It is important to understand this characteristic of a mortgage security. One way to understand it is to look at what happens to the change in price when interest rates move up and down by the same number of basis points. This can be seen in Exhibit 2 which shows the price-yield relationship for a security that exhibits positive convexity such as a Treasury security and a security that exhibits negative convexity. Let’s look at what happens to the price change in absolute terms when interest rates change. From Exhibit 2 we observe the following property:

This is not the case for a security that exhibits negative convexity. Instead, we also observe from Exhibit 2 the following property:

Why will a mortgage security exhibit negative convexity at some yield level? The explanation is the homeowner’s prepayment option. The value of a mortgage security declines as the value of the prepayment option increases. As mortgage rates in the market decline, the value of the prepayment option increases. As a result, the appreciation due to a decline in interest rates that would result if there had not been a prepayment option will be reduced by the increase in the value of the option.

We can see this by thinking about an agency mortgage security as equivalent to a position in a comparable-duration Treasury security and a call option. We can express the value of the mortgage security as follows:

When interest rates rise, we see the opposite effect of the prepayment option. A rise in interest rates results in a decline in the value of the Treasury security component of the mortgage security’s value. At the same time, the value of the prepayment option declines. The net effect is that while the value of a mortgage security deceases, it does not decline in value as much as a same-duration Treasury security because the decline in the value of the prepayment option offsets part of the depreciation.

Another way of viewing positive and negative convexity is how the duration changes when interest rates change. Exhibit 3 shows tangent lines to the price/yield relationship for two securities, one exhibiting positive convexity and the other negative convexity. The tangent line is related to the duration. The steeper the tangent line, the higher the duration. Notice in Exhibit 3 the following:

While a mortgage security can exhibit both positive and negative convexity, a Treasury security exhibits only positive convexity. Look at the problem of hedging the interest rate risk associated with a mortgage security by either shorting Treasury securities or selling Treasury futures (i.e., hedging an instrument that has the potential for negative convexity with an instrument that only exhibits positive convexity). As explained in Chapter 22, the hedging principle is that the change in the value of the mortgage security position for a given basis point change in interest rates will be offset by the change in the value of the Treasury position for the same basis point change in interest rates. When interest rates decline, prepayments cause the value of a mortgage security to increase less in value than that of a Treasury position with the same initial duration. Thus, when interest rates decline, simply matching the dollar duration of the Treasury position with the dollar duration of the mortgage security will not provide an appropriate hedge when the mortgage security exhibits negative convexity.

For this reason, many investors consider mortgages to be market-directional investments that should be avoided when one expects interest rates to decline.³⁵⁸ Fortunately, when properly managed, mortgage securities are not market-directional investments. Proper management begins with separating mortgage valuation decisions from decisions concerning the appropriate interest-rate sensitivity of the portfolio. In turn, this separation of the value decision from the duration decision hinges critically on proper hedging. Without proper hedging to offset the changes in the duration of mortgage securities caused by interest rate movements, the portfolio’s duration would drift adversely from its target duration. In other words, the portfolio would be shorter than desired when interest rates decline and longer than desired when interest rates rise.

To calculate the OAS for any mortgage security, a prepayment model is employed that assigns an expected prepayment rate every month—implying an expected cash flow—for a given interest rate path. These expected cash flows are discounted at U.S. Treasury spot rates to obtain their present value. This process is repeated for a large number of interest rate paths. Finally, the average present value of the cash flows across all paths is calculated. Typically, the average present value across all paths is not equal to the price of the security. However, we can search for a unique “spread” (in basis points) that, when added to the U.S. Treasury spot rates, equates the average present value to the price of the security. This spread is the OAS.

Historical comparisons are of only limited use for making judgments about current OAS levels relative to the past, because option-adjusted spreads depend on their underlying prepayment models. As a model changes, so does the OAS for a given mortgage security. There are periods where prepayment models change significantly, making comparisons to historical OASs tenuous. A portfolio manager should augment OAS analysis with other tools to help identify periods when spreads on mortgage securities are attractive, attempting to avoid periods when spread widening will erase the yield advantage over Treasuries with the same interest rate risk. The risk that the OAS may change, or spread risk, is managed by investing heavily in mortgage securities only when the initial OAS is large.

