In a few recent columns, we’ve talked about duration and convexity in the context of changing market prices. They are some of the most misunderstood and misused terms in finance, and clarifying them is probably worth a few column inches. I won’t focus here on how to calculate the measures, but rather helping readers understand what they tell us about a bond’s expected price performance.
As I described last week, portfolio managers holding MBS need to undertake convexity buying and selling as interest rate levels change; the duration of their portfolio is changing with rates, although the changes are the opposite of what a portfolio manager would normally do as rates move. In a bond market rally, a rational PM would look to extend his/her duration in order to get the maximum benefit from rising bond prices; in a selloff, the PM would shorten the portfolio’s duration in order to minimize his/her exposure to declining bond prices. Because of prepayments, however, the opposite is happening. Rising rates imply lower levels of prepayments and longer MBS durations, meaning that without adjustments the portfolio’s value would become more sensitive to changing rates. A decline in interest rates, alternatively, means that prepayments are increasing, shortening the duration of MBS and forcing PMs to buy bonds in the face of higher prices.
Let’s take a more technical look at these two measures. Duration is the sensitivity of a bond’s price to changes in its interest rate or, more precisely, the yield to which it is being valued. The easiest way to envision a bond’s duration is as a downward-sloping line where the horizontal axis shows the bond’s yield and the vertical axis its price. As the yield rises, the bond’s price declines, and vice versa. If charted, the duration is approximately a straight line; this is the case when the bond being evaluated does not have any type of option attached to it.
Convexity is a second-order measure of a bond’s expected price performance. The chart below shows the durations of two different bonds. One is a fixed-cash-flow bond (such as a Treasury note or non-callable corporate debenture), and the other is a mortgage-backed security, where the bondholder is short a series of options (due to the borrowers’ option to refinance their mortgages). Note that the line is curved for the MBS. The degree of curvature of a duration function is its convexity; since MBS are negatively convex, the line is lower than the straight line at higher and/or lower rate levels, i.e., the “wings.” (This relationship occurs because of changing prepayment expectations for the MBS. Note that if you owned a “put-able” bond, its price performance would exceed the straight line as rates, and the bond would be considered positively convex.)
The key implication is that a bond’s duration changes, to different degrees, as rates change. Convexity describes how much the value of a bond (or a bond portfolio) will deviate from the roughly straight line represented by its duration. A good way to get a handle on the concepts is to compare them to examples in the physical world. Duration is the equivalent of speed, which is a first-order measure of movement—it is measured in miles per hour, feet per second, etc. Convexity is the equivalent of acceleration; it is measured in feet per second per second, and is the rate of change of the rate of change.
A fun example is to pretend to be a judge in traffic court. Imagine that you have two cases being argued before you (bad drivers need to hire lawyers) where the drivers got speeding tickets for being timed at 60 MPH in a 55 MPH zone. If you’re inclined to give drivers a 5 mph “fudge factor,” do you have enough information to decide these cases?
The answer is that you don’t have enough information. In this example, one car was traveling at a constant 60 MPH, and never drove faster than 5 MPH above the speed limit. The other began from a standing stop, and was going over 100 MPH when it passed the policeman. Since “speed” is calculated as a function of time (remember from high school that Rate =Distance/Time), being told that the drivers were traveling at 60 MPH doesn’t tell you much about their actual behavior. Like speed, duration is a useful but incomplete measure for describing a complex phenomenon.
The other thing to note is taking duration and convexity into account still does not entirely explain a bond’s price performance. Both duration and convexity are calculated using fairly rigid assumptions—unchanged spreads, no changes in the shape of the yield curve, etc. Therefore, even the inclusion of convexity in the conversation does not account for all the reasons that a bond’s price changes and out- or underperforms a benchmark. A common mistake is to observe a bond underperforming its hedge ratio in a rally and assume that it underperformed because of “negative convexity.” In reality, the underwhelming relative performance may be due to spread widening rather than convexity. There’s a simple way to view this difference: Price changes due to duration and convexity mean that you’re moving up and down the price line shown above; spread widening means that the line itself is shifting up or down. That’s why models have so-called attribution functions that will analyze data from two different dates and tell the user how much of the price change is attributable to different factors such as duration, convexity, spread, etc.