I’m having a hard time figuring out how Schweser derived the calculations for this problem (e.g. where did the 0.066 come from?). Any help would be greatly appreciated!
Use the following probability distribution to calculate the standard deviation for the portfolio.
State of the Economy
Probability
Return on Portfolio
Boom
0.30
15%
Bust
0.70
3%
A) 6.0%. B) 6.5%. C) 5.5%.
Your answer: A was incorrect. The correct answer was C) 5.5%.
[0.30 × (0.15 − 0.066)2 + 0.70 × (0.03 − 0.066)2]1/2 = 5.5%.
The problem should read like this:
State of the Economy Probability Return on Portfolio
Boom 0.3 15%
Bust 0.7 3%
(0.3 × 15%) + (0.7 × 3%) = 6.6%; it’s the expected return.
The formula is:
σ² = Σ(P(Xi) × (Xi – μ)²)
They just used the formula.
Try and understand the meaning of variance first. It is the summation of “deviation from the mean value” squared. That said, ask yourself the following questions: What is the mean value in this case? How will you sum the deviations from the mean? Will you give all of them an equal weight? Probably, yes. You will do that only if these deviations had an equal likelihood of occurence. However, you are given the value of “how likely these deviations are”? What is that value? Do you get the drift?
Hope that helps!
I’d say it’s more accurate to say that the variance is the “expected” squared deviation from the mean". If you’re given probabilities, the probabilities are the weights. So, calcualte the mean, then calcualte the deviations. Square the deviations and mulitple times the probability of each deviation occurring. Add these up and you’re done.
In the case of sample or population variance, it’s still a weighted average squared deviation. It’s just that the weights are all the same (1/n for population variance, and 1/(n-1) for sample variance.)
Whenever we want to calculate any moment, we are interested in calculated in average (expected) moment. In case of second moment which is deviation, where we are interested in knowing the average (expected) deviation. So, in case we have probability/weights we calculate weighted by multiplying weights with the required term. In case of variance it is (Xi – μ)², so:
σ² = Σ(W(Xi) × (Xi – μ)²)
In case, it is equally weighted: it will be σ² = Σ(1/n × (Xi – μ)²) which is a basic formula