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Downside risk was first modeled by Roy (1952), who assumed that an investor's goal was to minimize his/her risk. This mean-semivariance, or downside risk, model is also known as “safety-first” technique, and only looks at the lower standard deviations of expected returns which are the potential losses.
The ratio is calculated as =, where is the asset or portfolio average realized return, is the target or required rate of return for the investment strategy under consideration (originally called the minimum acceptable return MAR), and is the target semi-deviation (the square root of target semi-variance), termed downside deviation.
Downside risk (DR) is measured by target semi-deviation (the square root of target semivariance) and is termed downside deviation. It is expressed in percentages and therefore allows for rankings in the same way as standard deviation. An intuitive way to view downside risk is the annualized standard deviation of returns below the target.
Roy's ratio is also related to the Sortino ratio, which also uses MAR in the numerator, but uses a different standard deviation (semi/downside deviation) in the denominator. In 1966, William F. Sharpe developed what is now known as the Sharpe ratio. [1]
The comparison of upside to downside risk is necessary because “modern portfolio theory measures risk in terms of standard deviation of asset returns, which treats both positive and negative deviations from expected returns as risk.” [1] In other words, regular beta measures both upside and downside risk.
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In financial mathematics, a deviation risk measure is a function to quantify financial risk (and not necessarily downside risk) in a different method than a general risk measure. Deviation risk measures generalize the concept of standard deviation .
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