# 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)[1]. 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[2]. 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[3]. 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[4]. 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[5].

## 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.

## Footnotes

1. Feibel B. J., 2003, p.163
2. Kim W. C., Kim J. H., Fabozzi F. J., 2016, p.196
3. Sortino F. A., Satchell S., 2001, p.179
4. Bali T., Atilgan Y., Demirtas O., 2013, p.121
5. Sortino F. A., Satchell S., 2001, p.178

## References

Author: Bartłomiej Olejniczak

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