The variance of a portfolio of two assets is calculated as
$Var(R_P)=w_a^2\sigma_a^2+w_b^2\sigma_b^2+2w_aw_b\sigma_{ab}$,
where $w_a,w_b$ are the respective weights of the two assets in the portfolio and $w_a+w_b=1$.
$\sigma_a$, $\sigma_b$: individual asset standard deviations (S.D.),
$\sigma_{ab}$: covariance between assets a and b
Let us take an example. Let $w_a=0.4$ and $w_b=0.6$, $\sigma_a=12.93\%$, $\sigma_b=8.21\%$ and $\sigma_{ab}=18.6\%$.
So, is $Var(R_P)=0.4^2\times12.93^2+0.6^2\times 8.21^2+2\times 0.4\times 0.6\times 18.6 = 59.94$
Hence, the S.D of the portfolio = $7.742\%$
This way is used in Principles of Corporate Finance (Brealey, Myers)
or should it be $Var(R_P)=0.4^2\times0.1293^2+0.6^2\times 0.0821^2+2\times 0.4\times 0.6\times 0.186 = 0.0938 $
Hence, the S.D of the portfolio = $0.3072 = 30.72\%$
This is used in CFA Institute(www.cfainstitute.org/learning/products/publications/inv/Documents/investments_chapter5.pptx) Slide 16/35
The two answers will be different. Subsequently, the portfolio standard deviation will differ even more for the above two ways.
I seek help to find which one is the correct way and why ?
Thanks.
EDIT: I prefer the second one as both the weights and the S.D.s are in decimals .. whereas in the first one, weights are in decimals and the S.D.s are taken at the "face value" of the percentage .. this seems inconsistent .. but our prof is adamant on using the first one .. because that way ensures that the SD < variance .. whereas the second way gives higher SD (square root of a fraction is higher) .. What is the correct way here ?
Answer:
Let us look at it differently.
When used as a percentage let us compute correlation coefficient.
$\rho = \frac{18.6}{12.93\times8.21} = .17521$
When used as decimals (as is) let us compute correlation coefficient.
$\rho = \frac{.186}{.1293\times.0821} = 17.521$
$\rho$ can never be more than 1. While this said, let us deeply investigate this
We also know that correlation coefficient is dimensionless. So Covariance is $\rho$ multiplied by two standard deviations.
When putting everything in decimal, you may have to divide covariance by the order of 10000. Not 100.
If you are representing everything in decimals the correct way is
$Var(R_P)=0.4^2\times0.1293^2+0.6^2\times 0.0821^2+2\times 0.4\times 0.6\times 0.00186 =0.005994$
$S.D = .0774 = 7.74$
Check out CFA website and make sure they say the covariance in percentage should be divided by 10000.
Your prof is spot on
Your example is also misleading, Covariance will not be mentioned percentages. Even in your CFA slide, it is given to be 0.0050 not as 50%. Quoting that would mislead everyone. CFA is also correct. Just in case Covariance is mentioned as a whole number, divide it by 10000.
Sorry for misleading earlier.
Good luck.