Intro to Finance: Risk and Return - Variance and Standard, Deviation Measures of Dispersion

What This Is

Variance and standard deviation are measures of dispersion, which quantify the spread or dispersion of a set of data points. In finance, these measures are crucial for assessing risk and uncertainty associated with investments, assets, or portfolios. For instance, consider Apple's stock price over the past year. If the stock price has fluctuated between $150 and $200, the variance and standard deviation would help us understand the magnitude of these fluctuations.

Key Formulas & Symbols

• Variance (?^2) = ?(xi - ?)^2 / (n - 1) where ?^2 = variance, xi = individual data points,-= mean, n = number of data points.
• Standard Deviation (?) = ?(?^2) where-= standard deviation.
• Population Standard Deviation = ?(?(xi - ?)^2 / N) where N = total population size.
• Sample Standard Deviation = ?(?(xi - ?)^2 / (n - 1)) where n = sample size.
• Coefficient of Variation (CV) =-/ ? where CV = coefficient of variation.
• Range = Maximum Value - Minimum Value where Range = range of data points.
• Interquartile Range (IQR) = Q3 - Q1 where Q3 = third quartile, Q1 = first quartile.

Step-by-Step Calculation

1. Calculate the mean (?) of the data points.
2. Subtract the mean from each data point (xi - ?) to find the deviations.
3. Square each deviation to find the squared deviations.
4. Sum the squared deviations and divide by the number of data points minus one (n - 1) to find the variance.
5. Take the square root of the variance to find the standard deviation.

Common Mistakes

• Mistake: Using the population standard deviation formula for a sample.
• Correction: Use the sample standard deviation formula (? = ?(?(xi - ?)^2 / (n - 1))) when working with a sample.
• Mistake: Failing to account for the degrees of freedom when calculating variance.
• Correction: Use n - 1 as the denominator when calculating variance to account for the degrees of freedom.
• Mistake: Confusing the coefficient of variation with the standard deviation.
• Correction: The coefficient of variation (CV) is the ratio of the standard deviation to the mean (CV =-/ ?).

Exam / CFA Tips

• Tip: Be careful with the formula for variance and standard deviation, as the test may try to trick you with the degrees of freedom.
• Tip: Make sure to understand the difference between the population and sample standard deviation formulas.
• Tip: Be prepared to calculate variance and standard deviation from scratch, as the test may not provide the mean or other intermediate values.

Quick Practice Problem

A bond has a face value of $1,000, a 5% coupon rate, and a yield to maturity of 8%. What is the bond's standard deviation of returns?

Answer: 0.05 (since the bond's returns are fixed at 5% per annum).

Explanation: The bond's returns are fixed at 5% per annum, so the standard deviation of returns is zero.

Last-Minute Cram Sheet

• The variance formula requires n - 1 as the denominator to account for the degrees of freedom.
• The standard deviation is the square root of the variance.
• The coefficient of variation is the ratio of the standard deviation to the mean.
• The range is the difference between the maximum and minimum values.
• The interquartile range (IQR) is the difference between the third and first quartiles.
• The sample standard deviation formula is used when working with a sample.
• The population standard deviation formula is used when working with the entire population.
• The standard deviation is a measure of dispersion, while the mean is a measure of central tendency.
• The variance is sensitive to outliers, while the standard deviation is not.
• The coefficient of variation is a relative measure of dispersion.