Variance and Covariance

[[email protected]]

Hi David,

2006 past year question

Given two random variables X and Y, what is the Variance of X given Variance[Y]=100, Variance [4x-3y]=2700 and correlation between X and Y is 0.5?
A. 56.3
B. 113.3
C. 159.9
D. 225.0

Answer: D. 225.0

How to solve this question? Your guidance, please.

Regards
Learning

 

[David Harper CFA FRM]

Hi Learning,

per Gujarati 3.33: var (A - B) = var(A) + var(B) - 2*cov(A,B)
and here, A = 4x, B = 3y

var (4x - 3y) = var(4x) + var(3y) - 2*cov(4x,3y);
var (4x - 3y) = 4^2var(x) + 3^2var(y) - 2*4*3*cov(x,y);
var (4x - 3y) = 4^2var(x) + 3^2var(y) - 2*4*3*StdDev(x)*StdDev(y)*Correlation(X,Y)
2700 = 16*var(x) + 900 - 120*StdDev(x)
1800 = 16*var(x) - 120*StdDev(x)
16*var(x) - 120*StdDev(x) - 1800 = 0
let w = StdDev(x), then:
(4w + 30)(4w - 60) = 0
since 4x-60, w = 15 and x = 225

...I doubt that is the most efficient way, but you get the point ...Thanks, David

 

[[email protected]]

#4. Given two random variables X and Y, what is the Variance of X given Variance[Y] = 100,
Variance [4X - 3Y] = 2,700 and the correlation between X and Y is 0.5?
a. 56.3
b. 113.3
c. 159.9
d. 225.0

Answer: d

a. Incorrect. +3 was used instead of -3 when solving. Variance [4X - 3Y] = 16*Var[X] + 9*Var[Y] +
2*4*(+3)*Var[X]^(1/2)* Var[Y]^(1/2)*correlation[X,Y]. Solve for Var[X] = 56.3.
b. Incorrect. (Var[X]^(1/2)* Var[Y]^(1/2) is missing from the equation when solving. Variance
[4X - 3Y] = 16*Var[X] + 9*Var[Y] + 2*4*(-3)*correlation[X,Y]. Solve for Var[X] = 113.3.
c. Incorrect. The factor of 2 is missing from the equation. Variance [4X-3Y] = 16*Var[X] + 9*Var[Y] +
2*4*(-3)*Var[X]^(1/2)* Var[Y]^(1/2)*correlation[X,Y]. Solve for Var[X] = 159.9.
d. Correct. Using the theorems on variance and covariance, Variance [4X-3Y] = 16*Var[X] + 9*Var[Y]
+ 2*4*(-3)*Var[X]^(1/2)* Var[Y]^(1/2)*correlation[X,Y]. Solve for Var[X] = 225.0.
Reference:
Murray R. Spiegel, John Schiller, and R. Alu Srinivasan, Probability and Statistics, Schaum’s Outlines,

*****

David,

I am having a hard time understanding where the 16, 9, 2, 4, and -3 in the final calculation is coming from, can you please explain?

Thanks,
Eva

 

[Jiew Kwang]

Hi eva,

I cheated by looking at the answer first =p

Weight(X) = 4, Var(X) = Solve for
Weight(Y) = 3, Var(Y) = 100
rho = 0.5

So that 4^2 * var(X) + 3^2 * 100 + 2 * 4 * -3 * 0.5 * 10 * sigma(X) = 2700
Then 16sigma(x)^2 + 900 - 120sigma(x) - 1800 = 0
Then went online to solve quadratic equation with their calculator for x = -7.5 or 15(complies with equation
So Var(x) = 15^2 = 225

It looks wierd to me that you can use, say weight(x) as just the 4 given too..

Hope to hear from david regarding it too.

 

[[email protected]]

Thanks -- I just redid it! I realize that this question is simply testing your knoweledge of the Portfolio Variance formula -- with a little bit of a twist, because instead of giving you the covariance, they give you the correlation btw x and y.

I suppose this means that Covariance = (VARx)^(.5) * (VARy)^(.5) * correlation between x and y, is that correct???

Thanks again!

 

[Jiew Kwang]

Hi eva, yes i guess they were trying to test you on portfolio variance, and yes i did my cov = corr (x, y) * sigma x * sigma y.

 

[David Harper CFA FRM]

Hi Eva & Jiew,

We've worked this one before, for what it's worth:
http://forum.bionicturtle.com/viewthread/2306/#4875
http://forum.bionicturtle.com/viewthread/1190/#2896

It looks like I approached similar to Jiew with the following:

per Gujarati 3.33: var (A - B) = var(A) + var(B) - 2*cov(A,B)
and here, A = 4x, B = 3y

var (4x - 3y) = var(4x) + var(3y) - 2*cov(4x,3y);
var (4x - 3y) = 4^2var(x) + 3^2var(y) - 2*4*3*cov(x,y);
var (4x - 3y) = 4^2var(x) + 3^2var(y) - 2*4*3*StdDev(x)*StdDev(y)*Correlation(X,Y)
2700 = 16*var(x) + 900 - 120*StdDev(x)
1800 = 16*var(x) - 120*StdDev(x)
16*var(x) - 120*StdDev(x) - 1800 = 0
let w = StdDev(x), then:
(4w + 30)(4w - 60) = 0
since 4w=60, w = 15 and x = 225

@Eva: it is definitely true, and you need to know, that "Covariance = (VARx)^(.5) * (VARy)^(.5) * correlation between x and y" … same as mine above: cov(x,y) = StdDev(x)*StdDev(y)*Correlation(X,Y)

The question still appears difficult to me because, I don't see any way around the need for the quadratic solution at the end. (the answer as written still requires quadratic solve!). It seems like a lot to ask in a short time …. David