Downside deviation

What Is Downside Deviation?

Downside deviation is a measure of investment risk that focuses specifically on the negative fluctuations of an asset's returns. Unlike traditional measures of volatility that treat both positive and negative movements as equally risky, downside deviation quantifies only the dispersion of returns below a specified target, often a minimum acceptable return (MAR) or zero. This makes downside deviation particularly relevant within the field of risk management and portfolio theory, as investors are generally more concerned with losses than with unexpected gains. By isolating the "bad" volatility, downside deviation provides a more intuitive picture of an investment's potential for undesirable outcomes. It is a critical component in calculating the Sortino ratio, a popular metric for evaluating risk-adjusted return.

History and Origin

The concept of distinguishing between upside and downside volatility has roots in early investment theory. Even Harry Markowitz, the Nobel laureate and father of Modern Portfolio Theory (MPT), recognized in 1959 that investors were primarily concerned with downside variability, proposing a measure called semivariance. However, the computational complexity of semivariance at the time led MPT to largely rely on standard deviation, which treats all deviations from the mean symmetrically²²,²¹.

Downside deviation gained prominence with the development of Post-Modern Portfolio Theory (PMPT) in the early 1990s by Brian Rom and Kathleen Ferguson. PMPT aimed to address the perceived flaws of MPT by incorporating investor preferences and focusing explicitly on downside risk. Frank Sortino further popularized the use of downside deviation as a crucial component of the Sortino ratio, which was designed to offer an improved measure of risk-adjusted performance by penalizing only returns below a chosen target,²⁰.

Key Takeaways

• Downside deviation measures the volatility of an investment's returns that fall below a specific threshold, typically zero or a minimum acceptable return.
• It differentiates "bad" volatility (losses) from "good" volatility (gains), aligning more closely with investor concerns about capital loss.
• This metric is a core component of the Sortino ratio, providing a risk-adjusted return measure focused solely on downside risk.
• Downside deviation offers a more nuanced view of an investment's risk profile compared to standard deviation, especially for non-normally distributed returns.
• It is used in investment performance evaluation, asset allocation strategies, and regulatory disclosures to help investors understand potential losses.

Formula and Calculation

Downside deviation is calculated similarly to standard deviation, but only includes returns that fall below a predefined target return (T) or the average return.

The formula for downside deviation ((\sigma_D)) is:

Where:

• (R_t) = The return for period (t)
• (T) = The target return (e.g., zero, risk-free rate, or minimum acceptable return)
• (N) = The number of periods where (R_t < T) (some definitions use total number of periods, n)
• (\sum_{t=1}^{{n} (R_t - T)}2 \quad \text{for all } R_t < T) = The sum of the squared differences between the actual return and the target return, only for those periods where the actual return is below the target.

It is crucial to note that some variations of the formula use the total number of observations ((N_{total})) in the denominator instead of just the number of downside observations ((N)). The interpretation of the resulting downside deviation can vary based on this choice¹⁹,¹⁸. Typically, for the Sortino ratio, the sum is divided by the total number of observations, (n), not just the number of observations below the target.

Interpreting the Downside Deviation

Interpreting downside deviation involves understanding that a lower number indicates less dispersion of negative investment returns below the chosen target. This means the investment has historically experienced smaller or less frequent losses relative to that target. Conversely, a higher downside deviation suggests greater and more frequent negative deviations.

For instance, two investments might have similar overall volatility as measured by standard deviation, but vastly different downside deviations. An investment with consistent positive returns and occasional large positive spikes could have a high standard deviation but a very low downside deviation (if the spikes increase overall volatility without going below the target). An investment with modest positive returns and occasional significant drops would show a higher downside deviation, which is often what investors are most concerned about. Investors use this metric to gauge the potential for losses and to align their risk tolerance with an investment's historical behavior.

Hypothetical Example

Consider two hypothetical portfolios, Portfolio A and Portfolio B, over five months, with a target return of 0% (meaning we are only concerned with negative returns).

To calculate the downside deviation for Portfolio A:

1. Identify returns below 0%: -3% and -5%.
2. Calculate the squared difference from 0% for these returns:
   • ((-3 - 0)^2 = 9)
   • ((-5 - 0)^2 = 25)
3. Sum these squared differences: (9 + 25 = 34).
4. Divide by (N-1) (where (N) is the total number of observations, 5) if using the total observations in the denominator for the Sortino calculation, or by (N-1) for (N) being downside observations (2-1=1) which is less common. For consistency with Sortino, we typically divide by total periods. So, (34 / (5-1) = 34 / 4 = 8.5).
5. Take the square root: (\sqrt{8.5} \approx 2.92).

For Portfolio B:

1. Identify returns below 0%: -1% and -2%.
2. Calculate the squared difference from 0% for these returns:
   • ((-1 - 0)^2 = 1)
   • ((-2 - 0)^2 = 4)
3. Sum these squared differences: (1 + 4 = 5).
4. Divide by (N-1) ((5-1 = 4)): (5 / 4 = 1.25).
5. Take the square root: (\sqrt{1.25} \approx 1.12).

