1 See the preface to this book regarding rounding.
2 LIBOR is the interest rate which major international banks offer each other on Eurodollar certificates of deposit.
3 In the fixed income market, market participants refer to changes in interest rates or differences in interest rates in terms of basis points. A basis point is defined as 0.0001, or equivalently, 0.01%. Consequently, 100 basis points are equal to 1%. (In our example the coupon formula can be expressed as 1-month LIBOR + 1%.) A change in interest rates from, say, 5.0% to 6.2% means that there is a 1.2% change in rates or 120 basis points.
4 In the agency, corporate, and municipal markets, inverse floaters are created as structured notes. We discuss structured notes in Chapter 3. Inverse floaters in the mortgage-backed securities market are common and are created through a process that will be discussed in Chapter 10.
5 The issuer hedges by using financial instruments known as derivatives, which we cover in later chapters.
6 In Chapter 3, we will describe other types of floating-rate securities.
7 These offbeat coupon bond formulas are actually created as a result of inquiries from clients of dealer firms. That is, a salesperson will be approached by fixed income portfolio managers requesting a structure be created that provides the exposure sought. The dealer firm will then notify the investment banking group of the dealer firm to contact potential issuers.
8 As explained in Chapter 2, high credit quality issuers are referred to as “investment grade” issuers and low credit quality issuers are referred to as “non-investment grade” issuers. The reason why high credit quality issuers have reduced their issuance of callable bonds while it is still the more popular structure for low credit quality issuers is explained later.
9 At this stage, we will use the terms interest rate and yield interchangeably. We’ll see in Chapter 6 how to compute a bond’s yield.
10 We’ll see how to compute the price of a bond in Chapter 5.
11 The arrow symbol in the expressions means “therefore.”
12 Recall from Chapter 1 that an embedded option is the feature in a bond issue that grants either the issuer or the investor an option. Examples include call option, put option, and conversion option.
13 As explained in Chapter 1, the coupon reset formula is set at the reset date at the beginning of the period but is not paid until the end of the period.
14 Robert I. Gerber, “A User’s Guide to Buy-Side Bond Trading,” Chapter 16 in Frank J. Fabozzi (ed.), Managing Fixed Income Portfolios (New Hope, PA: Frank J. Fabozzi Associates, 1997, p. 278.)
15 It should be noted that the classification used here is by no means universally accepted. Some market observers refer to the external bond market as consisting of the foreign bond market and the Eurobond market.
16 The calculation of the monthly mortgage payment is simply an application of the present value of an annuity. The formula as applied to mortgage payments is as follows:
where
MP = monthly mortgage payment
B = amount borrowed (i.e., original loan balance)
r = monthly mortgage rate (annual rate divided by 12)
n = number of months of the mortgage loan In our example,
B = $100, 000 r = 0.0067708 (0.08125/12) n = 360
Then
17 Factors affecting prepayments will be discussed in later chapters.
18 Freddie Mac previously issued passthrough securities that guaranteed the timely payment of interest but guaranteed only the eventual payment of principal (when it is collected or within one year).
19 “Tranche” is from an old French word meaning “slice.” (The pronunciation of tranche rhymes with the English word “launch,” as in launch a ship or a rocket.)
20 We will explain what is meant by “prepayment rate” later.
21 “Industrial Company Rating Methodology,” Moody’s Investors Service: Global Credit Research (July 1998), p. 6.
22 “Industrial Company Rating Methodology,” p. 7.
23 “Industrial Company Rating Methodology,” p. 3.
24 Edward I. Altman and Scott A. Nammacher, Investing in Junk Bonds (New York: John Wiley, 1987) and Edward I. Altman, “Research Update: Mortality Rates and Losses, Bond Rating Drift,” unpublished study prepared for a workshop sponsored by Merrill Lynch Merchant Banking Group, High Yield Sales and Trading, 1989.
25 Paul Asquith, David W. Mullins, Jr., and Eric D. Wolff, “Original Issue High Yield Bonds: Aging Analysis of Defaults, Exchanges, and Calls,” Journal of Finance (September 1989), pp. 923-952.
26 As a parallel, we know that the mortality rate in the United States is currently less than 1% per year, but we also know that 100% of all humans (eventually) die.
27 Moody’s Investors Service, Corporate Bond Defaults and Default Rates: 1970-1994, Moody’s Special Report, January 1995, p. 13.
28 The primary market for bonds is described in Section IX A.
29 SEC Rule 415 permits certain issuers to file a single registration document indicating that it intends to sell a certain amount of a certain class of securities at one or more times within the next two years. Rule 415 is popularly referred to as the “shelf registration rule” because the securities can be viewed as sitting on the issuer’s “shelf” and can be taken off that shelf and sold to the public without obtaining additional SEC approval. In essence, the filing of a single registration document allows the issuer to come to market quickly because the sale of the security has been preapproved by the SEC. Prior to establishment of Rule 415, there was a lengthy period required before a security could be sold to the public. As a result, in a fast-moving market, issuers could not come to market quickly with an offering to take advantage of what it perceived to be attractive financing opportunities.
30 There are other advantages to the corporation having to do with financial accounting for the assets sold. We will not discuss this aspect of financing via asset securitization here since it is not significant for the investor.
31 Interest rate risk is the risk of an adverse movement in the price of a bond due to changes in interest rates.
32 Duration is a measure of a bond’s price sensitivity to a change in interest rates.
33 The Treasury no longer issues callable bonds. The Treasury issued callable bonds in the early 1980s and all of these issues will mature no later than November 2014 (assuming that they are not called before then). Moreover, as of 2004, the longest maturity of these issues is 10 years. Consequently, while outstanding callable issues of the Treasury are referred to as “bonds,” based on their current maturity these issues would not be compared to long-term bonds in any type of relative value analysis. Therefore, because the Treasury no longer issues callable bonds and the outstanding issues do not have the maturity characteristics of a long-term bond, we will ignore these callable issues and simply treat Treasury bonds as noncallable.
34 Term structure means the same as maturity structure—a description of how a bond’s yield changes as the bond’s maturity changes. In other words, term structure asks the question: Why do long-term bonds have a different yield than short-term bonds?
35 Later, we provide a more mathematical treatment of these theories in terms of forward rates that we will discuss in Chapter 6.
36 In the liquidity preference theory, “liquidity” is measured in terms of interest rate risk. Specifically, the more interest rate risk, the less the liquidity.
37 One of the principles of finance is the “matching principle:” short-term assets should be financed with (or matched with) short-term liabilities; long-term assets should be financed with (or matched with) long-term sources of financing.
38 For a further discussion and evidence regarding business cycles and credit spreads, see Chapter 10 in Leland E. Crabbe and Frank J. Fabozzi, Managing a Corporate Portfolio (Hoboken, NJ: John Wiley & Sons, 2002).
39 The mortgage-backed securities sector is often referred to as simply the “mortgage sector.”
40 Global Relative Value, Lehman Brothers, Fixed Income Research, June 28, 1999, COR-2 AND 3.
41 As explained in Chapter 3, some municipal bonds are taxable.
42 Some maturities for Treasury securities shown in the exhibit are not on-the-run issues. These are estimates for the market yields.
43 The marginal tax rate is the tax rate at which an additional dollar is taxed.
44 The amount of the payment is found by dividing the annual dollar amount by four because payments are made quarterly. In a real world application, both the fixed-rate and floating-rate payments are adjusted for the number of days in a quarter, but it is unnecessary for us to deal with this adjustment here.
45 We say primarily because there are also technical factors that affect the swap spread. For a discussion of these factors, see Richard Gordon, “The Truth about Swap Spreads,” in Frank J. Fabozzi (ed.), Professional Perspectives on Fixed Income Portfolio Management: Volume 1 (New Hope, PA: Frank J. Fabozzi Associates, 2000), pp. 97 - 104.
46 As explained in Chapter 3, the on-the-run Treasury issues are the most recently auctioned Treasury issues.
47 Note that in our earlier illustration, we computed the present value of the semiannual coupon payments before the maturity date and then added the present value of the last cash flow (last semiannual coupon payment plus the maturity value). In the presentation of how to use the short-cut formula, we are computing the present value of all the semiannual coupon payments and then adding the present value of the maturity value. Both approaches will give the same answer for the value of a bond.
48 “Accrued” means that the interest is earned but not distributed to the bondholder.
49 The settlement date is the date a transaction is completed.
50 Notice that in computing the full price the present value of the next coupon payment is computed. However, the buyer pays the seller the accrued interest now despite the fact that it will be recovered at the next coupon payment date.
51 In its simple form, arbitrage is the simultaneous buying and selling of an asset at two different prices in two different markets. The arbitrageur profits without risk by buying cheap in one market and simultaneously selling at the higher price in the other market. Such opportunities for arbitrage are rare. Less obvious arbitrage opportunities exist in situations where a package of assets can produce a payoff (expected return) identical to an asset that is priced differently. This arbitrage relies on a fundamental principle of finance called the “law of one price” which states that a given asset must have the same price regardless of the means by which one goes about creating that asset. The law of one price implies that if the payoff of an asset can be synthetically created by a package of assets, the price of the package and the price of the asset whose payoff it replicates must be equal.
52 This may seem like a small amount, but remember that this is for a single $100 par value bond. Multiply this by thousands of bonds and you can see a dealer’s profit potential.
53 The definition of reconstitute is to provide with a new structure, often by assembling various parts into a whole. Reconstitution then, as used here, means to assemble the parts (the Treasury strips) in such a way that a new whole (a Treasury coupon bond) is created. That is, it is the opposite of stripping a coupon bond.
