CH-17-18 Analysis of Bonds with Embedded Options – Bond Markets, Analysis and Strategies : Book Guide

This Document Contains Chapters 17 to 18 CHAPTER 17 ANALYSIS OF BONDS WITH EMBEDDED OPTIONS CHAPTER SUMMARY In this chapter we look at how to analyze bonds with embedded options. Because the most common type of option embedded in a bond is a call option, our primary focus is on callable bonds. We begin by looking at the limitations of traditional yield spread analysis. Although corporate bonds are used in our examples, the analysis presented in this chapter is equally applicable to agency securities and municipal securities. DRAWBACKS OF TRADITIONAL YIELD SPREAD ANALYSIS Traditional analysis of the yield premium for a non-Treasury bond involves calculating the difference between the yield to maturity (or yield to call) of the bond in question and the yield to maturity of a comparable-maturity Treasury. The drawbacks of this convention, however, are (i) the yield for both bonds fails to take into consideration the term structure of interest rates, and (ii) in the case of callable and/or put able bonds, expected interest rate volatility may alter the cash flow of a bond. STATIC SPREAD: AN ALTERNATIVE TO YIELD SPREAD In traditional yield spread analysis, an investor compares the yield to maturity of a bond with the yield to maturity of a similar maturity on-the-run Treasury security. Such a comparison makes little sense, because the cash flow characteristics of the corporate bond will not be the same as that of the benchmark Treasury. The proper way to compare non-Treasury bonds of the same maturity but with different coupon rates is to compare them with a portfolio of Treasury securities that have the same cash flow. The corporate bond’s value is equal to the present value of all the cash flows. The corporate bond’s value, assuming that the cash flows are riskless, will equal the present value of the replicating portfolio of Treasury securities. The corporate bond’s price is less than the package of zero-coupon Treasury securities, because investors in fact require a yield premium for the risk associated with holding a corporate bond rather than a riskless package of Treasury securities. The static spread, also referred to as the zero-volatility spread, is a measure of the spread that the investor would realize over the entire Treasury spot rate curve if the bond is held to maturity. It is not a spread off one point on the Treasury yield curve, as is the traditional yield spread. The static spread is calculated as the spread that will make the present value of the cash flows from the corporate bond, when discounted at the Treasury spot rate plus the spread, equal to the corporate bond’s price. A trial-and error procedure is required to determine the static spread. Notice that the shorter the maturity of the bond, the less the static spread will differ from the traditional yield spread. The magnitude of the difference between the traditional yield spread and the static spread also depends on the shape of the yield curve. The steeper the yield curve, the more the difference for a given coupon and maturity. CALLABLE BONDS AND THEIR INVESTMENT CHARACTERISTICS The presence of a call option results in two disadvantages to the bondholder. First, callable bonds expose bondholders to reinvestment risk. Second, the price appreciation potential for a callable bond in a declining interest-rate environment is limited. This phenomenon for a callable bond is referred to as price compression. Traditional Valuation Methodology for Callable Bonds When a bond is callable, the practice has been to calculate a yield to worst, which is the smallest of the yield to maturity and the yield to call for all possible call dates. The yield to call (like the yield to maturity) assumes that all cash flows can be reinvested at the computed yield—in this case the yield to call—until the assumed call date. Moreover, the yield to call assumes that (i) the investor will hold the bond to the assumed call date, and (ii) the issuer will call the bond on that date. Often, these underlying assumptions about the yield to call are unrealistic because they do not take into account how an investor will reinvest the proceeds if the issue is called. Price-Yield Relationship for a Callable Bond As yields in the market decline, the likelihood that yields will decline further so that the issuer will benefit from calling the bond increases. The exact yield level at which investors begin to view the issue likely to be called may not be known, but we do know that there is some level, say y*. At yield levels below y*, the price-yield relationship for the callable bond departs from the price-yield relationship for the noncallable bond. For a range of yields below y*, there is price compression—that is, there is limited price appreciation as yields decline. The portion of the callable bond price-yield relationship below y* is said to be negatively convex. Negative convexity means that the price appreciation will be less than the price depreciation for a large change in yield of a given number of basis points. For a bond that is option-free and displays positive convexity, the price appreciation will be greater than the price depreciation for a large change in yield. It is important to understand that a bond can still trade above its call price even if it is highly likely to be called. COMPONENTS OF A BOND WITH AN EMBEDDED OPTION To develop a framework for analyzing a bond with an embedded option, it is necessary to decompose a bond into its component parts. A callable bond is a bond in which the bondholder has sold the issuer an option (more specifically, a call option) that allows the issuer to repurchase the contractual cash flows of the bond from the time the bond is first callable until the maturity date. In terms of price, a callable bond is equal to the price of the two components parts; that is, callable bond price = noncallable bond price – call option price. The reason the call option price is subtracted from the price of the noncallable bond is that when the bondholder sells a call option, the issuer receives the option price. The same logic applies to put able bonds. VALUATION MODEL The bond valuation process requires that we use the theoretical spot rate to discount cash flows. This is equivalent to discounting at a series of forward rates. For an embedded option the valuation process also requires that we take into consideration how interest-rate volatility affects the value of a bond through its effects on the embedded options. Depending on the structure of the security to be analyzed, three models can be used to account for the valuation effect of embedded options. The first model is for a bond that is not a mortgage-backed security or asset-backed security and which can be exercised at more than one time over its life. The second case is a bond with an embedded option where the option can be exercised only once. The third model is for a mortgage-backed security or certain types of asset-backed securities. Valuation of Option-Free Bonds The price of an option-free bond is the present value of the cash flows discounted at the spot rates. Consider an option-free bond with three years remaining to maturity and a coupon rate of 5.25%. The price of this bond can be calculated in one of two ways, both producing the same result. First, the coupon payments can be discounted at the zero-coupon rates: = $102.075 The second way is to discount by the one-year forward rates: = $102.075 Introducing Interest-Rate Volatility When we allow for embedded options, consideration must be given to interest-rate volatility. This can be done by introducing an interest-rate tree. This tree is nothing more than a graphical depiction of the one-period forward rates over time based on some assumed interest-rate model and interest-rate volatility. Interest-Rate Model As explained in the previous chapter, an interest-rate model is a probabilistic description of how interest rates can change over the life of a financial instrument being evaluated. An interest-rate model does this by making an assumption about the relationship between the level of short-term interest rates and interest-rate volatility (e.g., standard deviation of interest rates). The interest-rate models commonly used are arbitrage-free models based on how short-term interest rates can evolve (i.e., change) over time. The interest-rate models based solely on movements in the short-term interest rate are referred to as one-factor models. More complex models would consider how more than one interest rate changes over time. Interest-Rate Lattice An example of the most basic type of interest-rate lattice or tree is a binomial interest-rate tree. The corresponding model is referred to as the binomial model. In this mode, it is assumed that interest rates can realize one of two possible rates in the next period. Valuation models that assume that interest rates can take on three possible rates in the next period are called trinomial models. More complex models exist that assume in that more than three possible rates in the next period can be realized. For the binomial interest-rate tree, each node represents a time period that is equal to one year from the node to its left. Each node is labeled with an N, representing node, and a subscript that indicates the path that one-year forward rates took to get to that node. H represents the higher of the two forward rates and L the lower of the two forward rates from the preceding year. For example, node NHH means that to get to that node the following path for one-year forward rates occurred, i.e., the one-year rate realized is the higher of the two rates in the first year and then the higher of the one-year rates in the second year. Volatility and the Standard Deviation In the binomial model, it can be shown that the standard deviation of the one-year forward rate is equal to r0. The standard deviation is a statistical measure of volatility. For now it is important to see that the process that we assumed generates the binomial interest-rate tree (or equivalently, the forward rates) implies that volatility is measured relative to the current level of rates. Determining the Value at a Node In the binomial model, we find the value of the bond at a node is as follows. First calculate the bond’s value at the two nodes to the right of the node where we want to obtain the bond’s value. The cash flow at a node will be either (i) the bond’s value if the short rate is the higher rate plus the coupon payment, or (ii) the bond’s value if the short rate is the lower rate plus the coupon payment. To get the bond’s value at a node we follow the fundamental rule for valuation: The value is the present value of the expected cash flows. The appropriate discount rate to use is the one-year forward rate at the node. Constructing the Binomial Interest-Rate Tree To construct the binomial interest-rate tree, we use current on-the-run yields and assume a volatility, σ. The root rate for the tree, r0, is simply the current one-year rate. In the first year there are two possible one-year rates, the higher rate and the lower rate. What we want to find is the two forward rates that will be consistent with the volatility assumption, the process that is assumed to generate the forward rates, and the observed market value of the bond. There is no simple formula for this. It must be found by an iterative process (i.e., trial and error). The steps are described and illustrated following. Step 1: Select a value for r1. Recall that r1 is the lower one-year forward rate one year from now. In this first trial we arbitrarily selected a value of 4.5% for r1. Step 2: Determine the corresponding value for the higher one-year forward rate. This rate is related to the lower one-year forward rate as follows: r1(e2). This value is reported at node NH. Step 3: Compute the bond’s value one year from now. This value is determined as follows: 3a. The bond’s value two years from now must be determined. 3b. Calculate the present value of the bond’s value found in 3a using the higher rate. This value is VH. 3c. Calculate the present value of the bond’s value found in 3a using the lower rate. This value is VL. 3d. Add the coupon to VH and VL to get the cash flow at NH and NL, respectively. 3e. Calculate the present value of the two values using the one-year forward rate using r*, so we can compute: and . Step 4: Calculate the average present value of the two cash flows in step 3. This is the value at a node is . Step 5: Compare the value in step 4 with the bond’s market value. If the two values are the same, the r1 used in this trial is the one we seek. This is the one-year forward rate that would be used in the binomial interest-rate tree for the lower rate, and the corresponding rate would be for the higher rate. If, instead, the value found in step 4 is not equal to the market value of the bond, this means that the value r1 in this trial is not the one-period forward rate that is consistent with (1) the volatility assumption of 10%, (2) the process assumed to generate the one-year forward rate, and (3) the observed market value of the bond. In this case the five steps are repeated with a different value for r1. [Note. If we get a value less than $100, then the value for r1 is too large and the five steps must be repeated, trying a lower value for r1.] In this example, when r1 is 4.5% we get a value of $99.567 in step 4, which is less than the observed market value of $100.Therefore, 4.5% is too large and the five steps must be repeated, trying a lower value for r1. After we compute r1, we are still not done. Suppose that we want to “grow” this tree for one more year—that is, we want to determine r2. Now we will use the three-year on-the-run issue to get r2. The same five steps are used in an iterative process to find the one-year forward rate two years from now. But now our objective is as follows: Find the value for r2 that will produce an average present value at node NH equal to the bond value at that node and will also produce an average present value at node NL equal to the bond value at that node. When this value is found, we know that given the forward rate we found for r1, the bond’s value at the root—the value of ultimate interest to us—will be the observed market price. The binomial interest-rate tree constructed is said to be an arbitrage-free tree. It is so named because it fairly prices the on-the-run issues. Application to Valuing an Option-Free Bond To illustrate how to use the binomial interest-rate tree, consider a 5.25% corporate bond that has two years remaining to maturity and is option-free. Also assume that the issuer’s on-the-run yield curve is the one given earlier, and hence the appropriate binomial interest-rate tree is the one in Exhibit 17-12. Exhibit 17-13 shows the various values in the discounting process and produces a bond value of $102.075. It is important to note that this value is identical to the bond value found earlier when we discounted at either the zero-coupon rates or the one-year forward rates. We should expect to find this result because our bond is option free. This clearly demonstrates that the valuation model is consistent with the standard valuation model for an option-free bond. Valuing a Callable Corporate Bond The valuation process for a callable corporate bond proceeds in the same fashion as in the case of an option-free bond but with one exception: When the call option may be exercised by the issuer, the bond value at a node must be changed to reflect the lesser of its value if it is not called (i.e., the value obtained by applying the recursive valuation formula described previously) and the call price. Impact of Expected Interest Rate Volatility on Price Expected interest rate volatility is a key input into the valuation of bonds with embedded options. To see the impact on the price of a callable bond, Exhibit 17-15 shows the price of four 5%, 10-year callable bonds with different deferred call structures (six months, two year, five years, and seven years) based on different assumptions about the expected volatility of short-term interest rates. We observe the following from the exhibit: 1) The price of the option-free bond is the same regardless of the interest rate volatility assumed. This is expected since there is no embedded option that is affected by interest rate volatility. 