to cover both forum posts:
Actually this is question is a tricky one and I would like to dig a bit deeper here in order to be crystal-clear. The notation with Asset A and Asset B for the market portfolio and simultaneously being A and B as well for the investors weights is misleading!!
In a sense the most critical part to understand here is the fact that an investor wants to form a portfolio consisting only of the market portfolio (which consits only of two assets in our simplified world!) AND wants to determine the weights how he wants to split his wealth between these two assets which form the so called market (portfolio), call it the S&P500.
Let me explain in detail with a more practical example:
An investor wants to hold a portfolio of two assets; put differently: he wants to hold two assets which comprise the market and he thinks about splitting his wealth in one of the following combinations: e.g. 160% and -60%, 100% and 0% etc.
Then comes the more advanced part here: We are asked to compute the covariance of the investor's portfolio (how much he likes to devote to Asset A and Asset B in the market portolfio) with regard to the market portfolio (which in this simplified case consits of 2 assets; imagine that for simplification the S&P500 consists only of two assets: Exxon Mobile and American Airways.
The weights of these assets are given in % (let's say the % expresses the market cap as a reciprocal of the total market cap of the S&P500). In this case Exxon Mobile accounts for 56,82% and American Airways accounts for 43,18%.
Remember that we two variances (call them X and Y) for Exxon Mobile (Asset A of the market portfolio) and American Airways (Asset B of the market portfolio) are our two random variables.
Variance Exxon Mobile (X): 0,01
Variance American Airways (Y): 0,04
however, we do have four constants (2 weights for the investors portfolio - how he want's to split his wealth - and 2 weights - the so called market caps of Exxon and American Airways in %-terms - these weights add up to 1 and show how much of these two assets contribute to the whole market risk/reward relationship) given as:
a, b, c, d
We can then write the following covariance property:
Cov(aX + bY, cX + dY) >>> which splits up into
a*c*Cov(X,X) + b*d*Cov(Y,Y) + a*d*Cov(X,Y) + b*c*Cov(Y,X)
Remember: The covariance with itself Cov(X,X) and Cov(Y,Y) simplifies to the simple variance (sigma^2) of X and Y!
We write down the following:
a*c*Var(X) + b*d*Var(Y) + (a*d + b*c) * Cov(X,Y)
>>> this right-hand side is a simplification, it can also be written in the extended form given above: a*d*Cov(X,Y) + b*c*Cov(Y,X)
Remember: the covariance is invariant to whether it is written as Cov(X,Y) or Cov(Y,X). The covariance between (Exxon, American Airways) and the covariance between (American Airways, Exxon) yields the same result!
We therefore have the following covariance of the investor portfolio with the market portfolio:
Let's take the 50% wealth devoted to Asset A (Exxon) and 50% devoted to Asset B (American) case where we have:
a (50% wealth devoted to Exxon) = 0,5
b (50% wealth devoted to American) = 0,5
c (% weight of Exxon or call it %-market cap) = 0,5682
d (% weight of American or call it %-market cap)= 0,4318
variance of Exxon (X) = 0,01
variance of American Airways (Y) = 0,04
[Notice: the covariance is already given/computed in the spreadsheet: 0,006]
Following the above equation we compute:
covariance = 0,5*0,5682*0,01 + 0,5*0,4318*0,04 +(0,5*0,4318 + 0,5*0,5682) * 0,006 = 0,01448
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