Duration and Convexity

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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.

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.

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.

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]

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 swapInverse 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.

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.
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11 Responses to Duration and Convexity

  1. Marianna at #

    Hello, I wold like to ask how to assess duration of portfolio (currency of portfolio is EUR), when I have not only EUR denominated securities, but also USD, PLN, GBP, … thank you…
    I suppose it should be price sensitivity to yield for a parallel shift in yields. But what does duration mean for portfolio with securities in different currencies and different yield curves?

    • Whitney at #

      Duration is not defined in a multi-currency environment. I suppose you could use forward rates to convert all cash flows to EUR and then calculate duration. You would be treating exchange rates as fixed and interest rates as random. Due to the correlations between exchange rates and interest rates, that would be somewhat contrived. Another option would be to track a separate duration in each currency. If that is too messy, maybe the tool you are looking for is value-at-risk and not duration.

  2. Dear Glyn:

    In your article on duration and convexity, your claim in Figure 3 that duration is (the negative of) the slope of the curve is incorrect. The slope of that curve is d(Δp/p)/d(Δr) ≈ Δ(Δp/p)/Δ(Δr). Duration is -(dp/p)/dr ≈ -(Δp/p)/Δr; this would be the slope of a line through the origin (0,0) and the point (Δr,Δp/p).

    Please correct the article.

    Thanks!

    BCIII

    • Hi Bill,

      Thanks for commenting!

      You had me stumped for a moment, but I rebounded after a mug of coffee. You do raise a very interesting point, so readers may want to spend some time on this.

      What you have done is directly apply a partial derivative to calculate the slope of the tangent line in Exhibit 3 in terms of the notation used in Exhibit 3. Your result is straightforward and correct:

      ∂(Δp/p)/∂(Δr)

      You then notice that this looks different from (the negative of) the definition of duration in formula [3], which is

      (1/p)(∂p/∂r)

      But they are equal. The derivation of one from the other is a fun exercise in calculus, so I encourage everyone to give it a try. If you get stumped, see my derivation here

      http://riskencyclopedia.com/wp-content/uploads/files/duration_derivation.pdf

      A quick explanation of the derivation is that the derivative of the difference of a variable equals the derivative of the variable.

      This leads to the question of why I draw Exhibit 3 as I do. Wouldn’t this calculus confusion go away if I made the y-axis be Δp (or, even simpler, p)? Yes, but we would lose something.

      With that approach, I could not tell readers that duration is the (negative) slope of a tangent line. It would be the (negative) SCALED slope of a tangent line. That is confusing for most people. They ask what is the point of the scaling factor (1/p). After you have taken a derivative, why scale it?

      With the approach I take in the article, I apply the scaling in advance, directly in how I define the y-axis of Exhibit 3. My explanation that we are interested, not in the absolute change in p, but in the fractional change in p, makes sense at that point. Then I can present duration as the (negative) slope of the tangent line …. with no confusing after-the-fact scaling.

      The hope is that readers then accept the definition [3] of duration as intuitively obvious and not get bogged down in a formal derivation, which is what happened to you, Bill.

      Explaining duration is always difficult because you have to deal, somehow, with the two details of the negative sign in the definition and the scaling factor (1/p). To see how simple things would be without these, check out my article on delta and gamma, which closely parallels this one, but without those two bothersome details.

      http://riskencyclopedia.com/articles/delta_and_gamma/

      If you enjoyed this exercise in calculus, you may also enjoy my blog post “What Exactly is an Infinitesimal?” at

      http://glynholton.com/2008/06/what-exactly-is-an-infinitesimal/

      It is actually just an e-mail exchange between myself and another reader, but we stray into some fascinating foundational issues.

      Kind Regards,

      Glyn

      • Dear Glyn:

        Now that I see how you’re defining Δp and Δr, the explanation is clear.

        However, if you’re intent on showing the duration of a bond (or any other interest-rate sensitive asset or liability), mightn’t it be easier (and slightly less unconventional) to leave YTM on the x-axis and make the y-axis ln(price)?

        I’ve been teaching review courses for the CFA exams for ten years now, and I’ve never had any trouble explaining duration to candidates, whether it’s the slope of a particular curve or not. Nor have they had any trouble grasping my explanation.

