We valued an RMBS in class using Monte Carlo simulation. The RMBS has a par value of $1 million, a WAM of 10 years, a WAC of 8%, a pass-through rate of 8%, annual cash flows, a balloon payment at the end of the fourth year, and zero default risk. Figure 4.3 provides the binomial tree for the spot rates and the refinancing rates. Table 4.2 defines the 8 interest rate paths in terms of the refinancing rates. Table 4.4 provides the prepayment model.
(a) Calculate the cash flows for years 1, 2, 3, and 4 along path 6.
(b) The zero-coupon rates for years 1, 2, 3, and 4 along path 6 are 8.00%, 8.2996%, 8.1996%, and 8.2996%, respectively. Calculate the value of the RMBS for path 2.
Simulation Problem Homework Due at the beginning of class November 18. We valued an RMBS in class using Monte Carlo simulation. The RMBS has a par value of $1 million, a WAM of 10 years, a WAC of 8%, a pass-through rate of 8%, annual cash flows, a balloon payment at the end of the fourth year, and zero default risk. Figure 4.3 provides the binomial tree for the spot rates and the refinancing rates. Table 4.2 defines the 8 interest rate paths in terms of the refinancing rates. Table 4.4 provides the prepayment model. 1. Calculate the cash flows for years 1, 2, 3, and 4 along path 6. 1. The zero-coupon rates for years 1, 2, 3, and 4 along path 6 are 8.00%, 8.2996%, 8.1996%, and 8.2996%, respectively. Calculate the value of the RMBS for path 2. “London” — 2006/9/12 — 22:22 — page 171 — #193 C H A P T E R 4 MORTGAGE-BACKED SECURITIES SECTIONS 4.1 Prepayment Models 4.2 Numerical Example of Prepayment Model 4.3 MBS Pricing and Quoting 4.4 Prepayment Risk and Average Life of MBS 4.5 MBS Pricing Using Monte Carlo in C++ 4.6 Matlab Fixed-Income Toolkit for MBS Valuation 4.7 Collateralized Mortgage Obligations (CMOs) 4.8 CMO Implementation in C++ 4.9 Planned Amortization Classes (PACS) 4.10 Principal- and Interest-Only Strips 4.11 Interest Rate Risk 4.12 Dynamic Hedging of MBS Endnotes Mortgage-backed securities (MBSs) and mortgage pass-throughs (PT) are claims on a portfolio of mortgages. MBSs are created when a federal agency, mortgage banker, bank, or investment company buys up mortgages of a certain type—i.e., FHA (Federal Home Administration) or VA (Veterans’ Administration) insured—and then sells claims on the cash flows from the portfolio as MBSs, with the proceeds of the MBS sale being used to fi- nance the purchase of the mortgages. There are two types of MBS: agency and conventional (private-label).1 Agency MBS, such as a GNMA pass-through, are securities with claims on a portfo- lio of mortgages insured against default risk by FHM, VA, or FmHA (Farmers Mortgage Home Administration). A mortgage banker, bank, or investment company presents a pool of FHA, VA, or FmHA mortgages of a certain type (30-year fixed, 15-year variable rate, etc.) to GNMA (Ginnie Mae). If the mortgage pool is in order, GNMA will issue a separate guarantee that allows the MBSs on the mortgage pool to be issued as a GNMA PT. Other agency MBSs include the Federal Home Loan Mortgage Corporation (FHLMC) MBSs, which are claims on a portfolio of conventional mortgages. The FHLMC issues agency MBSs, whereby the FHLMC buys mortgages from the mortgage originator, and then cre- ates an MBS referred to as a participation certificate, which it issues through a network 171 “London” — 2006/9/12 — 22:22 — page 172 — #194 172 Mortgage-Backed Securities Chapter 4 of dealers. FHLMC has a swap program whereby FHLMC swaps MBSs for a savings and loan’s or commercial bank’s portfolio of mortgages of a certain type. Other government agencies such as FNMA (Fannie Mae) issue several types of MBS: participation certifi- cates, swaps, and PTs. With these certificates, homeowners’ mortgage payments pass from the originating bank through the issuing agency to the holds of the certificates. Conventional types, also known as private-label types, are issued by commercial banks (via their holding companies), S&Ls, mortgage bankers, and investment companies. Con- ventional issued MBSs include those issued by Prudential Home, Chase Mortgage, Citi- Corp Housing, Ryland/Saxon, GE Capital, and Countrywide. Conventional PTs must be registered with the SEC. These PTs are often insured with external insurance in the form of a letter of credit (LOC) of the private-label issuer, as well as internal insurance through the creation of senior and junior classes of the PT structured by the private-label issuer. There is both a primary and a secondary market for MBS. In the primary market, in- vestors buy MBSs issued by agencies or private-label investment companies either directly or through dealers. Many of the investors are institutional investors. Thus, the creation of MBS has provided a tool for having real estate financed more by institutions. In the pri- mary market, MBS issue denominations are typically between $25,000 to $250,000 (with some as high as $1M) and some have callable features. In the secondary market, MBSs are traded over-the-counter (OTC). OTC dealers are members of the Mortgage-Backed Secu- rities Dealer Association (MSDA). MBSs are some of the most complex securities to model and value due to their sensitiv- ity to prepayment and interest rates, which affects the timing, frequency, and size of cash flows