Financial engineering principles phần 6 pdf

bundle that comprises the MBS. Accordingly, if some homeowners happen to default on their mortgages, the excess supply of mortgages in the bundle will help to cover that event. Another way that MBS products are able to secure a triple-A rating is by virtue of their being supported by federal agencies. The three major agencies of the United States involved with supporting mortgages include Ginnie Mae, Fannie Mae, and Freddie Mac. 6 The key purpose of these governmental organizations is to provide assurance and confidence in the market for MBSs and other mortgage products. Table 4.3 summarizes key differences between an MBS and a callable bond. The most dramatic differences between MBSs and callable bonds are that the options embedded with the former are continuous while the single option embedded in the latter tends to be discrete, and the multiple options within an MBS can be triggered by many more variables. Figure 4.15 shows how an MBS’s cash flows might look; none of the cash flow boxes is solid because none of them can be relied on with 100 percent certainty. While less-than-100% certainty might be due partly to the vagaries of what precisely is meant by saying that the federal agencies issu- ing these debt types are “supported by” the federal government, more of the uncertainty stems from the embedded optionality. Although it may very well be unlikely, it is theoretically possible that an investor holding a mortgage- backed security could receive some portion of a principal payment in one of the very first cash flows that is paid out. This would happen if a home- 136 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT 6 Ginnie Mae pass-thrus are guaranteed directly by the U.S. government regarding timely payment of interest and principal. Fannie Mae and Freddie Mac pass-thrus carry the guarantee of their respective agency only; however, both agencies can borrow from the Treasury, and it is not considered to be likely that the U.S. government would allow any of these agencies to default. TABLE 4.3 MBS versus Callable Bond Optionality Mortgage Callable Callability Continuous Discrete (sometimes continuous) Call period Immediately Eligible after the passage of some time Call trigger Level of yields Level of yields, other cost considerations Homeowner defaults Homeowner sells property for any reason Property is destroyed as by natural disaster 04_200306_CH04/Beaumont 8/15/03 12:48 PM Page 136 owner decides or is forced to sell the home almost immediately after purchase and pays off the full principal of the loan. In line with what we would generally expect, principal payments will likely make their way more mean- ingfully into the mix of principal-coupon cash flows after some time passes (or, in the jargon of the marketplace, with some seasoning). How can probabilities be assigned to the mortgage product’s cash flows over time? While we can take the view that we adopted for our callable debenture — that at the start of the game every uncertain cash flow has a 50/50 chance of being paid — this type of evenly split tactic may not be very practical or realistic for mortgage products. For example, a typical home mortgage is a 30-year fixed-rate product. This type of product has been around for some time, and some useful data have been collected to allow for the evaluation of its cash flows over a variety of interest rate and eco- nomic environments. In short, various patterns can and do emerge with the nature of the cash flows. Indeed, a small cottage industry has grown up for the creation and maintenance of models that attempt to divine insight into the expected nature of mortgage product cash flows. It is sufficient here merely to note that no model produces a series of expected cash flows from year 1 to year 30 with a 50/50 likelihood attached to each and every pay- out. Happily, this conforms to what we would expect from more of an intu- itive or common sense approach. Given the importance of prepayment rates when valuing an MBS, several models have been developed to forecast prepayment patterns. Clearly, investors with a superior prepayment model are better equipped to identify fair market value. In an attempt to impose a homogeneity across prepayment assumptions, certain market conventions have been adopted. These conventions facilitate trades in MBSs since respective buyers and sellers know exactly what assumptions are being used to value various securities. Financial Engineering 137 O + Ϫ p 2 p 1 Time Cash Flow p 4 p 3 p 6 p 5 p 8 p 7 p 719 p 720 FIGURE 4.15 MBS cash flows over time. 04_200306_CH04/Beaumont 8/15/03 12:48 PM Page 137 One commonly used method to proxy prepayment speeds is the constant prepayment rate (CPR). A CPR is the ratio of the amount of mortgages pre- paid in a given period to the total amount of mortgages in the pool at the beginning of the period. That is, the CPR is the percentage of the principal outstanding at the beginning of a period that will prepay over the follow- ing period. For example, if the CPR for a given security in a particular month is 10.5, then the annualized percentage of principal outstanding at the beginning of the month that will repay during the month is 10.5 percent. As the name implies, CPR assumes that prepayment rates are constant over the life of the MBS. To move beyond the rather limiting assumption imposed by a CPR — that prepayments are made at a constant rate over the