April 2013 Question 4 iii)

[s16455441]

I am really struggling to understand how the question is calculating the covariances between the Return on the Asset and the Return on the Market

The way I normally approach these questions is similar to September 2012, Question9 where I do
Cov(Ri, Rm) = Cov(Ra, 0.5*Ra + 0.5Rb) = 0.5*Var(Ra) + 0.5Cov(RaRb). This worked as we knew the variance and covariance.

In this question I was hoping to construct the answer along the lines of:
Cov(Ra, Rm)
= Cov(Ra, 0.25Ra, 0.5Rb. 0.25Rc)
= 0.25*Var(Ra) + 0.5*Cov(Ra, Rb) + 0.25*Cov(Ra, Rc)

However as we don't have the Covariance between the assets I am assuming this isn't possible?
And there is the added complicated that we have different possible states of the market

My interpretation of the solution is

Cov(Ra, Rm) =
prob we are in state 1 * expected return in the market in state 1 * return of asset A in state 1 +
prob we are in state 2 * expected return in the market in state 2 * return of asset A in state 2 +
prob we are in state 3 * expected return in the market in state 3 * return of asset A in state 3 -
expected return of asset A * expected return of the market.

I don't understand how the above formula is constructed, e.g. is it from the core reading notes or is it just covariance rules we should know?

Many thanks in advance for help.

 

[John Potter]

Your interpretation of the solution is correct and it comes from the definition of covariance...

cov[X,Y] = E[XY] - E[X]E[Y]

where E[XY] = sum across states of { value of X * value of Y * prob }

However, this doesn't look the smartest way to go. I'd just use the security market line on each asset.

Mind you, Excel would make things quicker, especially with SUMPRODUCT to do all the expectations. This is good practice anyway, considering you could get something like this in Paper B?

John