1、Chapter 19 Analysis of Residential Mortgage-Backed Securities Learning ObjectivesAfter reading this chapter,you will understandthe cash flow yield methodology for analyzing residential mortgage-backed securitiesthe limitations of the cash flow yield methodologyhow the effective duration and convexit
2、y are calculated for the cash flow yield methodologyone measure for estimating prepayment sensitivitywhy the Monte Carlo simulation methodology is used to value residential mortgage-backed securitieshow interest-rate paths are simulated in a Monte Carlo simulation methodologyhow the Monte Carlo simu
3、lation methodology can be used to determine the theoretical value of a residential mortgage-backed securityLearning Objectives(continued)After reading this chapter,you will understandhow the option-adjusted spread,effective duration,and effective convexity are computed using the Monte Carlo simulati
4、on methodologythe complexities of modeling collateralized mortgage obligationsthe limitations of option-adjusted spreadmodeling risk and how it can be stress testedhow the total return is calculated for a residential mortgage-backed securitythe difficulties of applying the total return framework to
5、residential mortgage-backed securitiesStatic Cash Flow Yield Methodology The static cash flow yield methodology is the simplest to use,although we shall see that it offers little insight into the relative value of a residential mortgage-backed security(RMBS).Exhibit 19-1 summarizes cash flow yields
6、according to various PSA prepayment assumptions for the four tranches assuming different purchase prices.(See truncated version of Exhibit 19-1 in Overheads 19-5,19-6 and 19-7.)Notice that the greater the discount assumed to be paid for the tranche,the more a tranche will benefit from faster prepaym
7、ents.The converse is true for a tranche for which a premium is paid.The faster the prepayments,the lower the cash flow yield.Exhibit 19-1 Price Cash Flow Yield Table for the Four Tranches in FJF-06Tranche A:Orig.par:$194,500,000;type:sequential;coupon:6.00%(fixed)If PricePaid Is:50.00PSA100.00PSA165
8、.00PSA250.00PSA400.00PSA400.00PSA700.00PSA1000.00PSA902408.3709.0109.7610.6111.8712.5913.8815.6391248.098.669.3210.0711.1711.8112.9414.4792247.828.318.889.5310.4911.0312.0113.3393247.567.978.459.009.8110.2711.1012.22.106244.423.963.422.811.921.410.52-0.68107244.213.693.072.381.370.80-0.21-1.57108243
9、.993.412.731.960.830.20-0.93-2.44109243.783.142.391.540.30-0.40-1.64-3.29Average life:5.093.802.932.331.791.581.311.07Mod.duration:4.123.222.572.091.641.461.221.00Exp.maturity:9.407.155.404.153.072.652.241.82Exhibit 19-1 Price Cash Flow Yield Table for the Four Tranches in FJF-06Tranche B:Orig.par:$
10、36,000,000;type:sequential;coupon:6.50%(fixed)If PricePaid Is:50.00PSA100.00PSA165.00PSA250.00PSA400.00PSA400.00PSA700.00PSA1000.00PSA90317.858.128.498.959.6910.1310.8911.8391317.697.938.258.669.319.7010.3611.1892317.547.758.028.378.949.279.8410.5593317.397.577.808.098.578.859.339.92.106315.615.395.
11、104.744.163.823.232.51107315.485.244.914.503.853.462.801.99108315.365.084.724.273.543.112.381.48109315.244.934.544.043.242.761.960.97Average life:10.177.765.934.583.352.892.351.90Mod.duration:7.235.924.783.842.922.562.111.74Exp.maturity:7.235.924.783.842.922.562.111.74Exhibit 19-1 Price Cash Flow Yi
12、eld Table for the Four Tranches in FJF-06Tranche Z:Orig.par:$73,000,000;type:sequential;coupon:7.35%(fixed)If PricePaid Is:50.00PSA100.00PSA165.00PSA250.00PSA400.00PSA400.00PSA700.00PSA1000.00PSA90017.877.968.098.278.618.849.3310.1091017.827.898.008.168.478.689.119.7992017.767.837.938.078.338.518.89
13、9.4993017.717.767.857.978.208.358.689.20.106017.046.986.906.796.586.436.135.65107016.996.936.846.716.466.295.945.39108016.946.876.776.626.356.165.775.15109016.896.826.706.546.246.035.594.90Average life:22.3919.5716.2112.789.017.465.493.88Mod.duration:19.4216.6813.8111.088.066.785.113.67Exp.maturity:
14、29.7429.7429.7429.7429.7429.7429.7424.24Static Cash Flow Yield Methodology(continued)Vector Analysis One practice that market participants use to overcome the drawback of the PSA benchmark is to assume that the PSA speed can change over time.This technique is referred to as vector analysis.o A vecto
15、r is simply a set of numbers.In the case of prepayments,it is a vector of prepayment speeds.o Vector analysis is particularly useful for CMO tranches that are dramatically affected by the initial slowing down of prepayments,and then speeding up of prepayments,or vice versa.Static Cash Flow Yield Met
16、hodology(continued)Limitations of the Cash Flow Yield The same shortcomings found in the yield to maturity approach are also present in application of the cash flow yield measure:i.the projected cash flows are assumed to be reinvested at the cash flow yieldii.the RMBS is assumed to be held until the
17、 final payout based on some prepayment assumption The cash flow yield is dependent on realization of the projected cash flow according to some prepayment rate.If actual prepayments vary from the prepayment rate assumed,the cash flow yield will not be realized.Static Cash Flow Yield Methodology(conti
