Additional Praise for Fixed Income Securities Tools for Today’s Markets, 2nd Edition phần 10 pptx

TRADING CASE STUDY: November ’08 Basis into TYMO 453 explanation at the time for the cheapening of the futures contract from April 3, 2000, to April 10, 2000, was that many traders were forced to liquidate short basis positions. Since such liquidations entail selling futures and buying bonds, enough activity of this sort will cheapen the contract relative to bonds. By May 19, 2000, the forward yield curve had returned to the levels of February 28, 2001, but had flattened by between 3 and 4 ba- sis points. This yield curve move restored the 11/08s to CTD and re- duced their net basis to 3.51. Even though the futures contracts returned to their original levels, the options lost most of their time value. The total P&L of the trade to its horizon turned out to be $65,844. Note that this profit is substantially below the predicted P&L of about $153,532. First, the forward yield curve did flatten, making the shorter-maturity bonds closer to CTD than predicted by the parallel shift scenarios. Second, while the model assumed that the futures contract would be fair relative to the bonds on May 19, 2000, it turned out that the contract was still somewhat cheap to cash on that date. A quick way to quantify these effects is to notice that the net basis of the 11/08s on the horizon date was 3.51 while it had been predicted to be close to 1. This difference of 2.51 ticks is worth $100,000,000×( 2.51 / 32 )/100 or $78,438 in P&L. Adding this to the ac- tual P&L of $65,844 would bring the total to $144,282, much closer to the predicted number. By the way, a trader can, at least in theory, capture any P&L shortfall due to the cheapness of the futures con- tract on the horizon date by subsequent trading. Before concluding the case, the tail of this trade is described. By working with the net basis directly the case implicitly assumes that the tail was being managed. The conversion factor of the 11/08s was .9195, so, without the tail, the trade would have purchased about 920 contracts against the sale of $100,000,000 bonds. On February 28, 2000, there were 122 days to the last delivery date, and the repo rate for the 11/08s to that date was 5.55%. Hence, using the rule of Chap- ter 17, the tail was (20.20) 920 0555 122 360 17× × = . 454 NOTE AND BOND FUTURES contracts. In other words, only 920-17 or 903 contracts should have been bought against the bond position. On April 3, 2000, the re- quired tail had fallen to 13 contracts, or, equivalently, the futures po- sition should have increased to 920-13 or 907 contracts. Over that time period the futures price rose from 95-9 to 98-8 1 / 2 , making the tail worth about 2.98 per 100 face of contracts. Assuming an average tail of 15 contracts (i.e., $1,500,000 face), the tail in this trade turned out to be worth $44,765. In other words, had the tail not been man- aged, the P&L of the basis trade would have differed from the bond position times the change in net basis by about $44,765. 455 CHAPTER 21 Mortgage-Backed Securities A mortgage is a loan secured by property. Until the 1970s banks made mortgage loans and held them until maturity, collecting principal and interest payments until the mortgages were paid off. This primary market was the only mortgage market. During the 1970s, however, the securitiza- tion of mortgages began. The growth of this secondary market substan- tially changed the mortgage business. Banks that might otherwise restrict their lending, because of limited capital or because of asset allocation deci- sions, can now continue to make mortgage loans since these loans can be quickly and efficiently sold. At the same time investors have a new security through which to lend their surplus funds. Individual mortgages are grouped together in pools and packaged in a mortgage-backed security (MBS). In a pass-through security, interest and principal payments flow from the homeowner, through banks and servicing agents, to investors in the MBS. The issuers of these securities often guar- antee the ultimate payment of interest and principal so that investors do not have to face the risk of homeowner default. In striving to understand and value mortgage-backed securities, practi- tioners expend a great deal of effort modeling the aggregate behavior of homeowners with respect to their mortgages and analyzing the impact on a wide variety of MBS. This chapter serves as an introduction to this highly developed and specialized field of inquiry. 