One approach to quantifying yield curve risk for a security or a portfolio is to compute how changes in a specific spot rate, holding all other spot rates constant, affect the value of a security or a portfolio. The sensitivity of the change in value of a security or a portfolio to a particular spot rate change is called rate duration.³⁶⁰ In theory, there is a rate duration for every point on the yield curve. Consequently, there is not one rate duration, but a profile of rate durations representing each maturity on the yield curve. The total change in value if all rates change by the same number of basis points is simply the duration of a security or portfolio to a change in the level of rates. That is, it is the duration measure for level risk (i.e., a parallel shift in the yield curve).

Vendors of analytical system do not provide a rate duration for every point on the yield curve. Instead, they focus on key maturities of the spot rate curve. These rate durations are called key rate durations.

The impact of any type of yield curve shift can be quantified using key rate durations. A level shift can be quantified by changing all key rates by the same number of basis points and computing, based on the corresponding key rate durations, the affect on the value of a portfolio. The impact of a steepening of the yield curve can be found by (1) decreasing the key rates at the short end of the yield curve and determining the change in the portfolio value using the corresponding key rate durations, and (2) increasing the key rates at the long end of the yield curve and determining the change in the portfolio value using the corresponding key rate durations.

The value of an option-free bond with a bullet maturity payment (i.e., entire principal due at the maturity date) is sensitive to changes in the level of interest rates but not as sensitive to changes in the shape of the yield curve. This is because for an option-free bond whose cash flow consists of periodic coupon payments but only one principal payment (at maturity), the change of rates along the spot rate curve will not have a significant impact on its value. In contrast, while the value of a portfolio of option-free bonds is, of course, sensitive to changes in the level of interest rates, it is much more sensitive to changes in the shape of the yield curve than individual option-free bonds.

In the case of mortgage securities, the value of both an individual mortgage security and a portfolio of mortgage securities will be sensitive to changes in the shape of the yield curve, as well as changes in the level of interest rates. This is because a mortgage security is an amortizing security with a prepayment option. Consequently, the pattern of the expected cash flows for an individual mortgage security can be materially affected by the shape of the yield curve. To see this, look at Exhibit 4 which shows the key rate durations for a Ginnie Mae 30-year 10% passthrough, the current coupon passthrough at the time the graph was prepared. Exhibit 5 shows the key rate duration profile for a principal-only (PO) and an interest-only (IO) mortgage strip created from the Ginnie Mae passthrough whose key rate durations are shown in Exhibit 4.

From Exhibit 4 it can be seen that the passthrough exhibits a bell-shaped curve with the peak of the curve between 5 and 15 years. Adding up the key rate durations from 5 to 15 years (i.e., the 5-year, 7-year, 10-year, and 15-year key rate durations) indicates that of the total interest rate exposure, about 70% is within this maturity range. That is, the effective duration alone masks the fact that the interest rate exposure for this passthrough is concentrated in the 5-year to 15-year maturity range.

A PO will have a high positive duration. From the key rate duration profile for the PO shown in Exhibit 5 it can be seen that the key rate durations are negative up to year 7. Thereafter, the key rate durations are positive and have a high value. While the total risk exposure (i.e., effective duration) may be positive, there is exposure to yield curve risk. For example, the key rate durations suggest that if the long end of the yield curve is unchanged, but the short end of the yield curve (up to year 7) decreases, the PO’s value will decline despite an effective duration that is positive.

IOs have a high negative duration. However, from the key rate duration profile in Exhibit 5 it can be seen that the key rate durations are positive up to year 10 and then take on high negative values. As with the PO, this security is highly susceptible to how the yield curve changes.

