Semivariance

What Is Semivariance?

Semivariance is a statistical measure of dispersion that assesses the potential downside risk of an investment or portfolio. Unlike traditional risk measures that consider all deviations from the mean, semivariance focuses exclusively on negative fluctuations—those observations that fall below a specified mean or target return. This makes it a critical tool within Portfolio Theory for investors who are primarily concerned with potential losses rather than overall volatility. By isolating only the unfavorable deviations, semivariance offers a more intuitive representation of risk for risk-averse investors, providing valuable insights for risk management and portfolio optimization.

History and Origin

While the concept of downside risk has roots in earlier economic thought, the formal mathematical underpinning for semivariance within a portfolio context can be traced back to Harry Markowitz, the pioneer of Modern Portfolio Theory (MPT). Even as Markowitz introduced variance as the primary measure of risk in MPT, he also acknowledged the value of semivariance. In fact, Markowitz himself argued that semivariance was "the more plausible measure of risk" because investors generally worry about underperformance rather than overperformance. He noted that the primary reasons for its lesser adoption were "cost, convenience, and familiarity" compared to variance.

⁶Later, in the early 1990s, the concept gained renewed prominence with the development of Post-Modern Portfolio Theory (PMPT) by Brian M. Rom and Kathleen W. Ferguson. PMPT specifically sought to address the limitations of MPT by redefining risk to focus on downside deviations, with semivariance being a core component of this new framework. This evolution reflected a growing recognition that investors often perceive upside and downside volatility differently, preferring to mitigate losses while still benefiting from positive returns.

Key Takeaways

• Semivariance quantifies only the downside deviations of an asset's or portfolio's returns from a specified mean or target.
• It offers a more targeted measure of potential loss, making it particularly relevant for risk-averse investors.
• Unlike Standard Deviation and variance, semivariance ignores positive fluctuations, providing a clearer picture of downside risk.
• Semivariance can be used in asset allocation and portfolio construction to minimize the probability of significant losses.

Formula and Calculation

The semivariance formula measures the average of the squared deviations for observations that fall below the mean or a defined target value.

The formula for semivariance is:

$\text{Semivariance} = \frac{1}{n} \sum_{r_t < \text{Target}} (\text{Target} - r_t)^2$

Where:

• (n) = The number of observations where the return ((r_t)) is less than the target.
• (r_t) = The observed return at time (t).
• Target = The specified mean or benchmark return (e.g., the historical average expected return or a desired minimum return).

To calculate semivariance:

1. Determine the mean or desired target return for the dataset.
2. Identify all individual returns that fall below this target.
3. For each of these below-target returns, subtract the return from the target value.
4. Square each of these differences.
5. Sum all the squared differences.
6. Divide the sum by the number of observations that were below the target.

Interpreting the Semivariance

Interpreting semivariance is straightforward: a lower semivariance indicates less downside risk. Since it only considers deviations below a specific threshold (typically the mean or a target return), it provides a more focused view of unfavorable outcomes. For example, if two portfolios have the same overall risk as measured by standard deviation, the one with a lower semivariance would be preferred by an investor concerned primarily with capital preservation, as it implies smaller or fewer negative deviations from the desired return. This measure helps investors differentiate between "good" volatility (upside gains) and "bad" volatility (downside losses). An investor seeking to minimize the impact of negative market movements would prioritize a portfolio with a lower semivariance.

Hypothetical Example

Consider an investor evaluating a stock with the following monthly returns over six months:

• Month 1: +3%
• Month 2: -2%
• Month 3: +1%
• Month 4: -4%
• Month 5: +2%
• Month 6: -1%

First, calculate the average monthly return:
Average Return = (3 - 2 + 1 - 4 + 2 - 1) / 6 = -1 / 6 = -0.1667%

Now, identify returns below the average (-0.1667%):

• Month 2: -2%
• Month 4: -4%
• Month 6: -1%

Calculate the squared differences for these below-average returns:

• For Month 2: (-0.1667% - (-2%))^{2 = (1.8333%)}2 = 0.000336
• For Month 4: (-0.1667% - (-4%))^{2 = (3.8333%)}2 = 0.001469
• For Month 6: (-0.1667% - (-1%))^{2 = (0.8333%)}2 = 0.000069

Sum the squared differences: 0.000336 + 0.001469 + 0.000069 = 0.001874

There are 3 observations below the average.

Calculate semivariance:
Semivariance = 0.001874 / 3 = 0.00062467 or 0.062467%

This semivariance value provides a specific measure of the stock's downside volatility relative to its average return, indicating the magnitude of average squared deviations for only those periods when the stock underperformed. This information is particularly useful for an investor focused on controlling potential losses rather than overall fluctuations.

Practical Applications

Semivariance serves several practical applications in finance and investing, particularly for those prioritizing capital preservation and managing downside risk.

