Absolute variance drag: Meaning, Criticisms & Real-World Uses

What Is Absolute Variance Drag?

Absolute variance drag, a concept central to portfolio theory, quantifies the reduction in compounded returns of an investment or portfolio over time due to its volatility. It highlights the critical distinction between arithmetic average returns and geometric average returns. While the arithmetic mean represents the simple average of periodic returns, the geometric mean reflects the actual compounding growth of an investment over multiple periods. Absolute variance drag is the mathematical difference that arises because high volatility causes the geometric mean return to be lower than the arithmetic mean return, illustrating the "cost" of fluctuating returns on long-term wealth accumulation.

History and Origin

The concept of how volatility impacts compounded returns has been implicitly understood for a long time within financial mathematics. However, its formal articulation and widespread recognition in modern investment theory are often attributed to prominent financial economists, notably Roger Ibbotson. His foundational work, particularly in the "Stocks, Bonds, Bills, and Inflation (SBBI) Yearbook" series, extensively analyzed historical capital market returns and brought to light the practical implications of calculating average returns. Researchers have continued to explore the relationship between mean returns, volatility, and terminal wealth, demonstrating that while the mean of multi-period terminal wealth might be a compounding of the one-period mean, the median wealth is reduced by variance.⁵ This area of study underscores why the geometric mean is often considered a more accurate measure of actual wealth accumulation over long investment horizons compared to the arithmetic mean.

Key Takeaways

Absolute variance drag represents the inherent cost of volatility on compounded investment returns.
It is the difference between an investment's arithmetic mean return and its geometric mean return.
Higher volatility leads to a greater absolute variance drag, meaning a larger gap between simple average returns and actual compounded returns.
Understanding this drag is crucial for investors assessing long-term portfolio performance and setting realistic expected return forecasts.
The drag emphasizes that simply averaging returns (arithmetic mean) can overstate the true growth of capital over time, especially in volatile markets.

Formula and Calculation

Absolute variance drag is inherently linked to the mathematical relationship between the arithmetic mean and geometric mean of a series of returns. The difference between these two averages for a given set of returns is directly proportional to the variance of those returns.

The arithmetic mean return (R_A) for a series of (n) periodic returns (r_1, r_2, ..., r_n) is calculated as:
$R_A = \frac{1}{n} \sum_{i=1}^{n} r_i$

The geometric mean return (R_G) for the same series of returns is calculated as:
$R_G = \left[ \prod_{i=1}^{n} (1 + r_i) \right]^{\frac{1}{n}} - 1$

The approximate relationship between the geometric mean and arithmetic mean, especially for small returns, is given by:
$R_G \approx R_A - \frac{\sigma^2}{2}$
Where (\sigma^2) represents the variance of the periodic returns.

Therefore, the Absolute Variance Drag can be approximated as:
$\text{Absolute Variance Drag} \approx R_A - R_G \approx \frac{\sigma^2}{2}$
This approximation highlights that the magnitude of the drag is roughly half of the variance of the returns. A higher variance implies greater volatility and, consequently, a larger absolute variance drag.

Interpreting the Absolute Variance Drag

Interpreting absolute variance drag involves recognizing that it quantifies the "tax" that volatility imposes on wealth accumulation. A positive absolute variance drag indicates that the volatility of returns has reduced the actual compounded return (geometric mean) below the simple average return (arithmetic mean). For instance, if an investment has an arithmetic mean return of 10% and a geometric mean return of 8%, the absolute variance drag is 2%. This 2% difference represents the annual drag on the investment's compounding growth due to its fluctuating returns.

This concept is particularly relevant when evaluating investment horizons. Over longer periods, the impact of absolute variance drag becomes more pronounced because the effects of compounding are amplified. Investors and analysts use this understanding to distinguish between paper returns (arithmetic) and realized wealth accumulation (geometric). When comparing different investment strategies or asset allocation choices, a strategy with lower volatility, even if it has a similar arithmetic mean, may deliver higher actual compounded returns due to less absolute variance drag.

