Analysis of Return

In the evaluation of bond portfolio performance, the first step is the measurement of return on the portfolio over the evaluation period. The next step is an analysis of return. We can think of analysis of return as the identification of the factors that contributed to the realized performance and a quantitative assessment of the contribution of each factor to the total return. The total portfolio return is partitioned into components, each component representing the effect of the given factor.

The first level of this decomposition aims at distinguishing between the effect of the external interest rate environment and the management contribution. If we separate the effect of circumstances that are outside the control of the portfolio manager from the effect of the portfolio management process, we can gain valuable insight into the nature of the portfolio performance. Denoting the total realized portfolio return by R, such a partition can be written as:

1

where:

If the portfolio had no element of management, then the return would be I, or the return due to the environment. This portfolio can be considered to be randomly selected from an available universe of fixed-income securities. As a proxy for this management-free randomly selected portfolio, we can use the total of all default-free securities, best approximated by all outstanding U.S. Treasury issues. These are the only available securities that are truly fixed-income securities in the sense that the promised payments can be expected with virtual certainty. Including corporate, municipal, or agency issues constitutes an element of the management process: It involves a decision to accept a degree of default risk in exchange for higher yields typically expected on lower quality securities. The standard for identification of the effect of the internal interest rate environment is thus a value-weighted Treasury index.

One might argue that the relevant portfolio bogey should vary according to investor preference. In the determination of the investor's investment objectives, individual preferences are certainly appropriate. The intent here, however, is to measure the interest rate effect on a universe that involves no other aspect, such as credit risk or spread relationships. That does not mean that a comparison of the portfolio return to a broader bond market index is inappropriate. Such comparison is in fact an integral part of the performance analysis as discussed in this article. It is done by performing the return analysis for the chosen bogey as well, thus allowing a direct comparison of the resulting components of return between the actual portfolio and the specialized bogey.

We can achieve a more refined analysis of the external factor component by partitioning the actual holding-period return on the Treasury index into two sources: interest rate level and interest rate change. Higher interest rate levels mean higher holding-period returns, on which the effect of interest rate changes is then superimposed. The effect of the interest rate environment thus consists of two components: return that would be realized if interest rates did not change, and the return due to the actual interest rate change.

To assign a precise meaning to the assumption of no change in interest rates, we use the basic concepts of the term structure of interest rates. The discount rates that determine the present value of a unit payment at a given time in the future are called spot rates. Spot rates are essentially yields on pure discount bonds. The market value of a coupon bond can be considered the sum of the present values of its payments, each payment being discounted by the spot rate corresponding to the maturity of that payment. The yield to maturity, or the internal rate of return on the bond payments, is a mixture of spot rates of various maturities.

The future one-period rates implied by the current spot rates are called forward rates. Forward rates are defined by the property that we can obtain the spot rates by compounding the forward rates over the term of the spot rate. If the forward rates do not change, future spot rates will be formed by compounding the current forward rates over the appropriate future interval. This implies that an investment in a long security would realize the same return as rolling over a short-term security. As a consequence, forward rates exhibit the following property: Under the scenario of no change in the forward rates, the holding period returns over a given period are the same for securities of all maturities and coupons. No other scenario of interest rate development would make the holding period returns independent of the maturity of the security or portfolio. In this sense, no change in the forward rates is the most “neutral” forecast, since under this assumption no maturities or payment schedules would be ex ante preferred to others. This scenario is often referred to as the market implicit forecast.

One can therefore define the effect of the current level of interest rates as the return on Treasury bonds under the assumption of no change in the current forward rates. The effect of the interest rate change is then defined as the difference between the actual realized return on the Treasury index and the return under the market-implicit forecast. We can then decompose the effect of I, the external interest rate environment, in the following way:

2

where:

We can interpret the component E as the expected return on a portfolio of default-free Treasury securities. The component U is then the unexpected part of the actual return on the Treasury index, due to the forward rate change. The sum I of these components is then the actual return on the Treasury index. We can attribute the difference between the actual portfolio return and the actual Treasury index return, termed C in Eq. (1), to the management process.

