Downside deviation

Downside deviation is a downside risk calculation that aim on a returns which are below minimal barrier (MAR). Standard deviation uses all the deviations in average (positive and negative). While in risk management they can be interpreted differently. In case of investments with the same return, but different downside deviation, the one with normal downside deviation will be better in compare to the one with high deviation.

Model

To calculate downside deviation we need to take sum of the (returns - MAR) divided by total number of time periods. Final result is square rooted. An example formula can be: $DD={\sqrt {\frac {\sum _{t=1}^{n}min(0,r_{t}-MAR)}{n}}}$ where:

• DD - downside deviation
• MAR - minimum acceptable return
• n - total number of periods
• $r_{t}$ - return in period

Measure usage

The downside deviation is used in other risk-adjusted performance measure like Sortino Ratio. The Sortino ratio allows comparison of investments with different levels of downside risk. It focuses on normalized risk-adjusted return among programs. As in most ratios bigger means better. An example formula can be:

$SR={\frac {RP-TR}{DD}}$ where:

• SR - sortino ratio
• RP - portfolio return
• TR - target return
• DD - downside deviation

Comparison of downside deviation and standard deviation

There are two main differences between downside and standard deviation. Standard deviation uses in average positive and negative deviations while downside uses deviations below the reference point. Second difference is that downside deviation can be focused on a specific objective like minimal required return to avoid bankruptcy.

Assumptions

Downside deviation measure helps investor to asses risk for their investment. Can be used to define lowest possible investment thresholds. As all deviations based on historical data it does not guarantee success in the future.

Examples of Downside deviation

• Value-at-Risk (VaR): Value-at-Risk is a measure of market risk that measures the maximum potential loss on an investment over a specified period of time. It is calculated by taking the average of the worst possible outcomes, such as the lowest returns, over a period of time.
• Expected Shortfall (ES): Expected Shortfall is a measure of downside risk that quantifies the average amount of losses an investment may experience in case of extreme market movements. It is calculated by taking the average of the worst possible outcomes, such as the lowest returns, over a period of time.
• Downside Deviation: Downside Deviation is a measure of risk that measures the variability of an investment’s returns below a certain threshold, such as the minimum acceptable return (MAR). It is calculated by taking the average of the deviation from the MAR over a period of time.
• Tail Risk: Tail Risk is a measure of risk that measures the probability of extreme market movements, such as a large drop in the stock market or a sudden shift in the exchange rate. It is calculated by taking the average of the worst possible outcomes, such as the lowest returns, over a period of time.

Advantages of Downside deviation

A key advantage of downside deviation is that it provides an accurate risk calculation for returns below the minimal acceptable return (MAR) rate. This allows investors to better understand the potential risks associated with an investment, as well as the potential returns. Other advantages of downside deviation include:

• It helps to identify investments that may be too risky for an investor's portfolio.
• It allows for better diversification of investments, as it takes into account the downside risk associated with each individual position.
• It helps investors to identify potential opportunities that may have been overlooked as a result of traditional risk management practices.
• It allows investors to better understand the risk/reward profile of an investment and make more informed decisions.

Limitations of Downside deviation

The limitations of Downside Deviation include:

• Not accounting for upper outliers: Downside Deviation only considers returns that are below a certain minimum acceptable return (MAR) and does not account for any returns that are above this MAR.
• Not accounting for extreme events: Downside Deviation assumes that all returns are normally distributed, and so does not account for any extreme events that may occur.
• Difficult to compare to other strategies: Downside Deviation is a relative measure, which means that it is difficult to compare the performance of one portfolio to another with this measure.
• Not accounting for risk-free rate of return: Downside Deviation does not account for the risk-free rate of return, which can lead to overly conservative estimates of risk.
• Limited use for long-term investments: Downside Deviation is most useful for short-term investments, as it does not account for long-term trends or market forces that may affect returns over longer time periods.

Other approaches related to Downside deviation

• Semi-deviation: Semi-deviation is a measure of volatility which only takes into account negative returns, ignoring the positive returns. It is used to measure the downside volatility of a portfolio and can be used as a risk measurement.
• Value-at-Risk (VaR): VaR is a measure of portfolio risk which estimates the potential loss of an investment over a given horizon within a given confidence level. It is calculated by taking into account the probability of a market event that could affect the portfolio's value.
• Conditional Value-at-Risk (CVaR): CVaR is an extension of VaR and measures the expected loss of an investment given a certain event has occurred. It is calculated by taking into account both the probability and the magnitude of the event.
• Expected Shortfall (ES): ES is a risk measure that takes into account both the probability and the magnitude of potential losses. It is calculated by taking into account all potential losses and their likelihood of occurring.

In summary, downside deviation is one of several approaches for measuring risk and downside volatility of investments. Other approaches include semi-deviation, Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), and Expected Shortfall (ES), each of which measure the potential losses of an investment in different ways.

Footnotes

Downside deviation — recommended articles

Vomma — Calmar Ratio — Zomma — Risk measures — Capped Index — Implementation Shortfall — Capital market theories — Market Risk Premium — Look-Ahead Bias

References

• Bali T., Atilgan Y., Demirtas O., (2013) Investing in Hedge Funds: A Guide to Measuring Risk and Return Characteristics Elsevier Science, 2013
• Feibel B. J., (2003). Investment Performance Measurement. Wiley, 2003
• Kim W. C., Kim J. H., Fabozzi F. J., (2016) Robust Equity Portfolio Management: Formulations, Implementations, and Properties using MATLAB Wiley, 2016
• Melin M. H., (2001). High-Performance Managed Futures: The New Way to Diversify Your Portfolio. Wiley, 2010
• Rachev S. T., Stoyanov S. V., Fabozzi F. J., (2011). A Probability Metrics Approach to Financial Risk Measures. Wiley Blackwell, 2011
• Sortino F. A., (2001). The Sortino Framework for Constructing Portfolios: Focusing on Desired Target Return™ to Optimize Upside Potential Relative to Downside Risk. Elsevier Science, 2009
• Sortino F. A., Satchell S., (2001). Managing Downside Risk in Financial Markets. Butterworth-Heinemann, 2001

Author: Bartłomiej Olejniczak

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