life of an MBS — the industry proposed an alternative measure, the Public Securities Association (PSA) model. The PSA model posits that any given MBS will prepay at an annualized rate of 0.2 percent in the first month that an MBS is outstanding, and prepayments will increase by 0.2 percent per month until month 30. After month 30, it is assumed that prepayments occur at a rate of 6 percent per year for all succeeding months. Generally speaking, the PSA model provides a good description of prepayment patterns for the first several years in the life of an MBS and has proven to be a standard for comparing various MBSs. Figure 4.16 shows theoretical principal and coupon cash flows for a 9 percent Ginnie Mae MBS at 100 percent PSA. When an MBS is quoted at 100 percent PSA, this means that prepayment assumptions are right in line with the PSA model, above. An MBS quoted at 200 percent PSA assumes prepayment speeds that are twice the PSA model, and an MBS quoted at 50 percent PSA assumes a slower prepayment pattern. 138 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT 140 120 100 80 40 20 $1,000s 60 120 180 240 300 360 Month Interest Principal 9% 30-year Ginnie Mae, 100% PSA FIGURE 4.16 The relationship between pay-down of interest and principal for a pass- thru. 04_200306_CH04/Beaumont 8/15/03 12:48 PM Page 138 “Probability-weighted coupon” means the statistical likelihood of receiving a full coupon payment (equivalent to 100 percent of F times C). As principal is paid down from par, the reference amount of coupon payment declines as well (so that when principal is fully paid down, a coupon payment is equal to zero percent of F times C, or zero). “Probability-weighted principal” means the statistical likelihood of receiving some portion of principal’s payment. As is the case with a callable debenture, the initial price of an MBS is par, and YϭC. However, unlike our callable debenture, there is no formal lockout period with an MBS. While we might informally postulate that probability values for F should be quite small in the early stages of an MBS’s life (where maturities can run as long as 15 or 30 years), this is merely an edu- cated guess. The same would be true for postulating that probability values for C should be quite large in the early stages of an MBS’s life. Because an MBS is comprised of an entire portfolio of short call options (with each one linked to an individual mortgage), in contrast with the single short option embedded in a callable debenture, the modeling process for C and F is more complex; hence the existence and application of simplifying benchmark models, as with the CPR approach. At this stage we have pretty much defined the two extremes of optionality with fixed income products in the U.S. marketplace. However, there are gradations of product within these two extremes. For example, there are PACs, or planned amortization class securities. Much like a Thanksgiving turkey, an MBS can be carved up in a variety of ways. At Thanksgiving, some people like drumsticks and others pre- fer the thigh or breast. With bonds, some people like predictable cash flows while others like a higher yield that comes with products that behave in less predictable ways. To satisfy a variety of investor appetites, MBS pass-thrus can be sliced in a variety of ways. For example, classes of MBS can be created. Investors holding a Class A security might be given assurances that they will be given cash distributions that conform more to a debenture than a pass-thru. Investors in a Class B security would have slightly weaker assurances, those in a Class C security would have even weaker assurances, and so forth. As a trade-off to these levels of assurances, the class yield levels would be progressively higher. A PAC is a prime example of a security type created from a pool of mortgages. What happens is that the cash flows of an MBS pool are combined such that separate bundles of securities are created. What essentially distin- guishes one bundle from another is the priority given for one bundle to be assured of receiving full and timely cash flows versus another bundle. For simplicity, let us assume a scenario where a pool of mortgages is assembled so as to create three tranches of cash flow types. In tranche 1, investors would be assured of being first in line to receive coupon cash flows 140 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT 04_200306_CH04/Beaumont 8/15/03 12:48 PM Page 140 generated by the underlying mortgages. In tranche 2, investors would be sec- ond in line to receive coupon cash flows generated by the underlying mortgages. If homeowners with mortgages in this pool decide to pay off their mortgage for whatever reason, then over time tranche 2 investors would not expect to receive the same complete flow of payouts relative to tranche 1 investors. If only for this reason, the tranche 2 investors should not expect to pay the same up-front price for their investment relative to what is paid by tranche 1 investors. They should pay less. Why? Because tranche 2 investors do not enjoy the same peace of mind as tranche 1 investors of being kept whole (or at least “more whole”) over the investment horizon. And finally, we have tranche 3, which can be thought of as a “residual” or “clean- up” tranche. The tranche 3 investors would stand last in line to receive cash flows, only after tranche 1 and tranche 2 investors were paid. And consistent with the logic presented