Federal Reserve Bank of San Francisco

FRBSF Economic Letter

2000-01 | January 1, 2000

More Economic Letters

- Duration
- The influence of prepayments on the modified duration of MBS
- Alternative approaches to interest rate risk measurement
- References

Measuring interest rate risk–that is, the risk that interest rate fluctuations might impair a firm’s profitability or viability–is important both to financial institutions and to their regulators. Generally, methods for measuring interest rate risk focus on the *duration *of financial instruments, which is one way to characterize the sensitivity of their values to interest rate changes. However, measuring duration can be quite difficult, particularly for assets and liabilities with embedded interest-related options.

This *Economic Letter* focuses on residential mortgage-related assets, which typically embed interest-related options through borrowers’ right to prepay their debts before the end of the mortgage term. It explains how these prepayment options affect duration and describes how some methods used to measure interest rate risk for mortgage-related assets incorporate this dependence of duration on prepayments.

Some early efforts to promulgate standardized methods for measuring interest rate risk (e.g., Basel 1993) emphasized *Macaulay duration*, which is a measure of the cash-flow-weighted time until maturity of an instrument. Currently, however, most interest rate risk measurement methods (e.g., those discussed in Basel 1997, 1999) emphasize *modified duration*, which is the price elasticity of an instrument with respect to a change in yields.* *Generally, modified duration is defined with respect to changes in the yield-to-maturity. To keep matters simple, typical computations of modified duration use parallel shifts in the yield curve, where yields at each maturity horizon change by the same amount. In this context, a duration of 5 would indicate that the price of an asset would change by 5% for a 1 percentage point change in yield. For a (zero-coupon, non-amortizing, default-free, noncallable) bond with all cash payments to the investor occurring at the maturity date, there is an unambiguous relationship between the bond’s yield-to-maturity, current price, and remaining term to maturity. For example, on such a bond a modified duration of 5 also would indicate that there are 5 remaining years to maturity.

However, more generally the modified duration of an instrument is sensitive to the timing and amounts of interim receipts of cash flows. The payment of interest and principal before the maturity of an instrument shortens its duration relative to zero-coupon, non-amortizing bonds of equal maturity. As the coupon rate increases, the modified duration of a noncallable bond decreases. Also, modified duration is lower for amortizing bonds, on which some principal is scheduled to be returned before final maturity, than for non-amortizing bonds, on which principal is returned only at final maturity.

The existence of an option for a borrower to return additional principal before the scheduled return dates also affects a bond’s value and its modified duration. The predicted effects on mortgage bond value and duration depend on how the valuation model specifies (default and prepayment) option-exercise behavior and specifies market valuation of these uncertain option-related cash flows.

**The influence of prepayments on the modified duration of MBS**

The prepayment of principal is particularly important to measuring the duration of mortgage-related assets. Bank portfolios tend to contain large exposures to residential mortgage-related assets, including whole mortgages held in portfolio, mortgage-servicing rights, and mortgage-backed securities (MBS). This *Letter* focuses on one of these major asset classes, (agency) 30-year fixed-rate MBS passthroughs, which distribute cash flows to MBS bondholders in proportion to the receipt of interest and principal from the underlying pool of mortgage borrowers. Interest payments are determined by an MBS passthrough coupon rate, which tends to be about 50 basis points above the mortgage rate on the underlying mortgage loans, with this difference diverted to cover the costs of servicing the mortgages and insuring against default.

Although the stated maturity of an MBS is the remaining term until maturity of the underlying mortgage loans (about 30 years at issuance of the MBS), almost all of the cash from an MBS will be received before the stated maturity date. In addition to the scheduled receipt of interest and principal (amortization) each month in the mortgage term, MBS tend to experience early return of principal as borrowers either voluntarily prepay or default and trigger mortgage insurance payoffs.

The magnitude of prepayments depends on current interest rates and other aspects of the economic environment. At a given MBS passthrough coupon, prepayments tend to increase as current yields decline. The increasing prepayments effect is particularly large near the refinancing “threshold,” which is where the spread between current yields and the given coupon begins to exceed the minimal level of refinancing transactions costs. Below that threshold, prepayments on residential mortgages largely are due to defaults and to early payoffs by households who sell their homes in order to move to other residences.

Figure 1 illustrates the relationship between refinancing incentive spreads and prepayments using data on the historical prepayment rates of 30-year fixed-rate Freddie Mac passthroughs. On average, only about 1/4% of the remaining mortgages per month prepaid for those MBS with mortgage coupons well below the prevailing primary mortgage rate, spreads below negative 1-1/2 percentage points. Prepayment rates remained well below 1% per month for other (narrower negative and slightly positive) spreads below 1/2 percentage point. Then, prepayment rates increased noticeably as this spread widened above 1/2 percentage point. For MBS with coupons about 2 percentage points above the current primary market rate, prepayments averaged about 5% per month.