2) For any given level of interest rate volatility, the longer the deferred call, the higher the price. Again, as expected the value of the option-free bond has the highest price. 3) The price of a callable bond moves inversely to the interest rate volatility assumed. Determining the Call Option Value (or Option Cost) The value of a callable bond is expressed as the difference between the value of a noncallable bond and the value of the call option. This relationship can also be expressed as follows: value of a call option = value of a noncallable bond – value of a callable bond. Extension to Other Embedded Options The bond valuation framework presented here can be used to analyze other embedded options, such as put options, caps and floors on floating-rate notes, and the optional accelerated redemption granted to an issuer in fulfilling its sinking fund requirement. Because the value of a non-put able bond can be expressed as the value of a put able bond minus the value of a put option on that bond, this means that value of a put option = value of a non-put able bond – value of a put able bond. Incorporating Default Risk The basic binomial model explained above can be extended to incorporate default risk. The extension involves adjusting the expected cash flows for the probability of a payment default and the expected amount of cash that will be recovered when a default occurs. Modeling Risk The user of any valuation model is exposed to modeling risk. This is the risk that the output of the model is incorrect because the assumptions upon which it is based are incorrect. Implementation Challenge To transform the basic interest rate tree into a practical tool requires refinements. For example, the spacing of the node lines in the tree must be much finer. While one can introduce time-dependent node spacing, caution is required; it is easy to distort the term structure of volatility. Other practical difficulties include the management of cash flows that fall between two node lines. OPTION-ADJUSTED SPREAD The option-adjusted spread (OAS) was developed as a measure of the yield spread (in basis points) that can be used to convert dollar differences between value and price. Thus, basically, the OAS is used to reconcile value with market price. The OAS is a spread over the spot rate curve or benchmark used in the valuation. The reason that the resulting spread is referred to as option-adjusted is because the cash flows of the security whose value we seek are adjusted to reflect the embedded option. Translating OAS to Theoretical Value Although the product of a valuation model is the OAS, the process can be worked in reverse. For a specified OAS, the valuation model can determine the theoretical value of the security that is consistent with that OAS. As with the theoretical value, the OAS is affected by the assumed interest rate volatility. The higher (lower) the expected interest rate volatility, the lower (higher) the OAS. Determining the Option Value in Spread Terms Earlier we described how the dollar value of the option is calculated. The option value in spread terms is determined as follows: option value (in basis points) = static spread – OAS. EFFECTIVE DURATION AND CONVEXITY There is a duration measure that is more appropriate for bonds with embedded options that the modified duration measure. In general, the duration for any bond can be approximated as follows: duration = . where P_ = price if yield is decreased by x basis points, P+ = price if yield is increased by x basis points, P0 = initial price (per $100 of par value), and ∆y (or dy) = change in rate used to calculate price (x basis points in decimal form). When the approximate duration formula is applied to a bond with an embedded option, the new prices at the higher and lower yield levels should reflect the value from the valuation model. Duration calculated in this way is called effective duration or option-adjusted duration. Similarly, the standard convexity measure may be inappropriate for a bond with embedded options because it does not consider the effect of a change in interest rates on the bond’s cash flow. KEY POINTS • The traditional yield spread approach fails to take three factors into account: (1) the term structure of interest rates, (2) the options embedded in the bond, and (3) the expected volatility of interest rates. The static spread measures the spread over the Treasury spot rate curve assuming that interest rates will not change in the future. • The potential investor in a callable bond must be compensated for the risk that the issuer will call the bond prior to the stated maturity date. The two risks faced by a potential investor are reinvestment risk and truncated price appreciation when yields decline (i.e., negative convexity). • The traditional methodology for valuing bonds with embedded options relies on the yield to worst. The limitations of yield numbers are now well recognized. Moreover, the traditional methodology does not consider how future interest-rate volatility will affect the value of the embedded option. • To value a bond with an embedded option, it is necessary to understand that the bond can be decomposed into an option-free component and an option component. The binomial method can be used to value a bond with an embedded option. It involves generating a binomial interest-rate tree based on (1) an issuer’s yield curve, (2) an interest-rate model, and (3) an assumed interest-rate volatility. The binomial interest-rate tree provides the appropriate volatility-dependent one-period forward rates that should be used to discount the expected cash flows of a bond. Critical to the valuation process is an assumption about expected interest-rate volatility. • The OAS converts the cheapness or richness of a bond into a spread over the future possible spot rate curves. The spread is option adjusted because it allows for future interest-rate volatility to affect the cash flows. • Modified duration and standard convexity, used to measure the interest-rate sensitivity of an option-free bond, may be inappropriate for a bond with an embedded option because these measures assume that cash flows do not change as interest rates change. • The duration and convexity can be approximated for any bond, whether it is option-free or a bond with an embedded option. The approximation involves determining how the price of the bond changes if interest rates go up or down by a small number of basis points. If interest rates are changed and it is assumed that the cash flows do not change, the resulting measures are modified duration and standard convexity. However, when the cash flows are allowed to change when interest rates change, the resulting measures are called effective duration and effective convexity. ANSWERS TO QUESTIONS FOR CHAPTER 17 (Questions are in bold print followed by answers.) 1. What are the two drawbacks of the traditional approach to the valuation of bonds with embedded options? Traditional analysis of the yield premium for a non-Treasury bond involves calculating the difference between the yield to maturity (or yield to call) of the bond in question and the yield to maturity of a comparable-maturity Treasury. The latter is obtained from the Treasury yield curve. The drawbacks of this convention, however, are (i) the yield for all bonds (Treasury versus either callable or noncallable non-Treasury bonds) fails to take into consideration the term structure of interest rates, and (ii) in the case of callable and/or put able bonds, expected interest rate volatility may alter the cash flow of a bond. 2. What is the static spread for a three-year 9% coupon corporate bond selling at 105.58, given the following theoretical Treasury spot rate values equal to 50, 100, or 120 basis points? Period Spot Rate (%) 1 4.0 2 4.2 3 4.9 4 5.4 5 5.7 6 6.0 In traditional yield spread analysis, an investor compares the yield to maturity of a bond with the yield to maturity of a similar maturity on-the-run Treasury security. This means that the yield to maturity of a three-year zero-coupon corporate bond and a 9% coupon three-year corporate coupon bond would both be compared to a benchmark three-year Treasury security. Such a comparison makes little sense because the cash flow characteristics of the two corporate bonds will not be the same as that of the benchmark Treasury. The proper way to compare non-Treasury bonds (of the same maturity but with different coupon rates) is to compare them with a portfolio of Treasury securities that have the same cash flow. In our problem above, we consider the 9% three-year corporate bond selling for 105.58 (per $100). The cash flow per $100 par value for this corporate bond, assuming that interest rates do not change (i.e., assuming static interest rates), is five six-month payments of (0.09 / 2)$100 = $4.50 and a payment in three years (6 six-month periods) of $100 + $4.50 = $104.50. A portfolio that will replicate this cash flow would include six zero-coupon Treasury securities with maturities coinciding with the amount and timing of the cash flows of the corporate bond. The corporate bond’s value (105.58 quote per $100) is equal to the present value of all the cash flows. The corporate bond’s value, assuming that the cash flows are riskless, will equal the present value of the replicating portfolio of Treasury securities. In turn, these cash flows are valued at the Treasury spot rates. The period one spot rate is 4%. Dividing by two gives our semiannual discount rate for period one of 4.0% / 2 = 2.0%. Similarly, we can get the remaining five semiannual discount rates for periods two through six which are 2.1%, 2.45%, 2.7%, 2.85%, and 3.0%, respectively. Using our six semiannual discount rates, we get: theoretical price = = $4.4118 + $4.3168 + $4.1848 + $4.0451 + $3.9101 + $84.5171 = $108.3857. Thus, the price given the Treasury spot rates is $108.3857. However, the corporate bond’s price is $105.58, which is less than the package of zero-coupon Treasury securities. This is because investors require a yield premium for the risk associated with holding a corporate bond rather than a riskless package of Treasury securities. The static spread, also referred to as the zero-volatility spread, is a measure of the spread that the investor would realize over the entire Treasury spot rate curve if the bond is held to maturity. It is not a spread off one point on the Treasury yield curve, as is the traditional yield spread. The static spread is calculated as the spread that will make the present value of the cash flows from the corporate bond, when discounted at the Treasury spot rate plus the spread, equal to the corporate bond’s price. A trial-and error procedure is required to determine the static spread. To illustrate how this is done for the corporate bond in our problem we select the first spread of 50 basis points (i.e., 0.5%). To each Treasury (semiannual) spot rate shown in the spot rate column 50 basis points are added. So, for example, the spot rate is 4% + 0.5% = 4.5%. Dividing by two gives our semiannual discount rate for period one of 4.5% / 2 = 2.25%. Similarly, we can get the remaining five semiannual spot rates for periods two through five which are 2.25%, 2.35%, 2.70%, 2.95%, 3.10%, and 3.25%, respectively. Using our six semiannual spot rates, we get: theoretical price = = $4.4010 + $4.2957 + $4.1543 + $4.0060 + $3.8630 + $86.2533 = $106.9733. Thus, the spot rate plus 50 basis points renders a present value of $106.9733, which is greater than the corporate bond’s price of $105.58. Thus, the static spread is not 50 basis points but must be a higher spread. We will try 100 basis points. Proceeding in our trial and error procedure, we add 100 basis points to each Treasury spot rate shown in the spot rate column. So, now the spot rate is 4% + 1% = 5%. Dividing by two gives our semiannual discount rate for period one of 5% / 2 = 2.5%. Similarly, we can get the remaining five semiannual spot rates for periods two through five which are 2.6%, 2.95%, 3.2%, 3.35%, and 3.5%, respectively. Using our six semiannual spot rates, we get: theoretical price = = $4.3902 + $4.2748 + $4.1241 + $3.9673 + $3.8165 + $85.0108 = $105.5838. Thus, the spot rate plus 100 basis points renders a present value of $105.5838, which rounded off equals that corporate bond’s price of $105.58. Thus, the static spread appears to be 100 basis points. Although, we have our answer, we will go ahead and try 120 basis points given in our problem. So proceeding in our trial and error procedure, we now add 120 basis points to each Treasury spot rate shown in the spot rate column. The spot rate is 4% + 1.2% = 5.2%. Dividing by two gives our semiannual discount rate for period one of 5.2% / 2 = 2.6%. Similarly, we can get the remaining five semiannual spot rates for periods two through five which are 2.7%, 3.05%, 3.3%, 3.45%, and 3.6%, respectively. Using our semiannual spot rates, we get: theoretical price = = $4.3860 + $4.2665 + $4.1121 + $3.9519 + $3.7980 + $84.5197 = $105.0343. As we expected, the spot rate plus 120 basis points renders a present value of $105.0343, which is less than the corporate bond’s price of $105.58. 3. Under what conditions would the traditional yield spread be close to the static spread? There are three conditions that will cause the traditional yield spread to be closer to the static spread: a short maturity, a flat yield curve, and bullet payment at maturity for a corporate bond. More details are given below. Exhibit 17-3 shows the static spread and the traditional yield spread for bonds with various maturities and prices, assuming the Treasury spot rates shown in Exhibit 17-1. Notice that the shorter the maturity of the bond, the less the static spread will differ from the traditional yield spread. The magnitude of the difference between the traditional yield spread and the static spread also depends on the shape of the yield curve. The steeper the yield curve, the more the difference for a given coupon and maturity. Another reason for the small differences in Exhibit 17-3 is that the corporate bond makes a bullet payment at maturity. The difference between the traditional yield spread and the static spread will be considerably greater for sinking fund bonds and mortgage-backed securities in a steep yield curve environment. 