Fixed-Income Return Attribution

Factor-Based Return Attribution

Factors refer to a driver of returns. There are myriad of investment factors in existence and many models with which to capture them and use them to decompose returns. The curriculum reviews the Carhart 4-Factor Model. This model shows a portfolio in terms of the portfolio's sensitivity to a market index (RMRF), a market-capitalization factor (SMB), a book-value-to-price factor (HML), and a momentum factor (WML).

$$
R_p – R_f = \alpha_p + f_1RMRF + f_2SMB + f_3HML + f_4WML + E_p $$

Where:

\(R_p\) and \(R_f\) = The return on the portfolio and the risk-free rate of return, respectively.

\(\alpha_p\) = “alpha” or return in excess of that expected given the portfolio's level of systematic risk (assuming the four factors capture all systematic risk).

\(f\) = The sensitivity of the portfolio to the given factor.

\(RMRF\) = The return on a value-weighted equity index in excess of the one-month T-bill rate.

\(SMB\) = Small minus big, a size (market-capitalization) factor (SMB is the average return on three small-cap portfolios minus the average return on three large-cap portfolios).

\(HML\) = High minus low, a value factor (HML is the average return on two high-book-to-market portfolios minus the average return on two low-book-to-market portfolios).

\(WML\) = Winners minus losers, a momentum factor (WML is the return on a portfolio of the past year's winners minus the return on a portfolio of the past year's losers).

\(E_p\) = An error term that represents the portion of the return to the portfolio, p, not explained by the model.

Candidates at level III are likely already familiar with factor models. Each term has a sensitivity factor, which is multiplied by the factor itself. There is an error term involved and the final output is meant the show where the return in the portfolio came from due to which factors? The following output comes from a hypothetical manager highlighted in the 2020 level III curriculum:

$$ \scriptsize{\begin{array}{cccc|ccc}
& & \textbf{Factor} & \textbf{Sensitivity} & & \textbf{Contribution} & \textbf{to Active} \\ \hline
& \textbf{Portfolio} & \textbf{Benchmark} & \textbf{Difference} & \textbf{Factor} & \textbf{Absolute} & \textbf{Proportion} & \\
& & & & \textbf{Return} & & \textbf{of Total} \\ \hline
\textbf{Factor} & 1 & 2 & 3 & 4 & 3 \times 4 & \\
\textbf{RMRF} & 0.95 & 1.00 & (0.05) & 0.06 & -0.28\% & -13.3\% \\
\textbf{SMB} & (1.05) & (1.00) & (0.05) & (0.03) & 0.17\% & 8.1\% \\
\textbf{HML} & 0.40 & – & 0.40 & 0.05 & 2.04\% & 98.4\% \\
\textbf{WML} & 0.05 & 0.03 & 0.02 & 0.10 & 0.19\% & 9.3\% \\ \hline
& & & & \textbf{Factor} & \bf{2.12\%} & \bf{102.4\%} \\
& & & & \textbf{Tilts} & & \\ \hline
& & & & \textbf{Security} & \bf{-0.05\%} & \bf{-2.4\%} \\
& & & & \textbf{Selection} & & \\ \hline
& & & & \textbf{Active} & \bf{2.07\%} & \bf{100\%} \\
& & & & \textbf{Return} & & \\ \hline
\end{array}} $$

Fixed Income Return Attribution

Fixed income attribution approaches are similar to those used in equity attribution problems. They come in three main types:

• Exposure decomposition – duration based.
• Yield curve decomposition – duration based.
• Yield curve decomposition – full repricing based.

Candidates do not need to make the underlying calculations in any of the approaches but do need to know how to interpret them.

Exposure Decomposition

In exposure decomposition, top-down attribution is used to explain how a portfolio's active management compares with its benchmark. Top-down decisions might include:

• Portfolio duration bets.
• Yield curve positioning.
• Sector bets.

The term “exposure decomposition” relates to the breakdown of portfolio risk exposures by grouping a portfolio's component bonds by specified characteristics (e.g., duration, bond sector). The term “duration based” relates to the use of duration to represent interest rate exposure decisions.

An exposure decomposition model is similar to Brinson-type equity attribution models, whereby the portfolio is grouped by its market value weights in different economic sectors. In this case, however, the portfolio is grouped by its market value weights in duration buckets (i.e., exposure to different ranges of duration). This approach simplifies the data requirements and allows straightforward presentation of results relative to other fixed-income approaches. For these reasons, the exposure decomposition approach is used primarily for marketing and client reports, due to their ease of being understood.

