fabozzi_BMAS9_IM_19.docx

CHAPTER 19 ANALYSIS OF RESIDENTIAL MORTGAGE-BACKED SECURITIES CHAPTER SUMMARY There are two approaches to the analysis of residential mortgage-backed securities including pass-throughs, collateralized mortgage obligations (CMOs), and stripped mortgage-backed securities. They are: (1) the static cash flow yield methodology, and (2) the Monte Carlo simulation methodology. In this chapter we review the static cash flow yield methodology and its limitations and then focus on the Monte Carlo simulation methodology. The framework provided in this chapter applies to agency and nonagency residential mortgage-backed securities. STATIC CASH FLOW YIELD METHODOLOGY METHODOLOGY

The static cash flow yield methodology begins with the computation of the cash flow yield measure used for pass-throughs and based on some prepayment assumption. 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 . A vector is simply a set of numbers. In the case of prepayments, it is a vector of prepayment speeds. Vector analysis anal ysis 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. 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 yield, and (ii) the RMBS is assumed to be held until the 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. Yield Spread to Treasuries The yield for a RMBS will depend on the actual prepayment experience of the mortgages in the pool. 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 principal over time makes it inappropriate to compare the yield of a RMBS to a Treasury of a stated maturity. Instead, market participants have used two measures: Macaulay duration and average life. Copyright © 2016 Pearson Education, Inc.

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Static Spread The static spread is the yield spread in a static 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. There are two ways to compute the static spread for RMBS. One way is to use today’s yield curve to discount future cash flows and keep the mortgage refinancing rate fixed at today’s mortgage rate. Use of this approach to calculate the static spread recognizes different prices today of dollars to be delivered at future dates.

The second way to calculate the static spread allows the mortgage rate to go up the curve as implied by the forward interest rates. This procedure is sometimes called the zero-volatility OAS or simply the Z-spread . In this case a prepayment model is needed to determine the vector of future prepayment rates implied by the vector of future refinancing rates. Effective Duration Modified duration is a measure of the sensitivity of a bond’s price to interest-rate changes, assuming that the expected 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, when interest rates fall (rise), duration may decrease (increase) rather than increase (decrease). This property is referred to as negative convexity. Negative convexity has h as the t he same impact on the price p rice performance of a RMBS as it does do es on the 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 rates 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. To illustrate, consider tranche data: P _ = 102.1875; P + = 98.4063; P 0 (initial price) = 100.2813; mod ified duration formula yields, we have: Δ y = 0.0025. Substituting into the modified modified duration (with P _ = 102.1875) =

P  P 

2 P 0  y 

=

102.1875  98.4063 2(100.2813)(0.0025)

= 7.54.

To further illustrate, consider tranche data: P _ = 101.9063 (at 200 PSA; basis points decrease); P + = 98.3438 (at 150 PSA; basis point increase); P 0 = 100.2813; Δ y = 0.0025. Substituting into the effective duration formula yields

Copyright © 2016 Pearson Education, Inc.

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effective duration ( P _ = 101.9063) =

P  P 

2 P 0  y 

=

101.9063  98.3438 2(100.2813)(0.0025)

= 7.11.

Notice that the effective duration (which allows for changing cash flow when interest rates change) is less than the modified duration. The divergence between modified duration and effective duration is much more dramatic for bond classes trading at a substantial discount from par or at a substantial premium over par. Effective Convexity To illustrate the convexity formula, consider the above tranche data. The standard convexity is approximated as follows:

P

 P  2P 0

P0  y 

2

=

98.4063  102.1875  2(100.2813) (100.2813)(0.0025)2

= 49.78.

The effective convexity (which allows for changing cash flow when interest rates change so that _ = 101.9063) is P P

 P  2P 0

P0  y 

2

=

98.3438  101.9063  2(100.2813) (100.2813)(0.0025)2

= 398.88.

The standard convexity indicates positive convexity, whereas the effective convexity indicates they have negative convexity. The difference is even more dramatic for bonds not trading near par. For a PO created from a tranche, the standard convexity 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)(Δ y)2 = 2,000(0.01)2 = 0.2000 or 20.00%. 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 sensitivity = ( P s – P 0)100 where P s = price (per $100 par value) assuming a 1% increase in prepayment speed and P 0 = initial price (per $100 par value) at assumed prepayment speed.


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Notice that a security that is adversely affected by an increase in prepayment speeds will have a negative prepayment sensitivity while a security 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 determined 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 path dependency in a CMO tranche’s cash flows. First, the collateral prepayments are path dependent. 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 these balances. Because of the path dependency of a RMBS’s 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. In practice, however, 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 because 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 prepayment risk and interest-rate risk than the collateral, whereas some of them are much less sensitive. Using Simulation to Generate Interest-Rate Paths and Cash Flows The typical model that Wall Street firms and commercial vendors use to generate random interest-rate paths takes as inputs today’s 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 today’s Treasury securities. The volatility assumption determines the dispersion of future interest rates in the simulation. The simulations should be normalized so that the average simulated price of a zero-coupon Treasury bond equals today’s actual price. The simulation works by generating many scenarios of future interest-rate paths. In each month of the scenario, a monthly interest rate and a mortgage refinancing rate are generated. The monthly interest rates are used to discount the projected cash flows in the scenario. The mortgage refinancing rate is needed to determine the cash flow because it represents the opportunity cost


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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. Given the projected prepayments (voluntary and involuntary), the cash flow along an interest-rate path can be determined. Calculating the Present Value for a Scenario Interest-Rate Path Given the cash flow on an interest-rate path, its present value can be calculated. The discount rate for determining the present value is the simulated spot rate for each month on the interest-rate path plus an appropriate spread. The spot rate on a path can be determined from the simulated future monthly rates. The relationship that holds between the simulated spot rate for month T on path n and the simulated future one-month rates is z T( n) =

1+ f (n)1 + f 1

1/T

2

(n)  . . . 1 + fT (n)

1

where z T( n) = simulated spot rate for month T on path n, and f j(n) = simulated future one-month rate for month j on path n. The present value of the cash flow for month T on interest-rate path n discounted at the simulated spot rate for month T plus some spread is PV[C T( n)] =

CT (n )

1  z

T

(n)  K 

T

where PV[C T( n)] = present value of cash flow for month T on path n, C T (n) = cash flow for month T on path n, z T (n) = spot rate for month T on path n, and K = appropriate risk-adjusted spread. The present value for path n is the sum of the present value of the cash flow for each month on path n. Determining the Theoretical Value The present value of a given interest-rate path can be thought of as the theoretical value of a pass-through assuming that the path was actually realized. The theoretical value of the pass-through can be determined by calculating the average of the theoretical value of all the interest-rate paths. Looking at the Distribution of the Path Values The theoretical value generated by the Monte Carlo simulation method is the average of the path values. There is valuable information in the distribution of the path values. If there is substantial dispersion of the path values then the investor is warned about the potential variability of the model’s value.


