The equations earlier suggest that calculating the summary statistics for a fixed-income portfolio is a straightforward matter. But it is not easy to do so, starting with YieldPORT, which is an input in the other equations. I assumed for convenience a nice, evenly spaced, semiannual pattern to the timing of cash flows. In reality, a typical portfolio of hundreds of bonds has coupon and principal payments occurring on many business days throughout the year, so N has to be measured in days, not semiannual periods. In reality, it's a really big internal rate of return calculation.

Imagine solving for Yield? ORT back in the olden days before computers. Picture a back-office analyst working all day to get the solution by slow trial-and-error search, only to be told that the trading desk just sold some bonds and bought some others. The analyst has to start all over again the next day – it's the bond math version of the Myth of Sisyphus!

In practice, summary statistics for a fixed-income portfolio typically are calculated as weighted averages of those for the individual bonds. As we've seen, these statistics are reported on various Bloomberg pages and become the inputs to the averaging process. Suppose that the portfolio is composed of J bonds, each having a market value (including accrued interest) denoted MVj and interest rate sensitivities denoted MacDurj, ModDurj, and Convex ityj. The market-value-weighted averages for Macaulay duration, modified duration, and convexity are shown in equations 9.8 to 9.10.

(9.8)

(9.9) (9.10)

How well do these estimate the “true” portfolio statistics, MacDurPORT, ModDurPORT, and ConvPORT? The answer depends on the shape of the yield curve. If ever the curve is perfectly flat, the estimations are perfect. Usually the weighted averages are lower because of the normal, upward slope to the yield curve: AvgMacDur < MacDurPORT, AvgModDur < ModDurPORT, and AvgConvexity < ConvPORT. The discrepancy is smaller when rates overall are lower, the yield curve is flatter, and when more of the market-value weights are farther out the curve where intra-portfolio yield differences usually are smaller. In the occasional circumstance of an inverted yield curve, the weighted averages are higher than the “theoretically correct” portfolio statistics.

Notice that I did not include a market-value-weighted average for the portfolio dispersion statistic, DispPORT. That's because it is not commonly calculated. One reason is that dispersion is not a statistic reported on Bloomberg. The Bloomberg Yield and Spread Analysis pages have lots of data but not cash flow dispersion, even though its inputs are the same as Macaulay duration. A second, and more important, reason is that the weighted average of individual dispersion statistics can be very misleading, even if the yield curve is flat. Suppose a portfolio is composed entirely of zero-coupon Treasury STRIPS having a range of maturities. Each individual zero has a dispersion of zero – and, obviously, a weighted average of zeros is zero. However, the portfolio overall clearly has positive dispersion; its DispPORT > 0.

The main reason for estimating portfolio statistics via a weighted average is that fixed-income bonds other than Treasuries often contain embedded call options and sometimes even put options. Then there is no way to project with confidence the future cash flows needed to get YieldPORT, MacDurPORT, and so on. Fortunately, the relevant statistics can be calculated (and are available on Bloomberg) for individual bonds, but it is important to factor in the correct numbers.

We saw in Chapter 6 that for callable bonds, you need to be careful in selecting the duration and convexity statistics off Bloomberg pages. Those calculated using the yield to first call, the yield to worst, or the yield to maturity are just data. More useful numbers are the curve duration and convexity (also called OAS and effective duration and convexity). Floating-rate notes also might be present in the portfolio. Once again you need to be careful about which statistics to use. Recall from Chapter 7 that Bloomberg reports modified duration and convexity to the next coupon payment date as well as OAS duration and convexity for floaters. The former essentially is the price sensitivity with respect to benchmark interest rates and the latter to credit risk. If there are C-Tinkers or P-Tinkers in the portfolio, you should think about summarizing real rate versus inflation durations.

Suppose you have a portfolio of investment-grade corporate bonds, including both callable and noncallable securities, and want statistics for average modified duration and convexity. Clearly, you use the curve durations and convexities on the callable bonds. But what do you use for the noncallable bonds? The Bloomberg Yield and Spread Analysis page reports both the yield and curve durations and convexities. To be consistent, you should use their curve durations and convexities as well. Then you would be averaging apples with apples, not apples with oranges.

There is another reason to aggregate the curve durations and convexities even on a portfolio of all (noncallable) Treasury notes and bonds. That is theoretical correctness. Often the risk management problem is framed as the gain or loss on the portfolio given a parallel shift in the yield curve, sometimes described as a shape-preserving shift because all yields are assumed to change by the same amount. For instance, the concern is how much the portfolio declines in market value if all yields go up by 25 basis points. You could solve that what-if scenario directly by summing the new prices after increasing each yield. Or you can estimate the result using average duration and convexity.

