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 wit the market portfolio:

Let's take the 50% wealth devoted to Asset A (Exx0n) 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 have:

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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