1. Portfolio Construction and Optimization: Investors can use semivariance as the risk input in portfolio optimization models, aiming to construct portfolios that minimize expected losses rather than total volatility. This is especially relevant for pension funds, endowments, or individual investors nearing retirement who have a low tolerance for significant drawdowns. It allows for more precise diversification strategies focused on protecting against negative market movements.
2. Performance Evaluation: Performance measures like the Sortino Ratio directly utilize semivariance (or downside deviation, its square root) in their calculation. Unlike the Sharpe Ratio, which penalizes both upside and downside volatility, the Sortino Ratio specifically rewards managers who generate positive investment performance while minimizing exposure to detrimental volatility. This provides a more accurate assessment for investors with asymmetrical risk preferences.
3. Risk Budgeting: Semivariance can be incorporated into risk budgeting frameworks, allowing institutions and individuals to allocate risk capital specifically to downside exposure across different asset classes or strategies. This helps in understanding and managing the concentrated sources of potential loss within a large portfolio.
4. Tail Risk Analysis: While not solely a tail risk measure, semivariance contributes to understanding the lower end of the return distribution. Researchers have developed concepts like "realised semivariance" which measure downside risk using high-frequency data, demonstrating its utility in assessing negative moves in asset prices. I⁵ts focus on negative deviations makes it a relevant consideration when evaluating assets during periods of market stress or for strategies designed to mitigate extreme negative outcomes, such as those related to hedging. A recent study, for instance, indicated how semivariance emerged as an effective tool in exploring the relationship between crude oil prices and the stock market of the G5 nations amidst geopolitical events, offering insights into portfolio hedging strategies.

⁴## Limitations and Criticisms

While semivariance offers a more intuitive approach to risk for many investors, it is not without its limitations and criticisms.

One notable challenge lies in the subjectivity of the target return. While often set to the mean, an investor could choose any benchmark (e.g., zero return, the risk-free rate, or a specific investment objective). Different target values will yield different semivariance figures, potentially leading to varied portfolio decisions and making direct comparisons between analyses difficult if different targets are used.

³Another criticism points to its computational complexity compared to variance, especially for large datasets. While modern software makes calculations feasible, optimizing portfolios based on semivariance can be more involved than mean-variance optimization, which has well-known, closed-form solutions. This complexity has historically contributed to its lesser adoption in widespread financial practice.

²Furthermore, statistical properties and data assumptions can pose issues. Some research suggests that semivariance, like variance, can produce non-normal estimates even when the underlying mean returns are normally distributed. This raises concerns about its reliability in certain portfolio optimization and risk estimation contexts, particularly if distributions are highly skewed or have "fat tails." I¹t also entirely ignores positive volatility, which, while desirable, can still represent unpredictability and might be a relevant factor for some investment strategies, such as those involving alpha generation.

Semivariance vs. Variance

Semivariance and variance are both measures of dispersion, but they differ fundamentally in what they quantify.

The key distinction lies in the concept of risk itself. While variance treats all volatility as "risk"—meaning both unexpectedly high gains and unexpectedly low losses contribute equally to the measure—semivariance isolates only the undesirable outcomes. For an investor, an unexpectedly high positive return is generally welcomed, not seen as a "risk" to be minimized. Therefore, semivariance provides a more accurate representation of risk for those whose primary concern is avoiding or minimizing losses.

FAQs

Why is semivariance considered a better measure of risk than standard deviation for some investors?

Semivariance is considered a better measure of risk for some investors because it aligns more closely with the intuitive understanding of risk as the potential for loss. Unlike standard deviation, which treats both positive and negative deviations from the mean as equally risky, semivariance focuses exclusively on the deviations that fall below a specified threshold, typically the average return or a target return. This makes it particularly valuable for risk-averse investors who prioritize avoiding losses over maximizing all forms of volatility.

Can semivariance be used for individual stocks or only portfolios?

Semivariance can be calculated and applied to both individual stocks and entire portfolios. For an individual stock, it measures the downside risk associated with that specific asset. When applied to a portfolio, it provides a holistic measure of the aggregated downside risk across all its holdings. This versatility makes it a useful tool for evaluating potential losses at various levels of investment analysis.

What is the Sortino Ratio and how does it relate to semivariance?

The Sortino Ratio is a risk-adjusted performance measure that is directly related to semivariance. It is calculated by dividing the difference between a portfolio's actual return and a chosen target return (or risk-free rate) by its downside deviation (the square root of semivariance). The Sortino Ratio is considered superior to the Sharpe Ratio by some because it only penalizes downside volatility, providing a more accurate assessment of how well a portfolio generates returns for the risk of negative outcomes that an investor truly cares about.

Is semivariance widely used in the financial industry?

While semivariance is recognized as a theoretically superior measure of risk for risk-averse investors and is foundational to Post-Modern Portfolio Theory, its adoption in widespread financial practice is not as prevalent as that of standard deviation or variance. This is partly due to its perceived computational complexity and the long-standing familiarity with traditional mean-variance optimization methods. However, with advancements in computing power and a growing emphasis on managing downside risk, its use is becoming more common, particularly in specialized areas like hedge fund management and alternative investments.

Does semivariance consider black swan events?

Semivariance, being a historical measure, reflects the downside volatility observed in past data. If "black swan" events (rare, unpredictable, and high-impact events) are present in the historical returns data used for its calculation, then their impact on downside risk would be captured. However, semivariance itself does not predict future black swan events or account for events outside the observed historical period. For forward-looking assessment of extreme tail risks, other measures like Value-at-Risk (VaR) or Expected Shortfall (ES), often combined with stress testing, are typically employed.