Doubts

Population Variance Formula in Finance

Population Variance Formula in Finance

by Julian Hildebrandt -
Number of replies: 2

In our Assignment, we’ve been reviewing the formulas for variance and standard deviation, knowing that the sample version divides by n−1 while the population version divides by n. 

My group member Adrian and I were discussing the practical application of these formulas, especially given that financial markets generate numerous prices throughout the day (open, high, low, close, and many more intra-day observations). 

Could you describe a situation or provide an example from financial practice where using the population formula (dividing by n) would be appropriate? In other words, under what circumstances might we consider our dataset of prices to represent the entire population rather than just a sample? 

Looking forward to your insights!

In reply to Julian Hildebrandt

Re: Population Variance Formula in Finance

by Julio Crego -
This is a very interesting question that is way beyond the scope of this course. For this course, we always use the "sample" version.

From here, it is outside of this course.

Although one is called "sample" and the other population, they are both sample versions of the population variance:

\(Var(x) = \mathbb{E}\left(\left(x-\mathbb{E}(x)\right)^2\right)\)

The sample version works "better" for small samples, while the "population" version works for very large samples. Actually, for very large samples, they are almost the same, so the difference only matters in some situations that you will not encounter; hence, always use "sample."

Regarding close, open, etc. You are right. We have methods to compute the variance that exploit all those prices. There are literally books on how to estimate variance. However, variance does not change much over time, so we rarely need sophisticated methods to construct optimal portfolios. These methods are more useful to price derivative securities for instance.  
In reply to Julio Crego

Re: Population Variance Formula in Finance

by Julian Hildebrandt -
Hi Julio, thanks for your clear explanation. Your mention of more sophisticated methods for derivative pricing adds useful context.