**The Semi-Standard Deviation (SD) measures the measure of dispersion of those observations that are less than the expected value of the variable. The objective is to control the results that by default are lower than the expected value. **

In other words, the SD looks for the worst cases (situations where the observations are below the average) and we can build risk indicators, in English, *downside risk metrics* .

If we transfer the SD to the stock prices, the returns below the expected value are considered negative and the returns above the expected value are considered positive for our investment. We are more interested in controlling negative returns as they hurt our earnings.

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### Mathematically

We define the variable Z as a discrete random variable formed by Z _{1} , …, Z _{N} observations. We can define the **SD** as:

Where *E (Z)* is the expected value (average value) of the variable Z.

The **Semi-Variance (SV)** is defined in the same way:

Although SD and SV seem very similar concepts, they should not be equal since

We can calculate the SV using historical data in the following way:

We can calculate the SD using historical data as follows:

Normally all the terms of the formula are expressed in annual terms. If the data is expressed in other terms, we will have to annualize the results.

### Interpretation

We define D as:

- MIN: we look for the minimum between D and 0.

If D <0 then the result is D ^{2} .

If D> 0 then the result is 0.

- MAX: we look for the maximum between D and 0.

If D> 0 then the result is D ^{2} .

If D <0 then the result is 0.

## Practical example

We assume that we want to carry out a study on the degree of dispersion of the *AlpineSki* price for 18 months (one and a half years). Specifically, we want to find the dispersion of the observations that are below their mean value.

### Process

0. We download the quotes and calculate the continuous returns.

Difference = | min (Z _{t} – Z ‘, 0) | ^{two}

Months | Returns (Z _{t} ) |
Difference |

Jan-17 | 2.75% | 0.00% |

Feb-17 | 4.00% | 0.00% |

mar-17 | 7.00% | 0.00% |

Apr-17 | 9.00% | 0.00% |

May-17 | 7.00% | 0.00% |

Jun-17 | -0.40% | 0.11% |

Jul-17 | -2.00% | 0.25% |

Aug-17 | -4.00% | 0.48% |

Sep-17 | 0.20% | 0.08% |

Oct-17 | 1.50% | 0.02% |

Nov-17 | 2.00% | 0.01% |

Dec-17 | 4.50% | 0.00% |

Jan-18 | 3.75% | 0.00% |

Feb-18 | 5.50% | 0.00% |

mar-18 | 7.00% | 0.00% |

Apr-18 | 9.00% | 0.00% |

May-18 | -1.50% | 0.20% |

Jun-18 | -2.00% | 0.25% |

Half | 2.96% | |

Summation | 1.40% | |

SV12 | 0.009307185 | |

SD12 | 9,647% |

- We calculate:

### Outcome

The annualized Semi-Standard Deviation (SD) is 9.64%. In other words, the degree of dispersion of the observations that are below the mean value is 9.64%. The annualized Semi-Variance (SV) is 0.0093.