Duration and Convexity

Explained:

convexity

duration

duration matching

duration-convexity matching

Macaulay duration

modified duration

Duration and convexity are factor sensitivities that describe exposure to parallel shifts in the spot curve. They can be applied to individual fixed income instruments or to entire fixed income portfolios.

 

Changes in the Spot Curve
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, including tilts and bends.

The idea behind duration is simple. Suppose a portfolio has a duration of 3 years. Then that portfolio's value will decline about 3% for each 1% increase in interest rates—or rise about 3% for each 1% decrease 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.

Example: Sensitivity of a Fixed Income Portfolio to Interest Rates
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, represents an immediate parallel shift in interest rates (see the notation conventions documentation). For example, = .015 corresponds to a 1.5% (or 150 basis point) parallel rise in the spot curve. The variable is the dollar change in the portfolio's value corresponding to the shift in interest rates. Accordingly, is the fractional change in the portfolio's value. Note that we use preceding superscripts 0 to indicate quantities are as of the current time 0—shifts in the spot curve are considered instantaneous.

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 can not 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 significant information Exhibit 2 provides us about this particular portfolio is the fact that its value will decline if interest rates rise—and rise if interest rates fall. 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.

Duration
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 , 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]

This leads to the approximation

[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 average maturity of the of the cash flows. Specifically, assume a portfolio has fixed cash flows , each occurring at some time years from time 0. Let 0pv() denote the present value at time 0 of the cash flow , then the duration is

[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.

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].

In formula [3], all present values should be calculated using the spot interest rate for the maturity of the cash flow it is discounting. In practice, people often calculate all present values with a non-continuously compounded yield to maturity 0y for the entire portfolio. If this is done, formula [3] must be modified slightly. It becomes

[4]

where m is the frequency of compounding for the yield to maturity. For example, if the yield to maturity is compounded quarterly, m = 4. This formula is called modified duration.

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, other means must be employed for calculating 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:

Convexity
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 the notion of duration as simply weighted average maturity. To him, "duration" was what we now call Macaulay duration. Later, people realized that, if Macaulay duration was calculated using continuously compounded interest rates, the result equaled the factor sensitivity [1] that is called "duration" in this article. 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." Other terms, including "effective duration" and "option-adjusted duration" are also used. As a rule of thumb, if someone speaks to you about some duration concept, ask what they mean.

Related Internal Links

asset-liability management Techniques for protecting a firm's solvency in the context of accrual accounting.

bond Securitized debt.

compounding Simple, compound and continuously compound interest.

delta and gamma Factor sensitivities used with derivatives that are analogous to duration and convexity.

directional strategy A trading or investment strategy that entails taking net long or short positions in a market.

financial risk management Practices by which a firm optimizes the manner in which it takes financial risk.

fixed income term structure Refers collectively to a spot curve, forward curve, discount curve, yield curve or any other curve that describes the time value of money at a particulate point in time.

gap analysis A technique of asset-liability management used to assess interest rate risk or liquidity risk.

Greeks A set of factor sensitivities used to measure risk exposures related to options or other derivatives.

interest rate risk Risk due to uncertain future interest rates.

interest rate spreads Spreads between two interest rates.

option-adjusted spread Yield spread not attributable to embedded options.

return Any of a number of metrics for the change in an asset's or portfolio's accumulated value

scenario analysis Formalized "what if" analysis typically performed as a part of asset-liability management or corporate risk management.

yield Any of several metrics of the income or return to be earned from an investment.

Related Books

Frederick R. Macaulay first introduced the notion of duration in his 1938 book The Movement of Interest Rates, Bond Yields and Stock Prices in the United States Since 1856.

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