        Anyway, my apologies for the mistake.

        BCIII

        • Hi Bill:

          As I indicated in the article, YTM is used in the modified duration formula [4], which is what I presume you consider conventional. But you should emphasize to your students that modified duration is not a MODIFICATION of duration but merely an APPROXIMATION of duration. The name “modified duration” is unfortunate for this reason.

          Any duration formula based on a single YTM is an approximation, which is why I would not introduce duration with YTMs. The definition of duration is based on parallel shifts in the spot curve, so introduce duration using the spot curve. Then introduce approximations based on YTM later, as I did in the article.

          I can’t imagine what you are using ln(price) for. Does that appear in CFA study materials? I hope not.

          Having said all the above, there is another approach you can take. I don’t recommend it, but you can define duration in terms of YTM y instead of the level of the spot curve r. That is, definition [1] becomes

          -(1/p)(∂p/∂y)

          instead of

          -(1/p)(∂p/∂r)

          In this case, modified duration [4] is no longer an approximation. It is exact.

          Maybe this is where the name “modified duration” comes from … you actually modify the definition of duration to obtain the formula. I don’t know …

          This approach is easier to teach, and modified duration is easier to derive from the new definition of duration based on YTM than Macaulay duration is from the old definition based on the spot curve.

          I don’t recommend the approach because it leaves duration a far less useful tool. YTM is so 20th century. Why use it in this age of computers in every pocket? Many (most) instruments in a typical fixed income portfolio don’t have a well defined YTM. What do you do with callable bonds? Use yield to maturity, yield to first call, etc.? Effective duration defined in terms of YTM is pretty much useless. What instruments would you apply it to?

          - Glyn

          • Dear Glyn:

            I’ve never seen a definition of duration (Macaulay, modified, or effective) based on a shift in the spot curve; every definition is based on a (parallel) shift in the (par) yield curve (e.g., Fabozzi, Fixed Income Analysis, 2nd edition, 2006 (21st century)).

            As for ln(price): I was simply giving an example of a curve for which the slope is the duration of the bond. Nobody uses that curve, but if you’re intent on displaying duration as the slope of a curve, that’s the curve to use.

            BCIII

            • I don’t suppose you are an actuary, are you? The actuarial literature is where the concept of duration was developed, several decades after Macaulay introduced his definition. You will find a host of definitions of duration in the actuarial literature. Actuaries are highly mathematical and did some interesting work.

              Fabozzi is a great guy. I used to review his books, so we exchanged our share of emails. He would be the first to admit his books target the buy side. Yes, some buy side folks still use yield, although they shouldn’t. I would never price anything off a par yield curve, except maybe a par bond.

              So, if you are pricing off a spot curve, why would you build a separate yield curve just for calculating duration? If only for consistency, I would calculate duration off the same spot curve I used for pricing. I want to know my portfolio’s sensitivity to that spot curve, not some yield curve.

              I now see what you are doing with ln(price). If you take the derivative, the chain rule gives you the (negative of) the definition of duration, formula [1]. Nice. I like that.

              But we have already seen that taking the derivative of Δp/p, you get the same (negative of the) definition of duration.

              This raises an interesting question i will let readers ponder: Calculus tells us functions may have multiple antiderivatives, but they will differ only by the addition of an arbitrary constant c. That doesn’t appear to be the case here, as ln(p) and Δp/p are very different ….. they don’t differ by merely a constant c.

              I won’t give you the answer, as it is a fun problem to work out yourself … you may gain insights into both calculus and duration.

              For teaching purposes, should we define duration as (negative) the derivative of ln(p) or (negative) the derivative of Δp/p? Obviously, we should use the latter. No one thinks in terms of the natural log of their portfolio’s value …

              • Dear Glyn:

                No, I’m not an actuary; it’s one of the few careers I’ve not pursued. I’ve taught university mathematics, I’ve taught review courses for the CFA exams, I’ve taught project risk management, cost management, quality management, cash flow analysis, financial mathematics, and financial modeling, I’ve written software for GPS navigation, DOT (deep-ocean transponder) navigation, cost estimation for building petroleum refineries, running numerically controlled machinery (milling machines, lathes, punch presses, coördinate measuring machines), analyzing investment portfolios, simulating bond markets, and controlling freeway traffic cameras, I was a warhead designer, I analyzed mortgage-backed securities for a fixed income portfolio management company in Newport Beach, CA (and rewrote the lion’s share of their analysis software), I’ve consulted in project risk management, I’m a professional magician, and I’ve competed in only one equestrian world championship.

