2. Interpreting the OAS and Relative Value Application What an investor or a portfolio manager seeks to do is to buy a mortgage-backed security where value is greater than price. By using a valuation model such as the Monte Carlo model, a portfolio manager could estimate the value of a security, which at this point would be sufficient in determining whether to buy a security. That is, the portfolio manager can say that this security is 1 point cheap or 2 points cheap, and so on. The model does not stop here. Instead, it converts the divergence between price and value into some type of spread measure since most market participants find it more convenient to think about spreads than price differences.

The OAS was developed as a measure of the spread that can be used to convert dollar differences between value and price. As we explained in Chapter 9, in the binomial model a spread is measured relative to the benchmark interest rates used to generate the interest rate tree and which is therefore used to make the tree arbitrage free. The same is true in the case of the Monte Carlo simulation model. The spread is measured relative to the benchmark interest rates that were used to generate the interest rate paths and to adjust the interest rate paths to make them arbitrage free. Typically, for a mortgage-backed security the benchmark interest rates are the on-the-run Treasury rates. The OAS is then measuring the average spread over the Treasury spot rate curve, not the Treasury yield as explained in Chapter 9. It is an average spread since the OAS is found by averaging over the interest rate paths for the possible Treasury spot rate curves. Of course, if the LIBOR curve is used, the OAS is the spread over that curve.

This spread measure is superior to the nominal spread which gives no recognition to the prepayment risk. As explained in Chapter 9, the OAS is “option adjusted” because the cash flows on the interest rate paths are adjusted for the option of the borrowers to prepay. While we may understand the mechanics of how to compute the OAS and why it is appropriate to use the OAS rather than the nominal spread or Z-spread, the question is what does the spread represent? In Chapter 9, a discussion of what compensation the OAS reflects for corporate and agency bonds with embedded options was presented. The compensation is for a combination of credit risk and liquidity risk. The compensation depends on the benchmark interest rates used in the analysis. For example, consider the use of Treasury interest rates and more specifically the Treasury spot rate curve since this is the benchmark typically used in the calculation of an OAS for a mortgage-backed security. If there is an OAS computed using the Treasury benchmark, then what is that OAS compensating?

Consider first Ginnie Mae mortgage passthrough securities. As explained earlier (Chapter 3), Ginnie Mae is an arm of the U.S. government. The securities it issues are backed by the full faith and credit of the U.S. government. Effectively, a Ginnie Mae mortgage-backed security is a Treasury security with prepayment risk. The OAS removes the prepayment risk (i.e., the option risk). Additionally, if the benchmark is Treasury rates, the OAS should not be compensation for credit risk. That leaves liquidity risk. While Ginnie Mae mortgage passthrough securities may not be as liquid as on-the-run Treasury issues, they are fairly liquid as measured by the bid-ask spread. Nevertheless, part of the OAS should reflect compensation for liquidity risk. There is one more risk that was not the focus of Chapter 9, modeling risk. In our explanation of the Monte Carlo model, there were several critical assumptions and parameters required. If those assumptions prove incorrect or if the parameters are misestimated, the prepayment model will not calculate the true level of risk. Thus, probably a good part of the compensation for a Ginnie Mae mortgage passthrough security reflects payment for this model uncertainty.

If we consider CMOs issued by Ginnie Mae rather than the mortgage passthrough securities, the OAS would reflect the complexity associated with a particular tranche. For example, a planned amortization class (PAC) tranche would be less exposed to modeling risk than a support tranche in the same CMO structure. Hence, compensation for modeling risk would be greater for a support tranche than a PAC tranche in the same structure. Moreover, PAC tranches have greater liquidity than support tranches, so compensation is less for the former relative to the latter.

As we move from Ginnie Mae issued mortgage products to those issued by Freddie Mac and Fannie Mae, we introduce credit risk. As explained in Chapter 3, Freddie Mac and Fannie Mae are government sponsored enterprises (GSEs). As such, there is no requirement that a GSE be bailed out by the U.S. government. The GSEs are viewed as triple A rated. Consequently, in addition to modeling risk and liquidity risk, a portion of the OAS reflects credit risk relative to Treasury securities.

