Duration and Convexity

[korchamp]

Hi,
I am looking at measurement of sensitivity of bond prices to interest rates. eg. Macaulay's Duration, Modified Duration, and Convexity. I am wondering are there problems that are associated with such measures, as in disadvantage or weakness. And can u please discuss the eventual complications engendered by embedded options.

Thank you very much.

 

[David Harper CFA FRM]

Hi korchamp,

I assume you'd let us know if this is a homework question (out of courtesy)? It's pretty broad query, sort of like asking, what are the drawbacks of P/E ratio (ie, where to start??), but as it is very thematic to FRM and highly testable, I'll gladly offer a start to an idea that took me a long time to grasp.

Setting aside Mac duration, they (duration and convexity) have in common: they are are single-factor analytical sensitivity measures [i.e., Taylor Series in the bond asset class. Almost all of our analytical approximations are Taylor. Duration is just the name it gets in bonds!]; i.e., they reduce a complex bond price dependence into a single measure, simplifying for convenience but distorting, based on a single factor, YIELD (most commonly). In short, they suggest, bond price change = f[yield]. While convexity overcomes the key weakness of duration (linear approximation), it nevertheless participates in the reduction of the entire term structure of rates into a single (effectively flat) yield. So, their weakness stems from the fact that a bond reacts to the entire set of rates along the term structure, the behavior of which does not realistically reduce to a single (yield) number.

Here is Tuckman on the implication of this single-factor approach (how to overcome? multi-factor approaches, which is why the FRM goes to KEY RATES next, which is the most obvious graduation from single to multi-factor). There are two closely-related assumptions, single-factor and parallel shift, that IMO it's okay to treat as one idea for an initial understanding:

"Chapter 5 defined various measures of price sensitivity in a general, one-factor framework. This chapter defines measures of price sensitivity in a more restricted setting, namely, that of parallel shifts in yield. These measures have two important weaknesses. First, they can be reasonably used only for securities with fixed cash flows. Second, the assumption of parallel yield shifts is not a particularly good one and, at times, is internally inconsistent. Despite these weaknesses, however, there are several reasons fixed income professionals must understand these measures. First, these measures of price sensitivity are simple to compute, easy to understand, and, in many situations, perfectly rea-sonable to use. Second, these measures are widely used in the financial industry. Third, much of the intuition gained from a full understanding of these measures carries over to more general measures of price sensitivity -- Tuckman, Chapter 6, page 115"