In evaluating the management contribution C, consider the means by which the management process can affect the portfolio. Three principal management skills that have an effect on performance include maturity management, sector/quality management, and selection of the individual securities. A partitioning of the management contribution is as follows:

3

where:

Maturity management (which might more correctly be called duration management) is an important tool of a bond portfolio manager and one that typically has the largest impact on performance. The successful application of this skill is related to the ability of the manager to anticipate interest rate changes. Holding long duration portfolios during periods of decreasing interest rates and short duration portfolios during periods of rate increases will typically result in superior performance. Being short when rates decline or long when rates go up will have a negative impact on performance.

Sector and quality management allocates the portfolio among the alternative issuing sectors and quality groups of the bond market. There may be spread relationships among the individual sector/quality groups that the manager may be able to exploit. Having a portfolio concentrated in high-quality industrial issues, for instance, during a period when high-quality industrials generally perform better than other sectors, would increase the portfolio return. The ability to select the right issuing sector and quality group at the right time constitutes the sector/quality management skill.

Selectivity, or individual bond picking, is the skill of selecting specific securities within a given sector/quality group to enhance the portfolio return. Individual securities show specific returns over and above the average performance of their sector/quality group. While sector/quality management means selecting the right market segment, selectivity means concentrating on the bonds, within that segment, whose specific returns are the most advantageous. As with the other two management skills, selectivity is involved in the initial portfolio construction as well as in subsequent activities such as purchases, sales, or swaps within a sector/quality group.

There is a fourth important skill of bond portfolio management, namely, timing. Timing is not a separate skill, but rather, an aspect of each of the skills identified earlier. Timing the shift of the portfolio from short to long duration or vice versa is really an element of maturity management, rather than an independently exercised ability. Without timing, there would be no maturity management. Similarly, timing is an essential part of sector/quality management and a part of choosing the proper bonds within a given sector/quality group. To provide a meaningful analysis of the portfolio return, the timing aspect must be included in the calculation of the return components.

Measurement of Return Components

We can measure the return components by security repricing. Consider maturity management first. If all securities held during the evaluation period were Treasury issues and if each issue were consistently priced exactly on the term structure of default-free rates (so that there would be no specific returns on any security), the maturity management component M of the total return would be equal to the difference between the realized total return R and the effect I of the external environment. In other words, if the sector/quality effect and the selectivity effect were eliminated, the total management contribution can be attributed to maturity management. This means that we can reprice each security as if it were a Treasury issue priced from the term structure, measure the total return under such pricing, and subtract the external effect component I to obtain the effect of maturity management.

Practically, this is accomplished by estimating the term structure of default-free rates from the universe of Treasury issues as of each valuation date throughout the evaluation period. The default-free price of each security held on that date is then calculated as the present value of its payments discounted by the spot rates corresponding to the maturity of that payment. The total return over the evaluation period is then calculated using the default-free prices, but otherwise maintaining all actual activity in the portfolio, including all transactions, contributions and withdrawals, cash account changes, and the like. Finally, the actual Treasury index return over the evaluation period is subtracted to arrive at the maturity management component M.

To determine the spread/quality management component S of the total return, we price each security as if it were exactly in line with its own sector/quality group (that is, with no specific returns), calculate the total return under such prices, and subtract the total of the external component I and the maturity management component M.

Here we have to be careful to determine the sector/quality prices correctly. It is not correct to base the sector/quality pricing on sector/quality indexes, since the differences in actual performance among various sector/quality indexes is primarily due to the different maturity composition of the market segments. For instance, the telephone issues would generally perform poorly during periods of increasing interest rates, not because they are telephones but because they are longer than the bond market as a whole.

We therefore adopt the following approach: First, we define a meaningful classification of the bond market by sector/quality groups. We then estimate the term structure of default-free rates from U.S. Treasury issues. Next, for each valuation date, we calculate the default-free prices for all securities existing in the market at that date. We then calculate the spreads, or yield premia, for each security as the difference between the actual yield and yield determined from the default-free price. These yield premia are then averaged over all securities in the given sector/quality group to determine the average yield premium for the sector/quality group as of the given date. After all this is done, we can calculate the sector/quality prices of the securities in the given portfolio by determining their default-free prices from the term structure, calculating the yield, adding the appropriate average yield premium depending on the sector/quality of that security, and converting this yield back to price. When all securities in the portfolio have been priced according to their sector/quality group at each of the valuation dates, we calculate the total portfolio return with the sector/quality prices. Again, the portfolio return with these prices is calculated including all actual purchases, sales, swaps, contributions, and withdrawals. We then obtain the sector/quality component S of the portfolio management by subtracting the external effect component and the maturity management component from the return calculated on the sector/quality prices.