Hypothetical Example

Consider two hypothetical portfolios, Portfolio A and Portfolio B, both starting with an initial investment of $10,000 over three years.

Portfolio A (High Volatility):

Year 1: +50% return
Year 2: -30% return
Year 3: +40% return

Portfolio B (Low Volatility):

Year 1: +15% return
Year 2: +10% return
Year 3: +12% return

Let's calculate the arithmetic and geometric mean returns for each:

Portfolio A:

Arithmetic Mean Return: ((0.50 - 0.30 + 0.40) / 3 = 0.60 / 3 = 0.20) or 20%
Geometric Mean Return: (((1 + 0.50) \times (1 - 0.30) \times (1 + 0.40))^{1/3} - 1)
- ((1.50 \times 0.70 \times 1.40)^{{1/3} - 1 = (1.47)}{1/3} - 1 \approx 1.137 - 1 = 0.137) or 13.7%
Absolute Variance Drag for Portfolio A: (20% - 13.7% = 6.3%)

Portfolio B:

Arithmetic Mean Return: ((0.15 + 0.10 + 0.12) / 3 = 0.37 / 3 \approx 0.123) or 12.3%
Geometric Mean Return: (((1 + 0.15) \times (1 + 0.10) \times (1 + 0.12))^{1/3} - 1)
- ((1.15 \times 1.10 \times 1.12)^{{1/3} - 1 = (1.4156)}{1/3} - 1 \approx 1.123 - 1 = 0.123) or 12.3%
Absolute Variance Drag for Portfolio B: (12.3% - 12.3% = 0%) (or very close to zero due to rounding)

Even though Portfolio A had higher individual yearly returns (50% and 40%) compared to Portfolio B, its significant volatility resulted in a much higher absolute variance drag. Portfolio B, with its consistent, lower volatility returns, achieved a geometric mean return equal to its arithmetic mean (effectively no drag), leading to smoother and more predictable return on investment. This example clearly illustrates how volatility negatively impacts actual compounded wealth.

Practical Applications

Absolute variance drag has several practical applications across various facets of finance:

Investment Performance Measurement: When evaluating portfolio performance, particularly for long-term investments like retirement funds or endowments, the geometric mean return is a more accurate representation of the actual wealth growth than the arithmetic mean. The absolute variance drag quantifies the degree to which volatility has eaten into those compounded returns, providing a clearer picture of real investment success.
Asset Allocation Decisions: Understanding variance drag informs asset allocation strategies. Assets with high individual volatility, even if they have attractive arithmetic average returns, may contribute significantly to overall portfolio variance drag. This can lead investors to favor diversification and assets with lower correlation to reduce overall portfolio volatility and, consequently, minimize drag.
Risk Management: Absolute variance drag is a direct consequence of risk management considerations. Portfolios designed to mitigate volatility, for example, through hedging or investing in less volatile assets, aim to reduce this drag, thereby enhancing long-term compounded returns.
Financial Planning: For financial planners, communicating the effect of volatility on long-term wealth is crucial. Explaining absolute variance drag helps clients understand why achieving a certain arithmetic return might not translate into the expected end-wealth, especially if the investment path was highly volatile. This insight helps set realistic expectations for future wealth accumulation. As discussed by Firstlinks, the gap between arithmetic and geometric means "is what volatility costs the investor."⁴
Benchmarking and Fund Analysis: When comparing the performance of different mutual funds or index funds, analysts often rely on both arithmetic and geometric returns. Funds with lower absolute variance drag, implying more consistent returns for a given average, can be seen as more efficient in translating raw returns into actual compounded growth.