Moving on to nonagency mortgage-backed securities and real estate backed asset-backed securities, the OAS is compensating for: (1) credit risk (which varies depending on the credit rating for the tranche under consideration), (2) liquidity risk (which is greater than for Ginnie Mae, Fannie Mae, and Freddie Mac mortgage products), and (3) modeling risk.

F. Option Cost

G. Illustrations

1. Simple Structure The simple structure analyzed is Freddie Mac (FHLMC) 1915. It is a simple sequential-pay CMO bond structure. The structure includes eight tranches, A, B, C, D, E, F, G, and S. The focus of our analysis is on tranches A, B, and C. All three tranches were priced at a premium.

The top panel of Exhibit 5 shows the OAS, the option cost, and effective duration¹⁶¹ for the collateral and the three tranches in the CMO structure. However, tranche A had the smallest effective duration and tranche C had the largest effective duration. The OAS for the collateral is 51 basis points. Since the option cost is 67 basis points, the zero-volatility spread is 118 basis points (51 basis points plus 67 basis points).

At the time this analysis was performed, March 10, 1998, the Treasury yield curve was not steep. We explained that when the yield curve is relatively flat the zero-volatility spread will not differ significantly from the nominal spread. Thus, for the three tranches shown in Exhibit 5, the zero-volatility spread is 83 basis points for A, 115 basis points for B, and 116 basis points for C.

Notice that the tranches did not share the OAS equally. The same is true for the option cost. Both the Z-spread and the option cost increase as the effective duration increases. Whether or not any of these tranches were attractive investments requires a comparison to other tranches in the market with the same effective duration. While not presented here, all three tranches offered an OAS similar to other sequential-pay tranches with the same effective duration available in the market. On a relative basis (i.e., relative to the other tranches analyzed in the deal), the only tranche where there appears to be a bit of a bargain is tranche C. A portfolio manager contemplating the purchase of this last cash flow tranche can see that C offers a higher OAS than B and appears to bear less of the risk (i.e., has lower option cost), as measured by the option cost. The problem portfolio managers may face is that they might not be able to go out as long on the yield curve as tranche C because of effective duration, maturity, and average life constraints relative to their liabilities, for example.

Now let’s look at modeling risk. Examination of the sensitivity of the tranches to changes in prepayments and interest rate volatility will help us to understand the interaction of the tranches in the structure and who is bearing the risk. How the deal behaves under various scenarios should reinforce and be consistent with the valuation (i.e., a tranche may look “cheap” for a reason).

We begin with prepayments. Specifically, we keep the same interest rate paths as those used to get the OAS in the base case (the top panel of Exhibit 5), but reduce the prepayment rate on each interest rate path to 80% of the projected rate. As can be seen in the second panel of Exhibit 5, slowing down prepayments increases the OAS and price for the collateral. The exhibit reports two results of the sensitivity analysis. First, it indicates the change in the OAS. Second, it indicates the change in the price, holding the OAS constant at the base case.

EXHIBIT 5 OAS Analysis of FHLMC 1915 Classes A, B, and C (As of 3/10/98) All three tranches were trading at a premium as of the date of the analysis.

To see how a portfolio manager can use the information in the second panel, consider tranche A. At 80% of the prepayment speed, the OAS for this tranche increases from 32 basis points to 40 basis points. If the OAS is held constant, the panel indicates that the buyer of tranche A would gain $0.17 per $100 par value.

Notice that for all of the tranches reported in Exhibit 5 there is a gain from a slowdown in prepayments. This is because all of the sequential tranches in this deal are priced over par. (An investor in a tranche priced at a premium benefits from a slowdown in prepayments because the investor receives the higher coupon for a longer period and postpones the capital loss from a prepayment.) Also notice that while the changes in OAS are about the same for the different tranches, the changes in price are quite different. This arises because the shorter tranches have less duration. Therefore, their prices do not move as much from a change in OAS as a longer average life tranche. A portfolio manager who is willing to go to the long end of the yield curve, such as tranche C, would realize the most benefit from the slowdown in prepayments.

