Advanced tracking error

What Is Advanced Tracking Error?

Advanced tracking error, often simply referred to as tracking error, is a core concept within portfolio theory that quantifies the divergence between the returns of an investment portfolio and its chosen benchmark index. It is typically measured as the standard deviation of the difference between the portfolio's returns and the benchmark's returns over a specified period. A lower tracking error indicates that a portfolio's performance closely mirrors its benchmark, which is a primary goal for passive investment strategies such as those employed by many exchange-traded funds (ETFs)) and mutual funds. Conversely, a higher tracking error suggests a greater deviation, often implying a more active approach to portfolio management. This metric is crucial for evaluating how effectively a manager replicates a benchmark and for assessing the level of active risk taken.

History and Origin

The concept of tracking error gained prominence with the rise of passive investing and indexing strategies. As investors sought to replicate market performance through index funds, the need arose for a metric to gauge how closely these funds actually tracked their intended benchmarks. While the precise origin of the term is not attributed to a single moment, its widespread adoption in investment management evolved alongside the growth of quantitative analysis in the latter half of the 20th century. The understanding and application of tracking error were further refined with academic research that explored the nuances of active versus passive management. A notable contribution came from Antti Petajisto and K. J. Martijn Cremers, whose 2009 research introduced the concept of "Active Share," complementing tracking error by examining portfolio holdings rather than just returns to characterize fund activeness¹⁴. This work helped to differentiate between various types of active management, such as diversified stock picking versus factor bets, providing a more comprehensive framework for evaluating investment strategies.

Key Takeaways

• Advanced tracking error measures the volatility of a portfolio's returns relative to its benchmark.
• It is a crucial metric for evaluating the effectiveness of passively managed funds in replicating their benchmarks.
• A low tracking error is generally desired for index funds, while active managers might accept or target higher tracking error for potential outperformance.
• Tracking error can be influenced by factors such as management fees, trading costs, and the liquidity of underlying securities.
• Understanding tracking error is essential for assessing a portfolio's risk management and alignment with investment objectives.

Formula and Calculation

The advanced tracking error is calculated as the standard deviation of the difference between the portfolio's return and the benchmark's return over a period. This is often referred to as the ex-post (realized) tracking error, as it uses historical returns.

The formula for tracking error ($\sigma_{TE}$) is:

Where:

• $R_{P,t}$ = Portfolio return at time t
• $R_{B,t}$ = Benchmark return at time t
• $D_t = R_{P,t} - R_{B,t}$ = Daily difference in returns
• $\bar{D}$ = Average difference in returns over the period
• $n$ = Number of observations (e.g., daily periods)

Alternatively, for simplicity, and often in practice, it's expressed as the standard deviation of the series of excess returns (portfolio return minus benchmark return). The formula for the standard deviation of these differences is commonly used.

(\sigma_{TE} = \text{Standard Deviation}(R_P - R_B))¹³

Interpreting the Advanced Tracking Error

Interpreting advanced tracking error depends largely on the investment strategy and objectives of the portfolio. For passively managed funds, such as index funds or ETFs, a low tracking error indicates success in replicating the performance of the benchmark. Investors in these funds expect returns to closely match the index, and a consistently low tracking error confirms this. In such cases, factors like management fees, sampling methods (rather than full replication), and cash holdings can contribute to deviations, increasing the tracking error¹².

For actively managed portfolios, tracking error takes on a different meaning. Active managers aim to generate alpha, or excess returns, above their benchmark. Therefore, a certain level of tracking error is expected and often embraced, as it reflects the manager's deliberate deviations from the benchmark in pursuit of superior performance. However, not all tracking error is "good." Tracking error can arise from both desired active bets (e.g., security selection or sector allocation) and unintended, uncompensated risks (e.g., exposure to unrewarded systematic risk or high idiosyncratic risk). It is also important to consider that tracking error is backward-looking and assumes a specific probability distribution, which may not always hold true for future performance¹¹.

Hypothetical Example

Consider a hypothetical actively managed equity fund, "Growth Opportunities Fund," benchmarked against the "Global Equity Index." Over a specific year, the fund and the index have the following monthly returns:

To calculate the advanced tracking error, first, determine the monthly difference in returns ($D_t = R_{P,t} - R_{B,t}$):

Next, calculate the average difference in returns ($\bar{D}$) for the year:
$\bar{D} = (0.5 + 0.2 + 0.5 - 0.3 - 0.2 - 0.2 + 0.3 + 0.3 + 0.5 - 0.1 + 0.2 + 0.3) / 12 = 2.0 / 12 \approx 0.1667%$

Finally, calculate the standard deviation of these differences (the advanced tracking error):

A tracking error of 0.29% suggests that the fund's monthly returns deviated from the benchmark by about 0.29 percentage points on average. This indicates a moderate level of active management for the fund over this period.

Practical Applications

Advanced tracking error is widely used across various facets of finance, providing insights into portfolio construction, risk assessment, and manager evaluation.

