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Variance is a measure of the distribution of observations around the mean. It is often used for descriptive statistics, inferential statistics and hypothesis testing.
For investment modeling, variance is a widely used descriptive statistic, as is its square-root, the standard deviation. Variance is also used heavily in valuation models of individual securities, portfolio choice and the measurement of risk-adjusted portfolio performance.
Variance is calculated by taking the sum of all squared differences between each observation and the mean over the measurement period. After summing those, next divide by the number of observations. Because it is the result of squaring, variance will always be zero or greater.
Variance is not easily interpreted because the scale is in units squared, like percent-return-squared. So instead, to interpret, translate to standard deviation by taking the square-root of variance. Its units would then be in percent-return, and that can easily be interpreted.
Synonym: second central moment
For context, variance provides a measure for how far from the average the whole group of obervations are spread out. Imagine a bell-shaped curve and here a higher variance will have a wider distribution. On the other hand, when the variance is low, the distribution will be narrow and tall.
Doc: Besides variance what else could
we use on the x-axis for MPT charts? And why?
Wes: Standard deviation, because it's more interpretable and the relationship still holds.
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Variance definition for investment modeling (4:22)
The script includes two sections where we visualize and demonstrate the calculation of variance using demeaned returns.
We're sitting right here in Excel, and this is a snippet from our boot camp course.
This is one depiction of variance, from a discussion on portfolio theory.
Think about each dot here as a stock or portfolio. Each has a return and a risk measure. Risk is on the x-axis and return is on the y-axis.
Risk here can be interpreted as either variance or standard deviation. As you will see shortly, they are both related. You go seven steps with the exact same calculation, until the final step.
The key with variance is the calculation, so let's head there now.
Let's walk through a calculation for two stocks, Microsoft and eBay. We have six monthly observations of return for each stock from April to September 2003. Column F is the return on Microsoft, eBay is Column G.
Next we compute the average of each, here 2.38% for Microsoft and 3.98% for eBay. Then move those over to columns H and I.
In column J we take the return minus the average which gives us 3.24%. That's 5.62% minus 2.38%, or 3.24%. For eBay it is 8.91% minus 3.98%, or 4.93%. Carry that formula down for all months. Next, square these in columns L and M.
Next, using the
=SUM() function, add
up the products for each stock to get 0.0062 for Microsoft and 0.0109
Next, divide by 6 observations to get the variance of 0.0010 for Microsoft and 0.0018 for eBay. Recall, these are in units of returns squared so aren't interpretable. So it is common to use standard deviation, which is the square root of variance.
To get that, use the
or take the variance to the one-half power, as I have done here.
Click box for answer.
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