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1. Basic Mortgage-Backed Analysis ________________________ 3

2. Basic Asset-Backed Analysis___________________________25

3. Mortgage-Backed/Asset-Backed Valuation ________________ 35

4. Key Formulas_______________________________________ 47

2010 Allen Resources, Inc. All rights reserved.Warning: Copyright violations will be prosecuted.

Any use of these materials without the express written consent of the publisher is a violat ion of federal and/or international copyright laws.

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Mortgage-Backed/Asset-Backed Valuation

2010 Allen Resources, Inc.15-

3.Mortgage-Backed/Asset-Backed

Valuation

Learning Objectives

This summary includes a review and an analysis of the principles set forth by CFA Institute.

Upon review of this summary, you should be able to:

Calculate the cash flow yield of a mortgage-backed or asset-backed security .............pg. 36 Discuss the limitations of the cash flow yield measure .................................................pg. 37 Discuss the limitations of the nominal spread and the zero-volatility spread for a

mortgage-backed security ...............................................................................................pg. 37

Discuss the Monte Carlo simulation model for valuing a mortgage-backed securityand its limitations ............................................................................................................pg. 38 Discuss how the option-adjusted spread is computed using the Monte Carlo

simulation model and how this measure is interpreted .................................................pg. 38

Utilize the option-adjusted spread analysis to value mortgage-backed securities ........pg. 40 Discuss the interest rate risk of a fixed income security, with attention to the

securitys duration, convexity, option-adjusted spread, and the price value of a

basis point ........................................................................................................................pg. 42

Discuss some major assumptions that result in differences in effective durationreported by dealers and vendors .....................................................................................pg. 43

Discuss other measures of duration, and their limitations, used by practitioners inthe mortgage-backed market such as: cash flow duration, coupon curve duration,

and empirical duration ....................................................................................................pg. 43

Discuss whether the nominal spread, zero-volatility spread, or option-adjustedspread should be used to assess the value of a specific fixed income security .............pg. 44

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Study Session 15

Study Guide for the Level II 2011 CFA Exam - Reading Highlights15-36

Introduction

Learning Objective: Calculate the cash flow yield of a mortgage-backed or asset-backed security.

The cash flow yield of a mortgage-backed security (or asset-backed security) is the rate at whichthe expected cash flows can be discounted and equated to the purchase price of the security.Usually the discount rate found is monthly, and is converted to a bond-equivalent yield by thefollowing formula:

Bond-equivalent yield = 2 [(1 + Ym)6 1]

Simplified Example

A $1,000,000 par value, mortgage-backed security purchased at 100% of par value producesreturns (after prepayments and defaults) of $90,000 per month for twelve months. Find the cash

flow yield.

Solution

Using a financial calculator or a spreadsheet, the cash flow yield can be determined by findingthe monthly internal rate of return (1.2%) and converting it into a bond-equivalent yield(14.89%) using the formula above.

Example - Variation 1

Suppose the MBS was purchased for 105% of par value. What would the cash flow yield be?

Solution

Simply change the first value in the stream of cash flows from -$1,000,000 to -$1,050,000. Thenew monthly IRR is 0.44%, so the cash flow yield on a bond-equivalent basis will be 5.29%.

Example - Variation 2

Suppose the MBS was purchased for 105% of par value and the prepayments are such that thetotal cash flows are $100,000 per month for eleven months. Find the cash flow yield.

Solution

Again, change the first value in the stream of cash flows from -$1,000,000 to -$1,050,000.Change the 2

ndthrough 12

thvalues to $100,000. Delete the 13

thvalue of $90,000. The monthly

IRR will be 0.78%, leading to a cash flow yield on a bond-equivalent basis of 9.59%.

In general, the lower the purchase price, the higher the cash flow yield. Also, the more rapid theprepayment, the higher the cash flow yield.

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Learning Objective: Discuss the limitations of the cash flow yield measure.

The cash flow yield measure has several shortcomings that are related to the shortcomings ofyield-to-maturity measures. Cash flow yield implicitly assumes that the projected cash flows canbe reinvested at the cash flow yield until maturity, and that the security itself is held until

maturity (or final payout).