54 A short summary reason is: mortgage-backed securities and certain asset-backed securities are interest rate path dependent securities and the binomial model cannot value such securities.
55 By compounding the semiannual yield it is meant that the annual yield is computed as follows: effective annual yield = (1 + semiannual yield)2 − 1
56 This can be verified by using the future value of an annuity. The future of an annuity is given by the following formula:
where
i is the interest rate and
n is the number of periods.
In our example,
i is 4%,
n is 16, and the amount of the annuity is the semiannual coupon of $3.50. Therefore, the future value of the coupon payment is
Since the coupon payments are $56, the reinvestment income is $20.38 ($76.38 − $56). This is the amount that is necessary to produce the dollar return shortfall in our example.
57 The future value of the coupon payments of $4 for 30 six-month periods is:
Since the coupon payments are $120 and the capital gain is $0, the reinvestment income is $104.34. This is the amount that is necessary to produce the dollar return shortfall in our example.
58 A call schedule shows the call price that the issuer must pay based on the date when the issue is called. An example of a call schedule is provided in Chapter 1.
59 Some firms such as Prudential Securities refer to this yield as yield to maturity rather than cash flow yield.
60 For a discussion of other traditional measures, see Chapter 3 in Frank J. Fabozzi and Steven V. Mann, Floating Rate Securities (New Hope, PA; Frank J. Fabozzi Associates, 2000).
61 Two points should be noted abut the yields reported in
Exhibit 4. First, the yields are unrelated to our earlier Treasury yields on September 5, 2003 that we used to show how to calculate the yield on interim maturities using linear interpolation. Second, the Treasury yields in our illustration after the first year are all shown at par value. Hence the Treasury yield curve in
Exhibit 4 is called a
par yield curve.
62 We will assume that the annualized yield for the Treasury bill is computed on a bond-equivalent basis. Earlier in this chapter, we saw how the yield on a Treasury bill is quoted. The quoted yield can be converted into a bond-equivalent yield; we assume this has already been done in
Exhibit 4.
63 If we had not been working with a par yield curve, the equation would have been set equal to whatever the market price for the 1.5-year issue is.
64 Option risk includes prepayment and call risk.
65 Actually, the semiannual forward rates are based on annual rates calculated to more decimal places. For example,
f1,3 is 5.15% in
Exhibit 12 but based on the more precise value, the semiannual rate is 2.577%.
66 The procedure used is principal component analysis.
67 This is because there is still some chance that interest rates will decline in the future and the issue will be called.
68 For readers who are already familiar with option theory, this characteristic can be restated as follows: When the coupon rate for the issue is below the market yield, the embedded call option is said to be “out-of-the-money.” When the coupon rate for the issue is above the market yield, the embedded call option is said to be “in-the-money.”
69 Mathematicians refer to this shape as being “concave.”
70 The difference between Δ
y in the duration formula given by
equation (1) and Δ
y∗ in
equation (2) to get the approximate percentage change is as follows. In the duration formula, the Δ
y is used to estimate duration and, as explained later, for reasonably small changes in yield the resulting value for duration will be the same. We refer to this change as the “rate shock.” Given the duration, the next step is to estimate the percentage price change for any change in yield. The Δ
y∗ in
equation (2) is the specific change in ield for which the approximate percentage price change is sought.
71 More specifically, this is the formula for the modified duration of a bond on a coupon anniversary date.
72 Frederick Macaulay, Some Theoretical Problems Suggested by the Movement of Interest Rates, Bond Yields, and Stock Prices in the U.S. Since 1856 (New York: National Bureau of Economic Research, 1938).
73 “Half-life” is the time required for an element to be reduced to half its initial value.
74 The reason it is a linear approximation can be seen in
Exhibit 15 where the tangent line is used to estimate the new price. That is, a straight line is being used to approximate a non-linear (i.e., convex) relationship.
75 See
footnote 5 for the difference between Δ
y in the formula for
C and Δ
y∗ in
equation (4).
76 Index constructors such as Lehman Brothers when constructing maturity sector indexes define “short-term sector” as up to three years, the “intermediate sector” as maturities greater than three years but less than 10 years (note the overlap with the short-term sector), and the “long-term sector” as greater than 10 years.
77 Robert Litterman and José Scheinkman, “Common Factors Affecting Bond Returns,” Journal of Fixed Income (June 1991), pp. 54-61.
78 For a further explanation of the coefficient of determination, see Richard A. DeFusco, Dennis W. McLeavey, Jerald E. Pinto, and David E. Runkle, Quantitative Methods for Investment Analysis (Charlottesville, VA: Association for Investment Management and Research, 2002), pp. 388-390.
79 At one time, the Department of the Treasury issued 3-year notes, 7-year notes, 15-year bonds, 20-year bonds, and 30-year bonds.
80 See, for example, Philip H. Galdi and Shenglin Lu, Analyzing Risk and Relative Value of Corporate and Government Securities, Merrill Lynch & Co., Global Securities Research & Economics Group, Fixed Income Analytics, 1997, p. 11.
81 There must also be an adjustment for what is known as the “specials effect.” This has to do with a security trading at a lower yield than its true yield because of its value in the repurchase agreement market. As explained, in a repurchase agreement, a security is used as collateral for a loan. If the security is one that is in demand by dealers, referred to as “hot collateral” or “collateral on special,” then the borrowing rate is lower if that security is used as collateral. As a result of this favorable feature, a security will offer a lower yield in the market if it is on special so that the investor can finance that security cheaply. As a result, the use of the yield of a security on special will result in a biased yield estimate. The 10-year on-the-run U.S. Treasury issue is typically on special.
82 See, Oldrich A. Vasicek and H. Gifford Fong, “Term Structure Modeling Using Exponential Splines,” Journal of Finance (May 1982), pp. 339-358.
83 Actually we will see in Chapter 14 that the payments are slightly different each quarter because the amount of the quarterly payment depends on the actual number of days in the quarter.
84 See Uri Ron, “A Practical Guide to Swap Curve Construction,” Chapter 6 in Frank J. Fabozzi (ed.), Interest Rate, Term Structure, and Valuation Modeling (NY: John Wiley & Sons, 2002).
85 For example, a government bond issue being on “special” in the repurchase agreement market.
86 The question is what yields are used to construct the swap rate curve. Practitioners use yields from two related markets: the Eurodollar CD futures contract and the swap market. We will not review the Eurodollar CD futures contract here. It is discussed in Chapter 14. For now, the only important fact to note about this contract is that it provides a means for locking in 3-month LIBOR in the future. In fact, it provides a means for doing so for an extended time into the future.
Practitioners. use the Eurodollar CD futures rate up to four years to get 3-month LIBOR for every quarter. While there are Eurodollar CD futures contracts that settle further out than four years, for technical reasons (having to do with the convexity of the contract) analysts use only the first four years. (In fact, this actually varies from practitioner to practitioner. Some will use the Eurodollar CD futures from two years up to four years.) For maturities after four years, the swap rates are used to get 3-month LIBOR. As noted above, there is a swap rate for maturities for each year to year 10, and then swap rates for 15 years and 30 years.
87 These formulations are summarized by John Cox, Jonathan Ingersoll, Jr., and Stephen Ross, “A Re-examination of Traditional Hypotheses About the Term Structure of Interest Rates,” Journal of Finance (September 1981), pp. 769-799.
88 F. Lutz, “The Structure of Interest Rates,” Quarterly Journal of Economics (1940-41), pp. 36-63.
89 Cox, Ingersoll, and Ross, pp. 774-775.
90 It has been demonstrated that the local expectations formulation, which is narrow in scope, is the only interpretation of the pure expectations theory that can be sustained in equilibrium. See Cox, Ingersoll, and Ross, “A Re-examination of Traditional Hypotheses About the Term Structure of Interest Rates.”
91 Eugene F. Fama, “Forward Rates as Predictors of Future Spot Rates,” Journal of Financial Economics Vol. 3, No. 4, 1976, pp. 361-377.
92 John R. Hicks, Value and Capital (London: Oxford University Press, 1946), second ed., pp. 141-145.
93 Franco Modigliani and Richard Sutch, “Innovations in Interest Rate Policy,” American Economic Review (May 1966), pp. 178-197.
94 Donald Chambers and Willard Carleton, “A Generalized Approach to Duration,” Research in Finance 7(1988).
95 See, for example, Robert R. Reitano, “Non-Parallel Yield Curve Shifts and Durational Leverage,” Journal of Portfolio Management (Summer 1990), pp. 62-67, and “A Multivariate Approach to Duration Analysis,” ARCH 2(1989).
96 Thomas S.Y. Ho, “Key Rate Durations: Measures of Interest Rate Risk,” The Journal of Fixed Income (September 1992), pp. 29-44.
97 This is the numerical example used by Ho, “Key Rate Durations,” p. 33.
98 The portfolios whose key rate durations are shown in
Exhibit 7 were hypothetical Treasury portfolios constructed on April 23, 1997.
99 See Chapter 2 in DeFusco, McLeavey, Pinto, and Runkle, Quantitative Methods for Investment Analysis.
100 For any probability distribution, it is important to assess whether the value of a random variable in one period is affected by the value that the random variable took on in a prior period. Casting this in terms of yield changes, it is important to know whether the yield today is affected by the yield in a prior period. The term serial correlation is used to describe the correlation between the yield in different periods. Annualizing the daily yield by multiplying the daily standard deviation by the square root of the number of days in a year assumes that serial correlation is not significant.