to investors. Cash flows (CFs) from MBSs are the monthly CFs from the portfolio of mortgages (referred to as the collateral). Cash flows include interest on principal, sched- uled principal, and prepaid principal. Cash flow analysis is essential in the valuation of any MBS given their impact by the underlying features of the MBS, including weighted aver- age maturity (WAM), weighted average coupon rate (WAC), pass-through rate (PT rate), and prepayment rate or speed. The WAM is effectively the duration, or weighted length of time, of all the payment of MBS cash flows to be paid out to investors. The WAC is the rate on a portfolio of mortgages (collateral) that is applied to determine scheduled principal. The PT rate is the interest on principal and is lower than the WAC, with the difference go- ing to the MBS issuer. The prepayment rate or speed is the assumed prepayment rate made by homeowners of mortgages in the pool. In this chapter, we discuss MBS pricing and modeling in detail. In §4.1, we discuss prepayment and PSA models for MBS pricing. In §4.2, we give numerical examples us- ing Excel of how the prepayment models work. In §4.3, we discuss MBS pricing, quoting, and the value and return to investors based on different prepayment and interest rate as- sumptions. In §4.4, we discuss prepayment risk and the average life of MBS. In §4.5, we review in detail a numerical implementation in C++ and Excel for valuation and cash flow analysis of MBS using Monte Carlo simulation. In §4.6, we give numerical examples using the Fixed-Income Toolbox in Matlab. In §4.7, we discuss MBS derivatives, including col- lateralized mortgage obligations (CMOs) and sequential-pay tranche structures. We give examples using Excel. In §4.8, we give an implementation of a CMO in C++. In §4.9, we discuss planned amortization classes (PAC) and their structures. In §4.10, we review stripped MBS, including interest-only (IO) and principal-only (PO) securities. In §4.11, “London” — 2006/9/12 — 22:22 — page 173 — #195 Section 4.1 Prepayment Models 173 we discuss interest rate risk of MBSs. Finally, in §4.12, we discuss hedging MBS and using MBS for balance sheet asset-liability management. 4.1 PREPAYMENT MODELS MBS valuation models typically assume a prepayment rate or speed. Investors and issuers apply different prepayment models in analyzing MBS. Most models, though, are compared to a benchmark model or rate. The benchmark model is the one provided by the Public Securities Association (PSA). PSA measures speed by the Conditional Prepayment Rate (CPR). CPR is the proportion of the remaining mortgage balance that is prepaid each month and is quoted on an annual basis. The monthly rate is referred to as the Single- Monthly Mortality rate (SMM) and is given by: SMM = 1 − (1 − CPR)1/12 (4.1) The estimated monthly prepayment is: Monthly prepayment = SMM · [Beginning of month balance− Sched. prin. for month] For example, if CPR = 6%, beginning-of-the-month balance = $100M, and scheduled principal for month = $3M, then the estimated prepaid principal for the month would be $0.499M: SMM = 1 − [1− .06]1/12 = .005143 Monthly prepaid principal = .005143[$100M − $3M ] = $0.499M In the PSA model, CPR depends on the maturity of the mortgages. PSA’s standard model assumes that for a 30-year mortgage (360 months), the CPR is equal to .2% the first month, grows at that rate for 30 months to equal 6%, and stays at 6% for the rest of the mortgage’s life. This model is referred to as the 100% PSA model. Figure 4.1 shows the prepayment rate as a function of time in months. The estimation of CPR for month t is: CPR = { 0.06 ( t 30 ) , if t ≤ 30 0.06, if t > 30 (4.2) As an example, the CPR for month five is: CPR = .06 ( 5 30 ) = .01 SMM = 1 − [1− .01]1/12 = .000837 PSA’s model can be defined in terms of different speeds by expressing the standard model (100% PSA) in terms of a higher or lower percentage, such as 150% or 50%. In a period of lower rates, the PSA model could be 150%, and in a period of higher rates, it “London” — 2006/9/12 — 22:22 — page 174 — #196 174 Mortgage-Backed Securities Chapter 4 CPR (%) 6.0 0.2 0 30 360 Month Figure 4.1 100% PSA Model could be 50%. For the 100% PSA model, the average time a 30-year mortgage is held is 17 years; for a 225% PSA model, it is 8 years. Figure 4.2 shows the different prepayment rates as a function of time in months. CPR (%) 6.0 9.0 0.2 3.0 0 30 360 150 PSA 100 PSA 50 PSA Month Figure 4.2 PSA Models “London” — 2006/9/12 — 22:22 — page 175 — #197 Section 4.2 Numerical Example of Prepayment Model 175 Suppose we want to compute the CPR and SMM for month five with 165 PSA speed. Then we compute the following based on (4.1) and (4.2): CPR = .06 ( 5 30 ) = .01 165CPR = 1.65(.01) = .0165 SMM = 1 − [1 − .0165]1/12 = .0001386 4.2 NUMERICAL EXAMPLE OF PREPAYMENT MODEL Let p = monthly scheduled mortgage payment, F0 = the face value of the underlying mortgage pool of the MBS, M = WAM = weighted average of the number of months remaining until maturity, I = interest rate payment, SP = scheduled principal payment, PP = prepaid principal, RA = annual interest rate (WAC), Bi, i = 1, . . . , 360, the remain- ing mortgage balance in month i, and CFi, i = 1, . . . , 360 , the cash flow in the ith month. Note that the balance in month 1 is