Back to Basics: Which Duration is Best?

Teri Geske
Senior Vice President, Product Development

Note: This Back-to-Basics column on Duration was first published in 1997. Based on a number of recent inquiries on this subject, we are republishing the article, which we’ve revised and updated for this issue.
Fixed income professionals have come to rely on Duration as the primary measure of interest rate risk for individual securities and portfolios. Yet this widely accepted measure is still subject to misinterpretation and misuse, partly because there are various forms of Duration one might encounter (some of them being far more informative than others). In this Back-to-Basics article, we explain the differences among these duration measures and the implications of relying on the wrong one when evaluating a bond or managing a portfolio’s exposure to interest rate risk. We also discuss whether or not Duration can be interpreted as a measure of time, and how Duration relates to Average Life.

First, we review three types of Duration that may be calculated for a bond and/or for a portfolio1 , namely Macaulay’s (also known as Modified Duration), Effective Duration (also known as Option-Adjusted Duration), and Duration-to-Worst. These are defined as follows 2 :

-Macaulay’s (Modified) Duration – the approximate percentage change in a bond’s price given a 1% change in its yield-to-maturity . The Macaulay’s duration formula is based on a pre-determined set of principal and interest cash flows computed to the bond’s final maturity date and does not recognize that those cash flows could be affected by changes in interest rates, including the exercise of one or more embedded options (calls, puts, optional prepayments, floating rate coupons, including any reset caps or floors, etc.).

-Duration-to-Worst– the approximate percentage change in a bond’s price given a 1% change in its yield-to-maturity or its yield-to-call, whichever is lower. Duration-to-Worst is the same as Macaulay’s duration except the pre-determined set of principal and interest cash flows are based on either the final maturity date, or a call date within the bond’s call schedule, whichever would result in the lowest yield to the investor – i.e., the Yield-to-Worst. (Note that for puttable bonds, one would use a “duration-to-best” computed from cash flows to the maturity date or to the put date, whichever results in the highest yield to the investor).

-Effective Duration – the average percentage change in a bond’s price, based on upward and downward parallel shifts in the underlying term structure of interest rates (typically the Treasury spot curve). By determining what the bond’s price would be, given higher/lower interest rate environments, the effective duration measure reflects the increasing or decreasing likelihood of any option exercise, including calls, puts, changes in prepayment speeds for mortgage-backed securities, and the higher probability of encountering any rate caps/floors for securities with adjustable coupons.

Given that the primary objective of duration is to explain a bond’s or portfolio’s price sensitivity to changes in interest rates, we can see that neither Macaulay’s (Modified) Duration nor Duration-to-Worst can be used for this purpose, because neither one reflects the fact that a bond’s cash flows can be affected by a change in interest rates. Macaulay’s Duration assumes a bond will always survive to the stated maturity date, regardless of any call or put options, or in the case of a mortgage-backed security, that prepayments will be constant, regardless of a change in interest rates. Consider a mortgage pass-through forecasted to prepay at a CPR% of 18% for the remainder of the mortgage pool’s life, and that these cash flows produce a Macaulay’s duration of 3.20. Can we reasonably estimate the impact of a 50bp change in interest rates on the pass-through using the approximation: (– Duration x interest rates) = 1.60%? No, because the duration of 3.20 ignores the fact that if interest rates fall, prepayments are likely to increase, and vice versa. A similar error occurs with callable and puttable bonds, where Macaulay’s duration fails to recognize the increasing value of the call option as rates fall (or the rise in the put option’s value as rates rise). For bonds with adjustable rate coupons, Macaulay’s duration doesn’t reflect the fact that as interest rates change, the coupon rate on the bond changes; in essence it treats all bonds as fixed rate instruments. If Macaulay’s Duration is used to compare a portfolio’s interest rate sensitivity relative to a benchmark and the portfolio (or the benchmark) contains securities with any type of embedded options, a significant tracking error is likely to occur.

What about Duration-to-Worst? Even though Duration-to-Worst seems to recognize the presence of an embedded call option, it does not reflect the fact that the value of the option, i.e., the likelihood the option will be exercised, fluctuates as interest rates change. Duration-to-Worst is like an On/Off switch – it either assumes the bond is definitely going to be called, or is definitely not going to be called, without allowing for uncertainty. Therefore, Duration-to-Worst either under- or overestimates a bond’s interest rate sensitivity by assuming that a call will or will not be exercised, regardless of the future interest rate environment and can be a highly unstable and misleading measure.