1 BASIC MORTGAGE MATHEMATICS The most typical mortgage structure is a fixed rate, level payment mort- gage. Say that to buy a home an individual borrows from a bank $100,000 1 For a book-length treatment see Hayre (2001). secured by that home. To pay back the loan the individual agrees to pay the bank $599.55 every month for 30 years. The payments are called level because the monthly payment is the same every month. This structure dif- fers from that of a bond, for example, which makes relatively small coupon payments every period and then makes one relatively large princi- pal payment. The interest rate on a mortgage is defined as the monthly compounded yield-to-maturity of the mortgage. In the example, the interest rate y is de- fined such that (21.1) Solving numerically, y=6%. The intuition behind this definition of the mortgage rate is as follows. If the term structure were flat at y, then the left-hand side of equation (21.1) equals the present value of the mortgage’s cash flows. The mortgage is a fair loan only if this present value equals the original amount given by the bank to the borrower. 2 Therefore, under the assumption of a flat term structure, (21.1) represents a fair pricing condition. Mortgage pricing with- out the flat term structure assumption will be examined shortly. While a mortgage rate can be calculated from its payments, the pay- ments can also be derived from the rate. Let X be the unknown monthly payment and let the mortgage rate be 6%. Then the equation relating X to the rate is (21.2) Applying equation (3.3) to perform the summation, equation (21.2) may be solved to show that (21.3) X = × − + () = $, . . $. 100 000 06 12 1 1 10612 599 55 360 X n n 1 10612 100 000 1 360 + () = = ∑ . $, $. $,599 55 1 112 100 000 1 360 + () = = ∑ y n n 456 MORTGAGE-BACKED SECURITIES 2 This section ignores the prepayment option and the possibility of homeowner de- fault. Both are discussed in the next section. The rate of the mortgage may be used to divide the monthly payments into its interest and principal components. These accounting quantities are useful for tax purposes since interest payments are deductible from income while principal payments are not. Let B(n) be the outstanding principal balance of the mortgage after the payment on date n. The interest compo- nent of the payment on date n+1 is (21.4) In words, the interest component of the monthly payment over a particular period equals the mortgage rate times the principal outstanding at the be- ginning of that period. The principal component of the payment is the re- mainder, namely (21.5) In the example, the original balance is $100,000. At the end of the first month, interest at 6% is due on this balance, implying that the interest component of the first payment is (21.6) The rest of the monthly payment of $599.55 pays down principal, imply- ing that the principal component of the first payment is $599.55–$500.00 or $99.55. This principal payment reduces the outstanding balance from the original $100,000 to (21.7) The interest payment for the end of the second month will be based on the principal amount outstanding at the end of the first month as given in (21.7). Continuing this sequence of calculations produces an amortization table, selected rows of which are given in Table 21.1. Early payments are composed mostly of interest, while later payments are composed mostly of principal. This is explained by the phrase “interest lives off principal.” Interest at any time is due only on the then outstanding $, $. $,.100 000 99 55 99 900 45−= $, . $.100 000 06 12 500 00×= XBn y − () × 12 Bn y () × 12 Basic Mortgage Mathemetics 457 principal amount. As principal is paid off, the amount of interest necessar- ily declines. The outstanding balance on any date can be computed through the amortization table, but there is an instructive shortcut. Discounting using the mortgage rate at origination, the present value of the remaining pay- ments equals the principal outstanding. This is a fair pricing condition un- der the assumption that the term structure is flat and that interest rates have not changed since the origination of the mortgage. To illustrate this shortcut in the example, after five years or 60 monthly payments there remain 300 payments. The value of these pay- ments using the original mortgage rate for discounting is (21.8) where the second equality follows from equation (3.3). Hence, the balance outstanding after five years is $93,054.36, as reported in Table 21.1. To this point all cash flows have been discounted at a single rate. But Part One showed that each cash flow must be discounted by the rate ap- propriate for that cash flow’s maturity. Therefore, the true fair pricing con- dition for a $100,000 mortgage paying X per month for N months is $. . $. . . $, .599 55 1 10612 599 55 111 0612 06 12 93 054 36 1 300 300 + () =× −+ () = = ∑ n n 458 MORTGAGE-BACKED SECURITIES TABLE 21.1 Selected Rows from an Amortization Table of a 6% 30-Year Mortgage Payment Interest Principal Ending Month Payment Payment Balance 100,000.00 1 500.00 99.55 99,900.45 2 499.50 100.05 99,800.40 3 499.00 100.55 99,699.85 36 481.01 118.54 96,084.07 60 465.94 133.61 93,054.36 120 419.33 180.22 83,685.72 180 356.46 243.09 71,048.84 240 271.66 327.89 54,003.59 300 157.27 442.28 31,012.09 360 2.98 596.57 0.00 (21.9) where d(n) is the discount factor applicable for cash flows on date n. It is useful to think of equation (21.9) as the starting point for mort- gage pricing. The lender uses discount factors or, equivalently, the term structure of interest rates, to determine the fair mortgage payment. Only then does the lender compute the mortgage rate as another way of quoting the mortgage payment. 