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Measures of Risk - Variance and Standard Deviation

Risk reflects the chance that the actual return on an investment may be very different than the expected return. One way to measure risk is to calculate the variance and standard deviation of the distribution of returns.

Consider the probability distribution for the returns on stocks A and B provided below.

State Probability Return on
Stock A
Return on
Stock B
1 20% 5% 50%
2 30% 10% 30%
3 30% 15% 10%
3 20% 20% -10%

The expected returns on stocks A and B were calculated on the Expected Return page. The expected return on Stock A was found to be 12.5% and the expected return on Stock B was found to be 20%.

Given an asset's expected return, its variance can be calculated using the following equation:


  • N = the number of states,
  • pi = the probability of state i,
  • Ri = the return on the stock in state i, and
  • E[R] = the expected return on the stock.

The standard deviation is calculated as the positive square root of the variance.

Variance and Standard Deviation on Stocks A and B

Note: E[RA] = 12.5% and E[RB] = 20%

Stock A

Stock B

Although Stock B offers a higher expected return than Stock A, it also is riskier since its variance and standard deviation are greater than Stock A's. This, however, is only part of the picture because most investors choose to hold securities as part of a diversified portfolio.

Example Problems
Find the Expected Return, Variance, and Standard Deviation on
a stock given the following probability distribution of returns
for the stock.
State Probability Return
1 % %
2 % %
3 % %
4 % %
Expected Return: %
Standard Deviation: %


© 2002 - 2010 by Mark A. Lane, Ph.D.