional benefit to the issuer if interest rates decline. A delivery option, which allows the issuer to satisfy a sinking fund payment by delivering bonds to the bond's trustee in lieu of cash. The bond's trustee is appointed by the issuer but acts in a fiduciary capac- ity with the bondholders. If the bonds are currently trading below par, say at 90% of par, it is more cost effective for the issuer to buy back bonds from investors to meet the sinking fund requirements than to pay par. The delivery option benefits the issuer if interest rates rise. Of course, the benefit can be materialized only if there is a liquid market for the bonds. Investors can take defensive action by accumulating the bonds and refusing to sell them at a discount. From the issuer's perspective, the combination of the call option and the delivery option is effectively a "long straddle"-an option strategy involving the purchase of a put option and a call option on the same underlying with the same exercise price and expiration date. At expiration, if the underlying price is above the exercise price, the put option is worthless but the call option is in the money. In contrast, if the underlying price is below the exercise price, the call option is worthless but the put option is in the money. Thus, a long straddie benefits the investor when the underlying price moves up or down. The greater the move up or down (ie, the greater the volatility), the greater the benefit for the investor. As a consequence, a sinking fund bond benefits the issuer not only if interest rates decline but also if they rise. Determining the combined value of the underlying bond and the three options is quite challenging. EXAMPLE 1 Types of Embedded Options 1. Investors in putable bonds most likely seek to take advantage of: A. higher interest rates. 8. improvements in the issuer's credit rating C. movemerits in the price of the issuer's common stock. Solution: A is correct. A putable bond offers the bondholder the ability to take ad- vantage of a rise in interest rates by putting back the bond to the issuer and reinvesting the proceeds of the retired bond in a higher-yielding bond. 2. The conversion option in a convertible bond is a right held by: A. the issuer. B. the bondholders. C. the issuer and the bondholders jointly. Solution: B is correct. A conversion option is a call option that gives the bondholders the right to convert their bonds into the issuer's common stock. OPTIMAL EXERCISE OF OPTIONS The holder of an embedded bond option can extinguish (or possibly modify the terms of) the bond. Assuming that the option is currently exercisable, the obvious question is does it pay to exercise? Assuming that the answer is affirmative, the follow-up question is whether it is better to exercise the option at present or to wait. Let us consider the first question: Would it be profitable to exercise the option? The answer is usually straightforward: Compare the value of exercising with the value of not exercising. For example, suppose that a bond is currently putable at 100. If the bond's market price is above 100, putting the bond makes no sense because the cash value from selling the band would exceed 100. In contrast, if the bond's market price is 100 putting the bond should definitely be considered. Note that the market price of the bond cannot be less than 100 because such a situation creates an arbitrage opportunity: Buy the bond below 100 and immediately put it at 100. The logic of a call decision by the issuer is similar. If a bond's market price is significantly less than the call price, calling is foolish because the bond could be simply repurchased in the market at a lower price. Alternatively, if the price is very close to the call price, calling may make sense. Assume that we have determined that exercising the option would be profitable. If the option under consideration is European style, it is obvious that it should in fact be exercised: There is no justification for not doing so. But if it is an American-style or Bermudan-style option, the chalenge is to determine whether it is better to act now or to wait for a better opportunity. The problem is that although circumstances may become more favorable, they may also get worse. So, option holders must consider the odds and decide to act or wait, depending on their risk preference. The approach presented here for valuing bonds with embedded options assumes that the option holders, be they issuers or investors, are risk-neutral. They exercise if, and only if, the benefit from exercise exceeds the expected benefit from waiting, In reality, option holders may be risk averse and may exercise early even if the option is worth more alive than dead. EXAMPLE 2 Valuation of Default-Free Callable and Putable Bonds George Cahill, a portfolio manager, has identified three five-year annual coupon bonds issued by a sovereign government. The three bonds have identical characteristics. The exceptions are that Bond A is an option-free bond: Bond B is callable at par two years and three years from today; and Bond C is also callable at par two years and three years from today as well as putable at par one year from today. 1. Relative to the value of Bond A, the value of Bond B is: A. lower. B. the same. C. higher. Solution: A is correct. Bond B is a callable bond, and Bond A is the underlying option-free (straight) bond. The call option embedded in Bond B is an issuer option that decreases the bond's value for the investor. If interest rates de- cline, bond prices usually increase, however, the price appreciation of Bond B will be capped relative to the price appreciation of Bond A because the issuer will call the bond to refinance at a lower cost. 2. Relative to the value of Bond B, the value of Bond Cis: A. lower. B. the same. C higher. Solution: C is correct. Relative to Bond B, Bond C includes a put option. A put option is an investor option that increases the bond's value for the investor. Thus, the value of Bond C is higher than that of Bond B. 3. Given an anticipation of rising interest rates, Bond C will be expected to: A. be called by the issuer. B. be put by the bondholders. C mature without exercise of any of the embedded options. Solution: B is correct. As interest rates rise, bond prices decrease. Thus, the bondholders will have an incentive to exercise the put option so that they can reinvest the proceeds of the retired bond at a higher yield. EXAMPLE 3 Valuation of a Callable Bond Assuming Interest Rate Volatility Return to the valuation of the Bermudan-style three-year 4.25% annual coupon bond callable at par one year and two years from now as depicted in Exhibit 12. The one-year, two-year, and three-year par yields are 2.500%, 3.000%, and 3.500%, respectively, and the interest rate volatility is 10%. 