The problem is that the benign-sounding assumption of a parallel shift to the yield curve is inconsistent with the principle of no arbitrage. That's because when all the yields go up by 25 basis points, the implied spot rates change as well but not by the same amount – recall the example at the end of Chapter 6. For instance, zero-coupon bonds in the portfolio would be overpriced (relative to their no-arbitrage value) because their implied spot rates go up by more than 25 basis points (assuming the yield curve is upward sloping). So, it's theoretically impossible for all yields to shift by the same amount and still preserve the no-arbitrage assumption.

The key point is that the curve durations and convexities are calculated using a common assumption, namely a parallel shift in the benchmark Treasury curve, based on a model that itself keeps the no-arbitrage assumption intact. The model calibrates the change in the price of the bond, which depends on its coupon rate and maturity, and backs out the effective duration and convexity, as demonstrated in Chapter 6. Therefore, aggregating the curve durations and convexities by calculating the market-value-weighted average is theoretically correct. On the other hand, yield durations and convexities are well-defined statistics for a fixed-income bond that you can verify for yourself. Curve durations and convexities come from a “black box” and you need to trust the numbers (unless you build or have access to the underlying term structure model).

Calculating the average yield for the portfolio is another interesting problem. An obvious choice for the summary statistic is the market-value- weighted average of the individual yields. Denote these Yield for each of the J bonds and the portfolio average AvgYieldMV.

(9.11)

For callable bonds, the option-adjusted yield can be used. This is the yield to maturity after increasing the price for the value of the embedded call option. That depends on the assumed volatility of interest rates and requires an option-pricing model.

As with average duration and convexity, this market-value-weighted average yield is an accurate estimator of the portfolio internal rate of return only when the yield curve is flat. Typically, it is an underestimate, that is, AvgYieldMV < YieldPORT, the more so the steeper the curve. Nevertheless, AvgYieldMV does offer information – it indicates the annual return on the portfolio over the next year assuming a static yield curve.

You might be wondering why I put the “MV” in AvgYieldMV. It's because there is another way of averaging the individual yields to maturity. Instead of using market-value weights, we can use risk-based weights, in particular, the basis point value. Define the basis point value for each security (BPV) to be the modified duration times the market value, times one basis point: BPV. = ModDur. * MV. * 0.0001. The portfolio basis point value (BPV) is the sum of the BPV over the J securities in the portfolio. The BPV-weighted-average yield, AvgYieldBPV, is shown in equation 9.12.

(9.12)

Sometimes this is expressed as the weighted average of money durations (or, as commonly said, dollar durations). Recall from Chapter 6 that money duration is modified duration times market value. The weights are the same, just scaled differently.

Here's the good news: AvgYieldBPV is a really good approximation for YieldPORT. You'll see this in the numerical example in the next section. This – and not the market-value-weighted average yield – is a reasonable measure of the rate of return for the portfolio on a hold-to-maturity basis. The usual caveats apply – no default and reinvestment of cash flow at the same rate. The advantage of AvgYieldBPV is that it typically is easier to calculate than YieldPORT because it uses readily available inputs (i.e., on Bloomberg) from the individual bonds that compose the portfolio, including callables.

There is an interesting application here to the liability side of the balance sheet – usually we think about bond portfolios as assets using the perspective of the investor. Try to remember what you learned in school about calculating the weighted average cost of capital (the WACC) used to get the net present value of an investment project expanding the current line of business. To get the after-tax cost of debt, you probably were taught to use the yields (not the coupon rates) on existing debt liabilities and market values (not the par values). Essentially, you calculate an after-tax version of AvgYieldMV – a market-value-weighted average of yields.

Suppose that you do the capital budgeting exercise. Assume that debt is the primary component in the cost of capital – it's a very highly leveraged firm. The problem is that if the yield curve is upward sloping (so that AvgYieldMV < YieldPORT) and you invest MV to earn AvgYieldMV each period, you do not earn enough to pay off your liabilities – you need to earn YieldPORT on the project. A better cost of debt to get your “hurdle rate” is to use an after-tax version of AvgYieldBPV because it's a much better approximation for YieldPORT and can be easily calculated with our bond math tools.

We can combine the various weighted averages to estimate the change in market value resulting from some instantaneous yield “event,” described summarily by a change in the average yield (dAvgYield) for the portfolio.

(9.13)

However, we know that there is more than one way to formulate this expression. We can use the yield durations and convexities or the curve durations and convexities for AvgModDur and AvgConvexity. We can use the change in the MV-weighted or the BPV-weighted average yield for dAvgYield.

It's time for a numerical example to elucidate these portfolio statistics in theory and practice.

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