Also shown in the second panel of the exhibit is the second part of our experiment to test the sensitivity of prepayments: the prepayment rate is assumed to be 120% of the base case. The collateral loses money in this scenario because it is trading above par. This is reflected in the OAS of the collateral which declines from 51 basis points to 40 basis points. Now look at the three tranches. They all lost money because the tranches were all at a premium and the speeding of prepayments adversely affects the tranche.

Before looking at the last panel that shows the effect of a change in interest rate volatility on the OAS, let’s review the relationship between expected interest rate volatility and the value of a mortgage-backed security. Recall that the investor in a mortgage-backed security has sold an option to homeowners (borrowers). Thus, the investor is short an option. As will be explained in Chapter 14, the value of an option depends on expected interest rate volatility. When expected interest rate volatility decreases, the value of the option embedded in a mortgage-backed security decreases and therefore the value of a mortgage-backed security increases. The opposite is true when expected interest rate volatility increases—the value of the embedded option increases and the value of a mortgage-backed security decreases.

Now let’s look at the sensitivity to the interest rate volatility assumption, 13% in the base case. Two experiments are performed: reducing the volatility assumption to 9% and increasing it to 17%. These results are reported in the third panel of Exhibit 5.

Reducing the volatility to 9% increases the dollar price of the collateral by $1.03 and increases the OAS from 51 in the base case to 79 basis points. However, this $1.03 increase in the price of the collateral is not equally distributed among the three tranches. Most of the increase in value is realized by the longer tranches. The OAS gain for each of the tranches follows more or less the effective durations of those tranches. This makes sense, because the longer the duration, the greater the risk, and when volatility declines, the reward is greater for the accepted risk. At the higher level of assumed interest rate volatility of 17%, the collateral is severely affected. The longer the duration, the greater the loss. These results for a decrease and an increase in interest rate volatility are consistent with what we explained earlier.

Using the Monte Carlo simulation/OAS analysis, a fair conclusion that can be made about this simple structure is: what you see is what you get. The only surprise in this structure is the lower option cost in tranche C. In general, however, a portfolio manager willing to extend duration gets paid for that risk in this structure.

2. PAC/Support Tranche Structure Now let’s look at how to apply the methodology to a more complicated CMO structure, FHLMC Series 1706. The collateral (i.e., pool of passthroughs) for this structure is Freddie Mac 7s (7% coupon rate). A partial summary of the deal is provided in Exhibit 6. That is, only the tranches we will be discussing in this section are shown in the exhibit.¹⁶²

While this deal looks complicated, it is relatively simple compared to many deals that have been issued. Nonetheless, it brings out all the key points about application of OAS analysis, specifically, the fact that most deals include cheap bonds, expensive bonds, and fairly priced bonds. The OAS analysis helps identify how a tranche should be classified. A more proper analysis would compare the OAS for each tranche to a similar duration tranche available in the market.

All of the tranches in Exhibit 6 were discussed in Chapter 10. At issuance, there were 10 PAC tranches, three scheduled tranches, a floating-rate support tranche, and an inverse floating-rate support. Recall that the “scheduled tranches” are support tranches with a schedule, referred to in Chapter 10 as “PAC II tranches.”

The first two PAC tranches in the deal, tranche A and tranche B, were paid off at the time of the analysis. The other PAC tranches were still available at the time of the analysis. The prepayment protection for the PAC tranches is provided by the support tranches. The support tranches in this deal that are shown in Exhibit 6 are tranches LA, LB, and M. There were other support tranches not shown in Exhibit 6. LA is the shortest average life support tranche (a scheduled (SCH) bond).

EXHIBIT 6 Summary of Federal Home Loan Mortgage Corporation—Multiclass Mortgage Participation Certificates (Guaranteed), Series 1706

The collateral for this deal was trading at a premium. That is, the homeowners (borrowers) were paying a higher mortgage rate than available in the market at the time of the analysis. This meant that the value of the collateral would increase if prepayments slow down but would decrease if prepayments increase. What is important to note, however, is that a tranche could be trading at a discount, par, or premium even though the collateral is priced at a premium. For example, PAC C had a low coupon rate at the time of the analysis and therefore was trading at a discount. Thus, while the collateral (which was selling at a premium) loses value from an increase in prepayments, a discount tranche such as tranche C would increase in value if prepayments increase. (Recall that in the simple structure analyzed earlier, the collateral and all the tranches were trading at a premium.)