Cash flow yield ignores several sources of variation that can cause realized yields to differ fromcash flow yields. First, it ignores reinvestment risk - the risk that interest rates will have fallen,reducing the amount of reinvestment income earned by the interim cash flows. Also, prepaymentspeeds may differ from expected and are often inversely related to market interest rates; thiswould change the amount of the cash flows themselves. So interest rate movements can affectboth the amount of the cash flows and the reinvestment income they can earn.

Learning Objective: Discuss the limitations of the nominal spread and the zero-volatility spreadfor a mortgage-backed security.

The nominal spread of a mortgage-backed security is simply the difference between its yield andthat of a Treasury, where the maturity of the Treasury matches the mortgage-backed securitysaverage life (or the interpolated yield, if an exact match is not found).

However, since principal is often amortized over the life of the tranche, it would be inappropriateto compare the cash flow yield of an MBS to that of a Treasury. Instead, it would be moreappropriate to compute the average spread over the entire theoretical Treasury spot rate curve,not just at one point in time. When it is assumed that interest rates have no volatility, this iscalled the zero-volatility spread.

The analyst may or may not wish to change prepayment assumptions for future dates in light of

the current yield curve. If the forward rate curve implies a future change in interest rates, it maybe desirable to incorporate a change in the mortgage refinancing rate (and the prepayment rate)over time.

While this zero-volatility (option-adjusted) spread is more sophisticated than the nominal spread,it still does not provide for the uncertainty of future interest rates. It relies on a static yield curveand its implied forward rates. The analysis is fully deterministic and provides only a single pointestimate of spread value. What is needed instead is a range of potential values and the relativeprobability of values falling into a given subset of that range.

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Study Session 15


Monte Carlo Approaches

Learning Objective: Discuss the Monte Carlo simulation model for valuing a mortgage-backedsecurity and its limitations.

Learning Objective: Discuss how the option-adjusted spread is computed using the Monte Carlosimulation model and how this measure is interpreted.

One way to obtain a range of values for the spreads of mortgage-backed securities is the MonteCarlo simulation model. It is stochastic, not deterministic. By this, we mean that the outcomereflects probabilities and incorporates elements of chance. In applying Monte Carlo simulationanalysis to valuing mortgage-backed securities, the simulation model generates a large numberof random interest rate and prepayment rate scenarios (using a computer), aggregates them, andcomputes statistics such as the average spread.

One reason the Monte Carlo technique is so useful in modeling mortgage-backed securities is

that their cash flows are very path-dependent. This means that to determine what may happen ata given point in time, it is necessary to know what happened prior to it. So, for example, if youwere modeling interest rates across a period of time, you would not have rates going from 5%one month to 7% the next month. Instead, the rate for a given month is going to be defined asthat from a prior month plus some random change. Thus, if interest rates moved from 5% to 7%,they would almost always take a continuous path to get there, such as 5.0%, 5.5%, 6.0%, 6.5%,7.0%.

Interest rates are not the only variable that is path-dependent. Prepayments will also be path-dependent. For example, the level of prepayment for mortgage interest rates dropping from 8%to 6% is going to be very different from the prepayments in a scenario when mortgage rates have

risen from 5% to 6%. Also, adjustable rate mortgages will be affected by the path of interestrates as well, since most incorporate annual as well as lifetime caps and floors (or, if both,collars). Finally, the cash flow that a given CMO tranche receives will depend, in part, on thepath of cash flows to other tranches. For example, the support CMO may be exhausted by year10 under one rate path, but still providing a cushion at year 10 under another rate path. MonteCarlo simulation gives a tool for evaluating many of these situations rapidly.

It is important that the model be set up properly. This would include a validation of the currentterm structure of interest rates, by comparing the models computed value of a given Treasurysecurity to the actual observed market price. Also, future simulated projections of the termstructure of interest rates need to be arbitrage-free. The model calculates the present value ofprojected cash flows along each interest rate path, discounting them back to the present time. Thesimulated spot rate for month Talong path n is found by its relationship to the simulated forwardrates:

( ) ( ) ( ) ( ){ }1

1 21 + 1 + 1 + 1T

T Tz n f n f n f n=

wheref2(n)represents the simulated one-month forward rate, for month 2, along simulated

interest rate path n.