While the key rate durations are helpful in understanding the exposure of a mortgage security or a portfolio to yield curve risk, we will present an alternative methodology for assessing the yield curve risk of a mortgage security when we discuss the hedging methodology in Section V.

When interest rates change we must offset the resultant change in mortgage durations in order to keep the overall interest rate risk of the portfolio at its desired target. Neglecting to do so would leave the portfolio with less interest rate risk than desired after interest rates decline and more risk than desired after rates increase. A portfolio manager should adjust for changes in durations of mortgage securities—or equivalently, manage negative convexity—either by buying options or by hedging dynamically.

Hedging dynamically requires lengthening duration—buying futures—after rates have declined, and shortening duration—selling futures—after rates have risen.³⁶¹ Whether a portfolio manager employs this “buy high/sell low” dynamic strategy, or buys options, the portfolio’s performance is bearing the cost associated with managing negative convexity by foregoing part of the spread over Treasuries.

A portfolio manager can manage volatility risk by buying options or by hedging dynamically. The selection depends on the following:

Current models calibrate to historical experience. Although a portfolio manager does not know the magnitude of model error going forward, he can measure sensitivity to model error by increasing the prepayment rate assumed by the model for mortgage securities that are hurt by faster-than-expected prepayments and decreasing the prepayment rate assumed by the model for mortgage securities that are hurt by slower-than-expected prepayments.

Over time it has become cheaper to refinance mortgages as technological improvements have reduced the costs associated with refinancing. We expect this type of prepayment innovation to continue in the years ahead. Models calibrated to past behavior will understate the impact of innovation. Therefore, when evaluating mortgage securities that are vulnerable to this type of risk, a portfolio manager should carefully consider the likelihood and the effect of prepayment innovation in determining the size of a portfolio’s mortgage securities. Although a portfolio manager cannot hedge model risk explicitly, he can measure it and manage it by keeping a portfolio’s exposure to this risk in line with that of the broad-based bond market indices.

Empirically, studies have found that yield curve changes are not parallel. Rather, when the level of interest rates changes, studies have found that short-term rates move more than longer-term rates. Some firms develop their own proprietary models that decompose historical movements in the rate changes of Treasury strips with different maturities in order to analyze typical or likely rate movements. The statistical technique used to decompose rate movements is principal components analysis.

Most empirical studies, published and proprietary, find that more than 95% of historical movements in rate changes can be explained by changes in (1) the overall level of interest rates and (2) twists in the yield curve (i.e., steepening and flattening). For example, Morgan Stanley’s proprietary model of the movement of monthly Treasury strip rates finds the following “typical” monthly rate change in basis points for three maturities:

Because of the importance of yield curve risk for mortgage securities, a hedging methodology should incorporate this information about historical yield curve shifts. We will see how this is done in the next section.

To properly hedge the interest rate risk associated with a mortgage security, the portfolio manager needs to incorporate his knowledge of the following:

Since two factors (the “level” and “twist” factors discussed in the previous section) have accounted for most of the changes in the yield curve, two Treasury notes (typically the 2-year and 10-year) can hedge virtually all of the interest rate risk in mortgage securities. Since two hedging instruments are used, the hedge is referred to as a two-bond hedge.

To create the two-bond hedge, we begin by expressing a particular mortgage security in terms of an “equivalent position” in U.S. Treasuries or “equivalent position” in Treasury futures contracts.³⁶⁵ We identify this equivalent position by picking a package of 2-year and 10-year Treasuries that—on average—has the same price performance as the mortgage security to be hedged under the assumed “level” and “twist” yield curve scenarios. For hedging purposes, the direction of the change—up or down in the case of the “level” factor, flattening or steepening in the case of the “twist” factor—is not known. (In calculating how the price will change in response to changes in the two factors, it is assumed that the OAS is constant.)

In this way, the portfolio manager can calculate the unique quantities of 2-year and 10-year Treasury notes or futures that will simultaneously hedge the mortgage security’s price response to both “level” and “twist” scenarios. This combination is the appropriate two-bond hedge for typical yield curve shifts and therefore defines the interest rate sensitivity of the mortgage security in terms of 2-year and 10-year Treasury notes or futures.

