Duration is a measure of a bond’s price sensitivity to interest rate changes.  With the inverse relationship between interest rates and the price of a bond, higher duration means that the price will decrease more as interest rates rise.  It is a linear approximation of this price change that applies for small parallel shifts in the yield curve.

Calculating the duration of an individual bond is simple: there is a formula, a somewhat intimidating one, but a formula nonetheless.  As long as you know the characteristics of the bond (yield to maturity, present value, and cash flows), you can calculate the duration.  The duration of a portfolio of bonds is simply the weighted average of the individual bond durations.

The difficulty arises when we wish to know the duration of a portfolio where we do not know exactly what bonds it holds.  This is often the case when we are looking at historical data; we may know the composition, duration, average yield to maturity, etc. of the current portfolio but have to rely on some form of back extrapolation to glean what was going on in the past.

Bond ETFs are a perfect example of this.  Consider the iShares 20+ Year Treasury Bond ETF (TLT).  As of 5/19/2014, its duration was 16.7 years, its weighted average maturity was 26.93 years, and its weighted average yield-to-maturity was 3.28%.  As a point of reference, the yield on 20 year U.S. Treasuries was 3.11%.  What if we would like to know the duration a decade ago when the 20 year U.S. Treasury rate was 5.54%?  Unfortunately, without knowing the characteristics of TLT at this time, we can’t use the formula for duration.

Intuitively, we would expect the duration to be less since bonds have positive convexity (i.e. prices are less sensitive to rate changes when rates are higher).  But quantifying “less” is tricky.

Mathematically, duration is equal to the negative derivative of the bond price with respect to the yield divided by the price:

A more thorough discussion on duration can be found in our previous post.  From a practical point of view, we can look at the derivative as the percentage change in bond price divided by the change in interest rates, which should hold as long as the change in the interest rate change is small:

One approach is to numerically calculate the duration from daily price and yield changes.  We can do this using finite difference methods. Backward differences use only the data available on a given date, which is necessary for online algorithms, and central differences are more accurate but use data from the future.  Using backward differences with the change in the yield equal to the average of the change in the 20 year and 30 year treasury rates (see our previous post on how to get these easily), we obtain 16.6 years on 5/19/2014.  This isn’t a bad approximation at all; however, the following chart shows how this method performed over the previous few weeks.

On 4/21/2014, with no rate change, we did not get any estimate, while some of the days varied substantially from 16.7.  Since, in theory, we could have noted the duration of TLT from the iShares website each day, using central differences to obtain a more accurate estimate should be legitimate.  However, the results are not much improved.  Using finite differences is just too noisy.

We are in luck, though, as the form of the duration equation suggests another method: since duration is the quotient of two differences, we can approximate the duration of the portfolio by plotting price differences versus yield differences and determining the slope of the regression line (think rise-over-run).

Using this method, the duration is estimated as 15.5 years, not far off of the 16.7 mark.

The largest benefit of using this method is that it produces results that are much smoother, and realistic, over time compared to finite differences.

Because this is an after-the-fact analysis when we had the chance to know the true value at the time, it is not subject to the same rules that one must adhere to when backtesting (check out our paper on Backtesting with Integrity).  Hence, we could do further smoothing on the duration to remove more noise.  After all, we expect that the duration would not change abruptly from day to day as long as the yield curve is changing smoothly.

In summary, although determining the duration of a bond portfolio (or ETF) with complete knowledge of the portfolio components is easy, it is much trickier to calculate the duration of an unknown bond portfolio.  In this case, we must rely on reactive methods using data observed over time.

### Nathan Faber

Nathan is a Portfolio Manager at Newfound Research, a quantitative asset manager offering a suite of separately managed accounts and mutual funds. At Newfound, Nathan is responsible for investment research, strategy development, and supporting the portfolio management team. Prior to joining Newfound, he was a chemical engineer at URS, a global engineering firm in the oil, natural gas, and biofuels industry where he was responsible for process simulation development, project economic analysis, and the creation of in-house software. Nathan holds a Master of Science in Computational Finance from Carnegie Mellon University and graduated summa cum laude from Case Western Reserve University with a Bachelor of Science in Chemical Engineering and a minor in Mathematics.