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This question is from a book (Active Portfolio Management), but unfortunately there are no provided solutions.

"Assume that residual returns are uncorrelated across stocks. Stock A has a beta of 1.15, volatility=35%. Stock B has beta of 0.95 and a volatility of 33%. If the market volatility=20%, what is the correlation of stock A with stock B? Which has higher residual volatility?"

The slope is related to the correlation coefficient by, m = r( s_B / s_A ), (for s=standard deviation, r is correlation coef), but that's the extent of my knowledge here :-|

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I just used the formula in below link and did some math. I have that book too but haven't looked at it yet really. Lots of maths to have fun with. Let me know if this is correct or needs fixing.

Beta(a) = 1.15
Beta(b) = 0.95
V(a) = .35
V(b) = .33
V(m) = .20

rV(a) = V(a)/V(m) = .35/.20 = 1.75
rV(b) = V(b)/V(m) = .33/.20 = 1.65 

rV(a)*(x)=Beta(a) = 1.75(x)=1.15 = x = 1.15/1.75 x = .6571
rV(b)*(x)=Beta(b) = 1.65(x)=0.95 = x = 0.95/1.65 x = .5758

Source: http://wiki.fool.com/How_to_Calculate_Beta_From_Volatility_%26_Correlation

  • In your last two lines, (x) should be different no? In one case it should be x(a-m) and in the other it's x(b-m) (where a-m means correlation of a with m). – Fred Hatt Sep 19 '15 at 14:07
  • Not 100% sure. I just used this step, where x is the correlation in both so you just solve for it since the other variables are known. Multiply the value from Step 2 by the correlation to calculate beta. – Ross Sep 21 '15 at 12:52
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Using the following equations from the book a stab at the correlation can be made.

enter image description here

enter image description here

BA = 1.15    vA = 0.35
BB = 0.95    vB = 0.33
             vM = 0.20

Calculating the residual volatilities from equation 2.4

wA2 = vA^2 - BA^2 * vM^2 = 0.0696
wB2 = vB^2 - BB^2 * vM^2 = 0.0728

pAB = (BA * BB * vM^2)/
   Sqrt[(BA^2 * vM^2 + wA2)*(BB^2 * vM^2 + wB2)] = 0.378355

The correlation of stock A with stock B is 0.378 and stock B has the higher residual volatility.

However, the correlation is given as a "simple model", which may suggest that it is an approximation. If I have applied it correctly, some testing shows that it is only approximate.

Also of interest

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