# Standard Deviation

## Introduction

Standard deviation is a common statistical calculation for how far a variable quantity, such as the price of a stock, moves above or below its average value. Standard deviation measures volatility and other technical indicators are often calculated using standard deviations. A wider range implies greater standard deviation and if the standard deviation is increased the investment is considered to be riskier.

Some financial traders and analysts are using standard deviation to make a prediction on how a particular stock investment or stock portfolio will perform. To make that prediction they have to estimate the range of the investment's possible future performances based on a history of past performance, and then calculate the probability of meeting each performance level within that range.

## Essential Assumptions

Standard deviation is a measure of variability and diversity used in mathematics’ statistics area and in the theory of probability. Standard deviation is the difference between the value we believe we will obtain (when forecasting a movement) and the value we actually get from real time movements. When standard variation is low, data points approach the mean (or expected value) and a high variation is the indicator of a wide area where data points are spread. The standard deviation of a statistical population, data set or probability distribution is the square root of its variance, where variance is a measure of how far a set of numbers are spread out from each other.

In finance, standard deviation is a statistical measure that helps the trader be more specific when making assumptions on the volatility of the market, as it shows historic patterns of an investment’s volatility. An insecure stock will present a high standard deviation and a safe and stable blue-chip stock will have a low standard deviation.

## Calculation

The standard deviation of a random variable X is defined as:

Ω = ( E(X2) - (E(X))2 )1/2

Where E(X) is the expected value of X

## Interpretation

Standard deviation is used by investors as a barometer for the amount of expected volatility and a large dispersion is the indication of an investment that deviated from the expected normal returns.

The interpretation of standard deviation in finance is it basically represents the risk associated with a security. Risk is an important factor when deciding how to manage a portfolio of investments because the variation in returns on the asset is determined by the risk degree. It also gives traders a mathematical benchmark for their investment decisions. The conceptuality of risk management is that, as risk increases, the return on the assets will also be high, as compensation for the risk taken. Therefore, traders that work with investments that carry a high level of risk should expect a high return, which is called a “risk premium”.

## Example

Stock A has an average return of 10% over the past 20 years, with a standard deviation of 20 percentage points (pp). Stock B has average return of 12% over 20 years, but has a higher standard deviation, equal to 30 pp. When having to choose between the two stocks, an investor might consider, on the basis of risk and return, that stock A is the safer choice, since stock B has only 2% points of return higher than stock A, and those are not worth the extra 10 pp of standard deviation. The risk of the return is greater with stock B, which is expected to lose its financial investment, but might exceed it as well. Stock B is estimated to do so more often than stock A and to return as little as 2% more on average. For this example, stock A is presumed to earn 10% with plus or minus 20 pp, so it moves between a range from -10% to 30%. As stock B has a higher risk, it also comes with a greater risk premium. When studying the market, an investor should consider more extreme returns, with variances from -50% to 70%, including as much as three standard deviations from the average return.

The expected return on asset is given by the arithmetic mean of the security’s returns over a period of time. For the certain period, the difference between the expected return and the actual return is the difference from the mean. The squared value is the overall variance of the return on asset. The risk of a security is directly proportional with the value of the variance.

And to follow the formula, the square root of the variance brings the value of the standard deviation; that is how standard deviation is used in finance.Standard deviation is not an investment indicator itself, but it is used in finance to determine the width of Bollinger Bands, as they are, in turn, a highly popular financial analysis tool. The calculation of the Bollinger Bands is done with the aid of standard deviation : the upper band is given as x + nsx, where n has usually the value of 2 and the standard deviation is of about five, ergo there’s 5% chance of going outside the normal return on asset.

This measure is a risk representation associated with the price changes of a certain asset (stock, bond, property etc) or of a portfolio of assets (ETFs, index mutual funds, actively managed mutual funds etc). As traders evaluate investments, they should be able to estimate the expected return on asset and the degree of certainty of future returns. Standard deviation provides a quantified estimate of the ambiguity of the return on the period to come.

## Conclusion

Mathematics is the science investors all over the world count on to bring forth formulas and indicators to help them identify market trends. Standard deviation is a measure used in statistics and probability theory and it gives traders a better grip on the market volatility degree. As the calculated values further away from the mean value, the more insecure the market is. The higher the standard deviation represents the higher the degree of the risk involved. As the standard deviation lowers in value, so does the risk on that certain security. This measure is used in the calculus and charting of several financial indicators, such as Bollinger Bands and the Coefficient of Variation.