Standard Deviation – Definition and Other Information

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Standard Deviation
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Definition Examples, Types, or Variations Formula Related Terms As Used in the Hedge Fund World Applications Misused & Abused Additional Sources of Information Books News Scholarly Papers

1.	Definition
	In finance, standard deviation is a representation of the risk associated with a given security (stocks, bonds, property, etc.), or the risk of a portfolio of securities. Risk is an important factor in determining how to efficiently manage a portfolio of investments because it determines the variation in returns on the asset and/or portfolio and gives investors a mathematical basis for investment decisions. The overall concept of risk is that as it increases, the expected return on the asset will increase as a result of the risk premium earned - in other words, investors should expect a higher return on an investment when said investment carries a higher level of risk. Other Resources: CDM Trading: The square root of the variance. A measure of dispersion of a set of data from their mean. A measure of the volatility of an underlying. More… Contribute to this section by clicking ▲ top
2.	Examples, Types, or Variations
	For example, you have a choice between two stocks: Stock A historically returns 5% with a standard deviation of 10%, while Stock B returns 6% and carries a standard deviation of 20%. On the basis of risk and return, an investor may decide that Stock A is the better choice, because the additional percentage point of return (an additional 20% in dollar terms) generated by Stock B is not worth double the degree of risk associated with Stock A. Stock B is likely to fall short of the initial investment more often than Stock A under the same circumstances, and will return only one percentage point more on average. In this example, Stock A has the potential to earn 10% more than the expected return, but is equally likely to earn 10% less than the expected return. Calculating the average return (or arithmetic mean) of a security over a given number of periods will generate an expected return on the asset. For each period, subtracting the expected return from the actual return results in the variance. Square the variance in each period to find the effect of the result on the overall risk of the asset. The larger the variance in a period, the greater risk the security carries. Taking the average of the squared variances results in the measurement of overall units of risk associated with the asset. Finding the square root of this variance will result in the standard deviation of the investment tool in question. Use this measurement, combined with the average return on the security, as a basis for comparing securities. Other Resources: MoneyTerms.co.uk: The most common use of the standard deviation in finance is to measure the risk of holding a security or portfolio. For securities prices this means that for each possible price on a future date of a security one needs to first calculate: (price × probability that that will be the actual price). Adding the numbers this gives will give you the average, or expected price. The for each possible price calculate: (price - mean price)²×probability that this will be the actual price. The sum of these numbers gives the variance which is one measure of volatility. Again, the square root of the variance is the standard deviation which is the most widely used measure of volatility. Contribute to this section by clicking ▲ top
3.	Formula
	The standard deviation of a random variable X is defined as: $\sigma = \sqrt{\operatorname{E}((X-\operatorname{E}(X))^2)}$ $= \sqrt{\operatorname{E}(X^2) - (\operatorname{E}(X))^2}$ where E(X) is the expected value of X. Not all random variables have a standard deviation, since these expected values need not exist. For example, the standard deviation of a random variable which follows a Cauchy distribution is undefined because its E(X) is undefined. If the random variable X takes on the values $\scriptstyle x_1,\dots,x_N$ (which are real numbers) with equal probability, then its standard deviation can be computed as follows. First, the mean of X, $\overline{x}$ , is defined as a summation: $\overline{x} = \frac{1}{N}\sum_{i=1}^N x_i = \frac{x_1+x_2+\cdots+x_N}{N}$ where N is the number of samples taken. Next, the standard deviation simplifies to $\sigma = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \overline{x})^2}.$ Contribute to this section by clicking ▲ top
4.	Related Terms
	Beta Bollinger Bands Mean Covariance Variance Tracking Error Volatility Black-Scholes Option Pricing Model Contribute to this section by clicking ▲ top
5.	As Used in the Hedge Fund World
	Other Resources: Canadian Investment Review: For hedge funds, historical standard deviation is helpful in predicting future risk. The correlation between pre-investment standard deviation, downside deviation and maximum drawdown during the subsequent period of investment is significant. More… Contribute to this section by clicking ▲ top
6.	Applications
	Other Resources: MoneyTerms.co.uk: The standard deviation is the square root of the sum of the squares of the differences between a series of numbers and their average. The standard deviation of a series of numbers is the square root of their variance. The variance is calculated by calculating: (number - mean of all the numbers)²for each number, adding the results together and dividing by the number of numbers. Contribute to this section by clicking ▲ top
7.	Misused & Abused
	Other Resources: Canadian Investment Review: Many investors dismiss return volatility among hedge funds that have only had positive returns as ‘upside’ volatility. More… Contribute to this section by clicking ▲ top
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