The change in value of the two-bond hedge for a change in the level of the yield curve is:

Step 8: Determine the change in value of the two-bond hedge portfolio for a twist in the yield curve. This value is

Step 9: Determine the set of equations that equates the change in the value of the two-bond hedge to the change in the price of the mortgage security. To be more precise, we want the change in the value produced by the two-bond hedge to be in the opposite direction to the change in the price of the mortgage security. Using our notation, the two equations are:

Step 10: Solve the simultaneous equations in Step 9 for the values of H₂ and H₁₀. Notice that for the two equations, all of the values are known except for H₂ and H₁₀. Thus, there are two equations and two unknowns.

In Step 9, a negative value for H₂ or H₁₀ represents a short position and a positive value for H₂ or H₁₀ represents a long position.

Step 2: From the prices found in Step 1, calculate the price changes:

Step 3: Calculate the average price change (using absolute values) for each instrument resulting from a level change:

Step 4: In computing the price resulting from a twist in the shape of the yield curve, the 2-10 slope is assumed to change by 13.8 bps. (Recall from Section IV that this is the typical monthly twist in the shape of the yield curve.) The dollar price per $100 of par value is shown below:

Step 5: From the prices found in Step 4, calculate the price changes:

Step 6: Calculate the average price change for each instrument resulting from a twist in the yield curve:

Step 7: The change in value of the two-bond hedge portfolio for a change in the level of the yield curve is:

Step 8: The change in value of the two-bond hedge portfolio for a twist of the yield curve is:

Step 9: The two equations that equate the change in the value of the two-bond hedge to the change in the price of the mortgage security are:

Step 10: Solve the simultaneous equations in Step 9 for the values of H₂ and H₁₀. This is done as follows:

The value of 0.612478 for H₂ means that the par amount in the 2-year Treasury note futures will be 0.612478 per $1 of par amount of the mortgage security to be hedged. So, if the par amount of the Fannie Mae 5% to be hedged against interest rate risk is $1 million, then 2-year Treasury note futures with a notional value of $612,478 (= 0.612478 × $1 million) should be long (i.e., buying 2-year Treasury note futures). Notice that we have an example here of going long in a futures contract despite that we are seeking to hedge a long position!

The value of -0.906945 for H₁₀ means that the par amount in the 10-year Treasury note futures will be 0.906945 per $1 of par amount of the mortgage security to be hedged. Assuming again that the par amount of the Fannie Mae 5% to be hedged is $1 million, then 10-year Treasury note futures with a notional value of $906,945 (= 0.906945 × $1 million) should be shorted.

b. Duration Hedge versus Two-Bond Hedge It is interesting to note the difference between the potential performance difference between the bond hedge using duration only and the two-bond hedge that takes into consideration changes in both level and twist changes.

At the time of the hedge, the Fannie Mae 5% passthrough had an effective duration of 5.5. Using only duration to obtain the hedge position, it can be demonstrated that if the yield curve shift is a level one, the following price changes would result:

The above results indicate that if the duration hedge is used, when interest rates increase, the Fannie Mae 5% will decline by 1.339 points but the gain from the duration hedge will be 1.36 points. Hence, using a duration hedge the gain will be greater than the loss, resulting in a profit on the hedge. As can be seen from the results above, the opposite occurs if interest rates decline. This is the reason there is a market belief that mortgage securities are “market-directional” investments.

