# Variance on a two stock portfolio (Probabilities)

An analyst develops the follwoing information for two stocks. 50% of the fundes are invested in each stock. …1…2…3. Probability--------.5-------.3-------.2 Stock A------------25%----10%-----(-25%) Stock B------------1%------(-5%)—35% The expected return and variance of this two-stock portfolio is closest to: A. Er = 8.25% Var = 23.31% B. Er = 8.25% Var = 48.09% C. Er = 10.50% Var = 23.31% D. Er = 10.50% Var = 48.09% The correlation coeffiecient between the two A. -.86 B. -.24 C. .24 D. .86 I have found the return but I am having trouble finding the variance using the probability method.

Calculating the variance of a two-asset portfolio can be time consuming, especially when covariance is not provided. Although it is important to know the formulas, one can also use the HP-12C as a short-cut. To use the calculator short-cut, first compute the return in the three scenarios. In the first, the return is 13%, in the second, it is 2.5%, in the third, it is 5% (these are simple averages based on the weighting). Then, to take account of the probabilities, let’s assume we have 10 trials. We’d type the following into the calculator’s statistical register: 13, 13, 13, 13, 13, 2.5, 2.5, 2.5, 5, 5. Then, have the calculator compute the mean. It is 8.25%. Add the mean to the set of trials, which will cause the calculator to compute population variance rather than sample variance. Now, have the calculator compute the standard deviation. It is 4.828. Square that to get the variance: 23.31. This kind of method cannot always be used, so it’s important to know the formulas. But sometimes the short-cut comes in handy to save time or check work.

Do you know how to do this for L1? Cov(A,B) = E(A*B) - E(A)*E(B) (what a pain) E(A*B) = 0.5*0.25*0.01 + 0.3*0.1*-0.05 + 0.2*-0.25*0.35 = -0.01775 E(A) = 0.5*0.25 + 0.3*0.1 + 0.2*-0.25 =0.105 E(B) = 0.5*0.01 + 0.3*-0.05 + 0.2*0.35 = 0.06 Cov(A,B) = -0.01775 - 0.105*0.06 = -0.02405 Var(A) = E(A^2) -(E(A))^2 E(A^2) = 0.5*0.25^2 + 0.3*0.1^2 + 0.2*-0.25^2 = 0.04675 E(B^2) = 0.5*0.01^2 + 0.3*-0.05^2 + 0.2*0.35^2 =0.0253 Var(A) = 0.04675 - 0.105^2 = 0.035725 Var(B) = 0.0253 - 0.06^2 = 0.0217 std(A) =Sqrt(0.03725) = 0.189 std(B) = Sqrt(0.0217) = 0.147 b) correlation = cov(a,b)/(std(a)*Std(b)) = -0.864 a) Var§ = 0.5^2*Var(a) + 0.5^2*Var(B) + 2*0.5*0.5*Cov(a,b) = 0.25*0.035725 + 0.25*0.0217 + 0.5* -0.02405 = .2331% (so answer is a bit messed -> note std dev here is about 5% which is reasonable for two negatively correlated stocks each with std dev between 10-20%)

The book instructed us to calculate covariance, stdev a little differently. E(a)=.105 E(b)=.06 Cov a,b=.5(.25-.105)(.01-.06)+.3(.1-.105)(-.05-.06)+.2(-.25-.105)(.35-.06) Cov a,b=-.02405 var(a)=.5(.25-.105)^2+.3(.1-.105)^2+.2(-.25-.105)^2=.03573 var(b)=.5(.01-.06)^2+.3(-.05-.06)^2+.2(.35-.06)^2=.217 etc… (edit typed the cov answer wrong)

Yep - the formulas are equivalent E(X -mu)^2 = E(X^2 - 2*Mu*EX + Mu^2) = E(X^2) - Mu^2

Thanks so much for that…that problem took me about 30 minutes to do…i really hope CFA test writers don’t put something like that in there… Cheby…the calculator worked great…in seconds literally…i think im going to try that way first if i see a problem like that on the test

there is no way a question of that magnitude will be on there. but it is good practice because we will probably have to do each one of those calcs seperately.

BTW _ I think chebyshev’s approach to this problem is cheesy but pretty clever.

hahaha, what makes it “cheesy”

Imagine if the probabilities were 0.57212, 0.13217, and [whatever]…

[pedant] shouldn’t the variance be in units of (%)^2 ? [/pedant]

yep

using E(XY) - E(X)E(Y) is much easier in general to calculate variance and covariance… (Y=X for variance)