Consider a bond with a 7.50% coupon, maturing in 10 years, callable a year from now at a price of 103, currently priced at 103.45, with the following measures: Yield-to-Maturity – 6.526%; Yield-to-Call – 6.355%; Macaulay’s Modified Duration – 6.99; Duration-to-Worst – 1.02; Effective Duration – 3.00. Since the yield to the first call date (which is the worst possible call date in this example), is lower than the yield-to-maturity of the bond, the bond is “trading to call”. The Macaulay’s Modified Duration, which ignores the presence of the call option entirely, predicts the bond’s price will increase by approximately 6.99% (from 103.45 to 110.70) if interest rates decline by 1%. However, we know the price cannot rise that far since the bond is callable at 103 in a year, so the Macaulay’s duration is not a useful approximation of price sensitivity.

Duration-to-Worst suffers from a related flaw – it assumes that the bond’s status (i.e., trading to call or trading to maturity) will never change until it actually does. If the bond is currently trading to call, the Duration-to-Worst assumes the bond will definitely be called, regardless of any future change in interest rates; if the bond is trading to maturity, the Duration-to-Worst assumes the bond will never be called. So, Duration-to-Worst can “jump” back and forth, from either a fairly short duration based on the call date, out to the duration based on the maturity date as the bond “crosses over” from trading to call to trading to maturity. Let’s say that a 20bp increase in rates would cause this bond to trade to maturity, rather than to the call date. If we use Duration-to-Worst, that 20bp rise in rates would cause us to restate the bond’s duration from 1.02 to 6.99, an unrealistically large jump in price sensitivity for a small change in interest rates3 . Of course, neither Duration-to-Worst nor Modified Duration provides a good indication of the actual change the bond’s price would experience given a shift in the yield curve; for this, we must use Effective Duration, which reflects the impact of the value of embedded options on the bond’s price sensitivity.

The Effective Duration of a callable bond will always be less than the Macaulay’s duration, for the following reason: As interest rates fall, the call becomes more important to the behavior of the security and the increase in price that a decline in rates would otherwise cause is restricted by the presence of the call. On a percentage basis, this means the price of a callable bond increases by a smaller amount than the price of an otherwise identical but non-callable bond for a given decline in rates. Conversely, as interest rates rise the value of the embedded call option declines and therefore has less and less impact on the price of the bond. On a percentage basis, the price of a callable bond begins to decline by almost as much as that of a non-callable bond. When we remember that Duration is used to estimate a percentage change in price, we can see that the Effective Duration value must be smaller than the Macaulay’s Duration, which ignores the impact of the call feature on the bond’s price. Similar logic holds true for mortgage-backed securities, where prepayments can be viewed as “partial calls” (that are exercised somewhat inefficiently).

Effective Duration should not be viewed as a measure of time, although it is often spoken of in terms of “years”. For securities with no embedded options (where the Macaulay’s Modified Duration and Effective Duration will be equal), duration can be viewed as the weighted-average time until cash flows are received, where the weights are the present values of the cash flows themselves. However, since securities with embedded options have uncertain cash flows (with respect to amount and/or timing), it is not appropriate to view duration in terms of time. In fact, some securities, most notably CMO Interest-Only (IO) tranches, have an Effective Duration that is negative, which certainly cannot be viewed as a time increment (leaving the theory of relativity aside!). Effective Duration can be longer than the Average Life of a bond if the Average Life is computed to a call date; otherwise, Effective Duration will be shorter than Average Life4 .

Effective Duration is the only one of the duration measures discussed here that reflects the impact of embedded options on a bond’s interest rate sensitivity. We devote a great deal of effort and resources to provide our clients with robust effective durations (and the various models required to derive them) for all types of fixed income securities, portfolios and benchmark indices. BondEdge provides all three durations discussed in this article, i.e. Modified, Effective and “To-Worst” - we hope this review has helped in making an informed decision about how to use them.

1 The duration of a portfolio is the weighted-average (market value-weighted) of the durations of each bond in the portfolio.
2 Note that each of these measures describes the percentage change in a bond’s (or portfolio’s) value for a given change in rates, not the dollar price change. For bonds priced at par, the percentage change and the dollar price change are the same; for bonds priced away from par, a so-called “dollar duration” may be computed that describes the bond’s dollar price change given a change in rates. However, unless otherwise noted, the term “duration” refers to “percentage change in price”.

3 Although Duration-to-Worst is not an accurate measure of interest rate risk for securities and portfolios that contain embedded options, it is often used in the municipal market. This may be due to the fact that municipal portfolios have traditionally been managed to maximize reported yield, rather than on a total return basis. In the mid-1980’s to early 1990s, years in which interest rates declined, the average tax-exempt bond mutual fund consistently underperformed muni market benchmarks. In an earlier On-the-Edge article, we proposed the hypothesis that relying on Duration-to-Worst caused a widespread mis-estimation of the interest rate sensitivity of these funds, leading to this pervasive underperformance.

4With the possible exception of certain CMO tranches with extreme extension or contraction risk.