# Duration and Convexity

Duration and convexity are factor sensitivities that describe exposure to parallel shifts in the term structure of interest rates. They can be applied to individual fixed income instruments or to entire fixed income portfolios.

Exhibit 1: Duration assesses exposure to parallel shifts in the spot curve. It cannot warn of exposure to more complex movements in the spot curve, such as tilts and bends.

The idea behind duration is simple. Suppose a portfolio has a duration of 3 years. Then that portfolio’s value will fall about 3% for each 1% rise in interest rates—or rise about 3% for each 1% decline in interest rates. Such a portfolio is less risky than one which has a 10-year duration. That portfolio is going to decline in value about 10% for each 1% rise in interest rates. Convexity provides additional risk information.

Exhibit 2 illustrates how the price of a fixed income portfolio might respond to parallel shifts in the spot curve.

Exhibit 2: The fractional change in a fixed income portfolio’s value is graphed as a function of parallel shift in the spot curve.

Here, Δr represents an immediate parallel shift in interest rates. For example, Δr = .015 corresponds to a 1.5% (or 150 basis point) parallel rise in the spot curve. The variable Δp is the dollar change in the portfolio’s value resulting from the shift in interest rates. Accordingly, Δp/p is the fractional change in the portfolio’s value.

Exhibit 2 fully describes the portfolio’s sensitivity to parallel shifts in the spot curve. There is no more information that we could add to the picture. What we try to do with duration and convexity is summarize the entire picture of Exhibit 2 with just two numbers. Certainly, two numbers cannot describe the wealth of detail contained in a picture, so what we do is take the two most important pieces of information in the picture. Those two pieces of information are duration and convexity.

Let’s start with duration. The most important thing Exhibit 2 tells us about this particular portfolio that its value will decline if interest rates rise—and rise if interest rates decline. This is the information that duration conveys, along with the magnitude of such sensitivity.

If we fit a tangent line to the curve in Exhibit 2, it will capture the direction and magnitude of the portfolio’s sensitivity to interest rates. For small changes in interest rates, the line and the curve almost overlap.

Exhibit 3: A tangent line is fit to the curve of Exhibit 2. Duration is the slope of the curve multiplied by minus one.

Duration is defined to be the slope of that tangent line, multiplied by negative one. For example, in Exhibit 3, the slope of the tangent line is –2.5 (for each .01 shift in Δr, Δp/p shifts about –.025). The portfolio’s duration is 2.5 years.

Tangent lines are the province of calculus, so we turn to calculus for the formal definition. Duration is a weighted partial derivative:

[1]

[2]

For example, suppose a portfolio has a duration of 5 years. That portfolio will appreciate about 5% for each 1% decline in rates. It will depreciated about 5% for each 1% rise in rates. It is as simple as that.

Suppose a portfolio has a duration of –2 years. The portfolio’s value will rise about 2% for every 1% rise in rates. It will decline about 2% for each 1% decline in rates.

Approximation [2] is the primary reason people use duration. With a single number, it summarizes a bond or a portfolio’s sensitivity to changes in interest rates.

Typically, a bond’s duration will be positive. However, instruments such as IO mortgage backed securities have negative durations. You can also achieve a negative duration by shorting fixed income instruments or paying fixed for floating on an interest rate swap. Inverse floaters tend to have large positive durations. Their values change significantly for small changes in rates. Highly leveraged fixed-income portfolios tend to have very large (positive or negative) durations.

For portfolios whose cash flows are all fixed (for example, a portfolio of non-callable bonds) there is a particularly simple way to calculate duration. For such portfolios, duration is just the weighted average maturity of the cash flows. Specifically, assume a portfolio has fixed cash flows ci, each occurring τi years from now. Let v(ci) denote the present value of cash flow ci. Duration then equals

[3]

Now the name “duration” should make more sense, as should the fact that duration is measured in years! When duration is calculated in this way, it is called Macaulay duration. One caveat: the Macaulay formula for duration is correct only if interest rates are continuously compounded. That is, formulas [1] and [3] are equal if Δr is a parallel shift in the continuously compounded spot curve.

Take, for example, a 5-year zero-coupon note. Because it pays no coupons, its average maturity is precisely 5 years. Hence, based on the Macaulay formula for duration, the bond’s duration will be 5 years. This means that a 5-year zero will appreciate about 5% in value for each 1% decline in continuously compounded interest rates based on approximation [2].

Continuous compounding is rarely used in practical applications, so it is desirable to modify Macaulay duration to reflect sensitivity to interest rates compounded m times a year rather than compounded continuously. A precise modification can be derived using calculus, but usually the following approximation based on a single yield to maturity y for the cash flows ci is used instead.

[4]

For portfolios containing instruments that do not pay fixed cash flows, such as callable bonds, mortgage-backed securities or interest rate caps, the Macaulay or modified formulas for duration will not work. For these portfolios, option pricing theory or other means must be employed for calculating duration based on definition [1]. The result is typically called effective duration.

Now let’s consider convexity. If duration summarized the most significant piece of information about a bond or a portfolio’s sensitivity to interest rates, convexity summarizes the second-most significant piece of information. Duration captured the fact that the graph in Exhibit 2 was downward sloping. It did not, however, capture its upward curvature. Convexity describes curvature.

Exhibit 4 shows the best-fit parabola for the graph of Exhibit 2:

Exhibit 4: Convexity reflects curvature. Here a “best fit” parabola is fit to the graph of Exhibit 2.

Note that the best-fit parabola does not exactly overlay the curve in Exhibit 4 because the curve is not itself a parabola. In general, the best-fit parabola will have the form

[5]

where convexity is defined as a weighted second partial derivative

[6]

The best fit parabola is simply a second order Taylor polynomial. Our first-order approximation [2] now becomes a second-order approximation:

[7]

The thing to remember about convexity is that it is a metric of curvature. In Exhibit 4, the curvature of the graph bends upward (like a bowl). The convexity is positive. If the curvature bends downward (like an inverted bowl), the convexity is negative.

Duration and convexity have traditionally been used as tools for asset-liability management. To avoid exposure to parallel spot curve shifts, an organization (such as an insurance company or defined benefit pension plan) with significant fixed income exposures might structure its assets so that their duration matches the duration of its liabilities—so the two offset. This technique is called duration matching. Even more effective (but less frequently practical) is duration-convexity matching, in which assets are structured so that durations and convexities match.

In closing, it is worth mentioning that terminology associated with the notion of duration is non-standardized. Different people will use terms in different ways. This is due to the history of the notion duration. Macaulay (1938) first introduced duration as simply weighted average maturity. To him, “duration” was what we now call Macaulay duration. Later, people realized that Macaulay duration equaled sensitivity to parallel shifts in the continuously compounded spot curve, as defined by [1]. Because people didn’t typically think in terms of continuously compounded rates, this lead to the modification of Macaulay’s formula, which is now called “modified duration.”

## References

• Macaulay, Frederick R. (1938). The Movement of Interest Rates, Bond Yields and Stock Prices in the United States Since 1856.