18、nued)Yield Spread to Treasuries The yield for a RMBS will depend on the actual prepayment experience of the mortgages in the pool.o Nevertheless,the convention in all fixed-income markets is to measure the yield on a non-Treasury security to that of a“comparable”Treasury security.The repayment of pr
19、incipal over time makes it inappropriate to compare the yield of a RMBS to a Treasury of a stated maturity.o Instead,market participants have used two measures:Macaulay duration and average life.Static Cash Flow Yield Methodology(continued)Static Spread The static spread is the yield spread in a sta
20、tic scenario(i.e.,no volatility of interest rates)of the bond over the entire theoretical Treasury spot rate curve,not a single point on the Treasury yield curve.In a relatively flat interest-rate environment,the difference between the traditional yield spread and the static spread will be small.Sta
21、tic Cash Flow Yield Methodology(continued)Static Spread There are two ways to compute the static spread for RMBS.i.One way is to use todays yield curve to discount future cash flows and keep the mortgage refinancing rate fixed at todays mortgage rate.oUse of this approach to calculate the static spr
22、ead recognizes different prices today of dollars to be delivered at future dates.i.The second way to calculate the static spread allows the mortgage rate to go up the curve as implied by the forward interest rates.oThis procedure is sometimes called the zero-volatility OAS.oIn this case a prepayment
23、 model is needed to determine the vector of future prepayment rates implied by the vector of future refinancing rates.Static Cash Flow Yield Methodology(continued)Effective Duration Modified duration is a measure of the sensitivity of a bonds price to interest-rate changes,assuming that the expected
24、 cash flow does not change with interest rates.Modified duration is consequently not an appropriate measure for mortgage-backed securities,because prepayments influence the projected cash flow as interest rates change.When interest rates fall(rise),prepayments are expected to rise(fall).As a result,
25、when interest rates fall(rise),duration may decrease(increase)rather than increase(decrease).This property is referred to as negative convexity.Static Cash Flow Yield Methodology(continued)Effective Duration Negative convexity has the same impact on the price performance of a RMBS as it does on the
26、performance of a callable bond.When interest rates decline,a bond with an embedded call option,which is what a RMBS is,will not perform as well as an option-free bond.Although modified duration is an inappropriate measure of interest-rate sensitivity,there is a way to allow for changing prepayment r
27、ates on cash flow as interest rates change.This is achieved by calculating the effective duration,which allows for changing cash flow when interest rates change.Static Cash Flow Yield Methodology(continued)Modified Duration Example.To illustrate the effective duration calculation,consider tranche da
28、ta:P_=102.1875;P+=98.4063;P0(initial price)=100.2813;y=0.0025.Substituting into the duration formula yields:0102 18752y102 187598 40632100 2813 0 00257 54 with P_ .P modified dura PP.ti.on.Static Cash Flow Yield Methodology(continued)Effective Duration Example.To illustrate the effective duration ca
29、lculation,consider tranche data:P_=101.9063(at 200 PSA;basis points decrease);P+=98.3438(at 150 PSA;basis point increase);P0=100.2813;y=0.0025.Substituting into the duration formula yields:0101 90632y101 906398 34382100 2813 0 00257 11 with P_ .P effective dura PP.tn.io.Static Cash Flow Yield Method
30、ology(continued)Effective Duration Notice that the effective duration(which allows for changing cash flow when interest rates change)is less than the modified duration(7.11 versus 7.54).The divergence between modified duration and effective duration is much more dramatic for bond classes trading at
31、a substantial discount from par or at a substantial premium over par.Static Cash Flow Yield Methodology(continued)Standard Convexity Example.To illustrate the convexity formula,consider the above tranche data:P+=98.4063;P_=102.1875;P0(initial price)=100.2813;y=0.0025.The standard convexity is approx
32、imated as follows:0022 298.4063 1 02.1875 2(100.2813)(100.2813)(0.0025)24.930PPPPyStatic Cash Flow Yield Methodology(continued)Effective Convexity Example.To illustrate the effective convexity calculation(which allows for changing cash flow when interest rates change so that P_ changes),consider the
33、 other above tranche data:P+=98.3438(at 150 PSA;basis point increase);P_=101.9063(at 200 PSA;basis points decrease);P0=100.2813;y=0.0025.Substituting into the duration formula yields:0022101 9063298 3438101 90632100 2813100 2813 0 0025249 299 with P_.P P PPy.effective convexit.y.Static Cash Flow Yie