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Simulated Average Life The average life reported in a Monte Carlo analysis is the average of the average lives along the interest-rate paths. The greater the range and standard deviation of the average life, the more uncertainty there is about the security’s average life. Option-Adjusted Spread The option-adjusted spread is a measure of the yield spread that can be used to convert dollar differences between value and price. It represents a spread over the issuer’s spot rate curve or benchmark. In the Monte Carlo model, the OAS is the spread K that when added to all the spot rates on all interest-rate paths will make the average present value of the paths equal to the observed market price (plus accrued interest). Option Cost The implied cost of the option embedded in any RMBS can be obtained by calculating the difference between the OAS at the assumed volatility of interest rates and the static spread: option cost = static spread – option-adjusted spread. Effective Duration and Convexity Effective duration and effective convexity can be calculated using the Monte Carlo method as follows. First, the bond’s OAS is found using the current term structure of interest r ates. Next, the bond is repriced holding OAS constant but shifting the term structure. Two shifts are used to get the prices needed to apply the effective duration and effective convexity formulas: (i) yields are increased, and (ii) yields are decreased. Selecting the Number of Interest-Rate Paths Let’s now address the question of the number of scenario paths or repetitions, N, needed to value a RMBS. A typical OAS run will be done for 512 to 1,024 interest-rate paths.

The number of interest-rate paths determines how “good” the estimate is, not relative to the truth but relative to the model. The more paths, the more average spread tends to settle down. It is a statistical sampling problem. Most Monte Carlo simulation models employ some form of variance reduction to cut down on the number of sample paths necessary to get a good statistical sample. Variance reduction techniques allow us to obtain price estimates within a tick. Drawbacks of the OAS Methodology The OAS methodology provides a measure for relative valuation of MBS products that is vastly


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superior to some of the ad hoc measures that were historically used (and unfortunately still used) by portfolio managers. Those traditional measures fail to recognize the embedded options in these complex products. Nevertheless, using the OAS as a relative value measure has drawbacks. The first drawback is the prepayment model uncertainty. The second drawback is that in pricing MBS, there are differing OAS levels. OAS Constancy Problem There is a problem with calculating price sensitivity measures such as interest-rate and prepayment sensitivity using the OAS methodology. Recall that in measuring effective duration and effective convexity, the calculation assumed that the OAS is held constant and the interestrate paths are shifted by a specified number of basis points. After adding the initial OAS to the new interest-rate paths, a security is then revalued to obtain the two values to be used in the effective duration and convexity formulas. The trouble in assuming a constant OAS in calculating interest-rate sensitivity measures is that the assumption cannot be justified on theoretical grounds. Thus, we have a serious problem with the notion of keeping OAS constant for measuring price sensitivity to changes to all sensitivity measures (e.g., interest-rate and prepayment sensitivity). One conclusion that can be drawn from this is to adjust (and use) a prepayment model so that it equates OAS levels. TOTAL RETURN ANALYSIS

Neither the static cash flow methodology nor the Monte Carlo simulation methodology will tell a money manager whether investment objectives can be satisfied. The performance evaluation of an individual RMBS requires specification of an investment horizon, whose length for most financial institutions is dictated by the nature of its liabilities. The measure that should be used to assess the performance of a security or a portfolio over some investment horizon is the total return. Horizon Price for CMO Tranches The most difficult part of estimating total return is projecting the price at the horizon date. In the case of a CMO tranche the price depends on the characteristics of the tranche and the spread to Treasuries at the termination date . The key determinants are the “quality” of the tranche, its average life (or duration), and its convexity. Quality refers to the type of CMO tranche. Option-Adjusted Spread Total Return The total return and OAS frameworks can be combined to determine the projected price at the horizon date. At the end of the investment horizon, it is necessary to specify how the OAS is expected to change. The horizon price can be “backed out” of the Monte Carlo simulation model. Assumptions about the OAS value at the investment horizon reflect the expectations of the


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money manager. It is common to assume that the OAS at the horizon date will be the same as the OAS at the time of purchase. A total return calculated using this assumption is sometimes referred to as a constant-OAS total return. KEY POINTS 

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There are two methodologies commonly used to analyze all RMBS (agency and nonagency): cash flow yield methodology and Monte Carlo simulation methodology. The cash flow yield is the interest rate that will make the present value of the projected cash flow from an RMBS equal to its market price. The cash flow yield assumes that (1) all the cash flows can be reinvested at a rate equal to the cash flow yield, (2) the RMBS is held to the maturity date, and (3) the prepayment speed used to project the cash flow will be realized. In addition, the cash flow yield methodology fails to recognize that future interest-rate changes will affect the cash flow. Modified duration is not a good measure of price volatility for RMBS because it assumes that the cash flow does not change as yield changes. Effective duration does take into consideration how yield changes will affect prepayments and therefore cash flow. An RMBS is a security whose cash flow is path dependent. This means that cash flow received in one period is determined 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. A methodology used to analyze path-dependent cash flow securities is the Monte Carlo simulation. This methodology involves randomly generating many scenarios of future interest-rate paths, where the interest-rate paths are generated based on some volatility assumption for interest rates. The random paths of interest rates should be generated from an arbitrage-free model of the future term structure of interest rates. The Monte Carlo simulation methodology applied to RMBS involves randomly generating a set of cash flows based on simulated future mortgage refinancing rates. The theoretical value of a security on any interest-rate path is the present value of the cash flow on that path, where the spot rates are those on the corresponding interest-rate path. The theoretical value of a security is the average of the theoretical values over all the interestrate paths. Information about the distribution of the path values is useful in understanding the variability around the theoretical value. The average life reported is the average of the average lives from all the interest-rate paths and information about the distribution of the average life is useful. In the Monte Carlo simulation methodology, the option-adjusted spread is the spread that when added to all the spot rates on all interest-rate paths will make the average present value of the paths equal to the observed market price (plus accrued interest). The effective duration and effective convexity are calculated using the Monte Carlo simulation methodology by holding the OAS constant and shifting the term structure up and down. Using the OAS as a relative value measure has drawbacks.