Finally, to determine the selectivity component of the management contribution, we use the actual prices, which reflect the specific returns on each security. The selectivity component B is thus calculated by subtracting the total of all previously determined components from the actual total portfolio return.

In this way, we partition the total portfolio return into five components as follows:

4

These components are the effect of interest rate level (E), the effect of interest rate change (U), the maturity management (M), sector/quality management (S), and selectivity (B). The first component can also be interpreted as the expected return on default-free securities, and the second as the unexpected component of the actual return on the default-free Treasury market index. The first two components are the effect of external factors beyond the control of the portfolio manager, namely, the interest rate environment. Their sum is the actual return on the Treasury index. The last three components reflect factors within the control of the manager, that is, management skill. Together, they add up to the total management contribution. The sum of all five components is the actual return on the portfolio.

An alternative way of looking at the composition of the total return given by Eq. (4), which will reflect the way these components are actually calculated, is to consider the cumulative totals. The first total, E, is the expected return on a randomly selected portfolio of Treasury issues, calculated assuming no change in interest rates. The second total, E + U, is the actual return on a randomly selected portfolio of Treasury issues. The third total, E + U + M, is the return on the actual portfolio (including all activity) as if all securities were Treasury issues priced on the term structure (i.e., no sector/quality effects and no specific returns). The fourth total, E + U + M + S, is the return on the actual portfolio as if all securities were priced according to their issuing sector and quality (i.e., no specific returns). Finally, the fifth total, E + U + M + S + B, is the actual portfolio return. The decomposition of the total return into its components as specified in Eq. (4) provides a meaningful and informative analysis of the portfolio performance.

The effect of transaction costs is also included by this analysis. As a transaction is made, the cost is reflected in the price paid for a purchase and the price received for a sale. This, in turn, is captured in the return due to the selectivity component. Hence, excessive turnover of the portfolio would be reflected in the selectivity component of the portfolio.

After we have calculated components of return for the portfolio being analyzed, we can repeat the same return decomposition for a total bond market index such as the Lehman Government/Corporate Bond Index. The return components of the bond index provide benchmarks against which we can compare the return components of the portfolio.

We will conclude our exposition of the performance measurement by a discussion of risk adjustments. For equity portfolios, it is customary to calculate a risk-adjusted return, defined as the actual portfolio beta. Crude attempts at a similar adjustment for bond portfolios have been made by substituting the bond portfolio duration relative to an index for the beta of a stock portfolio. This is incorrect, since duration would measure the portfolio response only if interest rates always changed by parallel shifts of the forward rates.

It turns out that the correct adjustment for interest rate risk would actually be the maturity management component M as defined previously. Similarly, the sector/quality component S would be an adjustment for the second source of risk in the bond market, namely, the default risk. If the investment policy of a fund constrains the manager as to the maturity composition and/or sector and quality composition of the portfolio, it may be appropriate to consider the maturity and/or the sector/quality return components risk adjustments. For instance, if both maturity and sector composition of the portfolio are specifically prescribed by policy, the risk-adjusted return is equal to the selectivity component B. In general, however, interpretation of the maturity and sector/quality components as risk adjustments would mean removing the principal sources of return from the observed performance.

Summary

This paper has described a framework for comprehensively measuring and understanding the performance of a fixed-income portfolio. Macro components include the external interest rate environment and the managerial contribution to total returns. A more refined perspective is achieved by partitioning the external interest rate environment into expected and unexpected components. The managerial contribution is further partitioned into the return components of maturity, sector/quality, and individual security selection. These components are then contrasted with those of a total bond market index. An example of the analysis is contained in Table 25.1.

References

1. Bank Administration Institute. (1968). Measuring the Investment Performance of Pension Funds. Park Ridge, Illinois: B.A.I.
2. Dietz, P.O., H.R. Fogler, and D.J. Hardy. (1980). “The Challenge of Analyzing Bond Portfolio Returns.” Journal of Portfolio Management, 6, Spring, 53–58.
3. Fama, E.F. (1972). “Components of Investment Performance.” Journal of Finance, 27, 551–567.
4. Vasicek, O.A., and H.G. Fong. (1982). “Term Structure Modeling Using Exponential Splines.” Journal of Finance, 37 (May), 339–348.