1. Assume that nothing changes relative to the initial setting except that the in- terest rate volatility is now 15% instead of 10%. The new value of the callable bond is: A. less than 101.540. B. equal to 101.540. C. more than 101.540. Solution: A is correct. A higher interest rate volatility increases the value of the call option. Because the value of the call option is subtracted from the value of the straight bond to obtain the value of the callable bond, a higher value for the call option leads to a lower value for the callable bond. Thus, the value of the callable bond at 15% volatility is less than that at 10% volatility-that is, less than 101.540. 2. Assume that nothing changes relative to the initial setting except that the bond is now callable at 102 instead of 100. The new value of the callable bond is closest to: A. 100.000. B. 102.000. C. 102.114. Solution: C is correct. Looking at Exhibit 12, the call price is too high for the call option to be exercised in any scenario. Thus, the value of the call option is zero, and the value of the callable bond is equal to the value of the straight bond that is, 102.114. EXAMPLE 4 Valuation of a Putable Bond Assuming Interest Rate Volatility Return to the valuation of the Bermudan-style three-year 4.25% annual coupon bond putable at par one year and two years from now, as depicted in Exhibit 13. The one-year, two-year, and three-year par yields are 2.500%, 3.000%, and 3.500%, respectively, and the interest rate volatility is 10% 1. Assume that nothing changes relative to the initial setting except that the in terest rate volatility is now 20% instead of 10%. The new value of the putable bond is: A. less than 102.522. B. equal to 102.522. Cmore than 102.522. Solution: C is correct. A higher interest rate volatility increases the value of the put option. Because the value of the put option is added to the value of the straight bond to obtain the value of the putable bond, a higher value for the put option leads to a higher value for the putable bond. Thus, the value of the putable bond at 20% volatility is more than that at 10% volatility-that is, more than 102.522 2. Assume that nothing changes relative to the initial setting except that the bond is now putable at 95 instead of 100. The new value of the putable bond is closest to A. 97.522. B. 102.114. C107.522 Solution: B is correct. Looking at Exhibit 13, the put price is too low for the put option to be exercised in any scenario. Thus, the value of the put option is zero, and the value of the putable bond is equal to the value of the straight bond- that is, 102.114. PUTABLE VS. EXTENDIBLE BONDS Putable and extendible bonds are equivalent, except that their underlying option-free bonds are different. Consider a three-year 3.30% bond putable in Year 2. Its value should be exactly the same as that of a two-year 3.30% bond extendible by one year. Otherwise, there would be an arbitrage opportunity. Clearly, the cash flows of the two bonds are identical up to Year 2. The cash flows in Year 3 are dependent on the one-year forward rate two years from now. These cash flows will also be the same for both bonds regardless of the level of interest rates at the end of Year 2 If the one-year forward rate at the end of Year 2 is higher than 3.30%, the putable bond will be put because the bondholder can reinvest the proceeds of the retired bond at a higher yield and the extendible bond will not be extended for the same reason. 50, both bonds pay 3.30% for two years and are then redeemed. Alternatively, if the one-year forward rate at the end of Year 2 is lower than 3.30%, the putable bond will not be put because the bondholder would not want to reinvest at a lower yield and the extendible bond will be extended to hold onto the higher interest rate. Thus, both bonds pay 3.30% for three years and are then redeemed. SCENARIO ANALYSIS OF BONDS WITH OPTIONS Another application of valuing bonds with embedded options is scenario analysis over a specified investment horizon. In addition to reinvestment of interest and principal, option valuation comes into play in that callable and putable bonds can be redeemed and their proceeds reinvested during the holding period. Making scenario-dependent, optimal option-exercise decisions involves computationally intensive use of OAS technology because the call or put decision must be evaluated considering the evolution of interest rate scenarios during the holding period. Performance over a specified investment horizon entails a trade-off between reinvestment of cash flows and change in the bond's value. Let us take the example of a 4.5% bond with five years left to maturity and assume that the investment horizon is one year. if the bond is option free, higher interest rates increase the reinvestment income but result in lower principal value at the end of the investment horizon. Because the investment horizon is short, reinvestment income is relatively insignificant and performance will be dominated by the change in the value of the principal. Accordingly, lower interest rates will result in superior performance. If the bond under consideration is callable, however, it is not at all obvious how the interest rate scenario affects performance. Suppose, for example, that the bond is first callable six months from now and that its current market price is 99,74. Steeply rising interest rates would depress the bond's price, and performance would definitely suffer. But steeply declining interest rates would also be detrimental because the band would be called and both interest and principal would have to be reinvested at lower interest rates. Exhibit 17 shows the return over the one-year investment horizon for the 4.5% bond first callable in six months with five years left to maturity and valued on a 4% flat yield curve. Exhibit 17 clearly shows that lower interest rates do not guarantee higher returns for callable bonds. The point to keep in mind is that the bond may be called long before the end of the investment horizon. Assuming that it is called on the horizon date would overestimate performance. Thus, a realistic prediction of option exercise is essential when performing scenario analysis of bonds with embedded options. CONTINGENT CONVERTIBLES Contingent convertible bonds, or "CoCos," pay a higher coupon than otherwise identical non-convertible bonds; however, they usually are deeply subordinated and may be con- verted into equity or face principal write-downs if regulatory capital ratios are breached. Convertible contingent convertible bonds, or "CoCoCos," combine a traditional convertible bond and a CoCo. They are convertible at the discretion of the investor, thus offering upside potential if the share price increases. They are also converted into equity or face principal write-downs in the event of a regulatory capital breach. CoCos and CoCoCos are usually issued by financial institutions, particularly in Europe. EXAMPLE 9 Valuation of Convertible Bonds Nick Andrews, a fixed-income investment analyst, has been asked by his super- visor to prepare an analysis of the convertible bond issued by Heavy Element Inc., a chemical industry company, for presentation to the investment committee. Andrews has gathered the following data from the convertible bond's prospectus and market information: Issuer: Heavy Element Inc. Issue Date: 15 September 2020