3 This discussion is analogous to the discussion of yield-to-maturity in Chapter 3. Bonds are priced under the term structure of interest rates and then the resulting prices are quoted using yield. The fair pricing condition (21.9) applies at the time of the mortgage’s origination. Over time discount factors change and the present value of the mortgage cash flows changes as well. Mathematically, with N ៣ payments re- maining and a new discount function d ៣ (n), the present value of the mort- gage is (21.10) The monthly payment X is the same in (21.10) as in (21.9), but the new discount function reflects the time value of money in the current economic environment. The present value of the mortgage after its origination may be greater than, equal to, or less than the principal outstanding. If rates have risen since origination, then the mortgage has become a loan with a below-mar- ket rate and the value of the mortgage will be less than the principal out- standing. If, however, rates have fallen since origination, then the mortgage has become an above-market loan and the value of the mortgage will ex- ceed the principal outstanding. PREPAYMENT OPTION A very important feature of mortgages not mentioned in the previous sec- tion is that homeowners have a prepayment option. This means that a Xdn n N ) ) () = ∑ 1 Xdn n N () = = ∑ 1 100 000$, Prepayment Option 459 3 The lender must also account for the prepayment option described in the next sec- tion and for the possibility of default by the borrower. homeowner may pay the bank the outstanding principal at any time and be freed from the obligation of making further payments. In the example of the previous section, the mortgage balance at the end of five years is $93,054.36. To be free of all payment obligations from that time on the borrower can pay the bank $93,054.36. The prepayment option is valuable when mortgage rates have fallen. In that case, as discussed in the previous section, the value of an existing mortgage exceeds the principal outstanding. Therefore, the borrower gains in a present value sense from paying the principal outstanding and being free of any further obligation. When rates have risen, however, the value of an existing mortgage is less than the principal outstanding. In this situation a borrower loses in a present value sense from paying the principal out- standing in lieu of making future payments. By this logic, the prepayment option is an American call option on an otherwise identical, nonpre- payable mortgage. The strike of the option equals the principal amount outstanding and, therefore, changes after every payment. The homeowner is very much in the position of an issuer of a callable bond. An issuer sells a bond, receives the proceeds, and undertakes to make a set of scheduled payments. Consistent with the features of the em- bedded call option, the issuer can pay bondholders some strike price to re- purchase the bonds and be free of the obligation to make any further payments. Similarly, a homeowner receives money from a bank in ex- change for a promise to make certain payments. Using the prepayment op- tion the homeowner may pay the principal outstanding and not be obliged to make any further payments. The fair loan condition described in the previous section has to be amended to account for the value of the prepayment option. Like the con- vention in the callable bond market, homeowners pay for the prepayment option by paying a higher mortgage rate (as opposed to paying the rate ap- propriate for a nonprepayable mortgage and receiving less than the face amount of the mortgage at the time of the loan). Therefore, the fair loan condition requires that at origination of the loan the present value of the mortgage cash flows minus the value of the prepayment option equals the initial principal amount. The mortgage rate that satisfies this condition in the current interest rate environment is called the current coupon rate. When pricing the embedded options in government, agency, or corpo- rate bonds, it is usually reasonable to assume that these issuers act in ac- cordance with the valuation procedures of Chapter 19. More