Active mean absolute deviation: Meaning, Criticisms & Real-World Uses

What Is Active Mean Absolute Deviation?

Active Mean Absolute Deviation (AMAD) is a metric within the realm of quantitative finance and portfolio management that quantifies the average absolute difference between the returns of an actively managed investment portfolio and its designated benchmark index. Unlike traditional mean absolute deviation which measures dispersion around a dataset's own mean, AMAD specifically focuses on the deviation from a benchmark, making it a key indicator of a portfolio manager's active investment decisions. It provides insight into how much an active portfolio's performance varies from the index it aims to outperform or track, without considering the direction of those deviations. This measure is a component of sophisticated performance measurement used to assess how closely an active strategy adheres to or diverges from its comparative index.

History and Origin

The concept of Mean Absolute Deviation (MAD) as a measure of risk in investment portfolio optimization gained prominence as an alternative to Harry Markowitz's seminal Modern Portfolio Theory (MPT). MPT traditionally used variance or standard deviation to quantify risk, leading to complex quadratic programming problems for portfolio optimization. In 1991, Hiroshi Konno and Hiroaki Yamazaki proposed a linear programming model that utilized mean absolute deviation as the risk measure, significantly simplifying the computational challenge for large-scale portfolios³⁰, ³¹. This development paved the way for more computationally efficient portfolio optimization strategies. While their original work focused on general MAD for portfolio risk, the application of this absolute deviation principle to measure "active" divergence from a benchmark is a natural extension in the evolution of active management analytics, enabling a clearer assessment of active bets by focusing purely on the magnitude of difference.

Key Takeaways

Active Mean Absolute Deviation (AMAD) measures the average absolute difference between an active portfolio's returns and its benchmark's returns.
It highlights the magnitude of divergence stemming from active management decisions, irrespective of whether the deviation is positive or negative.
AMAD offers a straightforward and intuitive way to understand how much an active portfolio is differing from its benchmark.
Lower AMAD generally indicates a closer resemblance to the benchmark, while higher AMAD signifies greater active positioning.
It is a valuable tool for investors to evaluate the active risk taken by fund managers.

Formula and Calculation

The formula for Active Mean Absolute Deviation (AMAD) extends the traditional calculation of mean absolute deviation by applying it to the difference between a portfolio's return and its benchmark's return over a specified period.

The formula is expressed as:

AMAD = \frac{1}{N} \sum_{i=1}^{N} |R_{P,i} - R_{B,i}|

Where:

(AMAD) = Active Mean Absolute Deviation
(N) = The total number of observation periods
(R_{P,i}) = The return of the portfolio in period (i)
(R_{B,i}) = The return of the benchmark in period (i)
(|...|) = Denotes the absolute value, ensuring that all differences are treated as positive magnitudes.

To calculate AMAD, one first determines the difference between the portfolio's return and the benchmark index's return for each observation period. Then, the absolute value of each of these differences is taken. Finally, the sum of these absolute differences is divided by the total number of periods to arrive at the average absolute deviation. This measure focuses purely on the size of the discrepancies, providing a clear figure of how much the portfolio's performance has deviated from the benchmark²⁸, ²⁹.

Interpreting the Active Mean Absolute Deviation

Interpreting Active Mean Absolute Deviation involves understanding that it represents the average magnitude of a portfolio's departure from its benchmark. A higher AMAD indicates that the portfolio's returns have, on average, diverged significantly from the benchmark's returns. This suggests that the fund manager has taken substantial active bets, resulting in a portfolio composition and performance notably different from the index²⁶, ²⁷. Conversely, a low AMAD implies that the portfolio's returns have closely mirrored the benchmark, suggesting a more passive or "closet indexing" approach, even if the fund is marketed as actively managed²⁵.

Investors typically consider AMAD in conjunction with the portfolio's expected return and actual performance. A high AMAD coupled with strong outperformance relative to the benchmark might suggest skillful active asset allocation by the manager. However, a high AMAD with underperformance could indicate that the active bets have not paid off, leading to unfavorable deviations. AMAD is thus a critical metric for evaluating the effectiveness of active strategies and the true extent of active decision-making.

Hypothetical Example

Consider a hypothetical actively managed fund, Fund Alpha, and its benchmark, the Diversification Equity Index, over five monthly periods.

| Month | Fund Alpha Return (%) | Benchmark Index Return (%) | Difference ($R_P - R_B$) (%) | Absolute Difference ($|R_P - R_B|$) (%) |
| :---- | :-------------------- | :------------------------- | :----------------------------- | :------------------------------------ |
| 1 | 2.5 | 2.0 | 0.5 | 0.5 |
| 2 | -1.0 | -1.2 | 0.2 | 0.2 |
| 3 | 3.0 | 2.8 | 0.2 | 0.2 |
| 4 | 0.8 | 1.5 | -0.7 | 0.7 |
| 5 | -2.2 | -1.8 | -0.4 | 0.4 |

To calculate the Active Mean Absolute Deviation for Fund Alpha:

Calculate the difference between Fund Alpha's return and the Benchmark Index return for each month.
Take the absolute value of each difference.
Sum the absolute differences: (0.5 + 0.2 + 0.2 + 0.7 + 0.4 = 2.0).
Divide the sum by the number of periods (N=5): (AMAD = 2.0 / 5 = 0.4).

In this example, Fund Alpha's Active Mean Absolute Deviation is 0.4%. This figure indicates that, on average, Fund Alpha's monthly returns deviated from its benchmark index by 0.4 percentage points, regardless of the direction of the deviation. This illustrates the fund manager's active positioning and how much their decisions led the portfolio to move independently of the broader market represented by the benchmark. Investors can use this to gauge the extent of risk-adjusted return.

