Calculating Duration

July 15, 2005

Question: Exactly how many types of duration are there? Also, is it correct to assume that modified and effective duration are calculated the same way and that you can use either one to estimate your price change?

In response to the assumption that modified and effective duration are calculated in the same way, simply put, this is not correct.

Before we discuss the difference in calculations, let’s discuss their definitions. Both effective and modified duration are the ratio (percent) of the proportional change in bond value (price) to the parallel shift of the spot yield curve. They both estimate bond price changes. Investment professionals use duration to quickly assess price volatility. To calculate the “real” fair value of a bond, an investor would present value its cash flows and value the options that are embedded in it. Durations estimate bond price changes by the use of the following formula:

Percentage price change = - Duration x Yield change x 100

For example, if a bond has a duration of four years and market yields fall by 10 basis points (b.p.) then the price of the bond can be expected to increase by 0.40 percent or -4.0 x -0.0010 x 100.

This formula works well for small changes in yield but; not quite as well for large changes because other factors come into play.

The difference between the two is that modified duration can only be used for fixed-rate bullet securities. The derivative of bond price volatility relative to yield changes that results in the “modified duration” formula has embedded in it the assumption that the cash flows of the bond do not change as rates change. This means that the modified duration formula is inappropriate for any bond where the cash flows change as interest rates change. That would include callables, step-ups, floaters, or any mortgage backed security.

Intuitively, for example, in a callable bond with a four-year maturity in the upward rate scenarios and one-year maturity in the downward rate scenarios, the durations will be different depending upon interest rates. Modified duration would be approximately 3.75 percent in all scenarios; however, effective duration would take the “call option” into consideration and could be 1 percent, 2 percent, or 3.75 percent depending upon the security’s coupon and market rates. In order to more accurately calculate price changes of these more complex non-bullet securities, the fixed-income markets developed the concepts of effective duration and option adjusted spread (OAS).

To calculate effective duration, it is first necessary to calculate OAS. To calculate the OAS of a bond, you begin by projecting a series of interest rate paths into the future and evaluate the bond along these paths. These interest-rate paths can become extremely complex depending upon how many paths are used, what change (volatility) of interest rates is used, and how the paths are developed. The most simplistic method is one that represents a “tree” where rates diverge up and down consistently from each “node” of the tree branch. As an oversimplified example, let’s assume we start in period one at 5 percent and that rates move up and down by a determined amount of 50 b.p. from one “node” of each “branch” to the next. You then have two “branches” in time period two. Rates will then move up and down from these two branches to produce four more branches and so on as shown below. Each forward branch uses projected forward rates. In this example, it is assumed that the yield curve is flat.

The OAS is basically determined by forcing a spread over the discount rates (which we will assume to be Treasuries) until the average present value of all these paths equal the current price of the bond. If this average spread is 20 b.p., then the 20 b.p. OAS tells us that the bond is trading, option free, 20 b.p. over a riskless security. In other words, if the security is yielding 4.5 percent and the relative Treasury is yielding 4.0 percent, the OAS has determined that the cost of the embedded option is 30 b.p. Therefore, the security at 4.5 percent offers a yield 20 b.p. over the like Treasury after consideration of option costs. Theoretically, the security has value!

You must then move the starting yields (keeping the security coupons constant) up and down to produce a new “tree.” The entire analysis is performed again holding the OAS constant and new prices are calculated. Once the OAS has been established, you then can estimate effective duration by backing into the price change formula.

Now that we have discussed “modified” and “effective” duration, let’s briefly tackle other duration measurements. There are many different types of duration and it seems as though more types are being developed every year. Although “duration” has historically been based upon explaining price volatility of a series of cash flows, the reason why more types of durations are being developed is to represent sensitivities that have nothing to do with interest rate changes. Below is a list with a brief explanation of the various types of duration. Although this list is not comprehensive, it represents the durations that we mainly use here, at ALM First, when performing investment and balance sheet analytics.

Macaulay Duration

The only duration that can accurately be quoted as length of “time.” Discovered in 1938, by Frederic Macaulay, this duration is calculated as the “present-value-weighted time to receipt of cash flows,” which is why it is quoted in “years.” Each cash flow “time” is multiplied by the present value of the associated cash flows and then the sum of all of these terms is divided by the sum of the present values.

In 1966, Larry Fisher presented a proof of the relationship between duration and bond price changes. Modified duration is a measurement of the change in value of an instrument in response to a change in interest basis (payment frequency). This "modifies" Macaulay duration. The relationship of duration and price volatility can be expressed as follows:

Percentage price change = -Modified duration x Yield change x 100

Portfolio Duration

The price sensitivity of an entire portfolio as an aggregated unit, versus the weighted average price sensitivity of each individual security.

Partial Duration (Key Rate Durations)

Measures the price sensitivity of a bond (or a portfolio of bonds) to changes in specific parts of the yield curve.

Spread Duration

A measure of the percentage price change to a change in the spread (OAS) of a bond. This is a very important duration measurement for floaters.

Empirical duration was developed to deal with how the security has been trading instead of estimating how a security will trade. In other words, it uses historical measurements that calculate actual price changes and changes in the level of the market to measure how the security is actually performing.

Constant dollar duration measures duration for securities in particular price ranges and is mainly used in mortgage backed securities.

Seems like a lot? Yes, but duration tools are important for any financial institution to quickly determine potential price changes and to fully understand risk versus reward. Although these tools are mainly used for managing investments, they are also used in ALM analyses as well. Effective duration is the most common duration used here, at ALM First. It is used in managing investments and determining risk in balance sheet market values. The end result of managing duration well is greater income within your risk profile.

Emily Hollis is a CFA and president of ALM First Financial Advisors, LLC in Dallas, Texas. Contact her at 800-752-4628 or ehollis@almfirst.com.

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