When do you use probabalistic variance?

CFA Level I
#1

Is that just variance?

#2

Would that simply be a probability weighed variance? Very similar to just calculating variance for a portfolio of multiple assets with different composition weights rather than probabilities of outcomes?

#3

when you’re forward looking and you’ve given the probabilities :wink: if they are historic, you use the other formula (… divided by n-1 if its a sample and N if its a population)

#4

Is there any intuition behind the choice? i.e. why you use two different formulas for calculating the same thing?

#5

Mathematically it makes sense. When dividing by N (population) or N-1 (Sample) essentially you are quantifying your own weights based on size. By using probabilities you are developing weights by multiplying each observation by a probability weight. Based on previous examples, I have noticed that most variance problems calculated by probabilities are utilized in forward looking statements when conditional macro-economic events are KNOWN. Calculating variance by dividing by N and N-1 is utilized in calculations corresponding to historical data. ( Take that last part with a grain of salt, just a personal observation of mine)

#6

What Akeg said. I would expect to see them set up in a matrix style with probaility weighted expected returns. An example is artithmetic mean versus probability wieghted mean, they are in essense two similar equations, one is more refined and rigourous. One thing I had trouble with initially remebering for probalistic variance was there is no denominator (i.e to not divide by n-1) I am not sure of the rational behind this, maybe someone else could elebroate.

#7

Consider computing a population variance as:

σ_² = [Σ(X_i – _μ_x)²] / N

A little algebra gives us:

_σ_² = [Σ(Xi – μx)²] / N

= Σ[(Xi – μx)² / N]

= Σ[(1 / N)(Xi – μx)²]

This is, essentially, the probabilistic variance formula in which every observation is given a probability of (1 / N). It’s a weighted average, so the sum of the weights is one.

In general, the probabilistic variance formula is:

_σ_² = Σ[P(Xi)(Xi – μx)²]

This has the same property: it’s a weighted average, and the sum of the weights is one. In essence, the division by N is already incorporated into the probabilities P(Xi).

#8

Ah, very interesting, starting to get it. When you don’t need to memorise a bunch of similar formulas… that’s when you know you are on the right track. This also made me realise that arithmetic mean is just a special case of weighted mean. Why are we using equal weightings in the population variance calculation and then unique probabilities for the random variable variance?

And the second question is… the calculator can’t calculate probabilistic variance can it? It seems to only work if you have equal weights? i.e. population and sample variances? In fact, which can it calculate? Population or Sample? I’m guessing it only divides by N so it can only calculate population variance?

#9

There is a way to calculate probailistic variance on the calculator. Someone posted a “How To” a while back. I currently don’t remember though off the top of my head.

#10

Thanks will have a search.