Yield Curve Decomposition – Duration Based

Fixed-income attribution can be executed either top-down or bottom-up using duration-based yield curve decomposition at the security level. Using the known relationship between duration and yield to maturity (YTM), this method estimates the return of securities, sector buckets, or years-to-maturity buckets, as follows:

$$ \% \text{ Total return} = \% \text{ Income return} + \% \text{ Price return} $$

Where:

$$ \% \text{ Price return} ≈ (–\text{Duration} \times \text{Change in YTM}) $$

The percentage price return of a bond will be approximately equal to the negative of its duration for each 100 bp change in yields. The change in yield to maturity of the portfolio or instrument can be broken down into 1) yield curve factors and 2) spread factors, to provide additional insights. These factors represent the changes in the risk-free government curve and in the premium required to hold riskier sectors and bonds. When they are combined and applied to the duration, we can determine a percentage price change for each factor.

For example, a manager may have a view as to how the yield curve factors will change over time. We can use the attribution analysis to determine the value of the yield curve views as they unfold over time.

This approach is applied to both the portfolio and the benchmark to identify contributions to total return from changes in the yield to maturity. Comparing the differences between the benchmark's return drivers and the portfolio's return drivers gives us the effect of active portfolio management decisions.

This group of models is quite different from the exposure decomposition. One consequence of this difference is that more data points are required to calculate the separate absolute attribution analyses for the portfolio and the benchmark. Thus, the yield decomposition approach exchanges better transparency for more operational complexity. These models are typically used when preparing reports for analysts and portfolio managers, rather than in marketing or client reports.

Yield Curve Decomposition – Full Repricing

A full repricing attribution approach provides more precise pricing and allows for a broader range of instrument types and yield changes. It also supports a greater variety of quantitative modeling beyond fixed-income attribution (e.g., ex ante risk). This approach is better aligned with how portfolio managers typically view the instruments. However, it requires the full capability to reprice all financial instruments in the portfolio and the benchmark, including the rates and the characteristics of the instrument. Its complex nature can make it more difficult and costly to administer operationally and can make the results more difficult to understand, particularly for non-fixed-income professionals.

Instead of estimating price changes from changes in duration and yields to maturity, bonds can be repriced from zero-coupon curves (spot rates). A bond's price is the sum of its cash flows discounted at the appropriate spot rate for each cash flow's maturity. The discount rate to calculate the present value depends on the yields offered on the market for comparable securities and represents the required yield an investor expects for holding that investment. Typically, each cash flow is discounted at a rate from the spot curve that corresponds to the time the cash flow will be received.

As with duration-based approaches, instruments can be repriced following incremental changes in spot rates, regardless of the overall interest rate, spread, or bond-specific factors. Ultimately, this bottom-up repricing can contribute to a security’s return and be aggregated for portfolios, benchmarks, and active management.

Example

The following example comes directly from the CFAI 2022 Level III Curriculum. It is important to note that none of the underlying calculations for the analysis are necessary, only the interpretations.

$$ \scriptsize{\begin{array}{ccccc|cccc}
& & \textbf{Portfolio} & \textbf{Weights} & & & \textbf{Portfolio} & \textbf{Duration} & \\ \hline
& \textbf{Short} & \textbf{Mid} & \textbf{Long} & \textbf{Total} & \textbf{Short} & \textbf{Mid} & \textbf{Long} & \textbf{Total} \\ \hline
\text{Government} & 10.00\% & 10.00\% & 20.00\% & 40.00\% & 4.42 & 7.47 & 10.21 & 8.08 \\
\text{Corporate} & 10.00\% & 20.00\% & 30.00\% & 60.00\% & 4.40 & 7.40 & 10.06 & 8.23 \\
\text{Total} & 20.00\% & 30.00\% & 50.00\% & 100.00\% & 4.41 & 7.42 & 10.12 & 8.17 \\ \hline
& & \textbf{Benchmark} & \textbf{Weights} & & & \textbf{Benchmark} & \textbf{Duration} & \\ \hline
& \textbf{Short} & \textbf{Mid} & \textbf{Long} & \textbf{Total} & \textbf{Short} & \textbf{Mid} & \textbf{Long} & \textbf{Total} \\ \hline
\text{Government} & 20.00\% & 20.00\% & 15.00\% & 55.00\% & 4.42 & 7.47 & 10.21 & 7.11 \\
\text{Corporate} & 15.00\% & 15.00\% & 15.00\% & 45.00\% & 4.40 & 7.40 & 10.06 & 7.29 \\
\text{Total} & 35.00\% & 35.00\% & 30.00\% & 100.00\% & 4.41 & 7.44 & 10.14 & 7.19 \\ \hline
& & \textbf{Portfolio} & \textbf{Weights} & & & \textbf{Portfolio} & \textbf{Return} & \\ \hline
& \textbf{Short} & \textbf{Mid} & \textbf{Long} & \textbf{Total} & \textbf{Short} & \textbf{Mid} & \textbf{Long} & \textbf{Total} \\ \hline
\text{Government} & 10.00\% & 10.00\% & 20.00\% & 40.00\% & –3.48\% & –5.16\% & –4.38\% & –4.35\% \\
\text{Corporate} & 10.00\% & 20.00\% & 30.00\% & 60.00\% & –4.33\% & –6.14\% & –5.42\% & –5.48\% \\
\text{Total} & 20.00\% & 30.00\% & 50.00\% & 100.00\% & –3.91\% & –5.81\% & –5.00\% & –5.03\% \\ \hline
& & \textbf{Benchmark} & \textbf{Weights} & & & \textbf{Benchmark} & \textbf{Return} & \\ \hline
& \textbf{Short} & \textbf{Mid} & \textbf{Long} & \textbf{Total} & \textbf{Short} & \textbf{Mid} & \textbf{Long} & \textbf{Total} \\ \hline
\text{Government} & 20.00\% & 20.00\% & 15.00\% & 55.00\% & –3.48\% & –5.16\% & –4.38\% & –4.34\% \\
\text{Corporate} & 15.00\% & 15.00\% & 15.00\% & 45.00\% & –4.33\% & –6.14\% & –5.86\% & –5.44\% \\
\text{Total} & 35.00\% & 35.00\% & 30.00\% & 100.00\% & –3.84\% & –5.58\% & –5.12\% & –4.83\%
\end{array}} $$

