Quant Q

Assume two stocks are perfectly negatively correlated. Stock A has a standard deviation of 10.2% and stock B has a standard deviation of 13.9%. What is the standard deviation of the portfolio if 75% is invested in A and 25% in B? A) 0.00%. B) 3.76%. C) 4.18%. D) 0.17% I thought one needed either covariance or expected return to calculate this but obviously not. The solution (in the next post) did not make sense to me.

The standard deviation of the portfolio is found by: [W12 σ12 + W22 σ22 + 2W1W2σ1σ2r1,2]0.5, or [(0.75)2(0.102)2 + (0.25)2(0.139)2 + (2)(0.75)(0.25)(0.102)(0.139)(–1.0)]0.5 = 0.0418, or 4.18%.

C? Correlation = -1 Covarance = -1 * .139 * .102 = .-014 Var = (.75* .102)^2 + (.25*.139)^2 - 2*.75*.25*.14 = .001757 SD = .0419

The key is knowing that covariance is equal to correlation multiplied by the two standard deviations.

Also perfectly negatively correlated means correlation = -1

right, gotta memorize corr=cov (a,b) / sigma (a)* sigma (b)…and corr*sigma*sigma=cov…gotta own that equation

or since you know that it’s perfectly negatively correlated, just need to change one of the std dev to negative and calculate the weighted avg: .75*10.2% + .25*-13.9% = 4.18 Since std dev can’t be negative, even though if you changed std dev A to negative instead of B, you’d still get -4.18 or 4.18… If it’s perfectly correlated, then it’s straight weighted avg of the two. Note: can’t apply the same logic if correlation is not perfectly negative or perfectly positive

it can be derived that: for PERFECT positive correlation: PLUS sigma(port)=w1*sigma1 + w2*sigma2 for PERFECT negative correlation: MINUS sigma(port)=w1*sigma1 - w2*sigma2