4. Why is the investor of a callable bond exposed to reinvestment risk? Reinvestment risk is the risk caused by reinvesting fixed payments at a lower rate due to an environment of declining interest rates. The investor of a callable bond is exposed to this risk because as interest rates fall there is a greater likelihood that the callable bond will be called by the issuing firm. More details are given below. The characteristics of a callable bond can work to the disadvantage of an investor because the holder of a callable bond has given the issuer the right to call the issue prior to the expiration date. In brief, callable bonds expose bondholders to loss in value because an issuer will call a bond at a price below its market value. It will do this when the current yield on bonds in the market is lower than the issue’s coupon rate. For example, if the coupon rate on a callable corporate bond is 12% and prevailing market yields are 8%, the issuer may find it economical to call the 12% issue and refund it with an 8% issue. From the investor’s perspective, the proceeds received will have to be reinvested at a lower interest rate. This is called reinvestment risk. Relatedly, the price appreciation potential for a callable bond in a declining interest-rate environment is limited. This is because the market will increasingly expect the bond to be redeemed at the call price as interest rates fall. This phenomenon for a callable bond is referred to as price compression. Because of the disadvantages associated with callable bonds, these instruments often feature a period of call protection, an initial period when bonds may not be called. Also, the investor receives compensation in the form of a higher potential yield. 5. What is negative convexity? Negative convexity means that the price appreciation will be less than the price depreciation for a large change in yield of a given number of basis points. For a bond that is option-free and displays positive convexity, the price appreciation will be greater than the price depreciation for a large change in yield. It is important to note that a bond can still trade above its call price even if it is highly likely to be called. 6. Does a callable bond exhibit negative or positive convexity? A bond that is option-free displays positive convexity which means the price appreciation will be greater than the price depreciation for a large change in yield. Positive convexity does not hold for a callable bond. This is because when interest rates fall below a certain yield level, y*, the shape of the yield curve begins turning inward instead of outward. Perfect parallel with a noncallable bond is not achieved even for yields a bit above the coupon rate. This is because there is still the chance the market yield may drop further, making investors hesitant to pay the noncallable price due to the possibility yields will fall. 7. Suppose that you are given the following information about two callable bonds that can be called immediately: Estimated Percentage Change in Price if Interest Rates Change by: –100 basis points +100 basis points Bond ABC +5% –8% Bond XYZ +22% –16% You are told that both of these bonds have the same maturity and that the coupon rate of one bond is 7% and of the other is 13%. Suppose that the yield curve for both issuers is flat at 8%. Based on this information, which bond is the lower coupon bond and which is the higher coupon bond? Explain why. If both bonds were noncallable then bond XYZ would be the bond with the lower coupon rate of 7%. This is because ceteris paribus lower coupon bonds undergo greater changes in yields when interest rates change. The question is whether the callable feature of the bond affects this conclusion. This does not appear to be case given the greater volatility and the fact the upward change of +22% for a 100 basis point fall is greater than the downward change of 16% for 100 basis points rise. Positive convexity means the bond will have greater price appreciation than price depreciation for a large change in yield. This describes the situation for bond XYZ indicating it is the bond with the lower coupon rate of 7%. This implies that bond ABC is the bond with the higher coupon rate of 13%. [Note. Bond XYZ displays positive convexity, and bond ABC displays negative convexity. Assuming the same strike (call) price and the same current market yield, bond XYZ is less likely to be called than is bond ABC. One might notice that comparing the percentage price change can result in the same conclusion, but the reasoning is different.] 8. The theoretical value of a noncallable bond is $103; the theoretical value of a callable bond is $101. Determine the theoretical value of the call option. Effectively, the owner of a callable bond is entering into two separate transactions. First, the owner buys a noncallable bond from the issuer. Second, the owner sells the issuer a call option at a designated option price. The payment received lowers the value of the callable bond. Thus, in terms of price, a callable bond is generally speaking equal to the price of the two components parts that we can express as: callable bond price = noncallable bond price – call option price. Rearranging, we have: call option price = noncallable bond price – callable bond price. Inserting in our values, we have: call option price = $103 – $102 = $2. In terms of a bond’s par value of $1,000, the value of the call option is $20.00. 9. Explain why you agree or disagree with the following statement: “The value of a put able bond is never greater than the value of an otherwise comparable option-free bond.” As described below, one would not agree with the statement. For a put able bond, the bondholder has the right to sell the bond to the issuer at a designated price and time. A put able bond can be broken into two separate transactions. First, the investor buys a noncallable bond. Second, the investor buys an option from the issuer that allows the investor to sell the bond to the issuer. The price of a put able bond is then: Put able bond price = non-put able bond price + put option price. Given that the value of a put option is never negative and in most (if not all) cases in positive, the value of a put able bond is always greater than or equal to the value of a non-put able or noncallable bond or option-free bond. Thus, the statement is false. 10. Explain why you agree or disagree with the following statement: “An investor should be unwilling to pay more than the call price for a bond that is likely to be called.” As shown below, one would disagree with the statement. Consider a callable bond with a 10-year 13% coupon rate that is callable in one year at a call price of 104. Suppose that the yield on 10-year bonds is 6% and that the yield on one-year bonds is 5%. In a 6% interest rate environment for 10-year bonds, investors will expect that the issue will be called in one year. Thus investors will treat this issue as if it is a one-year bond and price it accordingly. The price must reflect the fact that the investor will receive a 13% coupon rate for one year. The price of this bond would be the present value of the two cash flows, which are (i) $6.50 (per $100 of par value) of coupon interest six months from now, and (ii) $6.50 coupon interest plus the call price of $104 one year from now. Discounting the two cash flows at the 5% prevailing market yield (2.5% every six months) for one-year bonds, the price is = $111.52. The price is greater than the call price. Consequently, an investor will be willing to pay a higher price than the call price to purchase this bond. Thus, one would not agree with the statement: “An investor should be unwilling to pay more than the call price for a bond that is likely to be called.” 11. In Robert Litterman, Jose Scheinkman, and Laurence Weiss, “Volatility and the Yield Curve,” Journal of Fixed Income, Premier Issue, 1991, p. 49, the following statement was made: “Many fixed income securities (e.g., callable bonds) contain embedded options whose prices are sensitive to the level of volatility. Modeling the additional impact of volatility on the value of the coupons allows for a better understanding of the price behavior of these securities.” Explain why. The probability of a bond being called is a function of the volatility for of interest rates. If expectations are that interest rates will not change (e.g., a flat yield curve) for a prolonged period of time approaching the maturity of the bond, then the option to call a bond would not be highly valued. On the other hand, if interest rates are believed to be volatile (and thus have a high probability of decreasing and impacting the discounted value of coupon payments), then the value of a call option on a bond would be highly valued. Thus, modeling the additional impact of volatility on the value of the coupons allows for a better understanding of the price behavior of these securities. 12. If an on-the-run issue for an issuer is evaluated properly using a binomial model, how would the theoretical value compare to the actual market price? For an on-the-run issue that is option-free the theoretical value is identical to the bond value found when we discount at either the zero-coupon rates or the one-year forwards. Since the binomial model uses forward rates, it would be consistent with the actual market price given by the standard valuation model for an option-free bond. For example, the model uses two forward rates (a higher and lower rate each step in generating the binomial interest rate tree) that will be consistent with the volatility assumption, the process that is assumed to generate the forward rates, and the observed market value of the bond. 13. The current on-the-run yields for the Ramsey Corporation are as follows: Maturity (years) Yield to Maturity (%) Market Value 1 7.5 100 2 7.6 100 3 7.7 100 Assume that each bond is an annual-pay bond. Each bond is trading at par, so its coupon rate is equal to its yield to maturity. Answer the below questions. (a) Using the bootstrapping methodology, complete the following table: Year Spot Rate (%) One-Year Forward Rate (%) 1 2 3 The one-year spot rate for year one is its annualized yield of 7.50%. Using this value and the yield to maturity for year 2, we can solve for the one-year spot rate for year two as shown below. We do this below using the bootstrapping methodology. The price of a theoretical 2-year zero-coupon security should equal the present value of two cash flows from an actual 2-year coupon security, where the yield used for discounting is the spot rate corresponding to each cash flow. The coupon rate for a 2-year security is given as 7.6% (since yield to maturity of 7.6% is the same as coupon rate given the market value is 100). Using $100 as par, the cash flow for this security for year one is CF1 = $7.60 and for year two is CF2 = $7.60 + $100 = $107.60. Given the one year spot rate of s1 = 7.5%, we can now solve for the 2-year theoretical spot rate, s2, using the following expression: = $100. Below we insert values for CF1, CF2, and s1 to solve for s2. = $100  = $100  $7.06977 + = $100  = $92.93023  = (1 + s2)2  s2 = – 1  s2 = – 1  s2 = 1.0760381 – 1  s2 = 0.076038 or about 7.6038%. Next we solve for the 3-year theoretical spot rate. The price of a theoretical 3-year zero-coupon security should equal the present value of three cash flows from an actual 3-year coupon security, where the yield used for discounting is the spot rate corresponding to each cash flow. The coupon rate for a 3-year security is given as 7.7%. Using $100 as par, the cash flow for this security is $7.70 for years one and two. For year three, it is $7.70 + $100 = $107.70 for year two. We can now solve for the 3-year theoretical spot rate, s3, using the below expression with CF1 = CF2 = $7.70, CF3 = $107.70, s1 = 7.5%, and s2 = 7.6038%. We have: = $100  = $100  $7.16279 + $6.6502123 + = $100  = $86.186700  = (1 + s3)3  s3 = – 1  s3 = – 1  s3 = 1.0771049 – 1  s3 = 0.0771049 or about 7.7105%. Thus, the spot rates are 7.50%, 7.6038%, and 7.7105% for years one, two, and three. We now compute the one-year forward rates for years one, two, and three. The one-year forward rate for year one is the same as the one-year spot rate of 7.50%. The one-year forward rate for year two is given by: f2 = . Inserting in our values for s1 and s2 and solving, we have: f2 = = = 1.077077 – 1 = 0.077077 or about 7.7077%. The one-year forward rate for year three is given by: f3 = . Inserting in our values for s2 and s3 and solving, we have: f3 = = = 1.0792425 – 1 = 0.0792417 or about 7.9242%. As seen below, we can now complete our table by inserting in all spot rates and one-year forward rates. We have: Year Spot Rate (%) One-Year Forward Rate (%) 1 s1 = 7.5000 f1 = 7.5000 2 s2 = 7.6038 f2 = 7.7077 3 s3 = 7.7105 f3 = 7.9242 (b) Using the spot rates, what would be the value of an 8.5% option-free bond of this issuer? The value of an 8.5% three-year option-free bond is the present value of the cash flows using the spot rates as the discount rates. Thus, using the one-year spot rates, the value of an 8.5% coupon option-free bond is given by: option-free bond price = . where the coupon payments per period (year) are C1 = C2 = C3 = $8.50 per $100 and the spot rates for years one, two and three are s1 = 7.50%, s2 = 7.6038%, and s3 = 7.7105%. Inserting in our coupon values and spot rate values and solving, we get: option-free bond price = = $7.90698 + $7.34114 + $86.82720 = $102.07532 or about $102.08 per $100. (c) Using the one-year forward rates, what would be the value of an 8.5% coupon option-free bond of this issuer? It should be the same $102.08 per $100 which we just computed in part (b). Below we demonstrate it is $102.08 using the one-year forward rates. Using the one-year forward rates, the value of an 8.5% coupon option-free (annual pay) bond is given by: option-free bond price = where C1 = C2 = C3 = $8.50 per $100, the spot rate for year one is s1 = 7.50%, and the one-year forward rates for years two and three are f2 = 7.7077%, and f3 = 7.9242%, respectively. Inserting in our values, we have: option-free bond price = $7.90698 + $7.34114 + $86.82720 = $102.07532 or about $102.08 per $100. (d) Using the binomial model (which assumes that one-year rates undergo a lognormal random walk with volatility ), show that if  is assumed to be 10%, the lower one-year forward rate one year from now cannot be 7%. The point denoted N is the root of the binomial interest rate tree and is the current one-year rate, or equivalently, the one-year forward rate, denoted by r0. In our problem, we have r0 = 7.5%. Assuming a one-factor interest-rate model, the one-year forward rate can evolve over time based on a random process called a lognormal random walk with a certain volatility. The following notation is used to describe the tree in the first year:  = assumed volatility of the one-year forward rate; r1,L = the lower one-year rate one year from now; and, r1,H = the higher one-year rate one year from now. The relationship between r1,L and r1,H is as follows: r1,H = r1,L(e2) where e is the base of the natural logarithm 2.71828 or equivalently r1,L = r1,H / e2. To see how to construct the binomial interest-rate tree, let’s use the assumed current on-the-run yields given in part (a) which are yield to maturities of 7.5%, 7.6%, and 7.7% for maturities of 1, 2, and 3 years, respectively, for our annual pay bond. Given our volatility, , is 10%, we can construct a two-year model using the two-year bond with a coupon rate of 7.60% (yield to maturity equals the coupon rate since all bonds sell at par of 100). As noted above, the root rate for the tree, r0, is simply the current one-year spot rate or forward rate of 7.5%. In the first year there are two possible one-year rates, the higher rate and the lower rate. What we want to find is the two forward rates that will be consistent with the volatility assumption, the process that is assumed to generate the forward rates, and the observed market value of the bond. There is no simple formula for this. It must be found by an iterative process (i.e., trial and error). The steps are described and illustrated below. Step 1: Select a value for r1. Recall that r1 is the lower one-year forward rate one year from now. In this first trial we can arbitrarily select a value. In this problem, however, we are asked to take 7.00% so that will be our “arbitrary” value reported at node NL. Step 2: Determine the corresponding value for the higher one-year forward rate, r1,H. This rate is related to the lower one-year forward rate, r1 or r1,L, as follows: r1,H = r1(e2). For r1 = 7.00% and  = 10%, the higher one-year forward rate, r1,H = 7.00%(2.71828182(0.10)) = 7.00%(1.2214028) = 8.5498%. This value would be reported at node NH. Step 3: Compute the bond’s value one year from now. This value is determined as follows: 3a. The bond’s value two years from now must first be determined. We are using a two-year bond, so the bond’s value at the end of year one (after the first coupon has been paid) is its maturity value ($100) plus its final coupon payment = (coupon rate)($1,000) = 0.076($1,000) = $7.60. Thus, the bond value two years from now is $107.60. 