101 See Chapter 4 in DeFusco, McLeavey, Pinto, and Runkle, Quantitative Methods for Investment Analysis.
102 See Chapter 6 in DeFusco, McLeavey, Pinto, and Runkle, Quantitative Methods for Investment Analysis.
103 For a further discussion, see Frank J. Fabozzi and Wai Lee, “Measuring and Forecasting Yield Volatility,” Chapter 16 in Frank J. Fabozzi (ed.), Perspectives on Interest Rate Risk Management for Money Managers and Traders (New Hope, PA: Frank J. Fabozzi Associates, 1998).
104 Jacques Longerstacey and Peter Zangari, Five Questions about RiskMetricsTM , JP Morgan Research Publication 1995.
105 See Robert F. Engle, “Autoregressive Conditional Heteroskedasticity with Estimates of Variance of U.K. Inflation,” Econometrica 50 (1982), pp. 987-1008.
106 See Chapter 9 in DeFusco, McLeavey, Pinto, and Runkle, Quantitative Methods for Investment Analysis.
107 Market participants also refer to this characteristic of a model as one that “calibrates to the market.”
108 An excellent source for further explanation of many of these models is Gerald W. Buetow Jr. and James Sochacki, Term Structure Models Using Binomial Trees: Demystifying the Process (Charlottesville, VA: Association of Investment Management and Research, 2000).
109 For a discussion of the binomial model and the underlying theory, see Chapter 4 in Don M. Chance, Analysis of Derivatives for the CFA Program (Charlottesville, VA: Association for Investment Management and Research, 2003).
110 A lattice is an arrangement of points in a regular periodic pattern.
111 The model described in this chapter was first presented in Andrew J. Kalotay, George O. Williams, and Frank J. Fabozzi, “A Model for the Valuation of Bonds and Embedded Options,” Financial Analysts Journal (May-June 1993), pp. 35-46.
112 If we were using a trinomial model, there would be three possible interest rates shown in the next year.
113 In practice, much shorter time periods are used to construct an interest rate tree.
115 Technically, the standard textbook definition of conversion value given here is theoretically incorrect because as bondholders convert, the price of the stock will decline. The theoretically correct definition for the conversion value is that it is the product of the conversion ratio and the stock price after conversion.
116 If the conversion value is the greater of the two values, it is possible for the convertible bond to trade below the conversion value. This can occur for the following reasons: (1) there are restrictions that prevent the investor from converting, (2) the underlying stock is illiquid, and (3) an anticipated forced conversion will result in loss of accrued interest of a high coupon issue. See, Mihir Bhattacharya, “Convertible Securities and Their Valuation,” Chapter 51 in Frank J. Fabozzi (ed.), The Handbook of Fixed Income Securities: Sixth Edition (New York: McGraw Hill, 2001), p. 1128.
117 See, for example: Michael Brennan and Eduardo Schwartz, “Convertible Bonds: Valuation and Optimal Strategies for Call and Conversion,” Journal of Finance (December 1977), pp. 1699 - 1715; Jonathan Ingersoll, “A Contingent-Claims Valuation of Convertible Securities,” Journal of Financial Economics (May 1977), pp. 289 - 322; Michael Brennan and Eduardo Schwartz, “Analyzing Convertible Bonds,” Journal of Financial and Quantitative Analysis (November 1980), pp. 907 - 929; and, George Constantinides, “Warrant Exercise and Bond Conversion in Competitive Markets,” Journal of Financial Economics (September 1984), pp. 371 - 398.
118 Mihir Bhattacharya and Yu Zhu, “Valuation and Analysis of Convertible Securities,” Chapter 42 in Frank J. Fabozzi (ed.), The Handbook of Fixed Income Securities: Fifth Edition (Chicago: Irwin Professional Publishing, 1997).
119 The name of the passthrough issued by Ginnie Mae and Fannie Mae is a Mortgage-Backed Security or MBS. So, when a market participant refers to a Ginnie Mae MBS or Fannie Mae MBS, what is meant is a passthrough issued by these two entities. The name of the passthrough issued by Freddie Mac is a Participation Certificate or PC. So, when a market participant refers to a Freddie Mac PC, what is meant is a passthrough issued by Freddie Mac. Every agency has different “programs” under which passthroughs are issued with different types of mortgage pools (e.g., 30-year fixed-rate mortgages, 15-year fixed-rate mortgages, adjustable-rate mortgages). We will not review the different programs here.
120 It may seem strange that the term “mortality” is used to describe this prepayment measure. This term reflects the influence of actuaries who in the early years of the development of the mortgage market migrated to dealer firms to assist in valuing mortgage-backed securities. Actuaries viewed the prepayment of a mortgage loan as the “death” of a mortgage.
121 It is referred to as a “conditional” prepayment rate because the prepayments in one year depend upon (i.e., are conditional upon) the amount available to prepay in the previous year. Sometimes market participants refer to the CPR as the “constant” prepayment rate.
122 The derivation of the CPR for a given SMM is beyond the scope of this chapter. The proof is provided in Lakhbir S. Hayre and Cyrus Mohebbi, “Mortgage Mathematics,” in Frank J. Fabozzi (ed.), Handbook of Mortgage-Backed Securities: Fifth Edition (New York, NY: McGraw-Hill, 2001), pp. 844 - 845.
123 The formula is presented in Chapter 19 of Frank J. Fabozzi, Fixed Income Mathematics (Chicago: Irwin Professional Publishing, 1997).
124 “Tranche” is from an old French word meaning “slice.” In the case of a collateralized mortgage obligation it refers to a “slice of the cash flows.”
125 The issuer of a CMO wants to be sure that the trust created to pass through the interest and principal payments is not treated as a taxable entity. A provision of the Tax Reform Act of 1986, called the Real Estate Mortgage Investment Conduit (REMIC), specifies the requirements that an issuer must fulfill so that the legal entity created to issue a CMO is not taxable. Most CMOs today are created as REMICs. While it is common to hear market participants refer to a CMO as a REMIC, not all CMOs are REMICs.
126 Actually, a CMO is backed by a pool of passthrough securities.
127 The window is also specified in terms of the length of the time from the beginning of the principal pay down window to the end of the principal pay down window. For tranche A, the window would be stated as 81 months, for tranche B 20 months.
128 The same principle for creating a floating-rate tranche and inverse-floating rate tranche could have been accomplished using the 4-tranche sequential-pay structure without an accrual tranche (FJF-01).
129 Actually there were two other tranches, R and RS, called “residuals.” These tranches were not described in the chapter. They receive any excess cash flows remaining after the payment of all the tranches. The residual is actually the equity part of the deal.
130 This ignores the costs of repossession and selling the property.
131 Christopher Flanagan and Edward Reardon, European Structures Products: 2001 Review and 2002 Outlook, Global Structured Finance Research, J.P. Morgan Securities Inc. (January 11, 2002), pp. 12-13.
132 There are concerns that both the creditors to the seller of the collateral (QHT’s creditors in our illustration) and the investors in the securities issued by the SPV have about the assets. Specifically, QHT’s creditors will be concerned that the assets are being sold to the SPV at less than fair market value, thereby weakening their credit position. The buyers of the asset-backed securities will be concerned that the assets were purchased at less than fair market value, thereby weakening their credit position. Because of this concern, the attorney will issue an opinion that the assets were sold at a fair market value.
133 The way this is accomplished is that a copy of the transaction’s payment structure, underlying collateral, average life, and yield are supplied to the accountants for verification. In turn, the accountants reverse engineer the deal according to the deal’s payment rules (i.e., the waterfall). Following the rules and using the same collateral that will actually generate the cash flows for the transaction, the accountants reproduce the yield and average life tables that are put into the prospectus or private placement memorandum.
134 The major monoline insurance companies in the United States are Capital Markets Assurance Corporation (CapMAC), Financial Security Assurance Inc. (FSA), Financial Guaranty Insurance Corporation (FGIC), and Municipal Bond Investors Assurance Corporation (MBIA).
135 As noted earlier, the seller is not a party to the transaction once the assets are sold to the SPV who then issues the securities. Hence, if the seller provides a guarantee, it is viewed as a third-party guarantee.
136 As explained earlier, the calling of a portion of the issue is permitted to satisfy any sinking fund requirement.
137 Dale Westhoff and Mark Feldman, “Prepayment Modeling and Valuation of Home Equity Loan Securities,” Chapter 18 in Frank J. Fabozzi, Chuck Ramsey, and Michael Marz (eds.), The Handbook of Nonagency Mortgage-Backed Securities: Second Edition (New Hope, PA: Frank J. Fabozzi Associates, 2000).
138 Charles Schorin, Steven Weinreich, and Oliver Hsiang, “Home Equity Loan Transaction Structures,” Chapter 6 in Frank J. Fabozzi, Chuck Ramsey, and Michael Marz, Handbook of Nonagency Mortgage-Backed Securities: Second Edition (New Hope, PA: Frank J. Fabozzi Associates, 2000).
139 This illustration is from Schorin, Weinreich, and Hsiang, “Home Equity Loan Transaction Structures.”
140 For a more detailed analysis of this tranche, see Schorin, Weinreich, and Hsiang, “Home Equity Loan Transaction Structures.”
