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
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”.
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.
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.