3b. At t = 1, compute the present value of the bond’s value found in 3a using the higher rate, r1,H. In our example the appropriate discount rate is the one-year higher forward rate, 8.5498%. The present value is $107.60 / 1.085498 = $99.12499. This is the value of VH reported at node NH. 3c. At t =1, compute the present value of the bond’s value found in 3a using the lower rate, r1,L. The discount rate used is the lower one-year forward rate, 7.00%. The value is $107.60 / 1.07 = $100.56075. This is the value of VL reported at node NL. 3d. Add the coupon to VH and VL to get the respective cash flow at NH and NL at the end of the first period. In our example we have $99.12499 + $7.60 = $106.72499 for the higher rate and $100.56075 + $7.60 = $108.16075 for the lower rate. 3e. Calculate the present value of the two values using the one-year forward rate using r*. At this point in the valuation, r* is the root rate, 7.50%. Therefore, = = $99.27906 and = = $100.61465. Step 4: Calculate the average present value of the two cash flows in step 3. This is the value at node N. We have: V = . Inserting the value for our example, we have: value at node N = V = = $99.946856. Step 5: Compare the value in step 4 with the bond’s market value. If the two values are the same, the r1 used in this trial is the one we seek. This is the one-year forward rate at t = 1 that would then be used in the binomial interest-rate tree for the lower rate, and the corresponding rate would be for the higher rate. If, instead, the value found in step 4 is not equal to the market value of the bond, this means that the value r1 in this trial is not the one-period forward rate that is consistent with (1) the volatility assumption of 10%, (2) the process assumed to generate the one-year forward rate, and (3) the observed market value of the bond. In this case, when r1 is 7.00% we get a value of $99.946856 in step 4, which is less than the observed market value of $100. Therefore, 7.00% is too large and the five steps must be repeated, trying a lower value for r1. In conclusion, we have demonstrated that the lower one-year forward rate one year from now cannot be 7.00%. (e) Demonstrate that if σ is assumed to be 10%, the lower one-year forward rate one year from now is 6.944%. Given our volatility, , is 10%, we can construct a two-year model using the two-year bond with a coupon rate of 7.60%. As noted above, the root rate for the tree, r0, is simply the current one-year spot rate or forward rate of 7.5%. The steps for r1 = 6.944% are described and illustrated below. Step 1: Select a value for r1, which is the lower one-year forward rate one year from now. In this second trial we would select a value based on our results in our first trial. However, we are asked to try 6.944% so that will be our value reported at node NL. Step 2: Determine the corresponding value for the higher one-year forward rate, r1,H. This rate is related to the lower one-year forward rate, r1 or r1,L, as follows: r1,H = r1(e2). Inserting in our values of r1 = 6.944% and  = 10%, the higher one-year forward rate one year from now (at t =1) is r1,H = 6.944%(2.71828182(0.10)) = 6.944%(1.2214028) = 8.4814%. This value is reported at node NH. Step 3: Compute the bond’s value one year from now. This value is determined as follows: 3a. The bond’s value two years from now must first be determined. We are using a two-year bond, so the bond’s value at the end of year one (after the first coupon has been paid) is its maturity value ($100) plus its final coupon payment = (coupon rate)($1,000) = 0.076($1,000) = $7.60. Thus, the bond value two years from now is $107.60. 3b. At t = 1, compute the present value of the bond’s value found in 3a using the higher rate, r1,H. In our example the appropriate discount rate is the one-year higher forward rate, 8.4814%. The present value is $107.60 / 1.084814 = $99.18749. This value is called VH. 3c. At t =1, compute the present value of the bond’s value found in 3a using the lower rate, r1,L. The discount rate used is the lower one-year forward rate, 6.944%. The computed value is $107.60 / 1.06944 = $100.61341. This value is called VL. 3d. Add the coupon to VH and VL to get the respective cash flow at NH and NL at the end of the first period. In our example we have $99.18749 + $7.60 = $106.78749 for the higher rate and $100.61341 + $7.60 = $108.21341 for the lower rate. 3e. Calculate the present value of the two values using the one-year forward rate using r*. At this point in the valuation, r* is the root rate, 7.50%. We get: = = $99.33720 and = = $100.66363. Step 4: Calculate the average present value of the two cash flows in step 3. This is the value at node N using the formula: V = . Inserting in the values form 3e, we get: V = = $100.0004  $100.00. Step 5: Compare the value in step 4 with the bond’s market value. If the two values are the same, then the selected r1 is the one-year forward rate one year from now used in the binomial interest-rate tree for the lower rate, and the corresponding rate would be for the higher rate. For the selection of r1 = 6.944 % we got a value of $100.0004 in step 4, which is extremely close to the observed market value of $100. Thus, for practical purposes we have demonstrated that 6.944% is the one-year forward rate one year from now. (f) Demonstrate that if σ is assumed to be 10%, the lower one-year forward rate two years from now is approximately 6.437%. Now we want to “grow” our binomial tree for one more year—that is, we want to determine r2. Now we will use the three-year on-the-run issue, the 7.7% coupon bond, to get r2. The same five steps are used in an iterative process to find the one-year forward rate two years from now. The steps for r2 (or r2,LL) = 6.437% are described and illustrated below. Step 1: Select a value for r2. Recall that r2 is the lower one-year forward rate at t = 2. In this first trial we can arbitrarily select a value. In this problem, however, we are asked to try 6.437% so that will be our “arbitrary” value. This value would be reported at node NLL. Step 2: Determine the corresponding value for the higher one-year forward rate, r2,LH at t = 2. This rate is related to the lower one-year forward rate, r2 or r2,LL, as follows: r2,LH = r2(e2). For r2 = 6.437% and  = 10%, the higher one-year forward rate, r2,LH = 6.437%(2.71828182(0.10)) = 6.437%(1.221403) = 7.863%. This value is reported at node NLH and is also the value reported at NHL for r2,LH. For NHH, we get r2,HH = r2,LL(e4) = 7.863%(2.71828184(0.1)) = 9.604%. Step 3: Compute the bond’s value one year from t = 1. First, the bond’s value three years from now must first be determined. Since we are using a three-year bond, the bond’s value (after the coupon is paid) is its maturity value ($100) plus its final coupon payment given by taking the coupon rate times $1,000. Doing this, we get: (coupon rate)($1,000) = 0.077($1,000) = $7.70. Thus, the bond value three years from now is $107.70. At t = 2, compute the present value of the bond’s value found in 3a using the higher rate, r2,LH. In our example the appropriate discount rate is the one-year higher forward rate, 7.863%. The present value is $107.70 / 1.07863 = $99.8496 at NLH. This value is called VLH. This is also the same value at NHL and is called VHL. Next, we compute the present value of the bond’s value using the lower rate, r12,LL. The discount rate used is the lower one-year forward rate, 6.437%. The value is $107.70 / 1.06437 = $101.1866 at NLL. This value is called VLL. The value for VHH is $107.70 / 1.09604 = $98.2638 given our rate of 9.604% reported at NHH in Step 2. Add the coupon to VLH and VLL to get the respective cash flow at NLH and NLL at the end of the second period. Doing this gives $99.84965 + $7.70 = $107.54965 and $101.18662 + $7.70 = $108.88662. Similarly, add the coupon to VHL and VHH to get the respective cash flow at NHL and NHH at the end of the second period. We get: $98.263845 + $7.70 = $105.963845 and $99.84965 + $7.70 = $107.54965. Now calculate the present value for VLH and VLL using the root rate of r* = 6.944%. We get: = = $100.5663 and = = $101.8165. Now calculate the present value for VHL and VHH using r* = 6.944%(1.221402758) = 8.4814%. We get: = = $97.6793 and = = $99.1411. Step 4: Calculate the average present value of the two cash flows in step 3 for both NL and NH. For the value at node NL, with r* = 6.944%, we get: VL = = = $101.19111  $101.19. For the value at node NH, with r* = 8.4814%, we get: VH = = = $98.4102  $98.41. Calculating the average present value of $101.19 and $98.41 with r* = 7.5%, we get: V = = = = $100.0007  $100.00. Step 5: Compare the value in step 4 with the bond’s market value. If the two values are the same, the r2 used in this trial is the one we seek. If the value is greater than (lesser than) the bond’s market value then we need to try a larger (smaller) value for r2. Since the values are both $100.00, we have demonstrated that the lower one-year forward rate two years from now is approximately 6.437%. (g) Show the binomial interest-rate tree that should be used to value any bond of this issuer. At node N, we have: r0 = 7.5% At node NL, we have: r1 = 6.944% At node NH, we have: r1(e2) = 8.481%* *r1(e2) = 6.944(2.71828182(0.1)) = 8.481% At node NLL, we have: r2 = 6.437% At node NLH (and NHL), we have: r2(e2) = 7.863%** ** r2(e2) = 6.437%(2.71828182(0.1)) = 7.863% At node NHH, we have: r2(e2) = 9.604%*** ***r2(e4) = 7.863%(2.71828184(0.1)) = 9.604% (h) Determine the value of an 8.5% coupon option-free bond for this issuer using the binomial interest-rate tree given in part g. The value of an 8.5% coupon option-free bond for this issuer using the binomial interest-rate tree given in part (g) is $102.0763. Details are given below. At node N, we have: V = 102.0763, C = 0, r0 = 7.5% [NOTE: Divide the following by two: (99.826+8.5 / 1.075) + (102.638+8.5 / 1.075) = 100.7684 + 103.3842 = 204.1526. Dividing by two gives: 102.0763.] At node NL, we have: VL = 102.638, C = 8.5, r1,L = 6.944% [NOTE: Divide the following by two: (100.5915+8.5 / 1.06944) + ($101.938+8.5 / 1.06944) = 102.0081 + 103.2671 = 205.2752. Dividing by two gives: 102.6376.] At node NH, we have: VH = 99.826, C = 8.5, r1,H = 8.481%* [NOTE: Divide the following by two: ($98.9936+8.5 / 1.08481) + (100.5915+8.5 / 1.08481) = 99.08979 + 100.56277 = 199.65257. Dividing by two gives: 99.8263.] At node NLL, we have: VLL = 101.938, C = 8.5, r2,LL = 6.437% [NOTE: (100+8.5) / 1.064378 = $101.9382.] At node NLH, we have: VLH = 100.5915, C = 8.5, r2,LH = 7.862%** [NOTE: (100+8.5) / 1.07863 = 100.5915.] At node NHH, we have: VHH = 98.9936, C = 8.5, r2,HH = 9.603%*** [NOTE: (100+8.5) / 1.09604 = 98.9936.] *r1,H = r1,L(e2) = 6.944(2.71828182(0.1)) = 8.4814% **r2,LH = r2,LL(e2) = 6.4378%(2.71828182(0.1)) = 7.8632% ***r2,HH = r2,LL(e4) = 7.86315%(2.71828184(0.1)) = 9.6041% (i) Determine the value of an 8.5% coupon bond that is callable at par (100) assuming that the issue will be called if the price exceeds par. The value of an 8.5% coupon callable bond for this issuer using the binomial interest-rate tree given in part (g) is $100.723. Details are given below. The value of an 8.5% coupon option-free bond for this issuer using the binomial interest-rate tree given in part (g) is $102.723. Details are given below. At node N, we have: V = 100.722, C = 0, r0 = 7.5% [NOTE: Divide the following by two: (99.554+8.5 / 1.075) + (100+8.5 / 1.075) = 100.7684 + 103.3842 = 201.4456. Dividing by two gives: 100.7228.] At node NL, we have: VL = MIN(100;101.455) = 100, C = 8.5, r1,L = 6.944% [NOTE: Divide the following by two: (100+8.5 / 1.06944) + ($100+8.5 / 1.06944) = 101.45497 + 101.45497 = 202.90993. Dividing by two gives: 101.45497.] At node NH, we have: VH = 99.554, C = 8.5, r1,H = 8.481%* [NOTE: Divide the following by two: ($98.9936+8.5 / 1.08481) + (100+8.5 / 1.08481) = 99.08979 + 100.0175 = 199.1073. Dividing by two gives: 99.55365.] At node NLL, we have: VLL = MIN(100;101.938) = 100, C = 8.5, r2,LL = 6.437% [NOTE: (100+8.5) / 1.06437 = $101.9382.] At node NLH, we have: VLH = MIN(100;100.5915) = 100, C = 8.5, r2,LH = 7.862%** [NOTE: (100+8.5) / 1.07862 = 100.5915.] At node NHH, we have: VHH = 98.9936, C = 8.5, r2,HH = 9.603%*** [NOTE: (100+8.5) / 1.09603 = 98.9936.] *r1,H = r1,L(e2) = 6.944(2.71828182(0.1)) = 8.4814% **r2,LH = r2,LL(e2) = 6.4378%(2.71828182(0.1)) = 7.8632% ***r2,HH = r2,LL(e4) = 7.86315%(2.71828184(0.1)) = 9.6041% 14. Explain how an increase in expected interest-rate volatility can decrease the value of a callable bond. From an investor’s point of view, they have sold a call option to the firm. The firm will exercise its call option when interest rates fall by enough to make it worth the company’s costs to refinance its debt at a lower interest rate. The probability of this fall occurring increases when there is greater volatility in interest rates. If and when interest rates fall, investors will have to “sell” their bonds back to the firm. If they want to invest the money received in bonds, they will have to purchase bonds with lower coupon payments. Thus, as expected interest-rate volatility increases, then the value of holding a callable fond can decrease. 