Getting Started - Basic Statistical Concepts
When you read about portfolio management in Section 11, you will need to understand some basic statistical concepts. Specifically, you will need to be able to calculate variance and covariance to help you understand the capital asset pricing model (CAPM) and the whole concept of the security market line (SML), the relationship between the risk and expected return of a particular investment.
Variance is a measure of how broadly a group of numbers spreads out from its mean average. In the world of finance, variance has implications for how risky an investment is. At its simplest, if we define "all whole numbers from 3 through 5" as our population, the whole set is comprised of 3, 4 and 5. The average is going to be 4, and there will be very little variance there.
Now let's consider another population: "all scores recorded for the Series 7 exam in a year". There are bound to be a few showoffs who get perfect scores. There will also be a few people who sign in, suffer anxiety attacks, excuse themselves for a moment and never come back, and score zeroes. And there will be a wide variety in between. An informed guess might peg the average score somewhere around 75%, but whether or not you accept that guess, you would probably accept that the actual scores will be all over the map - from 0% to 100% - and that there will be a much higher variance than in the first example.
Now that you understand variance conceptually, let's see if you can measure it. Start with the first example:
From the raw data, you can see that if you add each value (X, pronounced "kai") from each observation, you get a sum (Σ, pronounced "sigma") of 12. If you divide this total by the number of observations, you get the arithmetic mean (μ, pronounced "myoo") of 4. Up until this point we have simply used the term "average" to describe this. Essentially, variance is the measure of how far the X values range from this mean. To compute variance, follow these steps:
- Compute the mean (μ), as you did just now.
- Subtract μ from each observed value (X).
- Square each X -minus-μ quantity.
- Add each of those X -minus-μ-squared quantities.
- Divide by the number of observations.
Here's how you apply these steps to the example above:
|Observation||X||X - μ||(X - μ)2|
|Mean (μ)||4||σ² =||0.67|
The symbol for variance, σ², is derived from the lower-case Σ and is pronounced "sigma squared".
The variance is 0.67. Let's compare that with a more complicated population - the scores of 20 hypothetical people sitting for the Series 7 exam:
|Observation||X||X - μ||(X - μ)2|
|Mean (μ)||74.15||σ² =||143.13|
The variance in this case is 143.13. As you would expect, the variance in this second example is greater in magnitude than the variance computed in the simple example used to open this discussion.
Now that you understand the concept of variance and know what the Greek symbols represent, here's a formula you can jot down on the scrap paper you will receive when you write the exam:
Variance = σ² = Σ (X - μ)2
Here are two more points about variance which you ought to know, even though they will not be on the test:
- This discussion is about variance for observations of an entire population. A whole other formula is used to calculate variance of a sample of a population.
- The square root of the variance, σ², is another measure of dispersion called the standard deviation, and it is represented by the symbol σ.