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Once this simulated spot rate is calculated, the cash flow for month Talong path n may bediscounted by this rate plus a spread, S:

( )( )

( )1

CFPV CF

1 + +

T

T

T

T

nn

z n S

=

The present value for all the cash flows along a given interest rate path are then calculated,assuming 360 monthly periods, as:

( ) ( ) ( ) ( ){ }1 2 3601

PV path PV CF + PV CF + + PV CF360

n n n n

=

Finally, the price of the security should equal:

( )1

1PV Path

N

n

n

N =

In valuing mortgage-backed securities, one solves for the option-adjusted spread (OAS) which,when added to the simulated spot rate, discounts the cash flows of the security (averaging acrossall interest rate paths) to its observed market price. Note that this method does not result intelling us whether the observed market price is fair. Instead, what it tells us is the OAS that isimplied by the observed market price. One can then take the OAS and judge whether it is largeenough to justify the additional risk over Treasuries.

Recall the impliedoption cost (in this case, the option that the borrower has to prepay) is equal tothe difference between the static spread and the OAS:

Option cost = Static spread OAS

The Monte Carlo method is not without its shortcomings. Like any model, it is subject to modelrisk, the risk that the model will fail to incorporate all the variables that affect the value of thesecurity being examined. Also, the variables and assumptions (volatility, prepayment) includedmay be misspecified. Lastly, the OAS obtained is assumed to be constant over a given interestrate path. If OAS itself has a term structure, then Monte Carlo analysis will not capture it.

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Learning Objective: Utilize the option-adjusted spread analysis to value mortgage-backedsecurities.

Example

Consider a simplified sequential pay (plain vanilla) CMO with 3 classes, A, B, and Z, where theZ tranche is an accrual bond. Assume there is also a residual tranche, R. Assume that in the basecase, interest rate volatility is 15% per year. You are also given the following information aboutthe OAS:

OAS (bp) Option Cost (bp) Static Spread (bp)

Collateral 75 54 129

Class A 25 31 56

Class B 55 49 104

Class Z 80 56 136

The weighted average OAS among the classes will equal the OAS for the collateral. Note howthe OAS increases as the average (expected) life of the tranche increases; this is a commonoccurrence. An analyst would look to an exhibit such as the one above and try to find anattractive tranche; for example, one that had an OAS larger than another class, yet with loweroption cost. So if there was a Class C above with an OAS of 60 and an option cost of 45 thatwould be comparatively attractive to Class B.

The next step in this OAS model is to vary the prepayment rate. The baseline scenario wouldhave a prepayment rate designated as 100. Perhaps it is 165 PSA, or 200 PSA; whatever it is, wedesignated it 100% of the baseline case. Next, we vary the prepayment rate, keeping interestrates steady, and see what happens to our spreads and prices:

New OAS (bp) Change in Price/$100 par

Change in prepayments 80% 120% 80% 120%

Collateral 75 77 $0.00 +$0.06

Class A 8 45 -$0.45 +$0.50

Class B 37 83 -$0.90 +$1.20

Class Z 72 100 -$0.35 +$2.80

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Here we have increased and decreased prepayments by 20% and compared the effect on the OASand on the tranche price. Note that the collateral is relatively insensitive to the change in theprepayment rate; it is assumed that the collateral trades at close to par. This can happen when thecollateral is trading at close to par. However, it is evident that slower prepayments will result in ashrinking OAS (at a given price), or, as shown in the columns on the right-hand side, a shrinking

price for a given OAS.

Generally, the longer the duration, the worse the effect of prepayments upon the price at a givenOAS. However, Class Z is not affected as badly; this is because it was already an accrualtranche, so it wouldnt receive any payments immediately anyway. Also note that the collateralhas not changed in value significantly for slower prepayments, though the tranches shown abovehave all declined. This can happen when there is a residual class, which benefits from theextension.