specifically, 460 MORTGAGE-BACKED SECURITIES they exercise an option if and only if the value of immediate exercise ex- ceeds the value of holding in some term structure model. If this were the case for homeowners and their prepayment options, the techniques of Chapter 19 could be easily adapted to value prepayable mortgages. In practice, however, homeowners do not seem to behave like these institu- tional issuers. One way in which homeowner behavior does not match that of insti- tutional issuers is that prepayments sometimes occur for reasons unre- lated to interest rates. Examples include defaults, natural disasters, and home sales. Defaults generate prepayments because mortgages, like many other loans and debt securities, become payable in full when the borrower fails to make a payment. If the borrower cannot pay the outstanding principal amount, the home can be sold to raise some, if not all, of the outstanding balance. Since issuers of mortgage-backed securites often guarantee the ul- timate payment of principal and interest, investors in MBS expect to expe- rience defaults as prepayments. More specifically, any principal paid by the homeowner, any cash raised from the sale of the home, and any balance contributed by the MBS issuer’s reserves flow through to the investor as a prepayment after the event of default. 4 Disasters generate prepayments because, like many other debt secu- rities with collateral, mortgages are payable in full if the collateral is damaged or destroyed by fire, flood, earthquake, and so on. Without sufficient insurance, of course, it may be hard to recover the amount due. But, once again, MBS issuers ensure that investors experience these disasters as prepayments. While defaults and disasters generate some prepayments, the most important cause of prepayments that are not directly motivated by inter- est rates is housing turnover. Most mortgages are due on sale, meaning that any outstanding principal must be paid when a house is sold. Since people often decide to move without regard to the interest rate, prepay- ments resulting from housing turnover will not be very related to the be- havior of interest rates. Practitioners have found that the age of a Prepayment Option 461 4 The investor is protected from default but the homeowner is still charged a default premium in the form of a higher mortgage rate. This premium goes to the issuer or separate insurer who guarantees payment. mortgage is very useful in predicting turnover. For example, people are not very likely to move right after they purchase a home but more likely to do so over the subsequent few years. The state of the economy, partic- ularly of the geographic region of the homeowner, is also important in understanding turnover. While housing turnover does not primarily depend on interest rates, there can be some interaction between turnover and interest rates. A homeowner who has a mortgage at a relatively low rate might be reluc- tant to pay off the mortgage as part of a move. Technically, paying off the mortgage in this case is like paying par for a bond that should be selling at a discount. Or, from a more pragmatic point of view, paying off a low- rate mortgage and taking on a new mortgage on a new home at market rates will result in an increased cost that a homeowner might not want to bear. This interaction between turnover and interest rates is called the lock-in effect. Another interaction between turnover and interest rates surfaces for mortgages that are not due-on-sale but assumable. If a mortgage is assum- able, the buyer of a home may take over the mortgage at the existing rate. If new mortgage rates are high relative to the existing mortgage rate, then the buyer and seller will find it worthwhile to have the buyer assume the mortgage. 5 In this case, then, the sale of the home will not result in a pre- payment. Conversely, if new mortgage rates are low relative to the existing mortgage rate, then the mortgage will not be assumed and the mortgage will be repaid. Having described the causes of prepayments not directly related to in- terest rates, the discussion turns to the main cause of prepayments, namely refinancing. Homeowners can exercise their prepayment options in re- sponse to lower interest rates by paying the outstanding principal balance in cash. However, since most homeowners do not have this amount of cash available, they exercise their prepayment options by refinancing their mort- gages. In the purest form of a refinancing, a homeowner asks the original lending bank, or another bank, for a new mortgage loan sufficient to pay off the outstanding principal of the existing mortgage. Ignoring transaction costs for a moment, the present value advantage 462 MORTGAGE-BACKED SECURITIES 5 In fact, a home with a below-market, assumable mortgage should be worth more than an identical home without such a mortgage. [...]