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One problem with assuming a constant OAS in calculating interest-rate sensitivity measures is that the assumption cannot be justified on theoretical grounds. Total return is the correct measure for assessing the potential performance of CMO tranches over a specified investment horizon. The static cash flow yield or Monte Carlo simulation methodology can be incorporated into a total return framework to calculate the mortgage- backed security’s price at the horizon date. Scenario analysis is one way to evaluate the risk associated with investing in an RMBS.


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ANSWERS TO QUESTIONS FOR CHAPTER 19 (Questions are in bold print followed by answers.) 1. Suppose you are told that the cash flow yield of a pass-through security is 9% and that you are seeking to invest in a security with a yield greater than 8.8%. Answer the below questions. (a) What additional information would you need to know before you might invest in this pass-through security?

To determine your chances of actually getting an 8.8% return, you would want to know the types of securities backing the pass-through, the safety of the cash flows, and the expected volatility of the cash flows. More details are given below. A mortgage pass-through security consists of a set of marketable shares in a portfolio of mortgages for which investors receive monthly payments of both interest and principal. Normally the package is secured by credit insurance so that investors are protected from the credit risks of the individual mortgages in the portfolio. However, no protection is provided against the cash flow and return volatility associated with unanticipated principal prepayments, which typically occur when interest rates drop and homeowners refinance their mortgages. A conventional pass-through is a mortgage pass-through security that is not guaranteed by a government agency. If applicable, you would want to know the prepayment rate for a particular tranche formed from the pass-through security as well as any stipulated guarantees in terms of interest and principal payments. One should note that the greater the discount assumed to be paid for a tranche, the more a tranche will benefit from faster prepayments. The converse is true for a tranche for which a premium is paid. The faster the prepayments, the lower the cash flow yields. The above factors along with an expected reinvestment rate will give you an idea as to whether or not the 9% return will be realized. (b) What are the limitations of the cash flow yield for assessing the potential return from investing in a RMBS?

The limitations for the cash flow yield in a RMBS are like the yield to maturity for a bond where it is assumed that the coupon payments can be reinvested at a rate equal to the yield to maturity and the bond is held to maturity. These shortcomings are equally present in application of the cash flow yield measure: (i) the projected cash flows are assumed to be reinvested at the cash flow yield, and (ii) the RMBS is assumed to be held until the final payout based on some prepayment assumption. The importance of reinvestment risk, the risk that the cash flow will be reinvested at a rate less than the cash flow yield, is particularly important for many RMBS because payments come as frequently as every month. The cash flow yield, moreover, 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.


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2. Using the cash flow yield methodology, a spread is calculated over a comparable Treasury security. How is a comparable Treasury determined?

Comparable treasury issue refers to a Treasury security that is selected (by an independent investment banker) to have an actual or interpolated maturity comparable to the remaining term of the debt securities to be redeemed that would be utilized, at the time of selection and in accordance with customary financial practice, in pricing new issues of corporate debt securities of comparable maturity to the remaining term of such debt securities. At the time of purchase it is not possible to determine an exact yield for an RMBS; the yield will depend on the actual prepayment experience of the mortgages in the pool. 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. One way of determining a comparable Treasury is by matching maturity. However, the repayment of principal over time makes it inappropriate to compare the yield of non-Treasury instruments (such as a RMBS) to a Treasury of a stated maturity. Instead, market participants have used two measures: Macaulay duration and average life. The practice of spreading the yield to the average life on the interpolated Treasury yield curve is improper for an amortizing bond even in the absence of interest-rate volatility. What should be done is to calculate the static spread. This is the yield spread in a static scenario (no volatility of interest rates) of the bond over the entire theoretical Treasury spot rate curve (and not a single point on the Treasury yield curve). Of interest, the Federal Financial Institutions Examination Council has developed the rate spread calculator that gives the spread between the APR and the comparable treasury security utilizing the "Treasury Securities of Comparable Maturity under Regulation C" table, action taken, lock-in date, APR, term (loan maturity), and lien status. The rate spread is the spread between the APR on a loan and the rate on Treasury securities with comparable maturity periods for loan originations in which the APR exceeds the applicable rate by a percentage specified by the Board. The reporting requirement applies to originations of: home purchase loans, dwellingsecured home improvement loans, and refinancings. 3. What is 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 to deal with this drawback is referred to as vector analysis. A vector is simply a set of numbers. In the case of prepayments, it is a vector of prepayment speeds. 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. 4. In the calculation of effective duration and effective convexity, why is a prepayment model needed?

A prepayment model will tell us how prepayments cause the projected cash flows to change as interest rates change. This is needed because to properly compute effective duration and effective convexity we must account for cash flows changing with interest rates.


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5. The following excerpt is taken from an article titled “Fidelity Eyes $250 Million Move into Premium PACs and I-Os” that appeared in the January 27, 1992, issue of B ondWeek , pp. 1 and 21:

“Three Fidelity investment mortgage funds are considering investing this quarter a total of $250 million in premium planned amortization classes of collateralized mortgage obligations and some interest- only strips, said Jim Wolfson, portfolio manager … Wolfson … will look mainly at PACs backed by 9-10% Federal Home Loan Mortgage Corp. and Federal National Mortgage Association pass-throughs. These have higher option-adjusted spreads than regular agency pass-throughs, or similar premium Government National Mortgage Association-backed, PACs, he said. He expects I-Os will start to perform better as prepayments start to slow later in this quarter.