The top panel of Exhibit 7 shows the base case OAS, the option cost, and the effective duration for the collateral and tranches in Exhibit 6. The collateral OAS is 60 basis points, and the option cost is 44 basis points. The Z-spread of the collateral to the Treasury spot curve is 104 basis points.

The 60 basis points of OAS did not get equally distributed among the tranches—as was the case with the simple structure analyzed earlier. Tranche LB, the scheduled support, did not realize a good OAS allocation, only 29 basis points, and had an extremely high option cost. Given the prepayment uncertainty associated with this tranche, its OAS would be expected to be higher. The reason for the low OAS is that this tranche was priced so that its cash flow yield is high. Using the Z-spread as a proxy for the nominal spread (i.e., spread over the Treasury yield curve), the 103 basis point spread for tranche LB is high given that this appears to be a short average life tranche. Consequently, “yield buyers” (i.e., investors with a preference for high nominal yield, who may not be attentive to compensation for prepayment risk) probably bid aggressively for this tranche and thereby drove down its OAS, trading off “yield” for OAS. From a total return perspective, however, tranche LB should be avoided. It is a rich, or expensive, tranche. The other support tranche analyzed, tranche M, had an OAS of 72 basis points and at the time of this analysis was similar to that offered on comparable duration tranches available in the market.

EXHIBIT 7 OAS Analysis of FHLMC 1706 (As of 3/10/98)

The analysis reported in the top panel of Exhibit 7 help us identify where the cheap tranches are in the deal. The long average life and effective duration tranches in the deal are the PAC tranches G, H, J, and K. These tranches have high OAS relative to the other tranches and low option cost. They appear to be the cheap tranches in the deal. These PAC tranches had well protected cash flows and exhibited positive convexity (i.e., these tranches lose less in an adverse scenario than they gain in a positive scenario).

The next two panels in Exhibit 7 show the sensitivity of the OAS and the price (holding OAS constant at the base case) to changes in the prepayment speed (80% and 120% of the base case) and to changes in volatility (9% and 17%). This analysis shows that the change in the prepayment speed does not affect the collateral significantly, while the change in the OAS (holding the price constant) and price (holding OAS constant) for each tranche can be significant.

Tranches C and D at the time of the analysis were priced at a discount with short average lives. The OAS and price of these two tranches were not affected by a slowing down or a speeding up of the prepayment model. Tranche H was a premium tranche with a medium-term average life at the time of the analysis. Because tranche H was trading at a premium, it benefits from a slowing in prepayments, as the bondholder will receive the coupon for a longer time. Faster prepayments represent an adverse scenario. The PAC tranches are quite well-protected. The longer average life PACs will actually benefit from a reduced prepayment rate because they will be earning the higher coupon interest longer. So, on an OAS basis, the earlier conclusion that the long PACs were allocated a good part of the deal’s value holds up under our first stress test (i.e., changing prepayments).

The sensitivity of the collateral and the tranches to changes in volatility are shown in the third panel of Exhibit 7. A lower volatility increases the value of the collateral, while a higher volatility reduces its value (This is consistent with our option cost equation in Section IV.E.) The long average life PACs continue to be fairly well-protected, whether the volatility is lower or higher. In the two volatility scenarios they continue to get a good OAS on a relative value basis, although not as much as in the base case if volatility is higher (but the OAS still looks like a reasonable value in this scenario). This reinforces the earlier conclusion concerning the investment merit of the long PACs in this deal. Note, however, that PAC tranches H, J, and K are more sensitive to the volatility assumption than tranches C, D, E, and G and therefore the investor is accepting greater volatility risk (i.e., the risk that volatility will change) with tranches H, J, and K relative to tranches C, D, E, and G.