141 Schorin, Weinreich, and Hsiang, “Home Equity Loan Transaction Structures.”
142 Information about the U.K. residential mortgage-backed securities market draws from the following sources: Phil Adams, “UK Residential Mortgage-Backed Securities,” and “UK Non-Conforming Residential Mortgage-Backed Securities,” in Building Blocks, Asset-Backed Securities Research, Barclays Capital, January 2001; Christopher Flanagan and Edward Reardon, European Structured Products: 2001 Review and 2002 Outlook, Global Structured Finance Research, J.P. Morgan Securities Inc., January 11, 2002; and, “UK Mortgages-MBS Products for U.S. Investors,” Mortgage Strategist, UBS Warburg, February 27, 2001, pp. 15-21.
143 For a more detailed discussion of this structure, see Adams, “UK Residential Mortgage-Backed Securities,” pp. 31-37.
144 Information about the Australian residential mortgage-backed securities market draws from the following sources: Phil Adams, “Australian Residential Mortgage-Backed Securities,” in Building Blocks; Karen Weaver, Eugene Xu, Nicholas Bakalar, and Trudy Weibel, “Mortgage-Backed Securities in Australia,” Chapter 41 in The Handbook of Mortgage-Backed Securities: Fifth Edition (New York, NY: McGraw Hill, 2001); and, “Australian Value Down Under,” Mortgage Strategist, UBS Warburg, February 6, 2001, pp. 14-22.
145 The five major ones are Royal and Sun Alliance Lenders Mortgage Insurance Limited, CGU Lenders Mortgage Insurance Corporation Ltd., PMI mortgage insurance limited, GE Mortgage Insurance Property Ltd., and GE Mortgage Insurance Corporation.
146 The foreign exchange risk for these deals is typically hedged using various types of swaps (fixed/floating, floating/floating, and currency swaps).
147 Flanagan and Reardon, European Structures Products: 2001 Review and 2002 Outlook, p. 9.
148 The only reason for the use of ABS rather than SMM/CPR in this sector is historical. Auto-loan backed securities (which were popularly referred to at one time as CARS (Certificates of Automobile Receivables)) were the first non-mortgage assets to be developed in the market. (The first non-mortgage asset-backed security was actually backed by computer lease receivables.) The major dealer in this market at the time, First Boston (now Credit Suisse First Boston) elected to use ABS for measuring prepayments. You may wonder how one obtains “ABS” from “absolute prepayment rate.” Again, it is historical. When the market first started, the ABS measure probably meant “asset-backed security” but over time to avoid confusion evolved to absolute prepayment rate.
149 Actually, depending on the origination date, the guarantee can be up to 100%.
150 In 1997 Sallie Mae began the process of unwinding its status as a GSE; until this multi-year process is completed, all debt issued by Sallie Mae under its GSE status will be “grandfathered” as GSE debt until maturity.
151 This creates a mismatch between the collateral and the securities. Issuers have dealt with this by hedging with the risk by using derivative instruments such as interest rate swaps (floating-to-floating rate swaps described in Chapter 14) or interest rate caps (described in Chapter 14).
152 Donna Faulk, “SBA Loan-Backed Securities,” Chapter 10 in Asset-Backed Securities.
153 Thompson, “MBNA Tests the Waters.”
154 There are other management fees that are usually made based on performance. But these payments are made after payments to the mezzanine tranches.
155 Rating agencies have developed measures that quantify the diversity of a portfolio. These measures are referred to as “diversity scores.”
156 It is for this reason that a nonsynthetic CDO structure is referred to as a “cash” CDO structure because cash is required to be raised to purchase all the collateral assets.
157 For a more detailed discussion of these advantages and how it impacts the economics of a CDO, see Chapter 8 in Laurie S. Goodman and Frank J. Fabozzi, Collateralized Debt Obligations: Structures and Analysis (New York, NY: John Wiley & Sons, 2002).
158 Most common spreadsheet programs offer this type of algorithm.
159 The variance reduction technique is described in books on management science and Monte Carlo simulation.
160 These illustrations are from Frank J. Fabozzi, Scott F. Richard, and David S. Horowitz, “Valuation of CMOs,” Chapter 6 in Frank J. Fabozzi (ed.), Advances in the Valuation and Management of Mortgage-Backed Securities (New Hope, PA: Frank J. Fabozzi Associates, 1998).
161 We will explain how to compute the effective duration using the Monte Carlo methodology in Section V. A.
162 This deal was described in Chapter 10.
163 Lakhbir Hayre and Hubert Chang, “Effective and Empirical Duration of Mortgage Securities,” The Journal of Fixed Income (March 1997), pp. 17-33.
164 Hayre and Chang, “Effective and Empirical Duration of Mortgage Securities.”
165 Sam Choi, “Effective Durations for Mortgage-Backed Securities: Recipes for Improvement,” The Journal of Fixed Income (March 1996), pp. 24-30; and, Hayre and Chang, “Effective and Empirical Duration of Mortgage Securities.”
166 Douglas Breeden, “Risk, Return, and Hedging of Fixed-Rate Mortgages,” The Journal of Fixed Income (September 1991), pp. 85-107.
167 Breeden, “Risk, Return, and Hedging of Fixed-Rate Mortgages.”
168 See Bennett W. Golub, “Towards a New Approach to Measuring Mortgage Duration,” Chapter 32 in Frank J. Fabozzi (ed.), The Handbook of Mortgage-Backed Securities (Chicago: Probus Publishing, 1995), p. 673.
169 This approach was first suggested in 1986 in Scott M. Pinkus and Marie A. Chandoha, “The Relative Price Volatility of Mortgage Securities,” Journal of Portfolio Management (Summer 1986), pp. 9-22 and then in 1990 by Paul DeRossa, Laurie Goodman, and Mike Zazzarino, “Duration Estimates on Mortgage-Backed Securities,” Journal of Portfolio Management (Winter 1993), pp. 32-37, and more recently in Laurie S. Goodman and Jeffrey Ho, “Mortgage Hedge Ratios: Which One Works Best?” The Journal of Fixed Income (December 1997), pp. 23-33, and Laurie S. Goodman and Jeffrey Ho, “An Integrated Approach to Hedging and Relative Value Analysis,” Chapter 15 in Frank J. Fabozzi (ed.), Advances in the Valuation and Management of Mortgage-Backed Securities (New Hope, PA: Frank J. Fabozzi Associates, 1999).
170 Golub, “Towards a New Approach to Measuring Mortgage Duration,” p. 672.
171 Golub, “Towards a New Approach to Measuring Mortgage Duration.”
172 For a further discussion of margin requirements and illustrations of how the margin account changes as the futures price changes, see Don M. Chance, Analysis of Derivatives for the CFA Program (Charlottesville, VA: Association for Investment Management and Research, 2003), pp. 86-91.
173 The conversion factor is based on the price that a deliverable bond would sell for at the beginning of the delivery month if it were to yield 6%.
174 Remember that the short can always unwind his position by buying the same futures contract before the settlement date.
175 Actually, the cost of the investment should be adjusted because the amount that the investor ties up in the investment is reduced if there is an interim coupon payment. We will ignore this adjustment here.
176 Exchanges have developed put and call options issued by their clearinghouse that are customized with respect to expiration date, exercise style, and strike price. These options are called flexible exchange options and are nicknamed “Flex” options.
177 There are well-established institutional arrangements for mitigating counterparty risk in not only OTC options but also the other OTC derivatives described in this chapter (swaps, caps, and floors). These arrangement include limiting exposure to a specific counterparty, marking to market positions, collateralizing trades, and netting arrangement. For a discussion of these arrangements, see Chance, Analysis of Derivatives for the CFA Program, pp. 595-598.
178 In the next chapter we will fine tune our calculation to take into consideration day count conventions when computing swap payments.
179 Don’t get confused here about the role of commercial banks. A bank can use a swap in its asset/liability management. Or, a bank can transact (buy and sell) swaps to clients to generate fee income. It is in the latter sense that we are discussing the role of a commercial bank in the swap market here.
180 More specifically, an interest rate swap is equivalent to a package of forward rate agreements. A forward rate agreement (FRA) is the over-the-counter equivalent of the exchange-traded futures contracts on short-term rates. Typically, the short-term rate is LIBOR. The elements of an FRA are the contract rate, reference rate, settlement rate, notional amount, and settlement date.
181 Robert F. Kopprasch, John Macfarlane, Daniel R. Ross, and Janet Showers, “The Interest Rate Swap Market: Yield Mathematics, Terminology, and Conventions,” Chapter 58 in Frank J. Fabozzi and Irving M. Pollack (eds.), The Handbook of Fixed Income Securities (Homewood, IL: Dow Jones-Irwin, 1987).
182 It is common for market participants to refer to one leg of a swap as the “funding leg” and the other as the “asset leg.” This jargon is the result of the interpretation of a swap as a leveraged position in the asset.
The payment of the floating-rate is referred to as the “funding leg” and the fixed-rate side is referred to as the “asset side.”
183 Interest rate caps and floors can be combined to create an interest rate collar. This is done by buying an interest rate cap and selling an interest rate floor. The purchase of the cap sets a maximum rate; the sale of the floor sets a minimum rate. The range between the maximum and minimum rate is called the collar.
184 Note that there are no interim coupon payments to be considered for potential reinvestment income because we assume that the next coupon payment is six months from the time the strategy is implemented.
185 Note that the short seller must pay the party from whom the bond was borrowed any coupon payments that were made. In our illustration, we assumed that the next coupon payment would be in six months so there are no coupon payments. However, the short seller must pay any accrued interest. In our illustration, since the investor purchases the underlying bond for $96 plus accrued interest, the investor has paid the accrued interest. When the bond is delivered to cover the short position, the bond includes the accrued interest. So, no adjustment to the arbitrage profit is needed in our illustration to take accrued interest into account.