The higher yields on I-Os and premium PACs compensate for their higher prepayment risk, said Wolfson. “You get paid in yield to take on negative convexity,” he said. He does not feel prepayments will accelerate …” Answer the below questions. (a) Why would premium PACs and interest-only strips offer higher yields if the market expects that prepayments will accelerate or are highly uncertain?

Prepayments will be expected to accelerate if interest rates are expected to decline or if there is a greater possibility of decline due to general uncertainty as to which way rates will change. For such a situation, investors would expect to deal with greater reinvestment rate risk. To compensate for this risk, investors would have to be given higher rates of return for investing in securities that will be retired earlier than desired. More details on how this affects both planned amortization tranches (PAC tranches) and interest-only strips (I-Os or just IOs) are given below. PAC tranches can reduce prepayment risk in a manner desired by an investor’s preference. However, despite the redistribution of prepayment risk with sequential-pay and accrual collateralized mortgage obligations (CMOs), there is still considerable prepayment risk. That is, there is still considerable average life variability for a given tranche. This problem is mitigated by the PAC tranche. The greater predictability of the cash flow for PAC bonds occurs because there is a principal repayment schedule that must be satisfied. PAC bondholders have priority over all other classes in the CMO issue in receiving principal payments from the underlying collateral. The greater certainty of the cash flow for the PAC bonds comes at the expense of the non-PAC classes, called support or companion bonds. It is these bonds that absorb the prepayment risk. Because PAC bonds have protection against both extension risk and contraction risk, they are said to provide two-sided prepayment protection.

In early 1987, stripped MBS began to be issued where all the interest is allocated to one class (the IO class) and the entire principal to the other class (the PO class). The IO class receives no principal payments. IOs and POs are referred to as mortgage strips . The PO security is purchased at a substantial discount from par value. The yield an investor will


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realize depends on the speed at which prepayments are made. The faster the prepayments, the higher the yield the investor will realize. When an IO is purchased there is no par value. In contrast to the PO investor, the IO investor wants prepayments to be slow. The reason is that the IO investor receives only interest on the amount of the principal outstanding. As prepayments are made, the outstanding principal declines, and less dollar interest is received. In fact, if prepayments are too fast, the IO investor may not recover the amount paid for the IO. (b) What does Wolfson mean when he says: “You get paid in yield to take on negative convexity”?

Negative convexity has the same impact on the price performance of a RMBS as it does on the 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. Thus, Wolfson is pointing out that investors will require higher yields for such investments or equivalently: “You get paid in yield to take on negative convexity.” (c) What measure is Wolfson using to assess the risks associated with prepayments?

Because Wolfson is looking at securities that can experience a decline when prepayment increases, Wolfson wants a measure that captures this. Thus, Wolfson appears to be cognizant of the prepayment sensitivity measure. More details on this measure are supplied below. The value of a RMBS will depend on prepayments. To assess prepayment sensitivity, market participants have used the prepayment sensitivity measure that determines the basis point change in the price of an RMBS for a 1% increase in prepayments. Specifically, this measure is defined as prepayment sensitivity = ( P s – P 0)100 where P s = price (per $100 par value) assuming a 1% increase in prepayment speed, and P 0 = initial price (per $100 par value) at assumed prepayment speed. For example, suppose that for some RMBS at 300 PSA the price is P 0 = 106.10. A 1% increase in the PSA prepayment rate means that PSA increases from 300 PSA to 303 PSA. Suppose that at 303 PSA the price is recomputed using a valuation model to be P 0 = 106.01. Therefore, prepayment sensitivity = (106.01 – 106.10)100 = 9. Notice that a security that is adversely affected by an increase in prepayment speeds will have a negative prepayment sensitivity while a security that benefits from an increase in prepayment speed will have a positive prepayment sensitivity. 

6. In an article titled “CUNA Mutual Looks for Noncallable Corporates” that appeared in the November 4, 1991, issue of BondWeek , p. 6, Joe Goglia, a portfolio manager for CUNA Mutual Insurance Group, stated that he invests in “ planned amortization class tranches,

which have less exposure to prepayment risk and are more positively convex than other mortgage-backeds.” Is this true? As seen below there are a lot of factors to consider before we assume that a PAC tranche will absolutely have less exposure to prepayment risk and will be more positively convex that other mortgage-backed securities.


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The creation of a mortgage-backed security cannot make prepayment risk disappear. This is true for both a pass-through and a CMO. Thus the reduction in prepayment risk (both extension risk and contraction risk) that a PAC offers must come from somewhere. It comes from the support bonds. If the support bonds are paid off quickly because of faster-than-expected prepayments, there is no longer any protection for the PAC bonds. Planned amortization class tranches are structured to have less exposure to prepayment risk. For a security that is option-free and displays positive convexity, the price appreciation will be greater than the price depreciation for a large change in yield. Negative convexity means that the price appreciation will be less than the price depreciation for a large change in yield. Generally, the market will take the greater convexity bonds into account in pricing them. How much should the market want investors to pay up for convexity? If investors expect that market yields will change by very little — that is, they expect low interest rate volatility — investors should not be willing to pay much for convexity. In fact, if the market prices convexity high, investors with expectations of low interest rate volatility will probably want to “sell convexity.” In the case of a CMO tranche the price depends on the characteristics of the tranche and the spread to Treasuries at the termination date. The key determinants are the “quality” of the tranche, its average life (or duration), and its convexity. Quality refers to the type of CMO tranche. Consider, for example, that an investor can purchase a CMO tranche that is a PAC bond but as a result of projected prepayments could become a sequential-pay tranche. As another example, suppose that a PAC bond is the longest-average-life tranche in a reverse PAC structure. Projected prepayments in this case might occur in an amount to change the class from a long-average-life PAC tranche to a support tranche. The converse is that the quality of a tranche may improve as well as deteriorate. For example, the effective collar for a PAC tranche could widen at the horizon date when prepayment circumstances increase the par amount of support tranches outstanding as a proportion of the deal. For a tranche, a standard convexity can indicate positive convexity, whereas the effective convexity can indicates negative convexity. The difference is even more dramatic for securities not trading near par. For a PO created from a tranche, the standard convexity can be close to zero whereas the effective convexity can be very large. 7. What is a path-dependent cash flow security?