However, evidence using the two-bond hedge suggests otherwise: because the yield curve seldom moves in a parallel fashion, properly hedged mortgage securities are not market-directional. The two-bond hedge would produce the following results:

Let’s take a look at the Fannie Mae 5% duration and the duration implied for the two-bond hedging package. The duration for the Fannie Mae 5% is 5.5. We can obtain the implied duration of the two-bond hedge by computing the dollar value of a basis point (DV01) for the 2-year and 10-year Treasury note futures contracts.³⁶⁷ The DV01 for the 2-year Treasury note futures is 0.0186. For the 10-year Treasury note futures the DV01 is 0.067. So the hedging package (i.e., the two-bond hedge) has a DV01 that is the weighted average of the two dollar durations as shown below:

We will not go through all the steps but just provide the following basic information so that the positions in the hedging instruments can be computed:

To obtain H₂, we can substitute H₁₀ = - 0.440605 into the “Level” or the “Twist” equation and solve for H₂. Substituting into the “Level” equation we get:

Thus, H₂ = -0.766739 and H₁₀ = -0.440605.

These values indicate that a short position will be taken in both the 2-year and 10-year Treasury note futures. The value of 0.766739 for H₂ means that the par amount in the 2-year Treasury note futures will be 0.766739 per $1 of par amount of the mortgage security to be hedged. So, if the par amount of the Freddie Mac 7.5% to be hedged against interest rate risk is $1 million, then 2-year Treasury note futures with a par amount of $766,739 (= 0.766739 × $1 million) should be shorted. Similarly, the value of 0.440605 for H₁₀ means that the par amount in the 10-year Treasury note futures will be 0.440605 per $1 of market value of the mortgage security to be hedged.

Once again we see the value of using the two-bond hedge rather than using a duration hedge. At the time of the hedge, the Freddie Mac 7.5% passthrough had an effective duration of 4.4. Using only duration to obtain the hedge position, it can be demonstrated that if the yield curve shift is a parallel one, the following price changes would result:

A mortgage security with this characteristic is referred to as a “cuspy-coupon” mortgage security. For such mortgage securities, averaging the price changes is not a good measure of how prices will change. In other words, the tangent line (labeled “Cuspy Coupon” in Exhibit 6) is not a good proxy for the price/yield curve. At times, cuspy-coupon mortgage securities offer attractive risk-adjusted expected returns; however, they have more negative convexity than current-coupon mortgages.

Hedging cuspy-coupon mortgage securities only with Treasury notes or futures contracts may leave the investor exposed to more negative convexity than is desired. The negative convexity feature, as has been mentioned, is the result of the option that the mortgage security investor has granted to homeowners to prepay. That is, the investor has effectively sold an option to homeowners. To hedge the sale of an option (i.e., a short position in an option), an investor can buy an option. Thus, a portfolio manager can extend the two-bond hedge by buying interest rate options to offset some or all of a cuspy-coupon mortgage security’s negative convexity.

We will use a Freddie Mac 8.5% passthrough, a cuspy-coupon mortgage security in February 1997, to illustrate how the two-bond hedging methodology is extended to include options. The price of the Freddie Mac 8.5% passthrough was 103.50.

Without going through the mechanics of how to construct the two-bond hedge, panel a in Exhibit 7 shows (1) the change in the price for the mortgage security and (2) the change in the value of the two-bond hedge for changes in the level and twist in the yield curve. The last column in panel a of the exhibit shows the error for the two-bond hedge. The error is due to the negative convexity characteristic of the Freddie Mac 8.5% passthrough (cuspy-coupon mortgage security).

Because the prepayment option of the Freddie Mac 8.5s is closer to the refinancing threshold than that of the Freddie Mac 7.5s at the time of the analysis, the two-bond hedge error is greater: -0.05 for the 8.5s versus -0.02 for the 7.5s. Buying calls and puts eliminates this drift. Specifically, we added the purchase of the following two option positions per $100 of the Freddie Mac 8.5s to be hedged:

EXHIBIT 7 Alternatives for Hedging a Cuspy-Coupon Mortgage Security

Panel b of Exhibit 7 shows that adding the two option positions on the 10-year note offsets the two-bond hedge error, making the total package (mortgage security + two-bond hedge + options) insensitive to likely interest rate movements. Of course, buying these options requires paying a premium which amounted to about 7 basis points per month. Since the yield advantage of the 8.5s versus Treasuries was about 11 basis points, the expected excess return over Treasuries was about 4 basis points per month after we hedge out the negative convexity.