Semi-deviation (SD) and Semi-variance (SV)

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: 

Half-Deviation Formula 1

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

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

Half-variance

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

Half-variance and half-variance are different

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

Semivariance Formula

We can calculate the SD using historical data as follows: 

Half Deviation Formula

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:

Definition of D
  • MIN: we look for the minimum between D and 0.  

If D <0 then the result is D 2 .

Min D Less Than Zero

If D> 0 then the result is 0. 

Min D Greater Than 0
  • MAX: we look for the maximum between D and 0. 

If D> 0 then the result is D 2 . 

Max D Greater Than Zero

If D <0 then the result is 0. 

Max D Less Than Zero

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%
  1. We calculate:
Example Half-Variation And Half-Variance

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. 

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