In contrast, when prepayments rise, the quicker return of principal benefits all the tranchesshown. The collateral does not increase in value much, assuming it trades just below par. ClassesA, B, and Z all benefited, and investors who took the longest tranche (Z) benefited the greatest.

Since the collateral did not change much in price, we can infer that the residual tranche declinedin value because of the speedup.

Next, we can look at what happens when the interest rate volatility assumption is changed,holding the prepayments at their baseline level.

New OAS (bp) Change in Price/$100 par

Interest Rate Volatility 10% 20% 10% 20%

Collateral 99 50 +$0.90 -$0.85

Class A 40 6 +$0.40 -$0.40

Class B 72 23 +$1.20 -$1.35

Class Z 107 54 +$3.20 -$3.00

Notice that the decline in interest rate volatility increases the value of the collateral; as expected,the increase is not evenly distributed among the tranches. The lower volatility disproportionatelyrewards those investors who took on longer duration risk (the later tranches). Similarly, wheninterest rate volatility increases, the collateral is adversely affected, and the longer duration

classes suffer disproportionate losses. Since the collateral itself is affected by the change involatility, we dont expect gains to be found in the residual class.

The identification of rich or cheap securities would be done by comparing the OAS and optioncost, tranche by tranche. Generally, an investor would expect to be rewarded for investing inlonger duration tranches. However, if there is a tranche that has a higher OAS for a given optioncost, or a lower option cost for a given OAS, then that tranche would be comparatively attractive(cheap).

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Duration Measures of Interest Rate Risk

Learning Objective: Discuss the interest rate risk of a fixed income security, with attention to thesecuritys duration, convexity, option-adjusted spread, and the price value of a basis point.

Interest rate risk is the risk that a security will decline in value due to a change in interest rates.There are many ways to measure interest rate risk: duration, duration with a convexityadjustment, and the price value of a basis point.

Recall that effective duration is the percentage price change in a bond for a 100 basis pointchange in interest rates, taking into account that expected cash flows may be altered by shifts ininterest rates.

Effective duration =0

1

2 D UP P

P r

In particular, mortgage-backed securities are sensitive to interest-rate shifts. When rates drop,prepayments and curtailments rise, causing negative convexity.

So, for example, consider a mortgage-backed security that is currently trading at par value (call it100) to yield 7.00%. Suppose for a 1% increase in interest rates, the price drops to 94, but for a1% decrease in interest rates, the price only rises to 105, due to the effects of negative convexity.The effective duration will then be:

1 105 945.5

2 100 0.01

=

This measure of effective duration tells us that the bond in question would change in price 5.5%for a 100-basis point move in interest rates. This price sensitivity is a measure of interest raterisk.

Note, however, that we used a fairly large shift in interest rates - a full percentage. One couldinstead check the price change for a one basis point shift in interest rates. The average pricechange would then be termed the PVBP, or price value of a basis point. For such a small shift ininterest rates, we would not expect much in the way of accelerated prepayments and negativeconvexity. Suppose the price of our illustrative bond at 7.01% was $99.95 and at 6.99% was$100.05. Then the effective duration would be:

1 100.05 99.95 52 100 0.0001

=

This would say that for very small changes in interest rates, the security would change at a rateof 5 basis points for every basis point change in interest rates.

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We can also obtain this estimate of interest rate risk if we employ convexity. Recall thatconvexity is given by:

( )0

2

0

+ 2 Convexity

2

U DP P P

P r

=

Here, the effective convexity of the bond turns out to be -50. Using the convexity adjustment, wecan recompute the duration measure (refining our estimate):

Convexity adjustment to duration = Estimated convexity (r)2

100

Convexity adjustment to duration = -50 (0.01)2

100 = -0.5

So our new duration estimate would be 5.5 0.5 = 5.0.

Effective duration (and effective convexity) can be computed using Monte Carlo simulation by atwo-step process. First, using the current term structure of interest rates, find the bonds OAS.

