# THOUGHTS ON BOND PORTFOLIO STATISTICS

To be frank, I have more expertise at * producing* and

*individual bond and bond portfolio statistics than*

**thinking about***them in practice. I've never built or managed an actual fixed-income portfolio. I offer some thoughts, but keep in mind that they come from someone looking at the world from an ivory tower. I'm going to imagine that I've been asked to manage, or at least advise, a large portfolio of investment-grade corporate bonds, made up of both noncallables and callables.*

**using**Obviously, my first step is to look at a variety of * descriptive statistics* for the portfolio, including the various weighted averages for yield, duration, and convexity. I'd want to see breakdowns by credit rating and industry. If not included in the spreadsheets provided to me, I'd ask someone to calculate the implied probabilities of default for each bond for a range of assumptions about recovery, as in Chapter 3. If my performance is measured against the return on some broad index of corporate bonds, I would want to see the descriptive statistics for that portfolio as well to identify any significant differences.

The key point here is that the descriptive statistics have informational value as long as they are measured consistently, even if they are theoretically flawed. I'll be watching for * changes* in those statistics over time. For instance, if average yield to maturity calculated for either MV or BPV weights is increasing, I'll want to know why. Is it due to higher yield on benchmark Treasuries? If so, is it because of higher real rates or higher expected inflation? Is it due to higher credit spreads over Treasuries? If so, is it due to a change in the probability of default or to projected recovery rates? Is the higher average yield due to a change in taxation or a loss in liquidity?

I'm going to use the average * curve*, rather than

*duration and convexity in estimating potential changes in market value due to volatile interest rates. There are several reasons for this. Being an academic, I like theoretical correctness when possible and prefer to aggregate apples with apples. But, more important, I think about interest rate risk in the context of the Treasury yield curve. To me, the more intuitive what-if question is about a 25-basis-point shift in the Treasury curve, not a 25-basis-point change in the portfolio yield as estimated by the BPV-weighted average. Even though this is a portfolio of corporate bonds, interest rate risk commonly is*

**yield,***with respect to benchmark Treasuries. Average*

**framed***duration and convexity are better thought of as primarily descriptive statistics, possibly useful in formulating portfolio strategy, as we see in Chapter 10.*

**yield**There is another benefit to aggregating curve duration and convexity for the portfolio: The statistics can be decomposed into * partial* durations and convexities. Recall that the curve statistics are estimated using effective duration and convexity, as in Chapter 6. The entire Treasury yield curve is raised and lowered by a certain amount. Importantly, that shift translates to a nonparallel shift in the implied spot curve. Those implied spots are then used to value the bond assuming no arbitrage – that is, getting

*and*

**MV(up)***the inputs to the calculations.*

**MV(down),**Rather than shift the entire Treasury yield curve, the same model can shift only one particular point, for instance, the 5-year or 10-year. That produces the sensitivity of the bond price (or portfolio) to an isolated shift in that particular Treasury yield. These are the partial (sometimes called * key rate)* durations and convexities. The sums of those partial durations and convexities are just the overall curve statistics. The point is that now it is possible to estimate the impact of various twists and turns to the shape of the Treasury yield curve. That is likely to be more insightful that just summarizing those “events” by the change in the overall average yield.

# CONCLUSION

It is surely an unfair comparison, but sometimes bonds to me are like middle-school-age children. On their own, as individuals, they are fine and usually well behaved. The same can be said for bonds. I understand them and their risk-return profile. I can measure their comparative static properties. The problem is when they run together in portfolios, like children in a shopping mall. It's hard to predict pack behavior using weighted averages of their individual statistics.

You need a really good strategy to build and manage a bond portfolio. That comes next.