Quant question on Std.Dev. Calculation

CFA Level I
#1

Hi,

Can anyone tell me what formula they used here and how they managed to get 0.066 in the calculation?

I wanted to use the Var(Rp) forumal but I know requires the question to provide with at least a Cov1,2 value or a corr value. I don’t think you calculate either from the info provided. I used my BA11Plus calculate to see if i can get a value for r if i was to input state of economoy value to Y and Return on Portfolio to X and I ended up with 1.

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) 5.5%. C) 6.5%.

The correct answer was B) 5.5%.

[0.30 × (0.15 − 0.066 )2 + 0.70 × (0.03 − 0.066 )2]1/2 = 5.5%.

#2

The formula appears to be

Var=P(state1)*(return in state 1 - expected return across states)^2 + P(state2)* (return in state 2 - expected return across states)^2

Std=squareroot(Var)=Var^(1/2)

Expected return across states = P(state1)*return in state 1 + P(state2)* return in state 2

= 0.3*0.15+0.7*0.03

= 0.066

#3

So you’re saying they used the following formula --> Var(X)= P(X)[X-E(x)]^2 but why is that the square root was taken?

#4

i am just inferring from the context…

conventional Variance formula = 1/n*sum[X-E(x)]^2

the problem specific Variance formula= P(x)*[X-E(X)]^2

P(x) could be 1/n if we are equal weighting the observations.

The squarerrot was taken because we are looking for the Standard deviation which is the square root of variance.

#5

Ohh that makes great sense haha! Thanks so much Alladin.

Can you use ur calculator to solve a problem such as this quicker?

#6

haha i am not aware of any calculator shortcuts here…its probably quicker to do it step by step

#7

On the Ba 2 plus you type it into data and get the values in 1-V mode (easier, faster and lot less mistakes for me…)

#8

Like I-will-pass said, using the calculator is easier. Enter the returns in the X column as percents (not decimal form) and the probabilities in the Y column as percents and you get a stdev of 5.5%