# Bond Portfolios

Finally, we get to bond portfolios. In reality, investors do not hold, and borrowers do not issue, bonds in isolation; they are clustered in asset and liability portfolios. The question for analysis is how well the statistics about an individual bond – its yield to maturity, its duration and convexity – translate to a portfolio. That is, how do we calculate the yield, duration, and convexity of the overall portfolio? It's not as obvious as it might seem because it is not just a matter of calculating the market-value-weighted averages of the individual statistics. Sometimes that produces a reasonable number; other times it is very misleading. Sometimes when we work with portfolio summary statistics we even venture into the realm of theoretical incorrectness and make assumptions that allow for arbitrage opportunities.

Once we understand the risk and return profile of a bond portfolio in theory and in practice, we can turn our attention to strategy. That comes in the next chapter.

## BOND PORTFOLIO STATISTICS IN THEORY

We can think of a portfolio of fixed-income bonds as just one big bond representing many promised payments on scheduled future dates. In doing this we focus on cash flow, not on how the payments are accounted for as interest income or redemption of principal. This big-bundle-of-cash-flow approach makes sense only if the bonds are fairly homogeneous with respect to credit risk and taxation. It would be hard to interpret the summary statistics on a bond portfolio made up of half low-yield, high-quality, federal tax-exempt municipals and half high-yield, non-investment-grade corporate bonds.

Suppose that our portfolio is composed of a homogeneous class of traditional fixed-income securities, for instance, semiannual payment U.S. Treasury notes and bonds. There are no floaters or linkers, and for now not even callables. In general, the current market value of the portfolio is MV, which includes both the flat price and the accrued interest of the constituent securities. However, to simplify, I assume in this chapter that all the bonds have either just been issued or have made a coupon payment so that the accrued interest is zero. In any case, all the results here extend to between-coupon dates and for any other periodicity (e.g., payments made monthly, quarterly, or annually).

The future cash flows are designated CF1, CF2, ..., CFN. The longest- maturity bond redeems its principal in N semiannual periods, where N is an integer. The cash flow for each period consists of coupon interest on all remaining securities and principal on any maturing bonds. Some of the cash flows can even be zero, for instance, if the longest-maturity bond in the portfolio is a zero-coupon Treasury STRIPS.

The portfolio yield is the internal rate of return on the cash flows: MV, CF1, CF2, ..., CFN. It's the solution for YieldPORT in equation 9.1.

(9.1)

This is the same as solving for the yield to maturity on an ordinary fixed-income bond, as in Chapter 3, where CF1 to CPN-1 are the coupon payments (PMT) and CFn is the final payment including the principal (PMT + FV). Moreover, I can make a similar statement about the portfolio yield as I made about a yield to maturity – it does need to assume a flat yield curve. That is, the underlying Treasury yield curve corresponding to the many bonds in the portfolio can be upwardly sloped, downwardly sloped, or perfectly flat. YieldPORT, which also is called the cash flow yield, is in a sense an “average” of the various yields to maturity, which are in turn “averages” of the implied spot rates.

Given the portfolio yield and the schedule of cash flows, we calculate the Macaulay duration of the portfolio as the weighted-average time to the receipt of cash flow, as in equation 6.14 in Chapter 6. This statistic is denoted MacDurPORT.

(9.2)

The denominator in equation 9.2 is the market value of the portfolio, MV. In the numerator, the times to the receipt of cash flow (i.e., 1 out to N periods) are each multiplied by the share of the portfolio market value corresponding to that period.

The equation for portfolio Macaulay duration can be written more compactly using the summation sign.

(9.3)

Here we see that Macaulay duration is a weighted average of the times to receipt of cash flow. Another statistic for the portfolio, one that we have not seen yet, is the dispersion of the cash flow. It is denoted DispPORT and is calculated in equation 9.4:

(9.4)

Portfolio dispersion is the variance of the times to the receipt of cash flow; like Macaulay duration, it uses the shares of market value for each period as the weights. The same formula can be used to calculate the cash flow dispersion for an individual bond.

A change in the market value of the portfolio (dMV) resulting from a change in the portfolio yield (dYieldPORT) is estimated in the same manner as in Chapter 6 for individual bonds.

(9.5)

The termin parenthesis is the modified duration of the portfolio (ModDurPORT). It is important to note once again that using equation 9.5 does not inherently assume a parallel shift in the yield curve. Just as many shapes to the underlying yield curve can produce the same portfolio yield, many shifts and twists to the yield curve can produce that same change in the portfolio yield. It's just an estimation that happens to be better if the underlying curve is flat and its shift is parallel. Moreover, the estimation can be improved by adding the convexity adjustment.

The convexity statistic for the portfolio (ConvPORT) is derived in the Technical Appendix.

(9.6)

This convexity statistic also can be expressed as a function of the portfolio cash flow yield, Macaulay duration, and dispersion.

(9.7)

This formula is also derived in the Technical Appendix. It is a general relation, and it holds between coupon payment dates and regardless of the shape of the underlying yield curve. It's a very neat result that might surprise you if you've never seen it before. We see in equation 9.7 that for a given Macaulay duration of the portfolio (or individual bond), convexity is directly linked to the dispersion of cash flow – the greater the dispersion, the higher the convexity. The more concentrated are the cash flows, the lower the dispersion and the convexity.

A fixed-income money manager might think of a “laddered” portfolio as a way of addressing liquidity needs. Hence, the maturities of the bonds are spread out along the yield curve (the rungs of the ladder), so that a portion is always soon to mature and provide cash. Laddering also increases convexity by spreading out the cash flows. Greater convexity, other things being equal, generally is a good thing for the portfolio, increasing gains when yields go down and reducing losses when yields go up.

In the extreme, a “barbell” portfolio maximizes convexity by putting all the weight in the “wings” (i.e., in the shortest-term and longest-term maturities). In contrast, a “bullet” portfolio in which the maturities are tightly clustered has less dispersion and lower convexity. After this dose of street lingo, we are ready to tackle the portfolio statistics in practice.