A. Duration Measures

1. Effective Duration To calculate effective duration, the value of the security must be estimated when rates are shocked up and down a given number of basis points. In terms of the Monte Carlo model, the yield curve used (either the Treasury yield curve or LIBOR curve) is shocked up and down and the new curve is used to generate the values to be used in the effective duration and effective convexity formulas. This is analogous to the process we used to compute effective duration and effective convexity using the binomial model in Chapter 9.

In generating the prices when rates are shocked up and down, there is an assumption that the relationships assumed in generating the initial price do not change when rates are shocked up and down. Specifically, the yield volatility is assumed to be unchanged to derive the new interest rate paths for a given shock (i.e., the new Exhibit 1), the spread between the mortgage rate and the 10-year Treasury rate is assumed to be unchanged in constructing the new Exhibit 2 from the newly constructed Exhibit 1, and the OAS is assumed to be constant. The constancy of the OAS comes into play because when discounting the new cash flows (i.e., the cash flows in the new Exhibit 3), the current OAS that was computed is assumed to be the same and is added to the new rates in the new Exhibit 1.

We’ll use an illustration by Lakhbir Hayre and Hubert Chang to explain the calculation of effective duration for a mortgage-backed security, a FNMA 7.5% TBA passthrough on May 1, 1996.¹⁶⁴ On that day, the base mortgage rate was 7.64%. The price of the issue at the time was 98.781 (i.e., 98-25). The OAS was 65 basis points. Based on a shock of 25 basis points, the estimated prices holding the OAS constant at 65 basis points were as follows:

V₋ = 99.949 for a decrease in the yield curve of 25 basis points

V₊ = 97.542 for an increase in the yield curve of 25 basis points

The effective duration based on a Δy of 0.0025 is then

There are differences in the effective durations for a given mortgage-backed security reported by dealers and vendors of analytical systems. Several practitioners have explained and illustrated why there are differences in effective duration estimates reported by dealers and vendors. The differences result from, among others:¹⁶⁵

1. differences in the amount of the rate shock used

2. differences in prepayment models

3. differences in option-adjusted spread

4. differences in the relationship between short-term interest rates and refinancing rates

Earlier, we discussed the first reason. As explained, the rate shock is the amount interest rates are increased and decreased to obtain the two values that are inserted into the effective duration formula. If the change is too large, there is the problem with picking up the effect of convexity.

Prepayment models differ across dealers and vendors. Some dealer models consistently forecast slower prepayments relative to other dealer models and others the reverse.

The effective duration is dependent on the OAS computed. Recall that the calculation of the OAS is a byproduct of the Monte Carlo model. Therefore, the computed value for the OAS depends on all of the assumptions in the Monte Carlo model. Specifically, it depends on the yield volatility assumed and the prepayment model employed. Dealers and vendors make different assumptions regarding yield volatility and use proprietary prepayment models. These can result in differences in OAS. Since the OAS is added to the new simulated short-term rates to compute the new values for V₋ and V₊, a different OAS will result in different effective durations.

Finally, recall that in explaining the Monte Carlo simulation model that we stated that in moving from Exhibit 1 (the simulated short-term rates) to Exhibit 2 (the refinancing rates), an assumption must be made about the relationship between short-term rates and the 10-year Treasury rate (i.e., the rate used as a proxy for refinancing). Differences in models about how large the spread between these rates will be affects the value of a mortgage-backed security and therefore the values used in the duration equation when rates are shocked.

2. Other Duration Measures There have been other measures proposed for estimating the duration of a mortgage-backed security. These measures include cash flow duration, coupon curve duration, and empirical duration. The first two duration measures are forms of effective duration in that they do recognize that the values that should be used in the duration formula should take into account how the cash flows may change due to changes in prepayments when interest rates change. In contrast, empirical duration is a duration that is computed statistically using observed market prices. Below we describe how each of these duration measures is calculated, as well as the advantages and limitations of each.

a. Cash Flow Duration Recall from the general duration formula that there are two values that must be substituted into the numerator of the formula—the value if rates are decreased (V₋) and the value if rates are increased (V₊). With effective duration, these two values consider how changes in interest rates change the cash flow due to prepayments. This is done through the Monte Carlo simulation by allowing for the cash flows to change on the interest rate paths.