186 As explained in the previous chapter, this is the option granted to the short in the futures contract to select from among the eligible issues the one to deliver.
187 As explained in the previous chapter, the timing option is the option granted to the short to select the delivery date in the delivery month. The wild card option is the option granted to the short to give notice of intent to deliver after the closing of the exchange on the date when the futures settlement price has been fixed.
188 We discussed forward rates in earlier chapters. The reason that we refer to “forward rates” rather than “3-month Eurodollar futures rates” is because we will be developing generic formulas that can be used regardless of the reference rate for the swap. The formulas we present later in this chapter will be in terms of “forward rate for the period” and “period forward rate.”
189 The formula is derived as follows. The fixed-rate payment for period
t is equal to:
The present value of the fixed-rate payment for period
t is found by multiplying the previous expression by the forward discount factor for period
t (FDF
t ). That is, the present value of the fixed-rate payment for period
t is equal to:
Summing the present values of the fixed-rate payment for all periods gives the present value of the fixed-rate payments. Letting
N equal the number of periods in the swap, then the present value of the fixed-rate payments can be expressed as:
The condition for no arbitrage is that the present value of the fixed-rate payments as given by the expression above is equal to the present value of the floating-rate payments. That is,
Solving for the swap rate gives the formula in the text.
190 These include amortizing swaps (swaps where the notional amount declines over time) and accreting swaps (swaps where the notional amount increases over time).
191 The Black-Scholes option pricing formula is:
C = SN (d1 ) − Xe−rt N (d2 )
where
d1 =
d2 =
ln = natural logarithm
C = call option price
S = current stock price
X = strike price
r = short-term risk-free interest rate
e = 2
.718 (natural antilog of 1)
t = time remaining to the expiration date(measured as a fraction of a year)
s = standard deviation of the stock returns
N (
.) = the cumulative probability density. The value for N(
.) is obtained from a normal
distribution function.
192 The value is derived as follows. The current price of $83.96 is the present value of the maturity value of $100 discounted at 6% (assuming a flat yield curve). We know
S = 83.96 X = 88.00 t = 2 s = 0.10 r = 0.06
Substituting these values into the formula in the previous footnote:
193 While we have illustrated the problem of using the Black-Scholes model to price interest rate options, it can also be shown that the binomial option pricing model based on the price distribution of the underlying bond suffers from the same problems. 12 Fischer Black, Emanuel Derman, and William Toy, “A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options,” Financial Analysts Journal (January-February 1990), pp. 24-32.
194 Notice that the binomial interest rate tree shown in
Exhibit 13a is the same as
Exhibit 5 in Chapter 9. In that chapter, the yield curve used was assumed to be that of the issuer’s yield curve. In the current chapter, we are assuming it is the Treasury yield curve.
195 Fischer Black, “The Pricing of Commodity Contracts,”
Journal of Financial Economics (March 1976), pp. 161-179. The value of a call and put based on the Black model is:
C = e−rt [FN (d1) − XN (d2 )]
P = e−rt [XN (−d2) − FN (−d1)]
where
ln = natural logarithm
C = call option price
P = put option price
F = futures price
X = strike price
r = short-term risk-free rate
e = 2.718(natural antilog of 1)
t = time remaining to the expiration date(measured as a fraction of a year)
s = standard deviation of the return
N (.) = the cumulative probability density. The value forN (.) is obtained from a normal
distribution function.
196 Profeesor Donald Smith of Boston University provides a framework for the adjustment. A presentation using his framework is provided in Chapter 28 of Frank J. Fabozzi,
Bond Markets, Analysis, and Strategies (Upper Saddle River, NJ: Pearson Prentice Hall, 2006).
16Mathematically, the decision at a node is expressed as follows:
$10, 000, 000 × Maximum[(Rate at node − 5.2%), 0]
197 Mathematically, the decision to exercise at a node is expressed as follows:
$10, 000, 000 × Maximum [(4.8% − Rate at node), 0]
198 These types of credit risk were discussed in detail earlier.
199 Counterparty risk is the risk that a party to a financial transaction will default on its obligation.
200 There are several ways that default rates can be measured. These are described in Chapter 3. 4The default loss rate is described in Chapter 3.
201 David T. Hamilton and Richard Cantor, Rating Transitions and Defaults Conditional on Watchlist, Outlook and Rating History, Moody’s Investors Service, February 2004. 6A rating “notch” is a rating based on the modified rating (i.e., in the case of Moody’s with the “1”, “2”, and “3” modifiers). For example, if an issue is rated Baa2, then a reduction of one rating notch would be a rating of Baa3. A reduction of two rating notches would be a rating of Ba1.
202 An illustration of a rating transition matrix and the calculation of the probability of downgrade were provided.
203 “Industrial Company Rating Methodology,” Moody’s Investors Services: Global Credit Research (July 1998), p. 3.
204 For example, a firm that produces and sells goods has an operating cycle comprising of four phases: (1) purchase of raw material and produce goods, investing in inventory; (2) sell goods, generating sales, which may or may not be for cash; (3) extend credit, creating accounts receivable; and, (4) collect accounts receivable, generating cash. The four phases make up the cycle of cash use and generation. The operating cycle would be somewhat different for companies that produce services rather than goods, but the idea is the same—the operating cycle is the length of time it takes to generate cash through the investment of cash.
205 J.A. Largay III and C.P. Stickney, “Cash Flows, Ratio Analysis and the W.T. Grant Company Bankruptcy,” Financial Analysts Journal (July-August 1980), pp. 51-54. 11 For the period investigated, a statement of changes of financial position (on a working capital basis) was required prior to 1988. 12Standard & Poor’s, Corporate Ratings Criteria, undated, p. 26. 13Corporate Ratings Criteria, p. 27.
206 Some firms use different labels. For example, Microsoft refers to these cash flows as: Net cash from operations, Net cash used for financing, and Net cash used for investing.
207 Corporate Ratings Criteria, p. 27.
208 One of the most popular measures of cash flow in equity analysis is the “free cash flow.” This cash flow measure is defined as “the cash flow available to the company’s suppliers of capital after all operating expenses (including taxes) have been paid and necessary investments in working capital (e.g., inventory) and fixed capital (e.g., equipment) have been made.” (See, John D. Stowe, Thomas R. Robinson, Jerald E. Pinto, and Dennis W. McLeavey, Analysis of Equity Investments: Valuation (Charlottesville, VA: Association for Investment Management and Research, 2002), p. 115.) Analysts will make different adjustments to the statement of cash flows to obtain the free cash flow depending on the accounting information that is available. The procedure for calculating free cash flow starting with net income or statement of cash flows is explained in Stowe, Robinson, Pinto, and McLeavey, Analysis of Equity Investments: Valuation, pp. 119-124.
209 Corporate Ratings Criteria, p. 27.
210 Corporate Ratings Criteria, p. 27.
211 “Industrial Company Rating Methodology,” p. 6.
212 Examples include Xerox, which was forced to restate earnings for several years because it had inflated pre-tax profits by $1.4 billion, Enron, which is accused of inflating earnings and hiding substantial debt, and Worldcom, which failed to properly account for $3.8 billion of expenses.
213 The seminal paper on the agency-principal relationship in corporate finance is Michael Jensen and William Meckling, “Theory and the Firm: Managerial Behavior, Agency Costs and Ownership Structure,” Journal of Financial Economics (October 1976), pp. 305-360.
214 Michael Jensen, “The Modern Industrial Revolution, Exist, and the Failure of Internal Control Systems,” in Donald H. Chew, Jr. (ed.), The New Corporate Finance: Second Edition (New York, NY: McGraw-Hill, 1999).
216 McKinsey & Company’s Global Investor Opinion Survey, May 2002.
217 For a review of the impact of various corporate governance mechanism on shareholder returns, as well as the experiences and perspective of the California Public Employee Pension Fund (CalPERS), see Chapter 22 in Mark J.P. Anson, Handbook of Alternative Assets (Hoboken, NJ: John Wiley & Sons, 2002).
218 Sanjeev Bhojraj and Partha Sengupta, “Effect of Corporate Governance on Bond Ratings and Yields: The Role of Institutional Investors and Outside Directors,” Journal of Business, Vol. 76, No. 3 (2003), pp. 455-476.
219 M. Beasley, “An Empirical Analysis of the Relation Between the Board of Director Composition and Financial Statement Fraud,” Accounting Review (October 1996), pp. 443-465.
220 Partha Sengupta, “Corporate Disclosure Quality and the Cost of Debt,” Accounting Review (October 1998), pp. 459-474.
221 Howard Sherman, “Corporate Governance Ratings,” Corporate Governance (January 2004), p. 6.
222 Sherman, “Corporate Governance Ratings,” p. 5.
223 Standard & Poor’s, Corporate Governance Evaluations & Scores, undated, p. 2.
224 Corporate Governance Evaluations & Scores, p. 2.
226 Indicators that best practice might suggest such as a split chairman of the board/CEO role or a lead independent director are not considered by TCL because the firm does not believe they are significant in improving board effectiveness.
227 Duff & Phelps was acquired by Fitch.
228 William A. Cornish, “Unique Factors in the Credit Analysis of High-Yield Bonds,” in Frank K. Reilly (ed.), High-Yield Bonds: Analysis and Risk Assessment (Charlottesville, VA: Association for Investment Management and Research, 1990).
229 Jane Tripp Howe, “Credit Considerations in Evaluating High-Yield Bonds,” Chapter 21 in Frank J. Fabozzi (ed.), Handbook of Fixed Income Securities (Burr Ridge, IL: Irwin Professional Publishing, 1997), p. 408.
