7
∑ ି ሺ∑ఽ ሻሺ∑ా ሻ √ሺ∑ మ ି ሺ∑ఽ ሻ మ ሻሺ∑ మ ି ሺ∑ా ሻ మ ሻ =
ସ ൫√ଶ൯ሺ√଼ሻ = 1
PMCC =
(indicates a perfect positive correlation)
Therefore;
V P
= S P
2 = W A
2 S A
2 + W B
2 S B
2 + 2W A
W B
S A
S B
Simplified;
V P
= S P
2 = (W A
S A
+W B
S B
) 2
So;
S P S B Or to put in it simply, if the returns on A and B are perfectly positively correlated, the risk of the portfolio return (in terms of standard deviation) is just the weighted average of the risks of the constituent assets’ returns. At first impressions then, this may seem to be a positive outcome as the risk is quantifiable and linear and as is the expected returns. Thus, you can model your risk (x-axis) against return (y-axis) as a uniform straight line and make informed decisions on where to invest. However, it is because the two securities have a PMCC of 1 that means investing in these two assets simultaneously brings the same reward as investing all your money in just A or B, hence, we can say that no risk is eliminated and that all this maths has reaped no reward. Or, we could look at the flip side, investment bankers may be interested in buying two stocks that have a PMCC of -1, i.e. as one increases in value, the other decreases in value by the same amount. If we plug PMCC = -1 into the ‘Portfolio Risk Equation’, we get; V P = S P 2 = W A 2 S A 2 + W B 2 S B 2 - 2W A W B S A S B Simplified; V P = S P 2 = (W A S A –W B S B ) 2 So; S P = W A S A – W B S B This equation (that is slightly more complex for non-integer negative values of the PMCC - but nevertheless the theory holds) is very interesting as it implies that risk can be reduced or even completely nullified and the standard deviation would therefore be 0. In fact, for a given set of data (again, let’s use the table on page 1) there is a unique combination of W A : W B that can be found because if the two securities (with a PMCC of -1) ‘are combined in inverse proportion to the ratio of the standard deviations of their respective returns, so the resultant portfolio will yield a constant = W A S A + W B
return and hence be riskless’ 1 . For example, using the data;
W W
ൌ S S
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