For cash flow duration, there is recognition that the cash flow can change but the analysis to obtain the cash flow is done following a static methodology. Specifically, the cash flow duration is calculated as follows:

Step 1: Calculate the cash flow based on some prepayment assumption.

Step 2: From the cash flow in Step 1 and the market price (V₀), compute the cash flow yield.

Step 3: Increase the cash flow yield by Δy and from a prepayment model determine the new prepayment rate at that higher cash flow yield. Typically, the prepayment rate will be lower than in Step 1 because of the higher yield level.

Step 4: Using the lower prepayment rate in Step 3 determine the cash flow and then value the cash flow using the higher cash flow yield as the discount rate. This gives the value (V₊).

Step 5: Decrease the cash flow yield by Δy and from a prepayment model determine the new prepayment rate at that lower cash flow yield. Typically, the prepayment rate will be higher than in Step 1 because of the lower yield level.

Step 6: Using the higher prepayment rate in Step 5 determine the cash flow and then value the cash flow using the lower cash flow yield as the discount rate. This gives the value (V₋).

From the change in basis points (Δy), the values for V₊ and V₋ found in Steps 4 and 6, and the initial value V₀, the duration can be computed.

We can use the hypothetical CMO structure in Chapter 10 to illustrate how to calculate cash flow duration. Specifically, in FJF-2, there were four tranches, A, B, C, and Z. Let’s focus on tranche C. Suppose that the price for this tranche is 100.2813. Then the cash flow duration is computed as follows:

Step 1: Suppose that the assumed prepayment rate for this tranche is 165 PSA.

Step 2: Based on the assumed prepayment rate of 165 PSA and the price of 100.2813, it can be demonstrated that the cash flow yield is 7%.

Step 3: Suppose that the cash flow yield is increased (i.e., shocked) by 25 basis points (from 7% to 7.25%) and suppose that some prepayment model projects a prepayment rate of 150 PSA. (Note that this is a slower prepayment rate than at 7%.)

Step 4: Based on 150 PSA, a new cash flow can be generated. The cash flow is then discounted at 7.25% (the new cash flow yield). It can be demonstrated that the value of this tranche based on these assumptions would be 98.3438. This is the value V₊.

Step 5: Suppose that the cash flow yield is decreased (i.e., shocked) by 25 basis points (from 7% to 6.75%) and suppose that some prepayment model projects a prepayment rate of 200 PSA. (Note that this is a faster prepayment rate than at 7%.)

Step 6: Based on 200 PSA, the new cash flow can be generated. The cash flow is then discounted at 6.75% (the new cash flow yield). It can be demonstrated that the value of this tranche based on these assumptions would be 101.9063. This is the value V₋.

Now we have the following information:

V₀ = 100.2813

V₊ = 98.3438

V₋ = 101.9063

Δ _y = 0.0025

Then using the general form of duration, we obtain:

What type of duration measure is the cash flow duration—effective duration or modified duration? Technically, it is a form of effective duration because notice that in Steps 3 and 5 when the rate is changed, the cash flow is allowed to change. However, as has been stressed throughout, the valuation model that is used to get the new values to substitute into the duration formula is critical. The valuation model in the case of the cash flow duration is based on the naive assumption that there is a single prepayment rate over the life of the mortgage-backed security for any given interest rate shock. This is in contrast to the values produced by the Monte Carlo simulation model that does more sophisticated analyses of how the cash flow can change when interest rates change.

Why bother discussing the cash flow duration if it is an inferior form of effective duration? The reason is that it is a commonly cited duration measure and practitioners should be aware of how it is computed. Likewise, we did discuss in detail yield calculations despite their limitations.

An interesting question is how does this form of duration compare to modified duration. Recall from Chapter 7 and Chapter 9 that modified duration does not assume that cash flows will change when rates are shocked. That is, in the steps discussed above to obtain cash flow duration, in Steps 3 and 5 it is assumed that the prepayment rate is the same as in Step 1.