                But I’ve never been an actuary.

                As for your question about antiderivatives differing only by a constant, that’s true when you’re integrating (having already differentiated) with respect to the same variable. Inasmuch as the two functions under consideration are being differentiated with respect to different variables (YTM in one case, Δr in the other), the rule to which you allude doesn’t apply.

                I agree that nobody would ever look at ln(price) for any useful information, and when I teach Level I CFA review courses I mention it only in jest (“You don’t want to think about it this way!”). There’s actually a good reason to mention it in such courses, but we needn’t go into the details here.

                By the way, what brought me to your site was a discussion I had with a CFA candidate on AnalystForum in which he was answering a question about duration, and, I found out later, was using your article on duration as his source material. He wrote that duration was the (negative of the) slope of the price/yield curve, and I corrected him on that. Then he wrote that he’d been mistaken and that he’d meant to have Δp/p on the y-axis. When I pointed out that that was still wrong, he corrected himself further by writing that he’d meant to have Δr on the x-axis. As you can see, he’d completely missed the differences between your approach and that of CFA Institute; even if he’d gotten it correct from the start, it wouldn’t have been helpful to the candidate who’d posed the question, as he needs to understand the approach on which he will be tested.

                I have no argument against using changes in spot rates for computing duration – that’s the typical approach for computing key-rate durations, for example – but it’s not the approach that the CFA candidates are taught for calculating modified or effective duration.

                BCIII

                • Hi Bill:

                  That is quite a resume.

                  Yes, you should teach what the students will be tested on.

                  But I won’t let you off so easily with the math question.

                  The variable of integration or differentiation is actually irrelevant. If you differentiate ln(p) or Δp/p with respect to Δr, in both cases the result is (1/p)(∂p/∂r), which is definition of duration [1]. If you differentiate them with respect to yield y, in both cases the result is (1/p)(∂p/∂y), which is the definition of modified duration I developed in one of my earlier replies.

                  So whether you are working with duration (1/p)(∂p/∂r) or modified duration (1/p)(∂p/∂y), you appear to have a function with two anti-derivatives differing by more than an arbitrary constant c, which contradicts an established result of calculus.

                  Note that this is a math puzzle, not a finance puzzle. It holds irrespective of the variable of differentiation. I could differentiate with respect to Barack Obama’s approval rating o and still get the same result in both cases: (1/p)(∂p/∂o).

                  Don’t feel obliged to provide a solution to the puzzle. Writing (or reading) an explanation could be as tedious as writing (or reading) an explanation of how someone solved Rubiks Cube. The fun is in solving the puzzle oneself, so I encourage all readers to take a stab at it.

                  Cheers,

                  Glyn

  3. Filipa at #

    Hallo,

    Thank you for this site, it is of great help to me.

    I have a concrete question, concerning an example I have been trying to understand..

    Asset has following characteristics: Coupon bond with 8% coupon, 100 face value and 2 years to maturity, flat term structure with rt= 10%.

    Task: compute the asset’s change in PV due to a rise in IR of 1500 basis points a) by means of linear aprox. and by quadratic aprox, using duration and convexity.

    Now, I have some formulas:

    PVt(rt=10%)= 8/1.1+ 108/1.1squared= 96.52

    the part which I do not understand is:

    Durt= – partialder of PV t/ der IR/PVt (r)* (1*rt)

    and the actual numbers provided in the solution are:

    Durt=1*8/ 1.1 + 1*108/1.1/ 96.52893= 1.925

    I don’t understand the part with the partial derivatives, if someone would be willing to explain this to me, I would be very greatful as I am only a beginner.

    Thank you in advance,

    Best regards,

    Filipa

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