15. Answer the below questions. (a) What is meant by the option-adjusted spread? The option-adjusted spread (OAS) measures the yield spread (in basis points) that can be used to convert dollar differences between value and price. Thus, basically, the OAS is used to reconcile value with market price. The reason that the resulting spread is referred to as option-adjusted is because the cash flows of the security whose value we seek are adjusted to reflect the embedded option. (b) What is the option-adjusted spread relative to? The option-adjusted spread (OAS) is a spread over the spot rate curve or benchmark used in the valuation. In the case of the binomial method, the OAS is a spread over the binomial interest rate tree. 16. “The option-adjusted spread measures the yield spread over the Treasury on-the-run yield curve.” Explain why you agree or disagree with this statement. As seen below, there are various ways to measure the option-adjusted yield spread. In terms of the option-adjusted spread (OAS), it measures the yield spread over the spot rate curve or benchmark used in the valuation. More details are supplied below. In traditional yield spread analysis, an investor compares the yield to maturity of a bond with the yield to maturity of a similar maturity on-the-run Treasury security. This means that the yield to maturity of a 25-year zero-coupon corporate bond and an 8.8% coupon 25-year corporate coupon bond would both be compared to a benchmark 25-year Treasury security. Such a comparison makes little sense, because the cash flow characteristics of the two corporate bonds will not be the same as that of the benchmark Treasury. The proper way to compare non-Treasury bonds of the same maturity but with different coupon rates is to compare them with a portfolio of Treasury securities that have the same cash flow. For example, consider the 8.8% 25-year corporate bond selling for 87.0798. The cash flow per $100 par value for this corporate bond, assuming that interest rates do not change (i.e., assuming static interest rates), is 49 six-month payments of $4.40 and a payment in 25 years (50 six-month periods) of $104.40. A portfolio that will replicate this cash flow would include 50 zero-coupon Treasury securities with maturities coinciding with the amount and timing of the cash flows of the corporate bond. The option-adjusted spread (OAS) is a spread over the spot rate curve or benchmark used in the valuation. In the case of the binomial method, the OAS is a spread over the binomial interest rate tree. Some market participants construct the binomial interest-rate tree using the Treasury spot rates. In this case the OAS reflects the richness or cheapness of the security, if any, plus a credit spread. Other market participants construct the binomial interest-rate tree from the issuer’s spot rate curve. In this case the credit risk is already incorporated into the analysis, and the OAS therefore reflects the richness or cheapness of a security. Therefore, it is critical to know the on-the-run issues that the modeler used to construct the binomial interest-rate tree. 17. What is the effect of greater expected interest-rate volatility on the option-adjusted spread of a security? As with the theoretical value of a security, the option-adjusted spread is affected by the assumed interest rate volatility. The higher (lower) the expected interest rate volatility, then the lower (higher) the OAS. 18. The following excerpt is taken from an article titled “Call Provisions Drop Off” that appeared in the January 27, 1992, issue of BondWeek, p. 2: “Issuance of callable long-term bonds dropped off further last year as interest rates fell, removing the incentive for many issuers to pay extra for the provision, said Street capital market officials. . . . The shift toward noncallable issues, which began in the late 1980s, reflects the secular trend of investors unwilling to bear prepayment risk and possibly the cyclical trend that corporations believe that interest rates have hit all time lows.” Answer the below questions. (a) What “incentive” is this article referring to in the first sentence of the excerpt? The call option embedded in a callable bond becomes more valuable when issuers expect interest rates to fall. The likelihood of this occurring is greater if interest rates are believed to be high. However, since interest rates have fallen (and even bottomed out), issuers have lost their incentive to issue callable bonds. (b) Why would issuers not be willing to pay for this incentive if they feel that interest rates will continue to decline? If issuers feel that interest rates will continue to decline, then they are willing to pay a higher call premium and/or issue callable bonds with a higher coupon rate. However, if they believe the decline is limited due to prior declines, then they would not be willing to pay much for the incentive to issue callable bonds. That is, they would not be willing to issue callable bonds with too high of a premium or with too high of a coupon rate. Also, to issue callable bonds, investors must demand them and be willing to take accept what issuers feel are reasonable terms. Issuers will not issue callable bonds if they cannot get the terms they want. 19. The following excerpt is taken from an article titled “Eagle Eyes High-Coupon Callable Corporates” that appeared in the January 20, 1992, issue of Bond Week, p. 7: “If the bond market rallies further, Eagle Asset Management may take profits, trading $8 million of seven- to 10-year Treasuries for high-coupon single-A industrials that are callable in two to four years according to Joseph Blanton, Senior V.P. He thinks a further rally is unlikely, however. . . . The corporates have a 95% chance of being called in two to four years and are treated as two- to four-year paper in calculating the duration of the portfolio, Blanton said.” Answer the below questions. (a) Why is modified duration an inappropriate measure for a high-coupon callable bond? Money managers want to know the price sensitivity of a bond when interest rates change. Modified duration is a measure of the sensitivity of a bond’s price to interest-rate changes, assuming that the expected cash flow does not change with interest rates. Consequently, modified duration may not be an appropriate measure for bonds with embedded options because the expected cash flows change as interest rates change. For example, when interest rates fall, the expected cash flow for a callable bond may change. In the case of a put able bond, a rise in interest rates may change the expected cash flow. (b) What would be a better measure than modified duration? Although modified duration may be inappropriate as a measure of a bond’s price sensitivity to interest rate changes, there is a duration measure that is more appropriate for bonds with embedded options. Because duration measures price responsiveness to changes in interest rates, the duration for a bond with an embedded option can be estimated by letting interest rates change by a small number of basis points above and below the prevailing yield, and seeing how the prices change. In general, the duration for any bond can be approximated as follows: duration = . where P_ = price if yield is decreased by x basis points, P+ = price if yield is increased by x basis points, P0 = initial price (per $100 of par value), and Δ y (or dy) = change in rate used to calculate price (x basis points in decimal form). We can show how the application of this formula to an option-free bond gives the modified duration because the cash flows do not change when yields change. When the approximate duration formula is applied to a bond with an embedded option, the new prices at the higher and lower yield levels should reflect the value from the valuation model. Duration calculated in this way is called effective duration or option-adjusted duration. (c) Why would the replacement of 10-year Treasuries with high-coupon callable bonds reduce the portfolio’s duration? The replacement of 10-year Treasuries with high-coupon callable bonds reduces the portfolio’s duration because the effective duration for callable bonds can be well below the modified duration. More details are given below on the relationships among duration, modified duration, and effective duration. Duration is a generic concept that indicates the responsiveness of a bond to a change in interest rates. Modified duration is a duration measure in which the cash flow is not assumed to change when interest rates change. In contrast, effective duration measures the responsiveness of a bond’s price taking into account that the expected cash flow will change as interest rates change due to the embedded option. The difference between modified duration and effective duration for a bond with an embedded option can be quite dramatic. For example, a callable bond could have a modified duration of 5 and an effective duration of 3. For certain highly leveraged mortgage-backed securities, the bond could have a modified duration of 7 and an effective duration of 50! CHAPTER 18 ANALYSIS OF RESIDENTIAL MORTGAGE-BACKED SECURITIES CHAPTER SUMMARY There are two approaches to the analysis of residential mortgage-backed securities including pass-throughs, collateralized mortgage obligations (CMOs), and stripped mortgage-backed securities. They are: (1) the static cash flow yield methodology, and (2) the Monte Carlo simulation methodology. In this chapter we review the static cash flow yield methodology and its limitations and then focus on the Monte Carlo simulation methodology. The framework provided in this chapter applies to agency and nonagency residential mortgage-backed securities. STATIC CASH FLOW YIELD METHODOLOGY The static cash flow yield methodology begins with the computation of the cash flow yield measure used for pass-throughs and based on some prepayment assumption. Vector Analysis One practice that market participants use to overcome the drawback of the PSA benchmark is to assume that the PSA speed can change over time. This technique is referred to as vector analysis. A vector is simply a set of numbers. In the case of prepayments, it is a vector of prepayment speeds. Vector analysis is particularly useful for CMO tranches that are dramatically affected by the initial slowing down of prepayments, and then speeding up of prepayments, or vice versa. Limitations of the Cash Flow Yield The same shortcomings found in the yield to maturity approach are also present in application of the cash flow yield measure: (i) the projected cash flows are assumed to be reinvested at the cash flow yield, and (ii) the RMBS is assumed to be held until the final payout based on some prepayment assumption. The cash flow yield is dependent on realization of the projected cash flow according to some prepayment rate. If actual prepayments vary from the prepayment rate assumed, the cash flow yield will not be realized. Yield Spread to Treasuries The yield for a RMBS will depend on the actual prepayment experience of the mortgages in the pool. Nevertheless, the convention in all fixed-income markets is to measure the yield on a non-Treasury security to that of a “comparable” Treasury security. The repayment of principal over time makes it inappropriate to compare the yield of a RMBS to a Treasury of a stated maturity. Instead, market participants have used two measures: Macaulay duration and average life. Static Spread The static spread is the yield spread in a static scenario (i.e., no volatility of interest rates) of the bond over the entire theoretical Treasury spot rate curve, not a single point on the Treasury yield curve. In a relatively flat interest-rate environment, the difference between the traditional yield spread and the static spread will be small. There are two ways to compute the static spread for RMBS. One way is to use today’s yield curve to discount future cash flows and keep the mortgage refinancing rate fixed at today’s mortgage rate. Use of this approach to calculate the static spread recognizes different prices today of dollars to be delivered at future dates. The second way to calculate the static spread allows the mortgage rate to go up the curve as implied by the forward interest rates. This procedure is sometimes called the zero-volatility OAS. In this case a prepayment model is needed to determine the vector of future prepayment rates implied by the vector of future refinancing rates. Effective Duration Modified duration is a measure of the sensitivity of a bond’s price to interest-rate changes, assuming that the expected cash flow does not change with interest rates. Modified duration is consequently not an appropriate measure for mortgage-backed securities, because prepayments influence the projected cash flow as interest rates change. When interest rates fall (rise), prepayments are expected to rise (fall). As a result, when interest rates fall (rise), duration may decrease (increase) rather than increase (decrease). This property is referred to as negative convexity. Negative convexity has the same impact on the price performance of a RMBS as it does on the performance of a callable bond. When interest rates decline, a bond with an embedded call option, which is what a RMBS is, will not perform as well as an option-free bond. Although modified duration is an inappropriate measure of interest-rate sensitivity, there is a way to allow for changing prepayment rates on cash flow as interest rates change. This is achieved by calculating the effective duration, which allows for changing cash flow when interest rates change. To illustrate the effective duration calculation, consider tranche data: P_ = 102.1875; P+ = 98.4063; P0 (initial price) = 100.2813; Δy = 0.0025. Substituting into the duration formula yields modified duration (with P_ = 102.1875) = = = 7.54. To further illustrate the effective duration calculation, consider tranche data: P_ = 101.9063 (at 200 PSA; basis points decrease); P+ = 98.3438 (at 150 PSA; basis point increase); P0 = 100.2813; Δy = 0.0025. Substituting into the duration formula yields effective duration (P_ = 101.9063) = = = 7.11. Notice that the effective duration (which allows for changing cash flow when interest rates change) is less than the modified duration. The divergence between modified duration and effective duration is much more dramatic for bond classes trading at a substantial discount from par or at a substantial premium over par. Effective Convexity To illustrate the convexity formula, consider the above tranche data. The standard convexity is approximated as follows: = = 24.930. The effective convexity (which allows for changing cash flow when interest rates change so that P_ = 101.9063) is = = 249.299. The standard convexity indicates positive convexity, whereas the effective convexity indicates they have negative convexity. The difference is even more dramatic for bonds not trading near par. For a PO created from a tranche, the standard convexity can be close to zero whereas the effective convexity can be very large. For example, if the effective convexity is 2,000, and the yields change by 100 basis points, the percentage change in price due to convexity is: (effective convexity)(Δy)2 = 2,000(0.01)2 = 0.2000 or 20.00%. Prepayment Sensitivity Measure The value of a RMBS will depend on prepayments. To assess prepayment sensitivity, market participants have used the following measure: the basis point change in the price of an RMBS for a 1% increase in prepayments. Specifically, we have: prepayment sensitivity = (Ps – P0)100 where Ps = price (per $100 par value) assuming a 1% increase in prepayment speed and P0 = initial price (per $100 par value) at assumed prepayment speed. Notice that a security that is adversely affected by an increase in prepayment speeds will have