To illustrate this, we’ll once again use tranche C in FJF-2. In Step 3, the prepayment rate assumed is still 165 PSA despite the fact that rates are assumed to increase. That is, the cash flow is not assumed to change. Based on a cash flow yield of 7.25% and a prepayment rate of 165 PSA, the value of this tranche would decline to 98.4063. When the cash flow yield is assumed to decline to 6.75%, the prepayment rate is still assumed to be 165 PSA and the value of this tranche would be 102.1875. Then to calculate the modified duration we know:

V₀ = 100.2813

V₊ = 98.4063

V₋ = 102.1875

Δ_y = 0.0025

The modified duration is then:

Thus, the modified duration is greater than the cash flow duration for this tranche.

It is important to reiterate that the modified duration is inferior to the cash flow duration because the former gives absolutely no recognition to how prepayments may change when interest rates change. While cash flow duration is commonly cited in practice, it is a form of effective duration that does give some recognition that prepayments and therefore cash flow may change when interest rates change, but it is based on a naive assumption about how prepayments may change. The effective duration as computed using Monte Carlo simulation is superior to cash flow duration.

b. Coupon Curve Duration The coupon curve duration uses market prices to estimate the duration of a mortgage-backed security. This approach, first suggested by Douglas Breeden,¹⁶⁶ starts with the coupon curve of prices for similar mortgage-backed securities. The coupon curve represents generic passthrough securities of a particular issuer with different coupon rates. By rolling up and down the coupon curve of prices, the duration can be obtained. Because of the way it is estimated, this approach to duration estimation was referred to by Breeden as the “roll-up, roll-down approach.” The prices obtained from rolling up and rolling down the coupon curve of prices are substituted into the duration formula.

To illustrate this approach, suppose that the coupon curve of prices for a passthrough security for some month is as follows:

Coupon	Price
6%	85.19
7%	92.06
8%	98.38
9%	103.34
10%	107.28
11%	111.19

Suppose that the coupon curve duration for the 8% coupon passthrough is sought. If the yield declines by 100 basis points, the assumption is that the price of the 8% coupon passthrough will increase to the price of the current 9% coupon passthrough. Thus, the price will increase from 98.38 to 103.34. Similarly, if the yield increases by 100 basis points, the assumption is that the price of the 8% coupon passthrough will decline to the price of the 7% coupon passthrough (92.06). Using the duration formula, the corresponding values are:

V₀ = 98.38

V₊ = 92.06

V₋ = 103.34

Δ_y = 0.01

The estimated duration based on the coupon curve is then:

Breeden tested the coupon curve durations and found them to be relatively accurate in estimating the interest rate risk of generic passthrough securities.¹⁶⁷ Bennett Golub reports a similar finding.¹⁶⁸

While the advantages of the coupon curve duration are the simplicity of its calculation and the fact that current prices embody market expectations, there are disadvantages. The approach is limited to generic mortgage-backed securities and difficult to use for mortgage derivatives such as CMOs.

c. Empirical Duration When computing effective duration and cash flow duration, the values to be substituted into the duration formula are those based on some valuation model. For coupon curve duration, the observed market prices are used in the duration formula. In contrast, empirical duration is estimated statistically using historical market prices and market yields.¹⁶⁹ Regression analysis is used to estimate the relationship. Some firms such as PaineWebber use empirical duration, also called implied duration, as their primary measure of the duration of an MBS.

There are three advantages to the empirical duration approach.¹⁷⁰ First, the duration estimate does not rely on any theoretical formulas or analytical assumptions. Second, the estimation of the required parameters is easy to compute using regression analysis. Finally, the only inputs that are needed are a reliable price series and Treasury yield series.

There are disadvantages.¹⁷¹ First, a reliable price series for the mortgage security may not be available. For example, there may be no price series available for a thinly traded mortgage derivative security or the prices may be matrix priced (i.e., priced by a pricing service based on issues with similar characteristics) or model priced rather than actual transaction prices. Second, an empirical relationship does not impose a structure for the options embedded in a mortgage-backed security and this can distort the empirical duration. This may occur after a sharp and sustained shock to interest rates have been realized. Finally, the volatility of the spread to Treasury yields can distort how the price of a mortgage-backed security reacts to yield changes.