The existence of such a high degree of prepayment sensitivity to the position of current market yields relative to the coupon rates on the outstanding mortgages makes the duration of MBS instruments shift with market conditions, with the degree of shifting dependent on whether the MBS have relatively high or relatively low coupons. The degree to which duration changes with yields is known as the *convexity* of the instrument. When current yields are low enough to make refinancing potentially attractive to a significant fraction of mortgage borrowers in the foreseeable future, MBS exhibit *negative convexity*; that is, as yields increase, MBS bond prices become more sensitive to changes in yields. This is the opposite of the case for noncallable bonds with nonzero coupons; such bonds demonstrate a mild degree of positive convexity.

The size of the negative convexity of MBS can be illustrated using empirical duration measures for Freddie Mac 30-year passthroughs. The *empirical duration* measures calculate duration from the observed average degree of co-movement between actual MBS market prices and Treasury yields in a particular sample period. As illustrated in Figure 2, for deep discount MBS with passthrough coupons about 2 percentage points below the current par coupon, the empirical duration is about 6-1/2%. To put this number in perspective, note that this is equivalent to the modified duration and Macaulay duration of a zero-coupon (non-amortizing, noncallable) bond with a term to maturity of 6-1/2 years. The historical relationships shown in the figure suggest that if current yields drop by about 2 percentage points, narrowing the refinancing incentive gap from -2 percentage points to zero, then duration will drop to about 3-1/2%. Furthermore, the duration of an MBS passthrough can shorten very substantially with a 4 percentage point change in current yields. An MBS with a 2 percentage point coupon premium over the current par coupon has tended to have a duration of less than 1%, about 5-1/2 percentage points less than the duration of an MBS with a 2 percentage point coupon discount.

Mortgage interest rates have been sufficiently variable in recent years to create significant heterogeneity in the duration of the stock of MBS passthroughs outstanding and to change the duration of specific vintages of MBS over time. At the beginning of the 1990s, primary conventional mortgage rates were slightly over 10% (Figure 3). Mortgage rates trended down to slightly below 7% in late 1993, were back up to above 9% by the end of 1994 and have bounced up and down in roughly the 7% to 8-1/2% range during the latter half of the 1990s. When primary mortgage rates were at their peaks in 1990 and 1995, most of the stock of outstanding MBS were from earlier years and bore passthrough coupons which were low relative to newly issued mortgages; the average spread was negative and exceeded 100 basis points. Such negative spreads between historical and prevailing rates depressed refinancing activity (also shown in Figure 3) and implied that most outstanding MBS had relatively high durations; Figure 1 implies that the empirical durations were around 5%. However, when primary mortgage rates dropped to troughs near 7% in each of 1993, 1996, and 1998, spreads between outstanding and newly issued mortgage rates typically were positive and wide; this boosted prepayments and in many cases lowered the durations of the outstanding MBS to below 2%.

**Alternative approaches to interest rate risk measurement**

The above calculations illustrate that the duration-gap approach to interest rate risk measurement has major shortcomings for mortgage-related assets because the duration of such assets can change significantly with changes in current market yields. Accordingly, financial institutions and regulators have developed alternative approaches to interest rate risk measurement which, in principle, are better able to evaluate the degree of interest rate risk in instruments with durations that change rapidly as market conditions interact with embedded options. For example, the Office of Thrift Supervision has developed a Net Portfolio Value (NPV) model that simulates the effects of interest rate changes on the values of MBS passthroughs and other mortgage-related assets by incorporating prepayment assumptions. Also, the Office of Federal Housing Enterprise Oversight (OFHEO) has developed an elaborate simulation model for predicting prepayments, defaults, and mortgage asset values to be used in “stress tests” of Fannie Mae and Freddie Mac, the two housing finance government-sponsored Enterprises for which OFHEO is developing risk-based capital regulations.

Relative to the NPV model, OFHEO’s model for conducting stress tests is innovative in using housing prices as an additional predictor of mortgage prepayment rates. This principle gains some support from recent research by Mattey and Wallace (1999) which confirms that housing price changes have had a statistically significant effect on prepayment rates of MBS passthroughs.

Joe Mattey*Research Officer*

Basel Committee on Banking Supervision. 1993. “Measurement of Banks’ Exposure to Interest Rate Risk.” Consultative proposal (April).

Basel Committee on Banking Supervision. 1997. “Principles for the Management of Interest Rate Risk” (September).

Basel Committee on Banking Supervision. 1999. “Consultative Paper on a New Capital Adequacy Framework” (June).

Mattey, Joe, and Nancy Wallace. 1999. “Housing Price Cycles and Prepayment Rates of U.S. Mortgage Pools.” Federal Reserve Bank of San Francisco Working Paper 99-12 (August).

Opinions expressed in FRBSF Economic Letter do not necessarily reflect the views of the management of the Federal Reserve Bank of San Francisco or of the Board of Governors of the Federal Reserve System. This publication is edited by Sam Zuckerman and Anita Todd. Permission to reprint must be obtained in writing.

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