230 Robert Levine, “Unique Factors in Managing High-Yield Bond Portfolios,” in High-Yield Bonds, p. 35.
231 Kenneth S. Choie, “How to Hedge a High-Yield Bond Portfolio,” Chapter 13 in Frank J. Fabozzi (ed.), The New High-Yield Debt Market (New York, NY: HarperBusiness, 1990).
232 Stephen F. Esser, “High-Yield Bond Analysis: The Equity Perspective,” in Ashwinpaul C. Sondhi (ed.), Credit Analysis of Nontraditional Debt Securities (Charlottesville, VA: Association for Investment Management and Research, 1995), p. 47.
233 Esser, “High-Yield Bond Analysis: The Equity Perspective,” p. 54.
234 Suzanne Michaud, “A Rating Agency Perspective on Asset-Backed Securities,” Chapter 16 in Anand K. Bhattacharya and Frank J. Fabozzi (eds.), Asset-Backed Securities (New Hope, PA: Frank J. Fabozzi Associates, 1997).
235 In the chapter on asset-backed securities, the role of the attorneys in a transaction was described.
236 Standard & Poor’s, “Rating Hybrid Securitizations,” Structured Finance (October 1999), p. 2.
237 “Rating Hybrid Securitizations,” p. 3.
238 “Rating Hybrid Securitizations,” p. 3.
239 “Rating Hybrid Securitizations,” p. 3.
240 For municipalities, the debt burden usually is composed of the debt per capita as well as the debt as percentages of real estate valuations and personal incomes.
241 Sylvan G. Feldstein and Frank J. Fabozzi, The Dow Jones-Irwin Guide to Municipal Bonds (Homewood, IL: Dow Jones-Irwin, 1987), p. 72.
242 David T. Beers and Marie Cavanaugh, “Sovereign Ratings: A Primer,” Chapter 6 in Frank J. Fabozzi and Alberto Franco (eds.), Handbook of Emerging Fixed Income & Currency Markets (New Hope, PA: Frank J. Fabozzi Associates, 1997).
243 Beers and Cavanaugh, “Sovereign Credit Ratings: A Primer,” p. 68.
244 See Chapters 8 and 9 in Edward I. Altman, Corporate Financial Distress and Bankruptcy: A Complete Guide to Predicting and Avoiding Distress and Profiting from Bankruptcy (Hoboken, NJ: John Wiley & Sons, 1993). For discussion of MDA applied to predicting municipal bond ratings, see Michael G. Ferri and Frank J. Fabozzi, “Statistical Techniques for Predicting the Credit Worthiness of Municipal Bonds,” Chapter 44 in Frank J. Fabozzi, Sylvan G. Feldstein, Irving M. Pollack, and Frank G. Zarb (eds.), The Municipal Bond Handbook: Volume I (Homewood, IL: Dow-Jones Irwin 1983).
245 Edward I. Altman, “Financial Bankruptcies, Discriminant Analysis and the Prediction of Corporate Bankruptcy,” Journal of Finance (September 1968), pp. 589-699.
246 Edward I. Altman, Robert G. Haldeman, and Paul Narayann, “Zeta Analysis: A New Model to Identify Bankruptcy Risk of Corporations,” Journal of Banking and Finance (June 1977), pp. 29-54.
247 Martin S. Fridson, Financial Statement Analysis: A Practitioner’s Guide, Second Edition (Hoboken, NJ: John Wiley & Sons, 1995), p. 195.
248 Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy (May-June 1973), pp. 637-654.
249 Robert Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management (Spring 1973), pp. 141-183, and “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance, Vol. 29, No. 2 (1974), pp. 449-470.
250 For a more detailed discussion of these models, see Chapter 8 in Mark J.P. Anson, Frank J. Fabozzi, Moorad Choudhry, and Ren Raw Chen, Credit Derivatives: Instruments, Applications, and Pricing (Hoboken, NJ: John Wiley & Sons, 2004).
251 For the underlying theory and an illustration, see Don M. Chance, Analysis of Derivatives for the CFA Program (Charlottesville, VA: Association for Investment Management and Research, 2003), pp. 588-591.
252 The name “reduced form” was first given by Darrell Duffie to differentiate these models from the structural form models of the Black-Scholes-Merton type.
253 Robert Jarrow and Stuart Turnbull, “Pricing Derivatives on Financial Securities Subject to Default Risk,” Journal of Finance, Vol. 50, No. 1 (1995), pp. 53-86.
254 Darrell Duffie and Kenneth Singleton, “Modeling Term Structures of Defaultable Bonds,” Review of Financial Studies, Vol. 12 (1999), pp. 687-720.
255 For a further discussion of reduced form models, see Chapter 9 in Anson, Fabozzi, Choudhry, and Chen, Credit Derivatives: Instruments, Applications, and Pricing ; Darrell Duffie and Kenneth J. Singleton, Credit Risk: Pricing Measurement, and Management (Princeton, NJ: Princeton University Press, 2003), or Srichander Ramaswamy, Managing Credit Risk in Corporate Bond Portfolios (Hoboken, NJ: John Wiley & Sons, 2003).
256 John L. Maginn and Donald L. Tuttle, “The Portfolio Management Process and Its Dynamics,” Chapter 1 in John L. Maginn and Donald L. Tuttle (editors), Managing Investment Portfolios: A Dynamic Process (New York: Warren, Gorham & Lamont, sponsored by the Institute of Chartered Financial Analysts, Second Edition, 1990), pp. 1-3 and 1-5.
257 Jeffrey V. Bailey, Thomas M. Richards, and David E. Tierney, “Benchmark Portfolios and the Manager /Plan Sponsor Relationship,” in Frank J. Fabozzi (ed.), Current Topics in Investment Management (New York, NY: Harper & Row Publishers, 1990), p. 70.
258 For example, for an equity benchmark, categorization might be value, growth, capitalization in terms of size. For a bond benchmark it might be the major sectors of the bond index. 4Frank K. Reilly and David J. Wright, “Bond Indexes,” Chapter 7 in Frank J. Fabozzi (ed.), The Handbook of Fixed Income Securities (New York: McGraw-Hill, 2000).
259 This is the same example used at earlier. Bond-equivalent basis is computed by doubling the semiannual yield.
6To see this, suppose $100,000 is invested in each alternative. For the 1-year Treasury bill alternative, the manager earns 1.65% (one half the 1-year rate of 3.3%) each 6-month period for two periods (i.e., one year). The total dollars at the end of one year will be:
$100,000 (1.0165)2 = $103, 327
For the 6-month Treasury bill alternative, the manager invests $100,000 for six months at 1.5% (one half the annual rate) and then reinvests the proceeds for another six months at 1.8% (one half the 3.6% rate). The total dollars at the end of one year will be
$100,000 (1.015)(1.018) = $103,327.
Thus, both alternatives provide the same future value.
260 Chapter 20 in this book.
261 Harry M. Markowitz, “Portfolio Selection,” Journal of Finance (March 1952), pp. 71-91.
262 Many of the concepts are covered in Richard A. DeFusco, Dennis W. McLeavey, Jerald E. Pinto, and David E. Runkle, Quantitative Methods for Investment Analysis (Charlottesville, VA: Association for Investment Management and Research, 2001).
263 For a further discussion of skewed distributions, see Chapter 3 in DeFusco, McLeavey, Pinto, and Runkle, Quantitative Methods for Investment Analysis, pp. 138-144.
264 Tests for serial correlation are explained and illustrated in Chapter 9 of DeFusco, McLeavey, Pinto, and Runkle, Quantitative Methods for Investment Analysis, pp. 450-453.
265 RiskMetrics™—Technical Document, JP Morgan, May 26, 1995, New York p. 48.
266 RiskMetrics™—Technical Document, p. 48.
267 Ronald N. Kahn, “Fixed Income Risk,” Chapter 1 in Frank J. Fabozzi (ed.), Managing Fixed Income Portfolios (New Hope, PA: Frank J. Fabozzi Associates, 1997), pp. 2-3.
268 For a further discussion of the measures of shortfall risk discussed here, see DeFusco, McLeavey, Pinto, and Runkle, Quantitative Methods for Investment Analysis, pp. 253-256.
269 Kahn, “Fixed Income Risk,” p. 3.
270 For a further discussion of value at risk, see Chapter 9 in Don M. Chance, Analysis of Derivatives for the CFA Program (Charlottesville, VA: Association for Investment Management and Research, 2003), pp. 576-578.
271 The calculation of the portfolio variance is described in Chapter 11 of DeFusco, McLeavey, Pinto, and Runkle, Quantitative Methods for Investment Analysis.
272 The formula for determining the number of variances and covariances is found as follows: [Number of bonds × (Number of bonds + 1)]/2.
273 See, for example, Example 3-8 in DeFusco, McLeavey, Pinto, and Runkle,
Quantitative Methods for Investment Analysis, p. 133. The format in
Exhibit 2 uses the format shown in that book.
274 I thank Vadim Konstantinovsky, CFA and Lev Dynkin of the Quantitative Portfolio Strategy Group in Fixed Income Research at Lehman Brothers for providing not only the sample portfolio and its characteristics, but all of the analysis of the portfolio.
275 The author uses this term. See also DeFusco, McLeavey, Pinto, and Runkle, Quantitative Methods for Investment Analysis, p. 236 and the glossary to the book, p. 657.