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 determined 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. 8. Why is a pass-through security a path-dependent cash flow security?


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In the case of mortgage pass-through securities, prepayments are path dependent because this month’s prepayment rate depends on whether there have been prior opportunities to refinance since the underlying mortgages were originated. Unlike mortgage loans, the decision as to whether a corporate issuer will elect to refund an issue when the current rate is below the issue’s coupon rate is not dependent on how rates evolved over time to the current level. Moreover, in the case of adjustable-rate pass-throughs (ARMs), prepayments are not only path dependent but the periodic coupon rate depends on the history of the reference rate upon which the coupon rate is determined. This is because ARMs have periodic caps and floors as well as a lifetime cap and floor. 9. Give two reasons why a CMO tranche is a path-dependent cash flow security.

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 path dependency in a CMO tranche’s cash flows. First, the collateral prepayments are path dependent. 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 these balances. 10. Explain how, given the cash flow on the simulated interest-rate paths, the theoretical value of a RMBS is determined.

Given the cash flow on an interest-rate path, its present value can be calculated. The discount rate for determining the present value is the simulated spot rate for each month on the interest-rate path plus an appropriate spread. The spot rate on a path can be determined from the simulated future monthly rates. The relationship that holds between the simulated spot rate for month T on path n and the simulated future one-month rates is z T( n) =

1+ f (n)1 + f 1

1/T

2 (n)  . . . 1 + fT (n)

1

where z T( n) = simulated spot rate for month T on path n, and f j(n) = simulated future one-month rate for month j on path n. The interest-rate path for the simulated future one-month rates can be converted to the interest-rate path for the simulated monthly spot rates. Thus, the present value of the cash flow for month T on interest-rate path n discounted at the simulated spot rate for month T plus some spread is: PV[C T( n)] =

CT (n )

1  z

(n)  K  T

T

where PV[C T( n)] = present value of cash flow for month T on path n, C T (n) = cash flow for month T on path n, z T( n) = spot rate for month T on path n, and K = appropriate risk-adjusted


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spread. The present value for path n is the sum of the present value of the cash flow for each month on path n. We have (assuming a maturity of 360 months that consists of a path of 360 simulated interest-rate path scenarios): PV[path(n)] = 1/ 360  PV  C1 (n)  PV  C2 (n)  + . . . +PV  C360 (n)  where PV[path(n)] is the present value of interest-rate path n. The present value of a given interest-rate path can be thought of as the theoretical value of a pass-through if that path was actually realized. The theoretical value of the pass-through can be determined by calculating the average of the theoretical valu e of all the interest-rate paths. That is, the theoretical value is equal to theoretical value = 1/ N  PV path(1)   PV path(2)  + . . . + PV path(N )  . This procedure for valuing a pass-through is also followed for a CMO tranche. The cash flow for each month on each interest-rate path is found according to the principal repayment and interest distribution rules of the deal. 11. Explain how, given the cash flow on the simulated interest-rate paths, the average life of a RMBS is determined.

Given the cash flow on the simulated interest-rate paths, the average life can be determined for each path. The average life reported in a Monte Carlo analysis is the average of the average lives along the interest-rate paths. As with the theoretical value, additional information is conveyed by the distribution of the average life. The greater the range and standard deviation of the average life, the more uncertainty there is about the security’s average life. 12. Suppose that a support bond is being analyzed using the Monte Carlo simulation methodology. The theoretical value using 1,500 interest-rate paths is 88. The range for the path present values is a low of 50 and a high of 115. The standard deviation is 15 points. How much confidence would you place on the theoretical value of 88?

The probability of actually achieving a theoretical value of 88 is not very likely. However, we can put a probability of occurrence for a range of values surrounding 88. For example, if we can assume a distribution resembling a normal distribution, then we kno w there is about a twothirds probability that we will find a range of values between 73 and 103. Nevertheless, given the range of low and high values it appears the distribution is not perfectly normal but skewed to the left, thus taking on more values below 88 than above 88. 13. In a well-protected PAC structure, what would you expect the distribution of the path present values and average lives to be compared to a support bond from the same CMO structure?


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PAC tranches are structured to meet the designs of the investor who prefers a reduction is risk. The greater predictability of the cash flow for PAC bonds occurs because there is a principal repayment schedule that must be satisfied. PAC bondholders have priority over all other classes in the CMO issue in receiving principal payments from the underlying collateral. The greater certainty of the cash flow for the PAC bonds comes at the expense of the non-PAC classes, called support or companion bonds. It is these bonds that absorb the prepayment risk. Because PAC bonds have protection against both extension risk and contraction risk, they are said to provide two-sided prepayment protection. In conclusion, with less risk and uncertainty in a well-protected PAC structure, we would expect the distribution of the path present values and average lives to be more stable compared to a support bond from the same CMO structure. 14. Suppose that the following values for a RMBS are correct for each assumption:

PSA Assumption Value of Security 192

112.10

194

111.80

200

111.20

202

111.05

210

110.70

Assuming that the value of the security in the market is 111.20 based on 200 PSA. What is the prepayment sensitivity of this security?

To assess prepayment sensitivity, market participants have used the following measure. This measure determines the basis point change in the price of an RMBS for a 1% increase in prepayments. We have: prepayment sensitivity = ( P s – P 0)100 where P s = price (per $100 par value) assuming a 1% increase in prepayment speed and P 0 = initial price (per $100 par value) at assumed prepayment speed. In our problem, we find a 1% increase from a PSA assumption from 200 to 202, i.e., (202 – 200) / 200 = 2 / 200 = 0.01 or 1.00%. For these PSA assumptions, the initial price (per $100 par value) is P 0 = 111.20 and the price assuming a 1% increase in prepayment speed is P s = 111.05. Inserting in these values in our formula gives: prepayment sensitivity = ( P s – P 0)100 = (111.05 – 111.20)100 = 15. The security in our problem has a negative prepayment sensitivity. This indicates that it is adversely affected by an increase in prepayment.