Limitations and Criticisms

While absolute variance drag provides valuable insight into the impact of volatility on compounded returns, it's essential to acknowledge its limitations and potential criticisms:

Reliance on Historical Data: The calculation of absolute variance drag is based on historical returns and volatility. Past performance is not indicative of future results, and future volatility may differ significantly from historical patterns. This means that the calculated drag for a past period may not accurately predict the drag in a future period.
Approximation for Smaller Returns: The common approximation that absolute variance drag is roughly half the variance of returns holds best for small or moderate returns. For extremely large positive or negative periodic returns, this approximation may become less accurate, and the exact calculation of the geometric mean alongside the arithmetic mean is necessary for precise measurement.
Not a Direct Investment Strategy: Absolute variance drag is an analytical concept, not a direct investment strategy. Investors cannot directly "reduce" variance drag without addressing the underlying volatility of their investment vehicles. Strategies aimed at reducing volatility, such as diversification or tactical asset allocation, are indirect methods of mitigating the drag.
Misinterpretation of "Loss": While the term "drag" implies a loss, it is not a direct financial fee or expense. Instead, it represents the mathematical difference between two ways of averaging returns, reflecting the impact of sequential returns on the total capital. As one academic paper notes, "median wealth is affected by volatility drag but mean wealth is not affected by volatility drag."³ This highlights that the drag primarily impacts the typical investor's experience, which aligns more with median outcomes, rather than the theoretical average of all possible paths.
Contextual Relevance: For very short investment horizons or for analyses focused on single-period expected returns, the arithmetic mean might be more relevant, and thus the concept of variance drag might seem less critical. However, for any multi-period investment, particularly those involving reinvestment of returns, the drag becomes highly significant.

Absolute Variance Drag vs. Geometric Mean Return

Absolute variance drag and geometric mean return are closely related but represent different aspects of investment performance.

Absolute Variance Drag is the difference between the arithmetic mean return and the geometric mean return. It quantifies the amount by which volatility reduces the effective compounded return over time. It is a measure of the "cost" of volatility on long-term wealth. A higher drag indicates that the investment's volatile nature has significantly eroded its compounding potential, even if its simple average returns appear high.

The Geometric Mean Return, also known as the time-weighted rate of return or compounded annual growth rate (CAGR), is the actual average rate at which an investment grows over multiple periods, assuming returns are reinvested. It inherently accounts for the effects of compounding and the sequence of returns. The geometric mean is considered the most accurate measure of historical return on investment for evaluating how a single dollar would have grown over time.²

The confusion often arises because absolute variance drag is directly derived from the geometric mean's relationship with the arithmetic mean. The geometric mean is the result after volatility has imposed its drag. Therefore, understanding absolute variance drag helps to explain why the geometric mean is typically lower than the arithmetic mean and by how much that difference is attributable to volatility. Without volatility, or with perfectly stable returns, the geometric mean and arithmetic mean would be identical, and the absolute variance drag would be zero.

FAQs

Why is Absolute Variance Drag important for long-term investors?

Absolute variance drag is crucial for long-term investors because it quantifies how volatility erodes the actual compounded growth of their investments over extended periods. While the arithmetic mean might show a high average return, the absolute variance drag reveals the true, lower geometric mean that represents what the investor actually earns after accounting for up and down market swings. For example, the Bogleheads community often emphasizes long-term, low-cost investing, where understanding the impact of volatility on compounded returns is paramount.¹

Can Absolute Variance Drag be negative?

No, absolute variance drag, by definition, is typically zero or positive. The geometric mean return will always be less than or equal to the arithmetic mean return for any series of returns with volatility (unless all returns are identical, in which case both means are equal, and the drag is zero). The greater the volatility, the larger the positive drag.

How does diversification affect Absolute Variance Drag?

Diversification can help reduce a portfolio's overall volatility. By combining assets that do not move in perfect sync, diversification can smooth out portfolio returns, leading to a lower overall portfolio variance. A reduction in portfolio variance directly translates to a lower absolute variance drag, allowing the compounded returns (geometric mean) to be closer to the arithmetic mean, thus enhancing long-term wealth accumulation.