In this example, Portfolio A has a downside deviation of approximately 2.92%, while Portfolio B has a downside deviation of approximately 1.12%. This indicates that Portfolio B has experienced significantly less "bad" volatility below the 0% target compared to Portfolio A, making it potentially more appealing for investors focused on capital preservation.

Practical Applications

Downside deviation is widely applied across various aspects of finance, offering a more investor-centric view of risk compared to traditional measures.

• Portfolio Management and Construction: Portfolio managers use downside deviation to construct portfolios that explicitly manage exposure to negative returns. It helps in optimizing asset allocation for investors with specific risk tolerance levels, particularly those who are more risk-averse or have short-term investment horizons¹⁷. Strategies that aim to reduce downside volatility while retaining upside potential are often evaluated using this metric¹⁶.
• Performance Measurement: Beyond the Sortino ratio, downside deviation can be integrated into other performance metrics or used independently to assess how well an investment or manager performs during market downturns. Investment research firms like Morningstar incorporate downside deviation into their proprietary risk ratings and also report downside capture ratios, which measure how much of a benchmark's negative performance an investment has captured¹⁵,¹⁴. A lower downside capture ratio indicates better performance in down markets¹³.
• Risk Disclosure and Compliance: Regulatory bodies, such as the Securities and Exchange Commission (SEC), emphasize clear and tailored risk disclosures for investment products, especially mutual funds¹²,¹¹. While not explicitly mandating downside deviation, the focus on understanding potential adverse effects encourages funds to consider various measures that highlight downside risk¹⁰. Financial institutions like the Financial Industry Regulatory Authority (FINRA) also provide guidance and resources on risk assessment and disclosure to protect investors⁹,⁸.
• Hedge Fund and Alternative Investment Analysis: Downside deviation is particularly useful for evaluating alternative investments, such as hedge funds, whose return distributions may be skewed or non-normal. For these investments, standard deviation might not accurately represent the true risk, making downside deviation a more appropriate tool for assessing potential losses⁷.

Limitations and Criticisms

Despite its advantages in providing a more intuitive measure of "bad" risk, downside deviation has several limitations and criticisms.

One key criticism is that downside deviation only considers returns below a certain threshold and provides no information about upside potential. An investment could have a low downside deviation because it consistently generates modest positive returns but never experiences large gains, while another might have a higher downside deviation due to significant positive swings that also introduce more volatility overall. For long-term investors or those seeking growth, ignoring upside volatility might provide an incomplete picture of total investment performance⁶.

The choice of the target return (T) can also be subjective, which can significantly influence the calculated downside deviation and, consequently, metrics like the Sortino ratio⁵,⁴. Different investors or analysts might select different target returns based on their risk tolerance or investment objectives, leading to varied interpretations of the same investment's risk.

Furthermore, like all historical risk measures, downside deviation relies on past data, which may not be indicative of future performance. Financial markets are dynamic, and future market conditions or unforeseen events can lead to different risk profiles than those observed historically³. While it offers a valuable perspective, it should not be the sole basis for investment decisions, and practitioners should consider it alongside other qualitative and quantitative assessments.

Downside Deviation vs. Standard Deviation

Downside deviation and standard deviation are both measures of volatility, but they differ fundamentally in what they quantify, making them suitable for different analytical purposes.

The key point of confusion often arises because standard deviation includes all deviations from the mean, even those that are positive and beneficial to investors. For example, a mutual fund that experiences large positive returns alongside stable, slightly positive returns would have a higher standard deviation, which might incorrectly suggest it's "riskier" in a way that investors desire. Downside deviation addresses this by penalizing only the undesirable movements, providing a more focused measure for investors primarily concerned with minimizing losses.

FAQs

Q: Why is downside deviation considered a better risk measure than standard deviation by some?

A: Downside deviation is considered better by some because it aligns more closely with human behavior; investors are generally more concerned about losing money than they are about the volatility of gains². By focusing only on negative deviations below a chosen target, it provides a clearer picture of an investment's potential for undesirable outcomes, which standard deviation does not differentiate.

Q: What is a "minimum acceptable return" (MAR)?

A: The minimum acceptable return (MAR) is a specific rate of return that an investor or fund manager considers the absolute lowest acceptable performance. Returns below this MAR are considered a "loss" or undesirable outcome for the purpose of calculating downside deviation. It can be set at 0%, the risk-free rate, or any other target an investor defines based on their risk tolerance¹.

Q: Can downside deviation be used for individual stocks?

A: Yes, downside deviation can be calculated and applied to individual stocks. However, its insights are often most valuable when evaluating portfolios or funds where the objective is to assess the collective downside risk and its impact on overall diversification and investment performance. For individual stocks, while it shows downside volatility, other metrics like beta might be more common for understanding market sensitivity.