Next, reprice the bond, holding OAS constant, but changing the term structure (up and down, toget a 100 basis point shift). With the bonds prices under increasing and decreasing interest rates,one then applies the duration and convexity measures to get an OAS-based measure of interestrate risk.

Learning Objective: Discuss some major assumptions that result in differences in effectiveduration reported by dealers and vendors.

Financial information from dealers and vendors may give different estimates of effectiveduration for a given security. There are explanations for these differences. The most obviousreason would be the size of the interest rate shift used in measuring duration. Larger shifts will

be misleading unless a convexity adjustment is incorporated.

Also, if dealers are using Monte Carlo simulation to price the securities, effective duration maydiffer due to differences in assumptions inherent in the simulation. For example, differentsimulation models may specify different prepayment assumptions, or different interest ratevolatility assumptions.

Learning Objective: Discuss other measures of duration, and their limitations, used bypractitioners in the mortgage-backed market such as: cash flow duration, coupon curve duration,and empirical duration.

Cash flow durationrefers to the sensitivity of a cash flows present value to changes in interestrates. A zero-coupon bond has a duration equal to its stated term to maturity. Duration declinesas coupon income increases; also, prepayments reduce duration as well.

Coupon curve durationrefers to a method of estimating duration by comparing the prices ofsecurities with different coupons. For example, if interest rates increase by 1%, the market priceof a 5% coupon, 30-year bond will drop in value close to what the price was for a 4% coupon

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bond. Thus, a lower coupon is similar to yields rising by the difference in the coupons, and ahigher coupon is similar to yields falling by the difference in the coupons.

To see this, assume the market requires a yield of 5% (2.5%, semiannually). A 30-year 5%coupon bond would then trade at par, $1,000. A 30-year, 4% coupon bond would trade at adiscount, $845.46. If interest rates rose to 6%, the 5% bond would drop in value to $861.62, orwithin 2% of the original value of the 4% bond.

We can apply this technique to mortgage-backed securities.

Example

Assume that the coupon curve for FNMAs is as follows: 5% trades at 94, 6% trades at 96.5, 7%trades at 99.5. What is the estimated duration of the 6s?

Solution

Using coupon curve duration, we simply estimate duration by looking at the other coupon bondsand their prices:

99.5 942.85

2 96.5 0.01

=

Coupon curve duration has some limitations. As was seen in the above example with the 5%, 30-year coupon bond, it is imprecise. Moreover, it fails to account for differences in prepaymentassumptions.

Empirical durationis a duration measure obtained by performing a regression on historical

price changes against yield changes in Treasuries. An implicit assumption in using these resultsis, of course, that historical relationships between the price changes of mortgage-backedsecurities and changes in Treasury yields will continue into the future. If they do not, then onewill not be able to predict with confidence the price movement of mortgage-backed securities.This is a significant limitation.

Choosing a Valuation Model

Learning Objective: Discuss whether the nominal spread, zero-volatility spread, or option-adjusted spread should be used to assess the value of a specific fixed income security.

As mentioned above, the nominal spread has significant limitations, particularly with respect tomortgage-backed securities, which amortize principal during their life. This creates distortionswhen computing the spread against a single Treasury spot rate; moreover, it does not take intoaccount interest rate volatility.

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The zero-volatility spread solves one of these problems: it takes the spread over the entire spotrate curve. However, it does not take into account interest rate volatility (hence its name).Consequently, it is a poor choice for securities with embedded options (such as puts, calls, orprepayment options), because valuing such securities requires some understanding of how thevalue is affected by changes in interest rates. However, it may be adequate for securities that

have no options.

The option-adjusted spread is appropriate for securities with embedded options (again, hence thename). The option-adjusted spread can be determined from the Monte Carlo valuation method,which is particularly appropriate for securities like mortgage-backs that have cash flows that aredependent upon interest rate paths.

2010 Allen Resources, Inc. All rights reserved.Warning: Copyright violations will be prosecuted.

Any use of these materials without the express written consent of the publisher is a violat ion of federal and/or international copyright laws.

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