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15. An analysis of a CMO structure using the Monte Carlo method indicated the following, assuming 12% volatility:

OAS (basis points) Collateral Tranche PAC I A PAC I B PAC I C PAC II Support

Static Spread (basis points)

80

120

40 55 65 95 75

60 80 95 125 250

(a) Calculate the option cost for each tranche.

The implied cost of the option embedded in any RMBS can be obtained by calculating the difference between the OAS at the assumed volatility of interest rates and the static spread. We use the below formula: option cost = static spread – option-adjusted spread. The reason that the option cost is measured in this way is as follows. In an environment of no interest-rate changes, the investor would earn the static spread. When future interest rates are uncertain, the spread is less, however, because of the homeowner’s o ption to prepay; the OAS reflects the spread after adjusting for this option. Therefore, the option cost is the difference between the spread that would be earned in a static interest-rate environment (the static spread) and the spread after adjusting for the homeowner’s option. Below we compute the option cost for each tranche. Tranche PAC I A: option cost = 60 basis points – 40 basis = 20 basis points . Tranche PAC I B : option cost = 80 basis points – 55 basis = 25 basis points . Tranche PAC I C: option cost = 95 basis points – 65 basis = 30 basis points . Tranche PAC II : option cost = 125 basis points – 95 basis = 30 basis points . Support : option cost = 250 basis points – 75 basis = 175 basis points .

In general, a tranche’s option cost is more stable than its OAS in the face of market movements. This interesting feature is useful in reducing the computational costs of calculating the OAS as the market moves. For small market moves, the OAS of a tranche may be approximated by recalculating the static spread (which is relatively cheap and easy to calculate) and subtracting its option cost. (b) Which tranche is clearly too rich?

The support tranche is rich relative to Treasuries. We might add that a typical OAS run will be done for 512 to 1,024 interest-rate scenario paths to value a RMBS. The scenarios generated using the simulation method look very realistic and, furthermore, reproduce today’s Treasury curve. By employing this technique, the money manager is effectively saying that Treasuries are


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fairly priced today and that the objective is to determine whether a specific RMBS is rich or cheap relative to Treasuries. The number of interest-rate paths determines how “good” the estimate is, not relative to the truth but relative to the model. The more paths, the more average spread tends to settle down. (c) What would happen to the static spread for each tranche if a 15% volatility is assumed?

At the higher level of assumed interest-rate volatility of 15%, the static spread would fall because of the homeowner ’s option to prepay. This would negatively affect the collateral with its loss distributed among the tranches with tranches with longer duration experiencing greater losses. Tranche Z and any residual tranche would be least affected. (d) What would happen to the OAS for each tranche if a 15% volatility is assumed?

The OAS would fall for each tranche causing a negative change in price per dollar. As stated in part (c), this would negatively affect the collateral with its loss distributed among the tranches with tranches. 16. Why would the option-adjusted spread vary across dealer firms?

As discussed below, the option-adjusted spread will vary across dealer firms because each dealer will make their own volatility assumptions. In the Monte Carlo model, the OAS is the spread K that when added to all the spot rates on all interest-rate paths will make the average present value of the paths equal to the observed market price (plus accrued interest). The spread among dealer firms will differ to the extent interest-rate paths and their volatility assumptions differ. The typical model that Wall Street firms and commercial vendors use to generate random interest-rate paths takes as input today’s term structure of interest rates and a v olatility assumption. The term structure of interest rates is the theoretical spot rate (or zero-coupon) curve implied by today’s Treasury securities. The volatility assumption determines the dispersion of future interest rates in the simulation. The simulations should be normalized so that the average simulated price of a zero-coupon Treasury bond equals today’s actual price. Each model has its own model of the evolution of future interest rates and its own volatility assumptions. Typically, there are no significant differences in the interest-rate models of dealer firms and vendors, although their volatility assumptions can be significantly different. 17. Explain how the number of interest-rate paths used in the Monte Carlo simulation methodology is determined.

The number of interest-rate paths used in the Monte Carlo simulation methodology is determined by the number of sample paths necessary to get a good statistical sample to obtain a price estimate within a tick. More details are given below. What is the number of scenario paths or repetitions, N, needed to value a RMBS? A typical OAS Copyright © 2016 Pearson Education, Inc.

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run will be done for 512 to 1,024 simulation method look very realistic employing this technique, the money priced today and that the objective is relative to Treasuries.

interest-rate paths. The scenarios generated using the and, furthermore, reproduce today’s Treasury curve. By manager is effectively saying that Treasuries are fairly to determine whether a specific RMBS is rich or cheap

The number of interest-rate paths determines how “good” the estimate is, not relative to the truth but relative to the model. The more paths, the more average spread tends to settle down. It is a statistical sampling problem. Most Monte Carlo simulation models use some form of variance reduction to cut down on the number of sample paths necessary to get a good statistical sample. Variance reduction techniques allow us to obtain price estimates within a tick. By this we mean that if the model is used to generate more scenarios, price estimates from the model will not change by more than a tick. For example, if 1,024 paths are used to obtain the estimated price for a tranche, there is little more information to be had from the model by generating more than that number of paths. (For some very sensitive CMO tranches, more paths may be needed to estimate prices within one tick.) 18. Explain why you agree or disagree with the following statement: “ When the Monte

Carlo simulation methodology is used to value a RMBS, a PSA assumption is employed for all interest-rate paths.” As seen in the illustration and Exhibit 19-9, the collateral value of a CMO is not always influenced by the PSA assumption when using the Monte Carlos simulation methodology. However, the value of the tranches is influenced so that there is a need to employ an accurate PSA assumption to gage how the price will change when prepayment assumptions change. 19. Answer the below questions. (a) What assumption is made about the OAS in calculating the effective duration and effective convexity of a RMBS?

There is a problem with calculating price sensitivity measures such as interest-rate and prepayment sensitivity using the OAS methodology. Recall that in measuring effective duration and effective convexity, the calculation assumed that the OAS is held constant and the interestrate paths are shifted by a specified number of basis points. After adding the initial OAS to the new interest-rate paths, a security is then revalued to obtain the two values to be used in the effective duration and convexity formulas. The trouble in assuming a constant OAS in calculating interest-rate sensitivity measures is that the assumption cannot be justified on theoretical grounds. Thus, we have a serious problem with the notion of keeping OAS constant for measuring price sensitivity to changes to all sensitivity measures (e.g., interest-rate and prepayment sensitivity). One conclusion that can be drawn from this is to adjust (and use) a prepayment model so that it equates OAS levels.