Practical Applications

Active Mean Absolute Deviation finds several practical applications within the financial markets and investment analysis. One primary use is in evaluating the true nature of actively managed funds. By quantifying the average absolute deviation from a benchmark index, investors and analysts can discern whether a fund is genuinely taking active positions or merely closely replicating its index while charging higher active management fees. This is particularly relevant in the context of discussions around "closet indexing"²⁴.

Furthermore, AMAD can serve as a component in risk measurement and portfolio construction. While standard deviation is more common, the mean absolute deviation's robustness to outliers can be beneficial in certain analyses²², ²³. It provides a clear, intuitive measure of dispersion for those concerned with the typical magnitude of deviations, rather than their squared values. Fund managers may also use AMAD internally as part of their performance measurement toolkit to monitor the effectiveness of their active strategies and to ensure their portfolios are deviating in a manner consistent with their investment mandates. The S&P Indices Versus Active (SPIVA) scorecards, for instance, frequently highlight the challenges faced by active managers in outperforming their benchmarks over various time horizons, reinforcing the importance of metrics like AMAD in assessing active fund efficacy²¹.

Limitations and Criticisms

While Active Mean Absolute Deviation offers a clear and intuitive measure of divergence from a benchmark, it has certain limitations and criticisms. A primary critique is its mathematical tractability compared to standard deviation. Unlike standard deviation, which relies on squared differences, AMAD uses absolute values, making it less amenable to certain advanced statistical analyses and optimizations that require differentiable functions¹⁹, ²⁰. This can limit its use in complex portfolio optimization models, where variance-based measures are often preferred due to their mathematical properties.

Another limitation is that, like its traditional counterpart, AMAD does not differentiate between positive and negative deviations. Both a positive outperformance and a negative underperformance of the same magnitude contribute equally to the AMAD value. For investors primarily concerned with downside risk, AMAD alone may not provide sufficient insight without being coupled with other metrics that specifically address negative deviations. Although AMAD is less sensitive to extreme outliers than standard deviation¹⁸, its infrequent use in broader statistical contexts compared to variance and standard deviation can lead to less familiarity and potentially misinterpretation for those not accustomed to it¹⁶, ¹⁷.

Active Mean Absolute Deviation vs. Tracking Error

Active Mean Absolute Deviation (AMAD) and Tracking Error are both measures used to quantify how much an actively managed portfolio deviates from its benchmark index. While their purpose is similar, their calculation and underlying statistical properties differ.

Feature	Active Mean Absolute Deviation (AMAD)	Tracking Error
Calculation Basis	Uses the average of the absolute differences between portfolio and benchmark returns.	Typically calculated as the annualized standard deviation of the differences between portfolio and benchmark returns.¹³, ¹⁴, ¹⁵
Sensitivity to Outliers	Less sensitive to extreme outliers because it uses absolute differences rather than squared differences.¹⁰, ¹¹, ¹²	More sensitive to extreme outliers due to the squaring of deviations in its calculation.⁹
Mathematical Properties	Less mathematically tractable for certain advanced statistical models and optimizations.⁸	More mathematically tractable, making it more widely used in quantitative models, especially those involving quadratic optimization.⁷
Interpretation	Provides a straightforward average magnitude of deviation.	Measures the volatility of the active return (the excess return over the benchmark), often referred to as active risk.⁵, ⁶ A higher tracking error indicates greater volatility of excess returns.⁴
Common Usage	Less commonly used than tracking error in mainstream portfolio analysis, but valued for its intuitive nature and robustness to outliers.	A widely adopted and traditional measure of active risk in the investment industry, used by fund managers, institutional investors, and consultants.³

The primary distinction lies in how each measure handles the magnitude of deviations. AMAD treats all differences linearly, focusing on the average absolute spread. Tracking Error, by squaring deviations, gives greater weight to larger deviations, reflecting the impact of more extreme differences in returns².

FAQs

What does a high Active Mean Absolute Deviation indicate?

A high Active Mean Absolute Deviation indicates that an investment portfolio's returns have, on average, significantly diverged from its benchmark index's returns. This suggests that the fund manager is taking substantial active positions and that the portfolio's performance is notably different from the index it aims to outperform or track.

Is Active Mean Absolute Deviation a measure of risk?

Yes, Active Mean Absolute Deviation can be considered a measure of active risk. It quantifies the degree of dispersion or variability of an actively managed portfolio's returns relative to its benchmark. However, it specifically measures the magnitude of deviation, not the direction, making it a measure of exposure to differing from the benchmark. Other measures, like standard deviation of active returns (which is tracking error), are also commonly used for active risk.

How does Active Mean Absolute Deviation differ from standard deviation?

The fundamental difference is how they treat deviations. Active Mean Absolute Deviation sums the absolute values of deviations from a benchmark and then averages them. In contrast, standard deviation squares the deviations before summing them, then takes the square root of the average. This squaring process means standard deviation gives disproportionately more weight to larger deviations, making it more sensitive to outliers than AMAD¹.

Why might an investor use Active Mean Absolute Deviation?

An investor might use Active Mean Absolute Deviation because it provides an intuitive and straightforward understanding of the average absolute difference between a portfolio and its benchmark. Its robustness to outliers can also be appealing in certain contexts where extreme data points might otherwise disproportionately influence other risk measurement metrics. It helps in assessing the true extent of active decision-making by a fund manager.