above for tranche 2, tranche 3 investors should not expect to pay the same up-front price for their investment as tranche 1 or 2 investors; they should pay less. Note that tranche 1 investors are not by any means guaranteed of receiving all cash flows in a complete and timely matter; they only are the first in line as laying priority to complete and timely cash flows. In the unlikely event that every mortgage within the pool were to be paid off at precisely the same time, then each of the three tranches would simply cease to exist. This com- ment helps to reinforce the idea that tranche creation does not create new cash flows where none existed previously; tranche creation simply reallocates existing cash flows in such a way that at one end of a continuum is a security type that at least initially looks and feels like a more typical bond while at the other end is a security type that exhibits a price volatility in keeping with its more uncertain place in the pecking order of all-important cash flow receipts. This illustration is a fairly simplified version of the many different ways in which products can be created out of mortgage pools. Generally speaking, PAC-type products are consistent with the tranche 1 scenario presented. Readers can refer to a variety of texts to explore this kind of product creation methodology in considerable detail. From PACs to TACs to A, B, C, and Z tranches (and much, much more), there is much to keep an avid mortgage investor occupied. Figure 4.18 applies the PAC discussion to our cash flow diagram. Notice that the cash flow boxes in the early part of the PAC’s life are drawn in with solid lines. PACs typically come with preannounced lockout periods. Here, lockout refers to that period of time when the PAC is protected from not receiving complete and timely cash flow payments owing to option-related phenomena. The term of lockouts varies, though is generally 5 to 10 years. Again, the PAC is protected in this lockout period because it stands first in line to receive cash flows out of the mortgage pool. Many times a PAC is specified as being protected only within certain bandwidths of Financial Engineering 141 04_200306_CH04/Beaumont 8/15/03 12:48 PM Page 141 option-related activity. Typically the activity of homeowners paying off their mortgages is referred to as prepayment speed (or speed). Moreover, a con- vention exists for how these speeds are quoted. Accordingly, often PAC bandwidths define an upper and lower bound within which speeds may vary without having any detrimental effect on the PAC’s cash flows. The wider the bandwidth, the pricier the PAC compared to PACs with narrower bands. Once a particular PAC has experienced a prepayment speed that falls out- side of its band, it is referred to as a “busted PAC.” A PAC also is “busted” once its lockout period has passed. Not surprisingly, once “busted,” a PAC’s value tends to cheapen. As perhaps the next logical step from a PAC, we have DUS, or delegated underwriting and servicing security. In brief, a DUS carries significant prepayment penalties, so borrowers do not have a great incentive to prepay their loans. Accordingly, a DUS can be thought of as having significant lockout protection. The formula for a PAC or DUS or a variety of other products created from pass-thru might very well look like our last price formula, and it is repeated below. What would clearly differ, however, are the values we insert for probability. While large bond fund investors might perform a variety of complex analyses to calibrate precise probability values across cash flows, other investors might simply observe whether respective yield levels appear to be in line with one another. That is, we would expect a 10-noncall-five to trade at a yield below a 10-year DUS, a 10-year DUS to trade at a yield below a 10-year PAC with a lockout of five years, and so forth. 142 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT O + – p 1 p 2 Time Cash Flow Lockout period Post lockout The p’s represent probability values that are assigned to each cash flow after purchase. FIGURE 4.18 Applying the PAC discussion to the cash flow diagram. 04_200306_CH04/Beaumont 8/15/03 12:48 PM Page 142 Figure 4.19 summarizes the yield relationship to the different callable bond structures presented in this section. Each successive layer represents a different and higher-yielding callable product. For another perspective on the relationships among products, cash flows, and credit, consider Figure 4.20, which plots the price volatility of a triple-A-rated pass-thru against four 10-year final maturity bonds. One of the bonds is a bullet, while the other three are different types of callables. Each ϩ C ϫ p 5 &F ϫ p 6 11 ϩ Y>22 3 ϩ ϭ $1,000 Price ϭ C ϫ p 1 &F ϫ p 2 11 ϩ Y>22 1 ϩ C ϫ p 3 &F ϫ p 4 11 ϩ Y>22 2 Financial Engineering 143 Mortgage-Backed Security Prepayment penalties are comparable with PACs, but there are no bands to limit exposure to changes in prepayment activity, and these uncertain changes contribute to the uncertainty in timing of both coupon and principal payments. Planned Amortization Class Security Prepayment penalties are not as severe as with DUS, and although there are bands intended to limit exposure to changes in prepayment activity, these changes are nonetheless uncertain and thus contribute to the uncertainty in timing of both coupon and principal payments. DUS Although relatively severe penalties exist for early prepayments, there is uncertainty associated with the timing of both coupon and principal payments. Callable Non-Treasury Coupon-Bearing Bond After an initial lockout period, there is uncertainty of timing of final coupon and principal. Non-Treasury Coupon-Bearing Bond Credit risk Coupon-Bearing Treasury Bond Market risk Layers of increasing option-related risks (on top of credit risk and market risk) FIGURE 4.19 Summary of the yield relationship to callable bond structures. 