Our last article focused on standard deviation as a central tool in risk management. Thank you for your positive feedback and the exciting suggestions!
One topic that was particularly frequently requested:
The semi-standard deviation (also downside deviation) measures the dispersion below a reference value (usually the mean or a target return such as 0%). It answers the question: How strongly do the unfavorable (negative) values deviate on average from a target value?
Unlike standard deviation, the semi-standard deviation is a one-sided risk measure. We recall that standard deviation is a two-sided risk measure that considers both positive and negative deviations from the expected value. When we want to measure only the downside potential, semi-variance or semi-standard deviation is appropriate.
STEP 1: DEFINE A REFERENCE VALUE OR THRESHOLD
The semi-standard deviation measures only the dispersion of values below a reference value, usually:
STEP 2: DETERMINE THE NEGATIVE DEVIATIONS FROM THE REFERENCE VALUE
Consider only those observations that fall below the reference value:
For each value , define:
These deviations capture the downside potential.
STEP 3: CALCULATE THE SQUARED NEGATIVE DEVIATIONS FROM THE REFERENCE VALUE
Since the deviations are negative, they are converted to a positive dimension. This is achieved by squaring the negative deviations from the expected value. This gives the negative deviations a positive value.
STEP 4: CALCULATE THE SUM OF THE SQUARED NEGATIVE DEVIATIONS FROM THE REFERENCE VALUE
In the next step, the sum of the squared negative deviations from the expected value is calculated.
STEP 5: CALCULATE THE SEMI-VARIANCE
Now we can calculate the semi-variance.
STEP 6: CALCULATE THE SEMI-STANDARD DEVIATION
To calculate the semi-standard deviation as the average deviation of the negative values from the reference value, the semi-variance must be returned to its original dimension. To do this, we take the square root of the variance.
Companies typically exhibit left-skewed distributions that represent downside potential. This is partly because many risks exist where there is no upside potential (e.g., cybercrime attack, warehouse fire). Furthermore, companies strive to utilize their capacities optimally. Therefore, a short-term increase in output with increased demand is hardly possible. This assessment is reinforced by the shortage of skilled workers. Therefore, risk management should focus more on the analysis of downside measures.
Investors/risk managers fear losses – not gains. The semi-standard deviation focuses only on the downside.
In metrics such as the Sortino Ratio (a variant of the Sharpe Ratio), the semi-standard deviation is used to capture “downside risk.”
Very high positive values can suggest high risk with standard deviation – which is not always rational.
There is no built-in Excel function – but here’s how it works (e.g., for a data series A3:A12):
The semi-standard deviation can also be calculated very simply and compactly using an array formula:
=SQRT(AVERAGE(IF(A3:A12<AVERAGE(A3:A12),(A3:A12-AVERAGE(A3:A12))^2)))
(Note: In older Excel versions, CTRL+SHIFT+ENTER is required)
The semi-standard deviation can also be calculated very simply and compactly using an array formula:
=SQRT
(AVERAGE(IF(A3:A12
<AVERAGE(A3:A12),(A3:A12-AVERAGE(A3:A12))^2)))
(Note: In older Excel versions, CTRL+SHIFT+ENTER is required)
We see that the SEMI-STANDARD DEVIATION measures the AVERAGE NEGATIVE DEVIATION FROM THE EXPECTED VALUE and thus describes the downside potential very well.
We send you our best regards until the next TIPS AND TRICKS FROM EIQF
Dietmar Ernst and Joachim Häcker
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The semi-standard deviation (also downside deviation) measures the dispersion below a reference value (usually the mean or a target return such as 0%).
Many reluctantly recall their statistics lectures: formulas, variance, standard deviation – and the question: What’s the point of it all? However, standard deviation holds a real treasure: it is a central tool in modern risk management.