• With a higher duration than the benchmark (8.17 compared with 7.19 for the benchmark), the manager likely expected the rates to fall and took a bullish position on long-term bonds (interest rates) by increasing exposure to the long end of the interest rate curve (e.g., investing 50% of the portfolio in the longest-duration bucket versus 30% for the benchmark).
• Based on the overweight in the corporate sector (60% versus the 45% benchmark weight), the manager likely expected credit spreads to narrow. 2 Notice that this bet increases the 4.94 contribution to the duration of the corporate sector in the portfolio compared with the 3.28 contribution to the duration for the benchmark. This allocation makes the portfolio more exposed to market yield fluctuations in the corporate sector.
• The total portfolio return is –5.03%, relative to a total benchmark return of –4.83%, showing an underperformance of –0.20% over the period.

$$ \scriptsize{\begin{array}{c|c|c|c|c|c|c|c}
\textbf{Duration} & \textbf{Sector} & \textbf{Duration} & \textbf{Curve} & \textbf{Total} & \textbf{Sector} & \textbf{Bond} & \textbf{Total} \\
\textbf{Bucket} & & \textbf{Effect} & \textbf{Effect} & \textbf{Interest rate} & \textbf{Allocation} & \textbf{Selection} & \\
& & & & \textbf{Allocation} & & & \\ \hline
\text{Short} & \text{Government} & & & & & 0.00\% & 0.00\% \\ \hline
& \text{Corporate} & & & & 0.04\% & 0.00\% & 0.04\% \\ \hline
& \text{Total} & 0.40\% & 0.12\% & 0.52\% & 0.04\% & 0.00\% & \bf{0.56\%} \\ \hline
\text{Mid} & \text{Government} & & & & & 0.00\% & 0.00\% \\ \hline
& \text{Corporate} & & & & -0.05\% & 0.00\% & -0.05\% \\ \hline
& \text{Total} & 0.23\% & 0.03\% & 0.26\% & -0.05\% & 0.00\% & \bf{0.21\%} \\ \hline
\text{Long} & \text{Government} & & & & & 0.00\% & 0.00\% \\ \hline
& \text{Corporate} & & & & -0.22\% & 0.13\% & -0.09\% \\ \hline
& \text{Total} & -1.25\% & 0.37\% & -0.88\% & -0.22\% & 0.13\% & \bf{-0.97\%} \\ \hline
\textbf{Total} & & \bf{-0.62\%} & \bf{0.52\%} & \bf{-0.10\%} & \bf{-0.23\%} & \bf{0.13\%} & \bf{-0.20\%}
\end{array}} $$

Using the results from the table above, we can draw the following conclusions about the investment decisions made by this manager:

• The portfolio underperformed its benchmark by 20 bps.
• 62 bps were lost by taking a long-duration position during a period when yields increased (benchmark returns were negative in each duration bucket).
• 52 bps were gained as a result of changes in the shape of the yield curve. Given the manager’s overweighting in the long-duration bucket, we can infer that the yield curve flattened.
• 23 bps were lost because the manager overweighted the corporate sector during a period when credit spreads widened (the benchmark corporate returns in each duration bucket were less than the government returns in those same duration buckets).
• 13 bps were added through bond selection.