431

(b) Is it warranted?

As just noted, the trouble in assuming a constant OAS in calculating interest-rate sensitivity measures is that the assumption cannot be justified or warranted on theoretical grounds. The OAS methodology provides a measure for relative valuation of MBS products that is vastly superior to some of the ad hoc measures that were historically used (and unfortunately still used) by portfolio managers. Those traditional measures fail to recognize the embedded options in these complex products. Nevertheless, using the OAS as a relative value measure has drawbacks. The first drawback is the prepayment model uncertainty. The second drawback is that in pricing MBS, there are differing OAS levels. 20. What are the limitations of the option-adjusted spread measure?

Although the OAS measure is much more useful than the static cash flow yield measure, it still suffers from major pitfalls. These limitations apply not only to the OAS for RMBS but also the OAS produced from a binomial model. First, the OAS is a product of the valuation model. The valuation model may be poorly constructed because it fails to capture the true factors that affect the value of particular securities. Second, in Monte Carlo simulation the interest-rate paths must be adjusted so that on-the-run Treasuries are valued properly. That is, the value of an on-the-run Treasury is equal to its market price or, equivalently, its OAS is zero. The process of adjusting the interest-rate paths to achieve that result is ad hoc. A third problem with the OAS is that it assumes a constant OAS for each interest rate path and over time for a given interest-rate path. If there is a term structure to the OAS, this is not captured by having a single OAS number. Finally, the OAS is dependent on the volatility assumption, the prepayment assumption in the case of RMBS, and the rules for refunding in the case of corporate bonds. In addition, there is a problem with calculating an OAS for a portfolio by taking a weighted average of the OAS of the individual portfolio holdings. Instead, if an OAS for a portfolio is sought, it is necessary to obtain the portfolio’s cash flow along each interest -rate path. The OAS is then the spread that will make the average portfolio value equal to the portfolio’s market value. 21. What assumptions are required to assess the potential total return of a RMBS?

The measure that should be used to assess the performance of a security or a portfolio over some investment horizon is the total return. The total dollars received from investing in a RMBS consist of (i) the projected cash flow from the projected interest payments and the projected principal repayment (scheduled plus prepayments), (ii) the interest earned on reinvestment of the projected interest payments and the projected principal prepayments, and (iii) the projected price of the RMBS at the end of the investment horizon. To obtain the cash flow, a prepayment rate over the investment horizon must be assumed. The second step requires assumption of a reinvestment rate. Finally, cash flow yield or Monte


432

Carlo simulation methodologies can be used to calculate the price at the end of the investment horizon under a particular set of assumptions. Either approach requires assumption of the prepayment rate and the Treasury rates (i.e., the yield curve) at the end of the investment horizon. The cash flow yield methodology uses an assumed spread to a comparable Treasury to determine the required cash flow yield, which is then used to compute the projected price. The Monte Carlo simulation methodology requires an assumed OAS at the investment horizon. From this assumption, the OAS methodology can produce the horizon price. 22. What are the complications of assessing the potential total return of a CMO tranched using the total return framework?

The most difficult part of estimating total return is projecting the price at the horizon date. In the case of a CMO tranche the price depends on the characteristics of the tranche and the spread to Treasuries at the termination date. The key determin ants are the “quality” of the tranche, its average life (or duration), and its convexity. Quality refers to the type of CMO tranche. Consider, for example, that an investor can purchase a CMO tranche that is a PAC bond but as a result of projected prepayments could become a sequential-pay tranche. As another example, suppose that a PAC bond is the longest-average-life tranche in a reverse PAC structure. Projected prepayments in this case might occur in an amount to change the class from a long-average-life PAC tranche to a support tranche. The converse is that the quality of a tranche may improve as well as deteriorate. For example, the effective collar for a PAC tranche could widen at the horizon date when prepayment circumstances increase the par amount of support tranches outstanding as a proportion of the deal. To test the sensitivity of total return to various alternative assumptions scenario analysis is helpful. Its limitation is that only a small number of potential scenarios can be considered, and it fails to take into consideration the dynamics of changes in the yield curve and the dynamics of the deal structure. 23. On July 1, 2013, the FHLMC 30-Year Generic 4% 2012 was analyzed using the Monte Carlo valuation model of FactSet. At the time of the analysis the security’s price was 104.644 with accrued interest of 0.300 (per $100 par value). Summary information about the security is as follows:

Coupon (%) Maturity Date Issue Date Original Balance (USD) Current Balance (USD) Factor Date

4.000 01-Oct-2041 01-May-2012 26,885,718,052.11 23,341,475,489.50 01-Jun-2013

The results of a simulation using 200 interest-rate path are reproduced as follows:


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YTM (%) Average Life Modified Duration Effective Duration Effective Convexity Partial Duration — 6 Month Partial Duration — 1 Year Partial Duration — 2 Year Partial Duration — 5 Year Partial Duration — 10 Year Partial Duration — 30 Year Spread Duration Spread (TRSY) Z-Spread OAS (TRSY) OAS (Libor) Projected CPR (PB WAVG) Projected PSA (PB WAVG)

2.981 5.307 4.551 4.135 – 1.905 – 0.088 0.097 0.516 1.500 1.806 0.304 4.338 1.585 97.232 95.158 77.909 14.870 278.342

(a) Explain the meaning of each of the measures above. YTM(%) is the percentage rate of return paid on a bond, note, or other fixed income security if the investor buys and holds it to its maturity date. The calculation for YTM is based on the coupon rate, length of time to maturity, and market price. It assumes that coupon interest paid over the life of the bond will be reinvested at the same rate. Yield to maturity is the most popular measure of yield in the market. It is the rate that will make the present value of a bond's cash flows equal to its market price plus accrued interest. To find YTM, one has to develop the cash flows and then, through trial and error, find the interest rate that makes the present value of cash flow equal to the market price plus accrued interest. Average Life is the length of time the principal of a debt issue is expected to be outstanding. Average life is an average period before a debt is repaid through amortization or sinking fund payments. To calculate the average life, multiply the date of each payment (expressed as a fraction of years or months) by the percentage of total principal that has been paid by that date, summing the results and dividing by the total issue size. Average Life is also called "weighted average maturity" and "weighted average life." Modified Duration is the ratio of Macaulay duration to 1 + y as the modified duration. The equation is Macaulay duration divided by one plus the yield to maturity. The Macaulay duration 1C 2C nC nM + +. . .+ + n n 1 2 1  y  1  y  1  y  1  y  is where P = price of the bond, C = semiannual P coupon interest, y = one-half the yield to maturity or required yield, n = number of semiannual periods (number of years times 2), and M = maturity value.