04_200306_CH04/Beaumont 8/15/03 12:48 PM Page 143 of the callables is a 10-noncall-2, but each has a different status with regard to the relationship between F and K. Namely, one has F ϭ K, the other has F much greater than K (deep in-the-money), and the last has F much less than K (deep out-of-the-money). The price volatility of an at-the-money 10-noncall-2 is the same as that for a generic double-A-rated corporate security. Accordingly, with all the shortcomings and limitations that a mapping process represents, it would appear that such a process might be used to find connectors between things like credit profiles and cash flow compositions. The particular relationship highlighted in the figure might be of special interest to an investor looking for an additional and creative way to identify value across various financial considerations inclusive of credit and structure types. Parenthetically, a financing market exists for MBS securities as well. An exchange of an MBS for a loan of cash is referred to as a dollar roll. A dollar roll works very much like the securities lending example described ear- lier in this chapter, though obviously there are special accommodations for the unique coupon and price risk inherent in an MBS as opposed to a generic Treasury Bond. A preferred stock is a security that combines characteristics of both bonds and equities (see Figure 4.21). Like bonds, a preferred stock usually has a predetermined maturity date, pays regular dividends, and does not convey voting rights. Like an equity, a preferred stock ranks rather low in 144 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT BAA AA A AAA Credit ratings Cash flow types Price volatility 2 1 3 4 The intersection of the price volatility of a triple-A rated 10- noncall-2 and a double-A rated 10- year bullet bond. 1 Deep in-the-money 2 At-the-money 3 Deep out-of-the-money 4 10-year bullet bond FIGURE 4.20 Mapping process. 04_200306_CH04/Beaumont 8/15/03 12:49 PM Page 144 priority in the event of a default, but typically it ranks above common stock. The hybrid nature of preferred stock is supported by the fact that while some investment banks and investors warehouse these securities in their fixed income business, others manage them in their equity business. One special type of preferred stock is known as a convertible. As the name suggests, the security can be converted from a preferred stock product into something else at the choice of the investor. The “something else” is usually shares of stock in the company that originally issued the preferred stock. A convertible typically is structured such that it is convertible at any time, the conversion right is held by the investor in the convertible, and the convertible sells at a premium to the underlying security. Investors accept the premium since convertibles tend to pay coupons that are much higher than the dividends of the underlying common stock. Generally speaking, as the underlying common stock of a convertible declines, the convertible will trade more like a bond than an equity. That is, the price of the convertible will be more sensitive to changes in interest rates than to changes in the price of the underlying common stock. However, as the underlying stock price appreciates, the convertible will increasingly trade much more in-line with the price behavior of the underlying equity than with changes in interest rates. Figure 4.22 shows a preferred stock’s potential evolution from more of a bond product into more of an equity product. A convertible’s increasingly equitylike behavior is entirely consistent with the way a standard equity option would trade. That is, as the option trades more and more in-the-money, the more its price behavior moves into lock- step with the price behavior of the underlying equity’s forward or spot price. Parenthetically, an option that can be exercised at any time is called an American option, while an option that can be exercised only at expiration is called a European option. In the case of a European option type structure, if the underlying equity price is above the convertible-equity conversion price Financial Engineering 145 = Preferred stock Buy Buy Option Equity Convertible structuure Spot Bond FIGURE 4.21 Use of spot and options to create a convertible. 04_200306_CH04/Beaumont 8/15/03 12:49 PM Page 145 . pattern. 138 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT 140 120 100 80 40 20 $1,000s 60 120 180 240 300 360 Month Interest Principal 9% 30-year Ginnie Mae, 100% PSA FIGURE 4. 16 The. to value various securities. Financial Engineering 137 O + Ϫ p 2 p 1 Time Cash Flow p 4 p 3 p 6 p 5 p 8 p 7 p 719 p 720 FIGURE 4.15 MBS cash flows over time. 04_2003 06_ CH04/Beaumont 8/15/03 12:48. first cash flows that is paid out. This would happen if a home- 1 36 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT 6 Ginnie Mae pass-thrus are guaranteed directly by the U.S. government

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