a negative prepayment sensitivity while a security that benefits from an increase in prepayment speed will have a positive prepayment sensitivity. MONTE CARLO SIMULATION METHODOLOGY For some fixed-income securities and derivative instruments, the periodic cash flows are path dependent. This means that the cash flows received in one period are determined not only by the current and future interest-rate levels but also by the path that interest rates took to get to the current level. Pools of pass-throughs are used as collateral for the creation of collateralized mortgage obligations (CMOs). Consequently, for CMOs there are typically two sources of path dependency in a CMO tranche’s cash flows. First, the collateral prepayments are path dependent. Second, the cash flow to be received in the current month by a CMO tranche depends on the outstanding balances of the other tranches in the deal. Thus we need the history of prepayments to calculate these balances. Because of the path dependency of a RMBS’s cash flow, the Monte Carlo simulation method is used for these securities rather than the binomial method. Conceptually, the valuation of pass-throughs using the Monte Carlo method is simple. In practice, however, it is very complex. The simulation involves generating a set of cash flows based on simulated future mortgage refinancing rates, which in turn imply simulated prepayment rates. Valuation modeling for CMOs is similar to valuation modeling for pass-throughs, although the difficulties are amplified because the issuer has sliced and diced both the prepayment risk and the interest-rate risk into smaller pieces called tranches. The sensitivity of the pass-throughs composing the collateral to these two risks is not transmitted equally to every tranche. Some of the tranches wind up more sensitive to prepayment risk and interest-rate risk than the collateral, whereas some of them are much less sensitive. Using Simulation to Generate Interest-Rate Paths and Cash Flows The typical model that Wall Street firms and commercial vendors use to generate random interest-rate paths takes as inputs today’s term structure of interest rates and a volatility assumption. The term structure of interest rates is the theoretical spot rate (or zero-coupon) curve implied by today’s Treasury securities. The volatility assumption determines the dispersion of future interest rates in the simulation. The simulations should be normalized so that the average simulated price of a zero-coupon Treasury bond equals today’s actual price. The simulation works by generating many scenarios of future interest-rate paths. In each month of the scenario, a monthly interest rate and a mortgage refinancing rate are generated. The monthly interest rates are used to discount the projected cash flows in the scenario. The mortgage refinancing rate is needed to determine the cash flow because it represents the opportunity cost the mortgagor is facing at that time. Prepayments are projected by feeding the refinancing rate and loan characteristics, such as age, into a prepayment model. Given the projected prepayments (voluntary and involuntary), the cash flow along an interest-rate path can be determined. Calculating the Present Value for a Scenario Interest-Rate Path Given the cash flow on an interest-rate path, its present value can be calculated. The discount rate for determining the present value is the simulated spot rate for each month on the interest-rate path plus an appropriate spread. The spot rate on a path can be determined from the simulated future monthly rates. The relationship that holds between the simulated spot rate for month T on path n and the simulated future one-month rates is zT(n) = where zT(n) = simulated spot rate for month T on path n, and fj(n) = simulated future one-month rate for month j on path n. The present value of the cash flow for month T on interest-rate path n discounted at the simulated spot rate for month T plus some spread is PV[CT(n)] = where PV[CT(n)] = present value of cash flow for month T on path n, CT(n) = cash flow for month T on path n, zT(n) = spot rate for month T on path n, and K = appropriate risk-adjusted spread. The present value for path n is the sum of the present value of the cash flow for each month on path n. Determining the Theoretical Value The present value of a given interest-rate path can be thought of as the theoretical value of a pass-through assuming that the path was actually realized. The theoretical value of the pass-through can be determined by calculating the average of the theoretical value of all the interest-rate paths. Looking at the Distribution of the Path Values The theoretical value generated by the Monte Carlo simulation method is the average of the path values. There is valuable information in the distribution of the path values. If there is substantial dispersion of the path values then the investor is warned about the potential variability of the model’s value. Simulated Average Life The average life reported in a Monte Carlo analysis is the average of the average lives along the interest-rate paths. The greater the range and standard deviation of the average life, the more uncertainty there is about the security’s average life. Option-Adjusted Spread The option-adjusted spread is a measure of the yield spread that can be used to convert dollar differences between value and price. It represents a spread over the issuer’s spot rate curve or benchmark. In the Monte Carlo model, the OAS is the spread K that when added to all the spot rates on all interest-rate paths will make the average present value of the paths equal to the observed market price (plus accrued interest). Option Cost The implied cost of the option embedded in any RMBS can be obtained by calculating the difference between the OAS at the assumed volatility of interest rates and the static spread: option cost = static spread – option-adjusted spread. Effective Duration and Convexity Effective duration and effective convexity can be calculated using the Monte Carlo method as follows. First, the bond’s OAS is found using the current term structure of interest rates. Next, the bond is repriced holding OAS constant but shifting the term structure. Two shifts are used to get the prices needed to apply the effective duration and effective convexity formulas: (i) yields are increased, and (ii) yields are decreased. Selecting the Number of Interest-Rate Paths Let’s now address the question of the number of scenario paths or repetitions, N, needed to value a RMBS. A typical OAS run will be done for 512 to 1,024 interest-rate paths. The number of interest-rate paths determines how “good” the estimate is, not relative to the truth but relative to the model. The more paths, the more average spread tends to settle down. It is a statistical sampling problem. Most Monte Carlo simulation models employ some form of variance reduction to cut down on the number of sample paths necessary to get a good statistical sample. Variance reduction techniques allow us to obtain price estimates within a tick. Limitations of the Option-Adjusted Spread Measures The OAS is a product of the valuation model. The valuation model may be poorly constructed because it fails to capture the true factors that affect the value of particular securities. In Monte Carlo simulation the interest-rate paths must be adjusted so that on-the-run Treasuries are valued properly. Another problem with the OAS is that it assumes a constant OAS for each interest rate path and over time for a given interest-rate path. Finally, the OAS is dependent on the volatility assumption, the prepayment assumption in the case of RMBS, and the rules for refunding in the case of corporate bonds. Illustration We can use a plain vanilla deal (i.e., relatively simple derivative instrument with standard features) to show how CMOs can be analyzed using the Monte Carlo simulation method. Using the OAS from the Monte Carlo simulation methodology, a fair conclusion can be made about this simple plain vanilla structure: what you see is what you get. In general, however, a money manager willing to extend duration gets paid for that risk. TOTAL RETURN ANALYSIS Neither the static cash flow methodology nor the Monte Carlo simulation methodology will tell a money manager whether investment objectives can be satisfied. The performance evaluation of an individual RMBS requires specification of an investment horizon, whose length for most financial institutions is dictated by the nature of its liabilities. The measure that should be used to assess the performance of a security or a portfolio over some investment horizon is the total return. Horizon Price for CMO Tranches The most difficult part of estimating total return is projecting the price at the horizon date. In the case of a CMO tranche the price depends on the characteristics of the tranche and the spread to Treasuries at the termination date. The key determinants are the “quality” of the tranche, its average life (or duration), and its convexity. Quality refers to the type of CMO tranche. Option-Adjusted Spread Total Return The total return and OAS frameworks can be combined to determine the projected price at the horizon date. At the end of the investment horizon, it is necessary to specify how the OAS is expected to change. The horizon price can be “backed out” of the Monte Carlo simulation model. Assumptions about the OAS value at the investment horizon reflect the expectations of the money manager. It is common to assume that the OAS at the horizon date will be the same as the OAS at the time of purchase. A total return calculated using this assumption is sometimes referred to as a constant-OAS total return. KEY POINTS • There are two methodologies commonly used to analyze all RMBS (agency and nonagency): cash flow yield methodology and Monte Carlo simulation methodology. • The cash flow yield is the interest rate that will make the present value of the projected cash flow from an RMBS equal to its market price. The cash flow yield assumes that (1) all the cash flows can be reinvested at a rate equal to the cash flow yield, (2) the RMBS is held to the maturity date, and (3) the prepayment speed used to project the cash flow will be realized. In addition, the cash flow yield methodology fails to recognize that future interest-rate changes will affect the cash flow. • Modified duration is not a good measure of price volatility for RMBS because it assumes that the cash flow does not change as yield changes. Effective duration does take into consideration how yield changes will affect prepayments and therefore cash flow. • An RMBS is a security whose cash flow is path dependent. This means that cash flow received in one period is determined not only by the current and future interest-rate levels, but also by the path that interest rates took to get to the current level. • A methodology used to analyze path-dependent cash flow securities is the Monte Carlo simulation. This methodology involves randomly generating many scenarios of future interest-rate paths, where the interest-rate paths are generated based on some volatility assumption for interest rates. The random paths of interest rates should be generated from an arbitrage-free model of the future term structure of interest rates. The Monte Carlo simulation methodology applied to RMBS involves randomly generating a set of cash flows based on simulated future mortgage refinancing rates. • The theoretical value of a security on any interest-rate path is the present value of the cash flow on that path, where the spot rates are those on the corresponding interest rate path. The theoretical value of a security is the average of the theoretical values over all the interest-rate paths. Information about the distribution of the path values is useful in understanding the variability around the theoretical value. • The average life reported is the average of the average lives from all the interest-rate paths and information about the distribution of the average life is useful. • In the Monte Carlo simulation methodology, the option-adjusted spread is the spread that when added to all the spot rates on all interest-rate paths will make the average present value of the paths equal to the observed market price (plus accrued interest). • The effective duration and effective convexity are calculated using the Monte Carlo simulation methodology by holding the OAS constant and shifting the term structure up and down. • Total return is the correct measure for assessing the potential performance of CMO tranches over a specified investment horizon. • The static cash flow yield or Monte Carlo simulation methodology can be incorporated into a total return framework to calculate the mortgage-backed security’s price at the horizon date. • Scenario analysis is one way to evaluate the risk associated with investing in an RMBS. ANSWERS TO QUESTIONS FOR CHAPTER 18 (Questions are in bold print followed by answers.) 1. Suppose you are told that the cash flow yield of a pass-through security is 9% and that you are seeking to invest in a security with a yield greater than 8.8%. Answer the below questions. (a) What additional information would you need to know before you might invest in this pass-through security? To determine your chances of actually getting an 8.8% return, you would want to know the types of securities backing the pass-through, the safety of the cash flows, and the expected volatility of the cash flows. More details are given below. A mortgage pass-through security consists of a set of marketable shares in a portfolio of mortgages for which investors receive monthly payments of both interest and principal. Normally the package is secured by credit insurance so that investors are protected from the credit risks of the individual mortgages in the portfolio. However, no protection is provided against the cash flow and return volatility associated with unanticipated principal prepayments, which typically occur when interest rates drop and homeowners refinance their mortgages. A conventional pass-through is a mortgage pass-through security that is not guaranteed by a government agency. If applicable, you would want to know the prepayment rate for a particular tranche formed from the pass-through security as well as any stipulated guarantees in terms of interest and principal payments. One should note that the greater the discount assumed to be paid for a tranche, the more a tranche will benefit from faster prepayments. The converse is true for a tranche for which a premium is paid. The faster the prepayments, the lower the cash flow yields. The above factors along with an expected reinvestment rate will give you an idea as to whether or not the 9% return will be realized. (b) What are the limitations of the cash flow yield for assessing the potential return from investing in a RMBS? The limitations for the cash flow yield in a RMBS are like the yield to maturity for a bond where it is assumed that the coupon payments can be reinvested at a rate equal to the yield to maturity and the bond is held to maturity. These shortcomings are equally present in application of the cash flow yield measure: (i) the projected cash flows are assumed to be reinvested at the cash flow yield, and (ii) the RMBS is assumed to be held until the final payout based on some prepayment assumption. The importance of reinvestment risk, the risk that the cash flow will be reinvested at a rate less than the cash flow yield, is particularly important for many RMBS because payments come as frequently as every month. The cash flow yield, moreover, is dependent on realization of the projected cash flow according to some prepayment rate. If actual prepayments vary from the prepayment rate assumed, the cash flow yield will not be realized. 