276 The cash flows are obtained as follows. For the option-free bonds in the portfolio and the benchmark index, the cash flows are to the maturity date. For the callable bonds in the portfolio and the benchmark index, a “cash flow to adjusted duration” was computed. This is a weighted blend of a cash flow to maturity and to call. The weight is selected so that the blended cash flow has a duration equal to the option-adjusted duration. For the MBS in the portfolio and the benchmark, a “cash flow to worst” is a zero volatility cash flow produced using a single path of interest rates from the current yield curve and the Lehman Brothers prepayment model. The yield of each security is used in computing the present value of any of the cash flows for the security.
277 In theory, there is a rate duration for every maturity. In practice, a rate duration is computed for certain “key” maturities. These durations are called key rate durations.
278 Calculation of the OAS was covered earlier.
279 The concept of an option delta is described in several chapters in Chance, Analysis of Derivatives for the CFA Program.
280 These securities were described earlier.
281 The Public Securities Association is now the Bond Market Association.
282 These types of mortgage passthrough securities were described earlier.
283 For a discussion of factor models, see Chapter 11 of DeFusco, McLeavey, Pinto, and Runkle, Quantitative Methods for Investment Analysis, pp. 598-618.
284 This model is described in Lev Dynkin, Jay Hyman, and Wei Wu, “Multi-Factor risk Models and Their Applications,” in Frank J. Fabozzi (ed.), Professional Perspectives on Fixed Income Portfolio Management: Volume 2 (New Hope, PA: Frank J. Fabozzi Associates, 2001).
285 Phil Galdi, Mark Anderson, and Arjun Kondamani, U.S. High Grade Bond Index, Merrill Lynch (December 5, 2001).
286 Kenneth E. Volpert, “Managing Indexed and Enhanced Indexed Bond Portfolios,” Chapter 3 in Frank J. Fabozzi (ed.), Fixed Income Readings for the Chartered Financial Analyst Program: First Edition (New Hope, PA: Frank J. Fabozzi Associates, 2000). Volpert uses the term “active management/full-blown active” instead of “unrestricted active management.” I have changed the classification based on the suggestion of David Zahn, CFA, of Citigroup Asset Management.
287 Actually, some might say that the manager may be constrained to be within “±1 year” of the duration of the index. However, as emphasized earlier, characterizing duration in temporal terms (i.e., years) should be avoided.
288 “Non-corporate issues” is Lehman Brothers’ terminology for non-U.S. corporates.
289 The total return is also referred to as the horizon return. 5An investor can use multiple reinvestment rates for cash flows from the bond over the investment horizon.
290 Some broker/dealers, vendors of analytical systems, and regulators refer to scenario analysis as “simulation” even though the two techniques are not equivalent. Simulation is a more powerful tool that takes into consideration the dynamics of interactions of the factors. For a discussion of Monte Carlo simulation applied to common stock portfolio management, see Chapter 5 in Richard A. DeFusco, Dennis W. McLeavey, Jerald E. Pinto, and David E. Runkle, Quantitative Methods for Investment Analysis (Charlottesville, VA: Association for Investment Management and Research, 2001), pp. 261-266.
291 Frank J. Fabozzi and Gifford Fong, Advanced Fixed Income Portfolio Management (Chicago, IL: Probus Publishing, 1994), p. 281. 8For an explanation and illustration of dollar-weighted rate of return and time-weighted rate of return, see Chapter 2 of DeFusco, McLeavey, Pinto, and Runkle, Quantitative Methods for Investment Analysis, pp. 81-90.
292 These measures are described in Frank K. Reilly and Keith C. Brown, Investment Analysis and Portfolio Management (South-Western College Publishing, 2002), pp. 1109-1117.
293 The illustration is provided by Frank Jones and Leonard Peltzman of Guardian Life. A model developed by another vendor is presented in one of the end of chapter questions.
294 A special type of repurchase agreement used in the mortgage-backed securities market is called a “dollar roll.” For a description of a dollar roll, see Chapter 9 in Frank J. Fabozzi and David Yuen, Managing MBS Portfolios (Hoboken, NJ: John Wiley & Sons, 1998).)
295 The classical theory of immunization is set forth in F.M. Reddington, “Review of the Principles of Life Insurance Valuations,” Journal of the Institute of Actuaries, 1952; and Lawrence Fisher and Roman Weil, “Coping with Risk of Interest Rate Fluctuations: Returns to Bondholders from Naive and Optimal Strategies,” Journal of Business (October 1971), pp. 408-431.
296 The value in this column is found as follows:
297 This value is found as follows: $10,004,508 (1 + new yield/2)9.
298 There is a mathematical proof that shows this. We will not present the proof here. In the illustration shown in
Exhibits 6 and
7, at 8% the accumulated value is slightly less than the target ($13,934,180 versus $13,934,413). This difference is due to rounding.
5The duration of a zero-coupon liability is equal to the number of years to maturity of the liability divided by 1 plus one-half the yield. In our illustration, it is 5 divided by (1 + 0
.075/2).
299 Fisher and Weil, “Coping with Risk of Interest Rate Fluctuations: Returns to Bondholders from Naive and Optimal Strategies.”
300 For a more complete discussion of these issues, see John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross, “Duration and the Measurement of Basis Risk,” Journal of Business (January 1979), pp. 51 - 61.
301 H. Gifford Fong and Oldrich A Vasicek, “A Risk Minimizing Strategy for Portfolio Immunization,” Journal of Finance (December 1984), pp. 1541-1546.
302 Fong and Vasicek, “A Risk Minimizing Strategy for Portfolio Immunization.”
303 More specifically, linear programming can be employed because the risk measure is linear in the portfolio payments.
304 Martin L. Leibowitz and Alfred Weinberger, “Contingent Immunization—Part I: Risk Control Procedures,” Financial Analysts Journal (November-December 1982), pp. 17-31.
305 Martin L. Leibowitz and Alfred Weinberger, “Contingent Immunization—Part II: Problem Areas,” Financial Analysts Journal (January-February 1983).
306 G.O. Bierwag, George G. Kaufman, and Alden Toevs, “Immunization for Multiple Planning Periods,” unpublished paper, Center for Capital Market Research, University of Oregon, October 1979.
307 H. Gifford Fong and Oldrich A Vasicek, “A Risk Minimizing Strategy for Portfolio Immunization,” Journal of Finance (December 1984), pp. 1541-1546.
308 Robert R. Reitano, “A Multivariate Approach to Immunization Theory,” Actuarial Research Clearing House, Vol. 2 (1990), and “Multivariate Immunization Theory,” Transactions of the Society of Actuaries, Vol. XLIII, 1991. For a detailed illustration of the relationship between the underlying yield curve shift and immunization, see Robert R. Reitano, “Non-Parallel Yield Curve Shifts and Immunization,” Journal of Portfolio Management (Spring 1992), pp. 36-43.
309 In a defined benefit plan, the projected benefit payments estimated by an actuary represent the legal obligation of the plan sponsor.
310 Ronald Ryan and Frank J. Fabozzi, “Rethinking Pension Liabilities and Asset Allocation,” Journal of Portfolio Management (Summer 2002), pp. 7-15.
311 Moreover, as of the end of 2003, the corporate bond index yield curves that had been advocated are not good proxies for corporate bond market yields. For a more detailed discussion of the problems with this approach, see Ronald Ryan and Frank J. Fabozzi, “Pension Fund Crisis Revealed,” Journal of Investing (Fall 2003), pp. 43-48.
312 This chapter is authored by Jack Malvey, CFA.
313 Based on absolute returns of key Lehman indices from 1973.
314 Credit derivatives are discussed in Chapter 24.
315 Shortfall risk is explained in Chapter 17. Shortfall risk is the probability that the outcome will have a value less than the target return.
316 These sectors are referred to as “vol” sectors because the value of the securities depends on expected interest rate volatility. These “vol” securities have embedded call options and the value of the options, and hence the value of the securities, depends on expected interest rate volatility.
317 The model is referred to as a one-factor model because only the short-term rate is the factor used to construct the tree.
318 Recall that the longer the maturity, the greater the convexity.
319 Credit derivatives are covered in Chapter 24.
320 This chapter is authored by Christopher B. Steward, CFA, J. Hank Lynch, CFA, and Frank J. Fabozzi, PhD, CFA, CPA.
321 Some investors were concerned that the diversification benefits of global bond investing would be substantially diminished by the commencement of European Monetary Union (EMU) in 1999. But, in fact, the economies of continental Europe were already very closely tied together before EMU with most European central banks following the interest rate policies of the German Bundesbank for several years before the move to a single currency. Thus, the impact on diversification of a global bond portfolio caused by EMU has been a small one. EMU, however, has created a much more robust credit market in Europe as issuers and investors, no longer confined to their home markets, have access to a larger, more liquid pan-European bond market. Corporate bond issuance has increased sharply in Europe, and seems likely to continue, building toward a broader range of credits and instruments similar to those available in the U.S. bond market. This was discussed in Chapter 20.
322 The Japanese bond market has historically offered lower yields than most other bond markets.
323 While in the equity market where growth in a company’s market capitalization generally indicates financial strength, a company or country that issues a large amount of debt (especially relative to its equity in the case of a company or gross national product in the case of a country) may find itself in a more precarious financial position.
324 See Philippe Jorion, “Mean/Variance Analysis of Currency Overlays,” Financial Analysts Journal (May/June 1994), pp. 48 - 56. Jorion argues that currency overlays, although they can add value, are inferior to an integrated approach to currency management.
