READING 26: HEDGING MOTGAGE SECURITIES TO CAPTURE RELATIVE VALUE

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1 READING 26: HEDGING MOTGAGE SECURITIES TO CAPTURE RELATIVE VALUE Introduction Because of the spread offered on residential agency mortgage-backed securities, they often outperform government securities with the same interest rate risk and therefore they can be used to generate enhanced returns relative to a benchmark when the yield advantage of mortgage securities is attractive. However, to execute this strategy successfully, the prepayment risk of mortgage securities must be managed carefully. In this reading, we will see how this is done. Specifically, we will see how to hedge the interest rate risk associated with a fixed rate mortgage security in order to capture the spread over Treasuries. Note that we use the terms mortgage-backed securities and mortgage securities interchangeably in this reading. The most basic form of mortgage-backed security is the mortgage passthrough security. Securities that are created from mortgage passthrough securities include collateralized mortgage obligations and mortgage strips (interest-only and principal-only securities). A- The Problem The yield for a mortgage security is the cash flow yield, the interest rate properly annualized, that makes the present value of the projected cash flows from a mortgage-backed security equal to its price. The price-yield relationship exhibits both positive and negative convexity. At yield levels above y*, the mortgage security exhibits positive convexity; at yield levels below y*, the mortgage security exhibits negative convexity. For a security that exhibits positive convexity, the price increase when interest rates decline is greater than the price decrease when interest rates rise. For a security that exhibits negative convexity, the price increase when interest rates decline is less than the price decrease when interest rates rise. 1

2 Mortgage securities exhibit negative convexity at some yield level because of the home-owner 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: Value of mortgage security = Value of treaury security Value of the prepayment option The reason for subtracting the value of the prepayment option is that the investor in a mortgage security has sold a prepayment option. 2

3 When interest rates decline, the value of the Treasury security component of the mortgage security s value increases. However, the appreciation is reduced by the increase in the value of the prepayment option which becomes more valuable as interest rates decline. The net effect is that while the value of a mortgage security increases, it does not increase by as much as a sameduration Treasury security because the increase in the value of the prepayment option offsets part of the appreciation. 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 decreases, 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. For a security that exhibits positive convexity, the duration changes in the desired direction; for a security that exhibits negative convexity, there is an adverse change in the duration. 3

4 The tangent line is related to the duration. The steeper the tangent line, the higher the duration. While a mortgage security can exhibit both positive and negative convexity, a Treasury security exhibits only positive convexity. Hedging the interest rate risk associated with a mortgage security by either shorting Treasury securities or selling Treasury futures leads to a loss if interest rates fall. The reason is that the loss of short position on the treasury security will steeper than the gain on the mortgage security. B- Mortgage Security Risks There are five principal risks: spread, interest-rate, prepayment, volatility, and model risk. The yield of a mortgage security the cumulative reward for bearing all five of these risks has two components: the yield on equal interest-rate risk Treasury securities plus a spread. This spread is itself the sum of the option cost, which is the expected cost of bearing prepayment risk, and the option-adjusted spread (OAS), which is the risk premium for bearing the remaining risks, including model risk. 1- Spread Risk Spread risk is the risk that the spread between the treasury security and the mortgage security widening thus lowering the value of the mortgage security. If the spread narrows, the relative value of the mortgage security increases. Because the OAS can be thought of as the risk premium for holding mortgage securities, a portfolio manager does not seek to hedge spread risk. 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. 2- Interest Rate Risk The interest rate risk of a mortgage security corresponds to the interest rate risk of comparable Treasury securities. This risk can be hedged directly by selling a package of Treasury notes or Treasury note futures. After netting the value of the option, the portfolio manager earns the Treasury bill rate plus the potential to capture the OAS. a. Yield curve Risk Yield curve risk is the exposure of a portfolio or a security to a nonparallel change in the yield curve shape. 4

5 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. Vendors of analytical systems 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. For a mortgage security, yield curve risk is considerably greater than for a Treasury security because of the embedded option. 3- Prepayment Risk Because of the prepayment option, Mortgage securities exhibit negative convexity which means their duration varies in an undesirable way as interest rates change: extending as rates rise and shortening as rates fall. A portfolio manager should 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. Either way, the portfolio s performance is bearing the cost associated with managing negative convexity by forgoing part of the spread over Treasuries. 4- Volatility Risk An investment characteristic of an option is that its value increases with expected volatility. A portfolio manager can manage volatility risk by buying options or hedging dynamically. A portfolio manager will hedge dynamically when the volatility implied in the option price is high and the portfolio manager believes that future realized volatility will be lower than implied volatility. A portfolio manager will hedge by purchasing options when the implied volatility in option prices is low and the portfolio manager believes that actual future volatility will be higher than implied. Implied volatility is computed using the Monte Carlo simulation valuation framework. 5- Model Risk Misguidedly projecting past patterns into the future Not recognizing technological advances that have made refinancing easier and cheaper which results in a higher prepayment risk over time. C- How Interest Rates Change Over Time While key rate duration is a useful measure for identifying the exposure of a portfolio to different potential shifts in the yield curve, it is difficult to employ this approach to yield curve risk in hedging a portfolio. An alternative approach is to investigate how yield curves have changed historically and incorporate typical yield curve change scenarios into the hedging process. 5