2. Using the cash flow yield methodology, a spread is calculated over a comparable Treasury security. How is a comparable Treasury determined? The repayment of principal over time makes it inappropriate to compare the yield of a RMBS to a Treasury of a stated maturity. Instead, market participants have used two measures: Macaulay duration and average life. The practice of spreading the yield to the average life on the interpolated Treasury yield curve is improper for an amortizing bond even in the absence of interest-rate volatility. What should be done instead is to calculate what is called the static spread. This is the yield spread in a static scenario (i.e., no volatility of interest rates) of the bond over the entire theoretical Treasury spot rate curve (and not a single point on the Treasury yield curve). 3. What is vector analysis? One practice that market participants use to overcome the drawback of the PSA benchmark is to assume that the PSA speed can change over time. This technique to deal with this drawback is referred to as vector analysis. A vector is simply a set of numbers. In the case of prepayments, it is a vector of prepayment speeds. Vector analysis is particularly useful for CMO tranches that are dramatically affected by the initial slowing down of prepayments, and then speeding up of prepayments, or vice versa. 4. In the calculation of effective duration and effective convexity, why is a prepayment model needed? A prepayment model will tell us how prepayments cause the projected cash flows to change as interest rates change. This is needed because to properly compute effective duration and effective convexity we must account for cash flows changing with interest rates. 5. The following excerpt is taken from an article titled “Fidelity Eyes $250 Million Move into Premium PACs and I-Os” that appeared in the January 27, 1992, issue of BondWeek, pp. 1 and 21: “Three Fidelity investment mortgage funds are considering investing this quarter a total of $250 million in premium planned amortization classes of collateralized mortgage obligations and some interest-only strips, said Jim Wolfson, portfolio manager … Wolfson … will look mainly at PACs backed by 9-10% Federal Home Loan Mortgage Corp. and Federal National Mortgage Association pass-throughs. These have higher option-adjusted spreads than regular agency pass-throughs, or similar premium Government National Mortgage Association-backed, PACs, he said. He expects I-Os will start to perform better as prepayments start to slow later in this quarter. The higher yields on I-Os and premium PACs compensate for their higher prepayment risk, said Wolfson. “You get paid in yield to take on negative convexity,” he said. He does not feel prepayments will accelerate …” Answer the below questions. (a) Why would premium PACs and interest-only strips offer higher yields if the market expects that prepayments will accelerate or are highly uncertain? Prepayments will be expected to accelerate if interest rates are expected to decline or if there is a greater possibility of decline due to general uncertainty as to which way rates will change. For such a situation, investors would expect to deal with greater reinvestment rate risk. To compensate for this risk, investors would have to be given higher rates of return for investing in securities that will be retired earlier than desired. More details on how this affects both planned amortization tranches (PAC tranches) and interest-only strips (I-Os or just IOs) are given below. PAC tranches can reduce prepayment risk in a manner desired by an investor’s preference. However, despite the redistribution of prepayment risk with sequential-pay and accrual collateralized mortgage obligations (CMOs), there is still considerable prepayment risk. That is, there is still considerable average life variability for a given tranche. This problem is mitigated by the PAC tranche. The greater predictability of the cash flow for PAC bonds occurs because there is a principal repayment schedule that must be satisfied. PAC bondholders have priority over all other classes in the CMO issue in receiving principal payments from the underlying collateral. The greater certainty of the cash flow for the PAC bonds comes at the expense of the non-PAC classes, called support or companion bonds. It is these bonds that absorb the prepayment risk. Because PAC bonds have protection against both extension risk and contraction risk, they are said to provide two-sided prepayment protection. In early 1987, stripped MBS began to be issued where all the interest is allocated to one class (the IO class) and the entire principal to the other class (the PO class). The IO class receives no principal payments. IOs and POs are referred to as mortgage strips. The PO security is purchased at a substantial discount from par value. The yield an investor will realize depends on the speed at which prepayments are made. The faster the prepayments, the higher the yield the investor will realize. When an IO is purchased, there is no par value. In contrast to the PO investor, the IO investor wants prepayments to be slow. The reason is that the IO investor receives only interest on the amount of the principal outstanding. As prepayments are made, the outstanding principal declines, and less dollar interest is received. In fact, if prepayments are too fast, the IO investor may not recover the amount paid for the IO. (b) What does Wolfson mean when he says: “You get paid in yield to take on negative convexity”? Negative convexity has the same impact on the price performance of a RMBS as it does on the performance of a callable bond. When interest rates decline, a bond with an embedded call option, which is what a RMBS is, will not perform as well as an option-free bond. Thus, Wolfson is pointing out that investors will require higher yields for such investments or equivalently: “You get paid in yield to take on negative convexity.” (c) What measure is Wolfson using to assess the risks associated with prepayments? Because Wolfson is looking at securities that can experience a decline when prepayment increases, Wolfson wants a measure that captures this. Thus, Wolfson appears to be cognizant of the prepayment sensitivity measure. More details on this measure are supplied below. The value of a RMBS will depend on prepayments. To assess prepayment sensitivity, market participants have used the prepayment sensitivity measure that determines the basis point change in the price of an RMBS for a 1% increase in prepayments. Specifically, this measure is defined as prepayment sensitivity = (Ps – P0)100 where Ps = price (per $100 par value) assuming a 1% increase in prepayment speed, and P0 = initial price (per $100 par value) at assumed prepayment speed. For example, suppose that for some RMBS at 300 PSA the price is P0 = 106.10. A 1% increase in the PSA prepayment rate means that PSA increases from 300 PSA to 303 PSA. Suppose that at 303 PSA the price is recomputed using a valuation model to be P0 = 106.01. Therefore, prepayment sensitivity = (106.01 – 106.10)100 = 9. Notice that a security that is adversely affected by an increase in prepayment speeds will have a negative prepayment sensitivity while a security that benefits from an increase in prepayment speed will have a positive prepayment sensitivity. 6. In an article titled “CUNA Mutual Looks for Noncallable Corporates” that appeared in the November 4, 1991, issue of Bond Week, p. 6, Joe Goglia, a portfolio manager for CUNA Mutual Insurance Group, stated that he invests in “planned amortization class tranches, which have less exposure to prepayment risk and are more positively convex than other mortgage-backeds.” Is this true? As seen below there are a lot of factors to consider before we assume that a PAC tranche will absolutely have less exposure to prepayment risk and will be more positively convex that other mortgage-backed securities. The creation of a mortgage-backed security cannot make prepayment risk disappear. This is true for both a pass-through and a CMO. Thus the reduction in prepayment risk (both extension risk and contraction risk) that a PAC offers must come from somewhere. It comes from the support bonds. If the support bonds are paid off quickly because of faster-than-expected prepayments, there is no longer any protection for the PAC bonds. Planned amortization class tranches are structured to have less exposure to prepayment risk. For a security that is option-free and displays positive convexity, the price appreciation will be greater than the price depreciation for a large change in yield. Negative convexity means that the price appreciation will be less than the price depreciation for a large change in yield. Generally, the market will take the greater convexity bonds into account in pricing them. How much should the market want investors to pay up for convexity? If investors expect that market yields will change by very little—that is, they expect low interest rate volatility—investors should not be willing to pay much for convexity. In fact, if the market prices convexity high, investors with expectations of low interest rate volatility will probably want to “sell convexity.” In the case of a CMO tranche the price depends on the characteristics of the tranche and the spread to Treasuries at the termination date. The key determinants are the “quality” of the tranche, its average life (or duration), and its convexity. Quality refers to the type of CMO tranche. Consider, for example, that an investor can purchase a CMO tranche that is a PAC bond but as a result of projected prepayments could become a sequential-pay tranche. As another example, suppose that a PAC bond is the longest-average-life tranche in a reverse PAC structure. Projected prepayments in this case might occur in an amount to change the class from a long-average-life PAC tranche to a support tranche. The converse is that the quality of a tranche may improve as well as deteriorate. For example, the effective collar for a PAC tranche could widen at the horizon date when prepayment circumstances increase the par amount of support tranches outstanding as a proportion of the deal. For a tranche, a standard convexity can indicate positive convexity, whereas the effective convexity can indicates negative convexity. The difference is even more dramatic for securities not trading near par. For a PO created from a tranche, the standard convexity can be close to zero whereas the effective convexity can be very large. 7. What is a path-dependent cash flow security? For some fixed-income securities and derivative instruments, the periodic cash flows are path dependent. This means that the cash flows received in one period are determined not only by the current and future interest-rate levels but also by the path that interest rates took to get to the current level. 8. Why is a pass-through security a path-dependent cash flow security? In the case of mortgage pass-through securities, prepayments are path dependent because this month’s prepayment rate depends on whether there have been prior opportunities to refinance since the underlying mortgages were originated. Unlike mortgage loans, the decision as to whether a corporate issuer will elect to refund an issue when the current rate is below the issue’s coupon rate is not dependent on how rates evolved over time to the current level. Moreover, in the case of adjustable-rate pass-throughs (ARMs), prepayments are not only path dependent but the periodic coupon rate depends on the history of the reference rate upon which the coupon rate is determined. This is because ARMs have periodic caps and floors as well as a lifetime cap and floor. 9. Give two reasons why a CMO tranche is a path-dependent cash flow security. Pools of pass-throughs are used as collateral for the creation of collateralized mortgage obligations (CMOs). Consequently, for CMOs there are typically two sources of path dependency in a CMO tranche’s cash flows. First, the collateral prepayments are path dependent. Second, the cash flow to be received in the current month by a CMO tranche depends on the outstanding balances of the other tranches in the deal. Thus we need the history of prepayments to calculate these balances. 10. Explain how, given the cash flow on the simulated interest-rate paths, the theoretical value of a RMBS is determined. Given the cash flow on an interest-rate path, its present value can be calculated. The discount rate for determining the present value is the simulated spot rate for each month on the interest-rate path plus an appropriate spread. The spot rate on a path can be determined from the simulated future monthly rates. The relationship that holds between the simulated spot rate for month T on path n and the simulated future one-month rates is zT(n) = where zT(n) = simulated spot rate for month T on path n, and fj(n) = simulated future one-month rate for month j on path n. The interest-rate path for the simulated future one-month rates can be converted to the interest-rate path for the simulated monthly spot rates. Thus, the present value of the cash flow for month T on interest-rate path n discounted at the simulated spot rate for month T plus some spread is: PV[CT(n)] = where PV[CT(n)] = present value of cash flow for month T on path n, CT(n) = cash flow for month T on path n, zT(n) = spot rate for month T on path n, and K = appropriate risk-adjusted spread. The present value for path n is the sum of the present value of the cash flow for each month on path n. We have (assuming a maturity of 360 months that consists of a path of 360 simulated interest-rate path scenarios): PV[path(n)] = where PV[path(n)] is the present value of interest-rate path n. The present value of a given interest-rate path can be thought of as the theoretical value of a pass-through if that path was actually realized. The theoretical value of the pass-through can be determined by calculating the average of the theoretical value of all the interest-rate paths. That is, the theoretical value is equal to theoretical value = . This procedure for valuing a pass-through is also followed for a CMO tranche. The cash flow for each month on each interest-rate path is found according to the principal repayment and interest distribution rules of the deal. 11. Explain how, given the cash flow on the simulated interest-rate paths, the average life of a RMBS is determined. Given the cash flow on the simulated interest-rate paths, the average life can be determined for each path. The average life reported in a Monte Carlo analysis is the average of the average lives along the interest-rate paths. As with the theoretical value, additional information is conveyed by the distribution of the average life. The greater the range and standard deviation of the average life, the more uncertainty there is about the security’s average life. 12. Suppose that a support bond is being analyzed using the Monte Carlo simulation methodology. The theoretical value using 1,500 interest-rate paths is 88. The range for the path present values is a low of 50 and a high of 115. The standard deviation is 15 points. How much confidence would you place on the theoretical value of 88? The probability of actually achieving a theoretical value of 88 is not very likely. However, we can put a probability of occurrence for a range of values surrounding 88. For example, if we can assume a distribution resembling a normal distribution, then we know there is about a two-thirds probability that we will find a range of values between 73 and 103. However, given the range of low and high values it appears the distribution is not perfectly normal but skewed to the left, thus taking on more values below 88 than above 88. 