434

Effective Duration is the approximate percentage change in price for a 100 basis point change in rates. It allows for changing cash flow when interest rates change. Using the following definitions of P _ = the bond's price if its yield falls by x basis points, P + = the bond's price if its yield rises by x basis points, P 0 = the bond's initial price per $100 of par value, and Δ y = change in yield in decimal, we can use the following formula:

effective duration =

P  P 

2 P 0  y 

.

Effective Convexity is the convexity of a bond calculated using cash flows that change with yields and it allows for changing cash flow when interest rates change. In equation form we have: P  P  2P 0 effective duration =  2 P0  y 

where the definition for the variables were just given. Partial Duration — 6 Month is the same as a key rate duration that estimate the sensitivity of a security or a portfolio to changes in the yield curve and measure the sensitivities to movements in the Treasury spot curve about the selected points for key maturities – in this case 6 months. The basic principle of key rate duration is to change the yield for a particular maturity of the yield curve and determine the sensitivity of a security or portfolio to that change holding all other yields constant. One application of partial duration is the calculation of option-adjusted duration by changing a variable such the OAS while holding all other variables constant. Partial Duration — 1 Year is same as above but the key maturity used is one year. Partial Duration — 2 Year is same as above but the key maturity used is two years. Partial Duration — 5 Year is same as above but the key maturity used is five years. Partial Duration — 10 Year is same as above but the key maturity used is ten years. Partial Duration — 30 Year is same as above but the key maturity used is thirty years. Spread Duration measures the interest-rate sensitivity to changes in the OAS. It can be defined as the sensitivity of the price of a bond to a 100 basis point change to its option-adjusted spread. As the rate of the Treasury security in the option-adjusted spread increases, the rate of the option-adjusted spread also increases. Spread (TRSY) is the spread for the broad U.S. Treasury Index Fund. Z-Spread is zero-volatility OAS or simply the constant spread that will cause the price of a security to equal the present value of its cash flows when added to the yield at each point on the spot rate Treasury curve where a cash flow is received. In essence, each cash flow is discounted


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at a suitable Treasury spot rate plus the Z-spread. The Z-spread is also referred to as a "static spread" or zero-volatility OAS. OAS (TRSY) is the Option-Adjusted Spread for the broad U.S. Treasury Index Fund. In general, OAS is a measurement of the spread of a fixed-income security rate and the risk-free rate of return, which is adjusted to take into account an embedded option. Typically, an analyst would use the Treasury securities yield for the risk-free rate. The spread is added to the fixed-income security price to make the risk-free bond price the same as the bond. OAS (Libor) is the Option-Adjusted Spread for LIBOR, which is the world’s most widely-used benchmark for short-term interest rates. LIBOR represents the rates that some of the world’s leading banks charge each other for short-term loans. LIBOR refers to the London Interbank Offered Rate and serves as the first step to calculating interest rates on various loans throughout the world. There are a total of 35 different LIBOR rates each business day. The most commonly quoted rate is the three-month U.S. dollar rate. Projected CPR (PB WAVG) is the estimated conditional prepayment rate that is equal to the proportion of the principal of a pool of loans that is assumed to be paid off prematurely in each period. The calculation of this estimate is based on a number of factors such as historical prepayment rates for previous loans that are similar to ones in the pool and on future economic outlooks. The constant CPR is the weighted arithmetic average of those monthly CPRs, where the weights are the remaining principal balance. Projected PSA (PB WAVG) is the estimated Public Securities Association rate from which the schedule of principal repayment is protected. An assumed PSA speed allows the cash flow to be projected. (b) Given the computed convexity measure, how is this pass-through security expected to perform compared to a comparable Treasury security if the term structure of Treasury rates decreases substantially in a parallel fashion?

The static spread is the yield spread in a static 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. The yield to maturity (using the FactSet term) is 2.981%. The yield to maturity is the static cash flow yield which is of questionable value given the assumptions regarding the realization of the cash flows and the ability to reinvest interim cash flows (principal payments and interest). The average life is 5.037, which is the average life based on the static cash flows. The Spread (TRSY) is 1.585 and the Z-Spread is 97.232. FactSet uses a term structure of the short-term rate volatility as the volatility assumption. It is derived from the interest-rate cap market. Two benchmark rates were used in the Monte Carlo simulation analysis to determine the OAS: Treasury rates and LIBOR. The OAS (TRSY) is 95.158 and the OAS (Libor) is 77.909. The following price sensitivity measures were reported by FactSet:


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Modified Duration Effective Duration Effective Convexity Partial Duration — 6 Month Partial Duration — 1 Year Partial Duration — 2 Year Partial Duration — 5 Year Partial Duration — 10 Year Partial Duration — 30 Year Spread Duration

4.551 4.135 – 1.905 – 0.088 0.097 0.516 1.500 1.806 0.304 4.338

For this security, the effective duration and modified duration are very similar at 4.551 and 4.135. The effective convexity is – 1.905. Thus, the security is expected to have negative convexity. Negative convexity means that the price appreciation will be less than the price depreciation for a large change in yield of a given number of basis points. The “partial durations” are the same as the key rate durations and measure the sensitivities to movements in the Treasury spot curve about the selected points. The partial duration is negative for six months and peaks at 1.806 for ten years. The spread duration of 4.338 measures the interest-rate sensitivity to changes in the OAS.


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