P1.T2.304. Covariance (Miller)

[Fran]

AIM: Define, calculate, and interpret the covariance and correlation between two random variables.

Questions:

304.1. Two assets, X and Y, produce only three joint outcomes: Prob[X = -3.0%, Y = -2.0%] = 30%, Prob[X = +1.0%, Y = +2.0%] = 50%, and Prob[X = +5.0%, Y = +3.0%] = 20%:

What is the correlation between X & Y? (bonus question: if we removed the probabilities and instead simply treated the three sets of returns as a small [tiny actually!] historical sample, would the sample correlation be different?)

a. 0.6330
b. 0.7044
c. 0.8175
d. 0.9286

304.2. Each of random variable X and Y can have two outcomes. The following probability matrix gives their joint probabilities:

For example, the joint Prob[X = 4.0, Y = 3.0] = 30%. What is the covariance between X and Y?

a. -0.9727
b. 0.3150
c. 1.4842
d. 4.9224

304.3. Let X be a discrete uniform random integer in the set {1, 2, 3, 4, 5} with equal probability of each outcome and let Y = (X+1)^2:

What is the covariance between X & Y?

a. 5.5
b. 9.0
c. 16.0
d. 25.0

Answers:

• Here in Forum

 

[Tabriz]

1.d
2.b
3.c

 

[sagrr]

304.1

Can someone explain why we can't employ the E[X^2] - E[X]^2 = Var[X] rule here?

The solution describes that E[X^2] = 0.820%, and E[X] = 0.600%

0.820 - .6^2 = -.278

which obviously can't be right.

Thanks

 

[David Harper CFA FRM]

Hi @sagrr It works (your formula is solid for the population variance; and this example does specify the population), but you want 0.082% - 0.60%^2 = 0.00082 - 0.006^2 = 0.000784 which is the variance and sqrt(0.000784) = 2.80% standard deviation for x. Thanks,

 

[sagrr]

I get it. I think you meant

"but you want 0.082% - 0.60% = 0.00082 - 0.006^2 = 0.000784"

Thanks

 

[JMars7424]

Hi, I keep getting close to but not correct answer for 304.1 could it be calculator related? -.03 x -.02 x .30 gives me .0002 instead of .0180

this throws me off

 

[David Harper CFA FRM]

HI @JMars7424 That looks to me like merely like the calculator is rounding (2nd FORMAT will show more decimals) the solution gives "-3.0%*-2.0%*30% = 0.0180%" and 0.0180% = 0.000180. I actually have my calculator set to 4 decimals and, indeed, this product returns 0.0002 but the calculator is storing 0.000180. I hope that helps!