... trees, 6 For a general overview of Monte Carlo methods for fixed income securities, see Andersen and Boyle (2000) Implementing Prepayment Models 469 values on a particular date assume that the cash flows on that date have just been made Example: Date 4: $0 Date 3: $80/(1+.035/2)=$78.62 Date 2: ($78.62+$15)/(1+.0375/2)=$91.90 Date 1: ($91.90+$12)/(1+.0425/2)= $101 .74 Date 0: ( $101 .74+ $10) /(1+.04/2)= $109 .55... 0s of 5/15/2002 7.5s of 5/15/2002 15s of 5/15/2002 Price 96-12 103 -1215/16 106 -2 Do these prices make sense relative to one another? Why or why not? CHAPTER 2 Bond Prices, Spot Rates, and Forward Rates 2.1 You invest $100 for two years at 5%, compounded semiannually How much do you have at the end of the two years? 2.2 You invested $100 for three years and, at the end of those three years, your investment... 30, 2001 1.2 Here is a list of bond transactions on May 15, 2001 For each transaction list the transaction price Bond 10. 75s of 5/15/2003 4.25s of 11/15/2003 7.25s of 5/15/2004 Price 112-25/8 99-14+ 107 -4 Face Amount $10, 000 $1,000 $1,000,000 1.3 Use this list of Treasury bond prices as of May 15, 2001, to derive the discount factors for cash flows to be received in 6 months, 1 year, and 1.5 years Bond... Ten-Year Rates and Spreads over Two Periods in 2001 Date 9/21/01 9 /10/ 01 Change 11/16/01 11/9/01 Change 8 10- Year Swap Rate 10- Year Treasury Rate Swap Spread (bps) 5.386% 5.644% –0.258% 5.553% 4.895% 0.658% 4.689% 4.839% –0.150% 4.893% 4.303% 0.590% 69.7 80.5 10. 8 66.0 59.2 6.8 Apply equation (6.26) at a yield of 5.75% 474 MORTGAGE-BACKED SECURITIES or about $30 billion.9 To summarize: Given the size... compounded rate of return? 2.3 Using your answers to question 1.3, derive the spot rates for 6 months, 1 year, and 1.5 years 2.4 Derive the relationship between discount factors and forward rates 2.5 Using your answers to either question 1.3 or 2.3, derive the six-month rates for 0 years, 5 years, and 1 year forward 2.6 Are the forward rates from question 2.5 above or below the spot rates of question 2.3? Why... exercises for this chapter are built around a spreadsheet exercise Set up a column of interest rates from 1.75% to 8.25% in 25 basis point increments In the next column compute the price of a perpetuity with a face of 100 and a coupon of 5%: 100 ×.05/y where y is the rate in the first column In the next column compute the price of a one-year bond with a face of 100 and an annual coupon of 5%: 105 /(1+y)... are 97.5 610, 95.0908, and 92.5069 Find the risk-neutral probabilities for the sixmonth rate process Assume, as in the text, that the risk-neutral probability of an up move from date 1 to date 2 is the same from both date 1 states As a check to your work, write down the price trees for the six-month, one-year, and 1.5-year zeros 9.3 Using the risk-neutral tree derived for question 9.2, price $100 face... Structure 10. 1 On February 15, 2001, the yields on a 5-year and a 10- year interest STRIPS were 5.043% and 5.385%, respectively Assuming that the expected yield change of each is zero and that the yield volatility is 95 basis points for both, use equation (10. 27) to infer the risk premium in the marketplace Hint: You will also need equations (6.24) and (6.36) On May 15, 2001, the yields on a 5-year and 10- year... have triggered a prepayment of outstanding principal 470 MORTGAGE-BACKED SECURITIES While the Monte Carlo technique of moving forward in time to generate cash flows has the advantage of handling path dependence, the approach is not suitable for all problems Consider trying to price an American or Bermudan option one period before expiration using the Monte Carlo technique Recall that this option price... 7.5’s of 05/15/2001 11.625’s of 11/15/2002 Price 101 -253/4 103 -1215/16 110- 211/4 1.4 Suppose there existed a Treasury issue with a 7.5% coupon maturing on November 15, 2002 Using the discount factors derived in question 1.3, what would be the price of the 7.5s of November 15, 2002? 1.5 Say that the 7.5s of November 15, 2002, existed and traded at a price of 105 instead of the price derived in question . rate ap- propriate for that cash flow’s maturity. Therefore, the true fair pricing con- dition for a $100 ,000 mortgage paying X per month for N months is $. . $. . . $, .599 55 1 106 12 599 55 111. is (21.2) Applying equation (3.3) to perform the summation, equation (21.2) may be solved to show that (21.3) X = × − + () = $, . . $. 100 000 06 12 1 1 106 12 599 55 360 X n n 1 106 12 100 000 1 360 + () = = ∑ . $, $ the short rates. As with price trees, 468 MORTGAGE-BACKED SECURITIES 6 For a general overview of Monte Carlo methods for fixed income securities, see Andersen and Boyle (2000). values on a particular