34、ld Methodology(continued)Effective Convexity The standard convexity indicates positive convexity(24.930),whereas the effective convexity indicates they have negative convexity(249.299)The difference is even more dramatic for bonds not trading near par.For a PO created from a tranche,the standard con
35、vexity can be close to zero whereas the effective convexity can be very large.For example,if the effective convexity is 2,000,and the yields change by 100 basis points,the percentage change in price due to convexity is:(effective convexity)(y)2=2,000(0.01)2=0.2000 or 20.00%Static Cash Flow Yield Met
36、hodology(continued)Prepayment Sensitivity Measure The value of a RMBS will depend on prepayments.To assess prepayment sensitivity,market participants have used the following measure:the basis point change in the price of an RMBS for a 1%increase in prepayments.Specifically,we have:prepayment sensiti
37、vity=(Ps P0)100where Ps=price(per$100 par value)assuming a 1%increase in prepayment speed and P0=initial price(per$100 par value)at assumed prepayment speed.Notice that a security that is adversely affected by an increase in prepayment speeds will have a negative prepayment sensitivity while a secur
38、ity that benefits from an increase in prepayment speed will have a positive prepayment sensitivity.Monte Carlo Simulation Methodology For some fixed-income securities and derivative instruments,the periodic cash flows are path dependent.This means that the cash flows received in one period are deter
39、mined not only by the current and future interest-rate levels but also by the path that interest rates took to get to the current level.Pools of pass-throughs are used as collateral for the creation of collateralized mortgage obligations(CMOs).Consequently,for CMOs there are typically two sources of
40、 path dependency in a CMO tranches cash flows.i.First,the collateral prepayments are path dependent.ii.Second,the cash flow to be received in the current month by a CMO tranche depends on the outstanding balances of the other tranches in the deal.Thus we need the history of prepayments to calculate
41、these balances.Monte Carlo Simulation Methodology(continued)Because of the path dependency of a RMBSs cash flow,the Monte Carlo simulation method is used for these securities rather than the binomial method.Conceptually,the valuation of pass-throughs using the Monte Carlo method is simple but,in pra
42、ctice,it is very complex.The simulation involves generating a set of cash flows based on simulated future mortgage refinancing rates,which in turn imply simulated prepayment rates.Valuation modeling for CMOs is similar to valuation modeling for pass-throughs,although the difficulties are amplified b
43、ecause the issuer has sliced and diced both the prepayment risk and the interest-rate risk into smaller pieces called tranches.The sensitivity of the pass-throughs composing the collateral to these two risks is not transmitted equally to every tranche.Some of the tranches wind up more sensitive to p
44、repayment risk and interest-rate risk than the collateral,whereas some of them are much less sensitive.Monte Carlo Simulation Methodology(continued)Using Simulation to Generate Interest-Rate Paths and Cash Flows The typical model that Wall Street firms and commercial vendors use to generate random i
45、nterest-rate paths takes as inputs todays term structure of interest rates and a volatility assumption.The term structure of interest rates is the theoretical spot rate(or zero-coupon)curve implied by todays Treasury securities.The volatility assumption determines the dispersion of future interest r
46、ates in the simulation.The simulations should be normalized so that the average simulated price of a zero-coupon Treasury bond equals todays actual price.Monte Carlo Simulation Methodology(continued)Using Simulation to Generate Interest-Rate Paths and Cash Flows The simulation works by generating ma
47、ny scenarios of future interest-rate paths.oIn each month of the scenario,a monthly interest rate and a mortgage refinancing rate are generated.o The monthly interest rates are used to discount the projected cash flows in the scenario.o The mortgage refinancing rate is needed to determine the cash f
48、low because it represents the opportunity cost the mortgagor is facing at that time.Prepayments are projected by feeding the refinancing rate and loan characteristics,such as age,into a prepayment model.oGiven the projected prepayments(voluntary and involuntary),the cash flow along an interest-rate
49、path can be determined.Monte Carlo Simulation Methodology(continued)Using Simulation to Generate Interest-Rate Paths and Cash Flows To make this more concrete,consider a newly issued mortgage pass-through security with a maturity of 360 months.o Exhibit 19-4(see Overhead 19-27)shows N simulated inte
50、rest-rate path scenarios.o Each scenario consists of a path of 360 simulated one-month future interest rates.o Just how many paths should be generated is explained later.Exhibit 19-5(see Overhead 19-28)shows the paths of simulated mortgage refinancing rates corresponding to the scenarios shown in Ex