325 See Gary L. Gastineau, “The Currency Hedging Decision: A Search for Synthesis in Asset Allocation,” Financial Analysts Journal (May-June 1995), pp. 8-17 for a broad overview of the currency hedging debate. For a full discussion of the benefits of utilizing a partially hedged benchmark, see the currency discussion in Steve Gorman, The International Equity Commitment (Charlottesville, VA: AIMR 1998).
326 Recall from modern portfolio theory the important role of correlation in determining the benefits from diversification.
327 The Sharpe ratio measures returns in excess of the risk-free rate, per unit of standard deviation.
328 Technical analysis is covered in equity management textbooks.
329 See Gastineau, “The Currency Hedging Decision,” pp. 13-14.
330 One suggestion as to why currency markets trend is that central banks attempt to smooth foreign exchange rate movements through intervention. Thus, because central bank participation in the foreign exchange market is not motivated by profit, their actions keep the market from being truly efficient. See Robert D. Arnott and Tan K. Pham, “Tactical Currency Allocation,” Financial Analysts Journal, (May/June 1993) pp. 47-52.
331 See Richard M. Levich and Lee R. Thomas, “The Merits of Active Currency Risk Management: Evidence from International Bond Portfolios,” Financial Analysts Journal, (September/October 1993) pp. 63-70.
332 See Chapter 18 for a discussion of this strategy and how it is analyzed.
333 See “OECD Public Debt Markets: Trends and Recent Structural Changes” OECD 2002.
334 The JP Morgan EMBI + Index is comprised of mostly U.S. dollar-denominated sovereign debt issued by emerging market countries. Therefore, credit risk, and to a lesser extent interest rate risk, are the predominate risks associated with the index. For a U.S. investor, currency risk is virtually zero.
335 However, tactical allocations can also be momentum following, especially if a breakout of a technical range appears likely. Again, such technical strategies are discussed in investment management textbooks.
336 Inflation-indexed bonds were explained earlier.
337 Nominal yield to maturity is composed of a real yield and an inflation expectations component (Yield to Maturity = Real Yield to Maturity + Expected Inflation to Maturity). In these markets the nominal government bond yield and the real yield offered by inflation-indexed debt of the same maturity can be used to calculate the expected inflation rate to the maturity, sometimes called the breakeven inflation rate.
338 The structure of this discussion is taken from Brian D. Singer and Denis S. Karnosky, The General Framework for Global Investment Management and Performance Attribution (Charlottesville, VA: The Research Foundation of the Institute of Chartered Financial Analysts, 1994). The notation used is consistent with that of the authors.
339 The relationship in
equation (1) is approximate because bond market and currency returns of a foreign investment is more accurately expressed as the compounded gain of the two components: (1 +
ri ) × (1 +
e$, i ) - 1.
340 For an explanation of currency swaps, see Chapter 5 in Don M. Chance, Analysis of Derivatives for the CFA Program (Charlottesville, VA: Association for Investment Management and Research, 2003).
341 The forward rate can also be derived by looking at the alternatives from the perspective of a portfolio manager in country i.
342 A portfolio manager in any country, not just one located in one of the countries whose currency is mispriced, could take advantage of this arbitrage opportunity.
343 As investors move to exploit this arbitrage opportunity, their very actions will serve to eliminate it. This can occur through the combination of a number of factors: (1) the U.S. dollar will depreciate relative to the local currency (i.e., the spot exchange rate expressed in U.S. dollars per local currency unit will rise) as investors sell dollars and buy the local currency; (2) interest rates will rise in the U.S. as investors borrow in the U.S. and invest in country i; (3) interest rates in country i will fall as more is invested in country i; and (4) the 1-year forward rate for U.S. dollars will show an appreciation relative to the local currency (i.e., the forward exchange rate expressed in U.S. dollars per local currency unit will fall) to eliminate the arbitrage opportunity as investors buy U.S. dollars forward. In practice, the last factor will dominate.
344 Equation (2) assumes that exchange rates are quoted in “direct terms,” i.e., the value of the home currency for one unit of the local currency, though quote conventions vary by market. Over-the-counter forward contracts use market convention, most of which for the U.S. dollar are in indirect terms (local currency units per one dollar). Using indirect terms, the forward discount
or premium in
equation (3) becomes
fH,i =
ci -
cH . To avoid the complexities of compounding, the time period is assumed to be one year.
345 The derivation of the relationships presented in this section are provided in the appendix to this chapter.
346 This is the modified duration for the issue. Since the Japanese bond and the U.S. bonds are option-free, the modified duration is close to the effective duration. Duration is only a first approximation of the approximate change in value when interest rates change. By only considering duration, the analysis above ignores the impact of convexity on returns.
347 This chapter is authored by Frank J. Fabozzi, PhD, CFA, CPA, Shrikant Ramamurthy, and Mark Pitts, PhD.
348 The cheapest-to-deliver issue is the one issue from among all those that are eligible for delivery on a contract that has the highest return in a cash and carry trade. This return is called the implied repo rate.
349 When hedging more than just the level of interest rates (e.g., hedging changes in the slope also), more than one hedging instrument is used. One of the hedging instruments could require a long position even though the instrument to be hedged is a long position.
350 These delivery options were explained earlier.
351 Forward rates were covered earlier.
352 Whether the size of the spread is adequate is not an issue to us in this illustration.
353 Futures options on Treasury bonds are more commonly used by institutional investors. The mechanics of futures options are as follows. If a put option is exercised, the option buyer receives a short position in the underlying futures contract and the option writer receives the corresponding long position. The futures price for both positions is the strike price for the put option. The exchange then marks the positions to market and the futures price for both positions is then the current futures price. If a call option is exercised, the option buyer receives a long position in the underlying futures contract and the option writer receives the corresponding short position. The futures price for both positions is the strike price for the call option. The exchange then marks the positions to market and the futures price for both positions is then the current futures price.
354 Note that this is based on the risk-return profile the manager is willing to accept. It is not derived analytically in this illustration.
355 This chapter is authored by Kennath B. Dunn, PhD, Roberto M. Sella and Frank J. Fabozzi, PhD, CFA, CPA.
356 In European countries where mortgage-backed securities are issued, the coupon rate is typically a floating rate.
357 The cash flow yield is the interest rate (properly annualized) that makes the present value of the projected cash flows from a mortgage-backed security equal to its price.
358 This perception is exacerbated by the common practice of comparing the returns of the mortgage index with the returns of the government and corporate indices without adjusting for differences in duration. Because the mortgage index typically has less duration than either the corporate or government index, it generally has better relative performance when interest rates rise than when interest rates fall.
359 Model risk was discussed earlier.
360 Rate duration was discussed previously.
361 How the buying and selling of futures affects duration is explained in Chapter 22.
362 This was explained earlier.
363 For example, the approach is used by Morgan Stanley as discussed in this chapter and by Smith Breeden Associates (see Michael P. Schumacher, Daniel C. Dektar, and Frank J. Fabozzi, “Yield Curve Risk of CMO Bonds,” Chapter 15 in Frank J. Fabozzi (ed.), CMO Portfolio Management (New Hope, PA: Frank J. Fabozzi Associates, 1994).
364 Scott F. Richard and Benjamin J. Gord, “Measuring and Managing Interest-Rate Risk,” Chapter 7 in Frank J. Fabozzi (ed.), Perspectives on Interest Rate Risk Management for Money Managers and Traders (New Hope, PA: Frank J. Fabozzi Associates, 1998).
365 Treasury futures contracts were discussed at earlier.
366 The results can be verified by substituting these values into the “Level” or “Twist” equation.
367 The dollar value of a basis point, also called the price value of a basis point, is the change in the value of a position for a 1 basis point change in interest rates.
368 This chapter is authored by Mark J.P. Anson, PhD, CFA, CDA, Esq., and Frank J. Fabozzi, PhD, CFA, CPA.
369 British Bankers Association, Credit Derivatives Report 2002. 2David Rule, “The Credit Derivatives Market: Its Development and Possible Implications for Financial Stability,” Financial Stability Review (June 2001), pp. 117-140. 3Edward Altman, “Measuring Corporate Bond Mortality and Performance,” The Journal of Finance (June 1991), pp. 909-922; and Gabriella Petrucci, “High-Yield Review—First-Half 1997,” Salomon Brothers Corporate Bond Research (August 1997).
370 See “Default and Recovery Rates of Corporate Bond Issuers,” Moody’s Investors Services, February 2002.
371 Moody’s Investor Service Global Credit Research, Special Comment, “Historical Default Rates of Corporate Bond Issuers, 1920-1999,” January 2000.
372 “Financial Firms Lose $8 Billion so Far,” The Wall Street Journal (September 3, 1998), p. A2.
373 For a discussion of the factors considered by rating agencies, see Chapter 15.
374 This provision requires the following in order to qualify for a restructuring: (1) there must be four or more holders of the reference obligation and (2) there must be a consent to the restructuring of the reference obligation by a supermajority (66 2/3%). In addition, the supplement limits the maturity of reference obligations that are physically deliverable when restructuring results in a payout triggered by the protection buyer.
375 An increase in interest rates is used because we are looking at the price sensitivity to an increase in the credit spread.
376 See Chapter 15 in Douglas J. Lucas, Laurie S. Goodman, and Frank J. Fabozzi, Collateralized Debt Obligations: Structures and Analysis, Second Edition (Hoboken, NJ: John Wiley & Sons, 2006).
377 The illustration and discussion in this section draws from “Nth to Default Swaps and Notes: All About Default Correlation,” CDO Insight (May 30, 2003) UBS Warburg.