• Index Fund Management: For managers of index funds and ETFs, minimizing tracking error is a primary objective. It signifies how accurately the fund replicates its target benchmark index. Fund managers employ various techniques, such as full replication or sampling, to keep this error low¹⁰.
• Active Portfolio Management: In active management, tracking error is a measure of active risk. Portfolio managers may set limits on their acceptable tracking error to control the extent of their deviation from the benchmark. Higher tracking error can be a result of deliberate investment decisions intended to generate excess return through strategic asset allocation or security selection⁹.
• Performance Attribution: Tracking error is a key component in performance attribution analysis, helping to break down a portfolio's deviation from its benchmark into various sources, such as sector allocation, stock selection, or currency exposure.
• Regulatory Compliance: Some regulatory bodies and client mandates impose constraints on the maximum allowable tracking error for certain types of funds, particularly those marketed as passively managed.
• Tax Efficiency: In some contexts, allowing for a higher tracking error may inadvertently lead to better after-tax returns for index funds. This is because strict benchmark replication can trigger taxable events, such as selling appreciated assets, while a more flexible approach might allow for tax-loss harvesting or deferral strategies⁸.

Limitations and Criticisms

Despite its widespread use, advanced tracking error has several limitations and criticisms that investors and managers should consider. One significant drawback is its backward-looking nature; it is calculated using historical data and thus may not be a reliable predictor of future deviations⁷. Market conditions can change rapidly, and past tracking behavior does not guarantee similar performance going forward.

Another criticism is that tracking error, by itself, does not differentiate between "good" (intentional) and "bad" (unintended) deviations from a benchmark. A high tracking error could result from skilled active management, leading to outperformance, or from poor decisions and uncompensated risks⁶. For instance, it doesn't distinguish between deviations caused by a manager's strong conviction in a particular stock (which might lead to positive alpha) versus those caused by operational inefficiencies or excessive trading costs. Academic research has highlighted that simply minimizing tracking error can be detrimental for skilled active managers, especially when market conditions or index characteristics change⁵.

Furthermore, tracking error focuses purely on the statistical difference in returns and does not provide insight into the underlying holdings or the true "activeness" of a portfolio. A fund could have a low tracking error by closely mirroring its benchmark's holdings but still be considered a "closet indexer" if it charges high active management fees without truly active positions. Conversely, a highly diversified stock-picking fund might have a relatively low tracking error because its diversification dampens the impact of individual stock deviations, even if its holdings are very different from the benchmark⁴.

Advanced Tracking Error vs. Active Share

While both advanced tracking error and active share are measures used to assess the activeness of a portfolio, they capture different dimensions of active management. Tracking error measures the volatility of the difference in returns between a portfolio and its benchmark. It is calculated based on historical price movements and indicates how closely the portfolio's return stream mirrors that of the benchmark. A low tracking error suggests the portfolio is behaving similarly to the benchmark, making it a key metric for evaluating passive strategies³.

In contrast, active share measures the percentage of a portfolio's holdings that differ from its benchmark index. It is calculated by comparing the actual securities held in the portfolio to those in the benchmark. A high active share indicates that the portfolio manager is making significant, conviction-based deviations from the benchmark's holdings. The distinction is crucial because a fund can have a high active share (different holdings) but a relatively low tracking error if its active bets are diversified and don't lead to significant return deviations. Conversely, a fund could have a low active share (similar holdings) but a high tracking error if its minor deviations are highly volatile or concentrated in specific factors. Together, these two metrics provide a more holistic view of a fund's active management style, characterizing it by both its holdings and its return behavior².

FAQs

What causes advanced tracking error?

Advanced tracking error can be caused by various factors, including management fees, trading costs, cash holdings within the portfolio, differing dividend policies between the fund and the index, and rebalancing activities. Additionally, if an index fund uses a sampling strategy (holding a subset of the benchmark's securities) rather than full replication, it can introduce tracking error¹.

Is a high or low advanced tracking error better?

It depends on the investment objective. For a passive investing strategy, such as an index fund, a low tracking error is desirable as it indicates the fund is successfully mimicking its benchmark. For an active management strategy, a higher tracking error is often expected, as it signifies the manager is making deliberate deviations from the benchmark in an attempt to generate superior returns.

Can tracking error be negative?

No, tracking error is a measure of volatility, specifically the standard deviation of return differences, which is always a positive value. While the differences in returns (portfolio return minus benchmark return) can be negative or positive, the standard deviation of these differences, which is the tracking error, is always non-negative.

How is advanced tracking error different from active return?

Advanced tracking error measures the volatility of the difference between portfolio and benchmark returns, indicating the consistency of deviation. Active return, also known as excess return, simply measures the absolute difference between the portfolio's return and the benchmark's return over a period. Active return tells you whether the portfolio outperformed or underperformed, while tracking error tells you how consistently it deviated.