13. In a well-protected PAC structure, what would you expect the distribution of the path present values and average lives to be compared to a support bond from the same CMO structure? PAC tranches are structured to meet the designs of the investor who prefers a reduction is risk. The greater predictability of the cash flow for PAC bonds occurs because there is a principal repayment schedule that must be satisfied. PAC bondholders have priority over all other classes in the CMO issue in receiving principal payments from the underlying collateral. The greater certainty of the cash flow for the PAC bonds comes at the expense of the non-PAC classes, called support or companion bonds. It is these bonds that absorb the prepayment risk. Because PAC bonds have protection against both extension risk and contraction risk, they are said to provide two-sided prepayment protection. In conclusion, with less risk and uncertainty in a well-protected PAC structure, we would expect the distribution of the path present values and average lives to be more stable compared to a support bond from the same CMO structure. 14. Suppose that the following values for a RMBS are correct for each assumption: PSA Assumption Value of Security 192 112.10 194 111.80 200 111.20 202 111.05 210 110.70 Assuming that the value of the security in the market is 111.20 based on 200 PSA. What is the prepayment sensitivity of this security? To assess prepayment sensitivity, market participants have used the following measure. This measure determines the basis point change in the price of an RMBS for a 1% increase in prepayments. We have: prepayment sensitivity = (Ps – P0)100 where Ps = price (per $100 par value) assuming a 1% increase in prepayment speed and P0 = initial price (per $100 par value) at assumed prepayment speed. In our problem, we find a 1% increase from a PSA assumption from 200 to 202, i.e., (202 – 200) / 200 = 2 / 200 = 0.01 or 1.00%. For these PSA assumptions, the initial price (per $100 par value) is P0 = 111.20 and the price assuming a 1% increase in prepayment speed is Ps = 111.05. Inserting in these values in our formula gives: prepayment sensitivity = (Ps – P0)100 = (111.05 – 111.20)100 = 15. The security in our problem has a negative prepayment sensitivity. This indicates that it is adversely affected by an increase in prepayment. 15. An analysis of a CMO structure using the Monte Carlo method indicated the following, assuming 12% volatility: OAS (basis points) Static Spread (basis points) Collateral 80 120 Tranche PAC I A 40 60 PAC I B 55 80 PAC I C 65 95 PAC II 95 125 Support 75 250 (a) Calculate the option cost for each tranche. The implied cost of the option embedded in any RMBS can be obtained by calculating the difference between the OAS at the assumed volatility of interest rates and the static spread. We use the below formula: option cost = static spread – option-adjusted spread. The reason that the option cost is measured in this way is as follows. In an environment of no interest-rate changes, the investor would earn the static spread. When future interest rates are uncertain, the spread is less, however, because of the homeowner’s option to prepay; the OAS reflects the spread after adjusting for this option. Therefore, the option cost is the difference between the spread that would be earned in a static interest-rate environment (the static spread) and the spread after adjusting for the homeowner’s option. Below we compute the option cost for each tranche. Tranche PAC I A: option cost = 60 basis points – 40 basis = 20 basis points. Tranche PAC I B: option cost = 80 basis points – 55 basis = 25 basis points. Tranche PAC I C: option cost = 95 basis points – 65 basis = 30 basis points. Tranche PAC II: option cost = 125 basis points – 95 basis = 30 basis points. Support: option cost = 250 basis points – 75 basis = 175 basis points. In general, a tranche’s option cost is more stable than its OAS in the face of market movements. This interesting feature is useful in reducing the computational costs of calculating the OAS as the market moves. For small market moves, the OAS of a tranche may be approximated by recalculating the static spread (which is relatively cheap and easy to calculate) and subtracting its option cost. (b) Which tranche is clearly too rich? The support tranche is rich relative to Treasuries. We might add that a typical OAS run will be done for 512 to 1,024 interest-rate scenario paths to value a RMBS. The scenarios generated using the simulation method look very realistic and, furthermore, reproduce today’s Treasury curve. By employing this technique, the money manager is effectively saying that Treasuries are fairly priced today and that the objective is to determine whether a specific RMBS is rich or cheap relative to Treasuries. The number of interest-rate paths determines how “good” the estimate is, not relative to the truth but relative to the model. The more paths, the more average spread tends to settle down. (c) What would happen to the static spread for each tranche if a 15% volatility is assumed? At the higher level of assumed interest-rate volatility of 15%, the static spread would fall because of the homeowner’s option to prepay. This would negatively affect the collateral with its loss distributed among the tranches with tranches with longer duration experiencing greater losses. Tranche Z and any residual tranche would be least affected. (d) What would happen to the OAS for each tranche if a 15% volatility is assumed? The OAS would fall for each tranche causing a negative change in price per dollar. As stated in part (c), this would negatively affect the collateral with its loss distributed among the tranches with tranches. 16. Why would the option-adjusted spread vary across dealer firms? As discussed below, the option-adjusted spread will vary across dealer firms because each dealer will make their own volatility assumptions. In the Monte Carlo model, the OAS is the spread K that when added to all the spot rates on all interest-rate paths will make the average present value of the paths equal to the observed market price (plus accrued interest). The spread among dealer firms will differ to the extent interest-rate paths and their volatility assumptions differ. The typical model that Wall Street firms and commercial vendors use to generate random interest-rate paths takes as input today’s term structure of interest rates and a volatility assumption. The term structure of interest rates is the theoretical spot rate (or zero-coupon) curve implied by today’s Treasury securities. The volatility assumption determines the dispersion of future interest rates in the simulation. The simulations should be normalized so that the average simulated price of a zero-coupon Treasury bond equals today’s actual price. Each model has its own model of the evolution of future interest rates and its own volatility assumptions. Typically, there are no significant differences in the interest-rate models of dealer firms and vendors, although their volatility assumptions can be significantly different. 17. Explain how the number of interest-rate paths used in the Monte Carlo simulation methodology is determined. The number of interest-rate paths used in the Monte Carlo simulation methodology is determined by the number of sample paths necessary to get a good statistical sample to obtain a price estimate within a tick. More details are given below. What is the number of scenario paths or repetitions, N, needed to value a RMBS? A typical OAS run will be done for 512 to 1,024 interest-rate paths. The scenarios generated using the simulation method look very realistic and, furthermore, reproduce today’s Treasury curve. By employing this technique, the money manager is effectively saying that Treasuries are fairly priced today and that the objective is to determine whether a specific RMBS is rich or cheap relative to Treasuries. The number of interest-rate paths determines how “good” the estimate is, not relative to the truth but relative to the model. The more paths, the more average spread tends to settle down. It is a statistical sampling problem. Most Monte Carlo simulation models use some form of variance reduction to cut down on the number of sample paths necessary to get a good statistical sample. Variance reduction techniques allow us to obtain price estimates within a tick. By this we mean that if the model is used to generate more scenarios, price estimates from the model will not change by more than a tick. For example, if 1,024 paths are used to obtain the estimated price for a tranche, there is little more information to be had from the model by generating more than that number of paths. (For some very sensitive CMO tranches, more paths may be needed to estimate prices within one tick.) 18. Explain why you agree or disagree with the following statement: “When the Monte Carlo simulation methodology is used to value a RMBS, a PSA assumption is employed for all interest-rate paths.” As seen in the illustration and Exhibit 18-9, the collateral value of a CMO is not always influenced by the PSA assumption when using the Monte Carlos simulation methodology. However, the value of the tranches is influenced so that there is a need to employ an accurate PSA assumption to gage how the price will change when prepayment assumptions change. 19. What assumption is made about the OAS in calculating the effective duration and effective convexity of a RMBS? Measures for the effective duration and effective convexity for any security can be calculated using the Monte Carlo method as follows. First, the bond’s OAS is found using the current term structure of interest rates. Next, the bond is repriced assuming OAS is constant but shifting the term structure. Two shifts are used to get the prices needed to apply the effective duration and effective convexity formulas: in one, yields are assumed to increase, and in the second, they are assumed to decrease. 20. What are the limitations of the option-adjusted spread measure? Although the OAS measure is much more useful than the static cash flow yield measure, it still suffers from major pitfalls. These limitations apply not only to the OAS for RMBS but also the OAS produced from a binomial model. First, the OAS is a product of the valuation model. The valuation model may be poorly constructed because it fails to capture the true factors that affect the value of particular securities. Second, in Monte Carlo simulation the interest-rate paths must be adjusted so that on-the-run Treasuries are valued properly. That is, the value of an on-the-run Treasury is equal to its market price or, equivalently, its OAS is zero. The process of adjusting the interest-rate paths to achieve that result is ad hoc. A third problem with the OAS is that it assumes a constant OAS for each interest rate path and over time for a given interest-rate path. If there is a term structure to the OAS, this is not captured by having a single OAS number. Finally, the OAS is dependent on the volatility assumption, the prepayment assumption in the case of RMBS, and the rules for refunding in the case of corporate bonds. In addition, there is a problem with calculating an OAS for a portfolio by taking a weighted average of the OAS of the individual portfolio holdings. Instead, if an OAS for a portfolio is sought, it is necessary to obtain the portfolio’s cash flow along each interest-rate path. The OAS is then the spread that will make the average portfolio value equal to the portfolio’s market value. 21. What assumptions are required to assess the potential total return of a RMBS? The measure that should be used to assess the performance of a security or a portfolio over some investment horizon is the total return. The total dollars received from investing in a RMBS consist of (i) the projected cash flow from the projected interest payments and the projected principal repayment (scheduled plus prepayments), (ii) the interest earned on reinvestment of the projected interest payments and the projected principal prepayments, and (iii) the projected price of the RMBS at the end of the investment horizon. To obtain the cash flow, a prepayment rate over the investment horizon must be assumed. The second step requires assumption of a reinvestment rate. Finally, either of the methodologies described in this chapter—cash flow yield or Monte Carlo simulation —can be used to calculate the price at the end of the investment horizon under a particular set of assumptions. Either approach requires assumption of the prepayment rate and the Treasury rates (i.e., the yield curve) at the end of the investment horizon. The cash flow yield methodology uses an assumed spread to a comparable Treasury to determine the required cash flow yield, which is then used to compute the projected price. The Monte Carlo simulation methodology requires an assumed OAS at the investment horizon. From this assumption, the OAS methodology can produce the horizon price. 22. What are the complications of assessing the potential total return of a CMO tranched using the total return framework? The most difficult part of estimating total return is projecting the price at the horizon date. In the case of a CMO tranche the price depends on the characteristics of the tranche and the spread to Treasuries at the termination date. The key determinants are the “quality” of the tranche, its average life (or duration), and its convexity. Quality refers to the type of CMO tranche. Consider, for example, that an investor can purchase a CMO tranche that is a PAC bond but as a result of projected prepayments could become a sequential-pay tranche. As another example, suppose that a PAC bond is the longest-average-life tranche in a reverse PAC structure. Projected prepayments in this case might occur in an amount to change the class from a long-average-life PAC tranche to a support tranche. The converse is that the quality of a tranche may improve as well as deteriorate. For example, the effective collar for a PAC tranche could widen at the horizon date when prepayment circumstances increase the par amount of support tranches outstanding as a proportion of the deal. To test the sensitivity of total return to various alternative assumptions scenario analysis is helpful. Its limitation is that only a small number of potential scenarios can be considered, and it fails to take into consideration the dynamics of changes in the yield curve and the dynamics of the deal structure. Solution Manual for Bond Markets, Analysis and Strategies Frank J. Fabozzi 9780132743549, 9780133796773