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Back-to-Basics

The following article is reprinted from the Quarter 2, 2004 issue of On the Edge,
the Interactive Data Fixed Income Analytics quarterly newsletter.

Back-to-Basics: Effective Duration and Floating Rate Securities

Teri Geske
Senior Vice President, Product Development

"What is the effective duration of a floating rate security?" We hear this question from time to time, often followed by the conjecture that the answer is, "the time to the security's next coupon reset date". In this Back-to-Basics column, we examine the effective duration of floating or adjustable rate securities, and offer some examples of why these securities can have durations that may challenge our expectations.

To begin, recall that Effective Duration is the percentage change in a security's price averaged over equal and opposite parallel shifts in the Treasury curve (or, for non-US$ securities, the appropriate underlying term structure of interest rates). Given this definition, the duration of a truly floating rate security, one whose coupon adjusts upward or downward immediately and in lock-step with changes in interest rates, would be 0.00. This result is obtained because price is simply the present value of expected future cash flows, and for a true floater any change in future coupon payments resulting from a change in interest rates is offset by the change in the discount rate used to determine that cash flow's present value. In other words, the change in the numerator of the present value calculation is offset by a change in the denominator and thus, there is no change in present value, so there is no change in price. If the price doesn't change when interest rates change, we're really saying the effective duration = 0.00.

Before we move on to a slightly more complicated example, there is another layer to this reasoning that should be acknowledged. We said that the price of a pure floater is unaffected as long as a change in the discount rate is offset by the change in the future cash flows. However, the discount rate used to value a cash flow has two components: the risk free-interest rate and a spread. If the change in the discount rate results from a change in spreads, rather than from a change in interest rates, the denominator in the present value calculation will change, but the cash flows in the numerator will not; in this case, there will be a change in the price of the floater - in other words, a floater's effective duration may be zero, but its spread duration certainly is not. Furthermore, if the floater's price is significantly different than par the effective duration will not be zero. This situation exists when the cash flows promised to the investor, as determined by the index value in the floater formula plus a spread, is significantly different from the rate investors are using to discount those cash flows. This could be the case where a floater was issued some time ago and the issuer's credit quality has deteriorated (or improved), such that the spread in the coupon formula is less than (or greater than) the risk premium currently demanded in the market.

Now consider a floater with a coupon that resets quarterly (and assume the price of the floater is approximately equal to par). If interest rates were to change immediately after the coupon reset date, the discount rate used to compute the present value of the floater's future cash flows would change immediately but would not be offset by an adjustment to the coupon cash flows until the next reset date, three months in the future. At the next reset date, the remaining coupon payments would "catch up" to the new rate environment and equilibrium would be restored. So, the price would only deviate from its initial value based on the change in the present value of the first cash flow, the one that is "out of synch" with the new rate environment. On a percentage basis (and percentage change determines effective duration), the magnitude of the price change is a function of how long the discount rate is out of synch with the coupon rate. If the asymmetry between the coupon rate and discount rate exists only for three months, the duration is about 0.25. If a floater resets only semi-annually it would take six months for the coupons to "catch up" to the discount rate and the duration would be about 0.50. Intuitively, the longer the time to reset, the bigger the potential change in price given a change in interest rates. In the limit, the next reset date could be the bond's maturity date - which would make the security a fixed rate bond, with an effective duration commensurate with its time to maturity.

We now introduce the impact of periodic or lifetime caps on a floater's effective duration. A cap limits the extent to which changes in the coupon rate are permitted to keep up with changes in interest rates. For example, the floater's coupon might reset semi-annually, based on 6 month LIBOR + 225 bps, subject to a periodic cap of 1.00. In this case, if 6 month LIBOR increased or decreased by more than 1% from one reset date to the next, the floater's coupon could not fully reset, based on the formula. A floater may have a periodic cap, such as this, which designates the amount by which the coupon can change on a single reset date, or it may have a lifetime cap, which the coupon rate may never exceed, or it may have both. Adjustable Rate Mortgages (ARMs) typically have both (except ARMs with coupons tied to the Cost of Funds Index - COFI). CMO floaters usually have a lifetime cap but no periodic cap.

These caps (periodic and/or lifetime) are embedded options that become more valuable with increased rate volatility . The investor is short the lifetime cap, but has both a long and short position in any periodic cap. If rates rise, the value of the lifetime cap increases because there is a greater likelihood that the coupon formula will hit the cap rate. This negatively impacts the investor because of the higher probability that the coupon rate will become fixed in a rising rate environment. The price of the floater will fall as the interest rate used to discount the cash flows increases but the coupon cash flows do not; thus the duration of the security will become much longer than the zero-ish value we expect from an uncapped floater. The closer the lifetime cap value is to the current coupon rate, the greater the impact on the security's duration. Even if the lifetime cap has not yet been reached, the increasing probability of encountering it in the future causes the value of that embedded option to increase, thereby causing the price of the floater to decline (because the investor is short the cap). Similarly, as rates rise a periodic cap becomes more valuable, and the investor is also short that option. In fact, a tight reset cap can have a substantial impact on a floater's duration - in 1994, when interest rates rose approximately 300bps over the course of a year, the effective duration on GNMA ARM securities, which have 1% annual reset caps, extended to 3.5 and higher, due to impact of a swift, sharp rise in interest rates on the value of the embedded caps.

Although we typically speak of the periodic limit on the change in a coupon as a cap, it is also a floor, because the rate on the security cannot decline by more than the specified amount. In the case of a falling rate environment, this works to the investor's advantage (the investor is "long" the floor), as the security pays an above-market rate for a while, until the coupon rate is allowed to "catch up" to the new level of interest rates. In computing the effective duration of floaters with embedded caps, we use a Monte Carlo simulation to generate a variety of future interest rate paths, some of which are likely to encounter the embedded caps and/or floors. So, even though the cap/floor level may not be breached for a small change in rates, the proximity of the caps/floors will be reflected in the effective duration. If we think of the cap or floor as being a barrier, we know we don't have to actually crash into the barrier to feel its presence - we can extend our arms and fingertips to see if it is close enough to touch. If it is, we might begin to modify our behavior. Similarly, the extent to which the Monte Carlo simulation detects the potential impact of caps and floors on a floater's cash flows will determine the sensitivity of price, and therefore the effective duration, to a change in interest rates.

Therefore, if the effective duration of a floating/adjustable rate security is notable different than time to the next reset date, there are either embedded caps or floors affecting the security's price, or there is a persistent mismatch between the way the coupon cash flows change and the discount rates used to discount those cash flows.

We hope this has been a useful review of this subject - if you'd like us to cover a particular topic in a future Back-to-Basics article, we'd like to hear from you. Please e-mail your suggestions to marketing at fia.marketing@interactivedata.com.