6 D- Hedging Methodology Hedging is a special case of interest rate risk control where the portfolio manager seeks to completely offset the dollar price change in the instrument to be hedged by taking an opposite position in an appropriate hedging instrument that will produce the same dollar price change for the same change in interest rates. To properly hedge the interest rate risk associated with a mortgage security, the portfolio manager needs to incorporate his knowledge of the following: how the yield curve changes over time, and the effect of changes in the yield curve on the homeowner s prepayment option. 1- Interest Rate Sensitivity Measure Interest rate sensitivity (IRS) measures a security s or a portfolio s percentage price change in response to a shift in the yield curve. 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. 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.) 2- Computing the Two Bond Hedge Step 1: For an assumed shift in the level of the yield curve, compute the following: price of the mortgage security for an assumed increase in the level of interest rates; price of the mortgage security for an assumed decrease in the level of interest rates; price of the 2-year Treasury note (or futures) for an assumed increase in the level of interest rates; price of the 2-year Treasury note (or futures) for an assumed decrease in the level of interest rates; price of the 10-year Treasury note (or futures) for an assumed increase in the level of interest rates; price of the 10-year Treasury note (or futures) for an assumed decrease in the level of interest rates. Step 2: From the prices found in Step 1, calculate the price change for the mortgage security, 2-year Treasury note (or futures), and 10-year Treasury note (or futures) for the assumed shift in the level of 6

7 interest rates. There will be two price changes for each of the mortgage security, 2-year hedging instrument, and 10-year hedging instrument. Step 3: Calculate the average price change for the mortgage security and the two hedging instruments for the assumed shift in the level of interest rates assuming that the two scenarios (i.e., increase and decrease) are equally likely to occur. The average price change will be denoted as follows: MBS price L = average price change for the mortgage security for a level shift 2 H price L = average price change for the 2 year Treasury hedging instrument for a level shift 10 H price L = average price change for the 10 year Treasury hedging instrument for a level shift Step 4: For an assumed twist (flattening and steepening) of the yield curve, compute the following: price of the mortgage security for an assumed flattening of the yield curve; price of the mortgage security for an assumed steepening of the yield curve; price of 2-year Treasury note (or futures) for an assumed flattening of the yield curve; price of 2-year Treasury note (or futures) for an assumed steepening of the yield curve; price of 10-year Treasury note (or futures) for an assumed flattening of the yield curve; price of 10-year Treasury note (or futures) for an assumed steepening of the yield curve. Step 5: From the prices found in Step 4, calculate the price change for the mortgage security, 2-year Treasury note (or futures), and 10-year Treasury note (or futures) for the assumed twist in the yield curve. There will be two price changes for each of the mortgage security, 2-year hedging instrument, and 10-year hedging instrument. Step 6: Calculate the average price change for the mortgage security and the two hedging instruments for the assumed twist in the yield curve assuming that the two scenarios (i.e., flattening and steepening) are equally likely to occur. The average price change will be denoted as follows: MBS price T = average price change for the mortgage security for a twist in the yield curve 2 H price T = average price change for the 2 year Treasury hedging instrument for a twist in the yield curve 10 H price T = average price change for the 10 year Treasury hedging instrument for a twist in the yield curve Step 7: Compute the change in value of the two-bond hedge portfolio for a change in the level of the yield curve. This is done as follows. Let H 2 = amount of the 2 year hedging instrument per $1 of market value of the mortgage security H 10 = amount of the 10 year hedging instrument per $1 of market value of the mortgage security Our objective is to find the appropriate values for H2 and H10 that will produce the same change in value for the two-bond hedge as the change in the price of the mortgage security that the portfolio manager seeks to hedge. The change in value of the two-bond hedge for a change in the level of the yield curve is H 2 (2 H price L ) + H 10 (10 H price L ) 7

8 Step 8: Determine the change in value of the two-bond hedge portfolio for a twist in the yield curve. This value is H 2 (2 H price T ) + H 10 (10 H price T ) 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: Level: H 2 (2 H price L ) + H 10 (10 H price L ) = MBS price L Twist: H 2 (2 H price T ) + H 10 (10 H price T ) = MBS price T Step 10: Solve the simultaneous equations in Step 9 for the values of H 2 and H 10. Notice that for the two equations, all of the values are known except for H 2 and H 10. Thus, there are two equations and two unknowns. In Step 9, a negative value for H 2 or H 10 represents a short position and a positive value for H 2 or H 10 represents a long position. 3- Underlying Assumptions The underlying assumptions for the two-bond hedge are: The yield curve shifts used in constructing the two-bond hedge are reasonable. The prepayment model used does a good job of estimating how the cash flows will change when the yield curve changes. Assumptions underlying the Monte Carlo simulation model are realized (e.g., the interest rate volatility assumption). The average price change is a good approximation of how the mortgage security s price will change for a small movement in interest rates. E- Hedging Cuspy-Coupon Mortgage Securities In many cases, the average price change is a good approximation of how a mortgage security s price will change for a small movement in interest rates. However, some mortgage securities are very sensitive to small movements in interest rates. For example, a mortgage security whose coupon is 100 basis points higher than the current coupon could be prepaid slowly if rates rise by 25 basis points but prepaid very quickly if rates fall by 25 basis points. Small changes in interest rates have large effects on prepayments for such securities and hence on their prices. 8

9 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. 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. A portfolio manager can extend the twobond hedge by buying interest rate options to offset some or all of a cuspy-coupon mortgage security s negative convexity. 9