$$ \scriptsize{\begin{array}{c|c|c|c|c|c|c|c|c|c}
\textbf{Bond} & \textbf{Yield} & \textbf{Roll} & \textbf{Shift} & \textbf{Slope} & \textbf{Curvature} & \textbf{Spread} & \textbf{Specific} & \textbf{Residual} & \textbf{Total} \\ \hline
\text{Gov’t. 5% June 21} & –0.19\% & –0.04\% & 0.43\% & 0.01\% & 0.15\% & 0.00\% & 0.00\% & –0.01\% & 0.35\% \\ \hline
\text{Gov’t. 7% 30 June 26} & –0.22\% & –0.03\% & 0.71\% & 0.04\% & 0.04\% & 0.00\% & 0.00\% & –0.03\% & 0.52\% \\ \hline
\text{Gov’t. 6% 30 June 31} & 0.12\% & 0.01\% & –0.48\% & 0.05\% & 0.09\% & 0.00\% & 0.00\% & –0.01\% & –0.22\% \\ \hline
\text{Corp. 5% 30 June 21} & –0.11\% & –0.02\% & 0.21\% & 0.05\% & 0.05\% & 0.04\% & 0.02\% & –0.02\% & 0.22\% \\ \hline
\text{Corp. 7% 30 June 26} & 0.12\% & 0.01\% & –0.35\% & –0.02\% & –0.02\% & –0.07\% & 0.00\% & 0.02\% & –0.31\% \\ \hline
\text{Corp. (B) 6% 30 June 31} & –0.39\% & –0.03\% & 1.41\% & –0.26\% & –0.11\% & 0.30\% & 0.00\% & –0.04\% & 0.88\% \\ \hline
\text{Corp. (p) 6% 30 June 31} & 0.78\% & 0.06\% & –2.82\% & 0.52\% & 0.33\% & –0.60\% & 0.15\% & –0.05\% & –1.63\% \\ \hline
\textbf{Total} & \bf{0.11\%} & \bf{–0.04\%} & \bf{–0.89\%} & \bf{0.39\%} & \bf{0.53\%} & \bf{–0.33\%} & \bf{0.17\%} & \bf{–0.14\%} & \\ \end{array}} $$

• Yield: The portfolio overweighted corporate bonds and longer-term maturities relative to the benchmark (from Exhibit 4), which generally offer higher yield than government bonds and short-term maturities. This decision contributed 11 bps to the excess return (from Exhibit 6).
• Roll: The portfolio overweighted longer maturities (from Exhibit 4). Because of the shape of the yield curve, bonds with longer maturities generally sit on a flatter part of the yield curve, where the roll return is limited. The overweighting of the longer maturities reduced the portfolio roll return by 4 bps.
• Shift: The portfolio overall duration of 8.17 is greater than the benchmark duration of 7.19 (from Exhibit 4), which, given the increase in yield of +1%, reduced the portfolio return by 89 bps.
• Slope: The slope flattening caused the long-term yields to increase less than yields on shorter terms to maturity. The overweight at the long end of the curve contributed 39 bps to the excess return.
• Curvature: The reshaping of the yield curve resulted in a larger yield increase at the five-year maturity point. The manager underweighted that part of the yield curve. This decision contributed 53 bps to the excess return.
• Spread: The manager overweighted the corporate sector, which resulted in a 33 bp reduction in return because corporate spreads widened.
• Specific spread: Looking at the bond-specific spreads in Exhibit 6, the corporate 5% 30 June 2021 bond added 2 bps of selection return and the corporate (P) 6% 30 June 2031 bond added 15 bps of selection return. These decisions added a total of 17 bps to active return.
• Residual: A residual of –0.14% is unaccounted for because duration and convexity can only estimate the percentage price variation. It is not an accurate measure of the true price variation. The residual becomes more important during large yield moves, which is the case here, with a +1% yield shift.

Question

A three-year Pakistani government bond has a duration of 3.2. If local yields drop by 88 bps, what will be the resulting price return for the bond?

1. 0.02816%.
2. 2.816%.
3. -0.086%.

Solution

The correct answer is B.

$$
\% \text{ Price return} \approx (–\text{Duration} \times \text{Change in YTM}) $$

$$
-3.2 \times 0.0088 = 0.02816 \text{ or } 2.816\% $$

A and C are incorrect. Based on the formula and calculations, the correct answer is 2.816%.

Performance Measurement: Learning Module 1: Portfolio Performance Evaluation; Los 1(f) Interpret the output from fixed-income attribution analyses