Sharpe Ratio

CFA Level III
#1

in order to increase the sharpe ratio, do you always add an atlernative investment (or any other asset) to a portfolio if the correlation coefficient is less than 1? or do you not add the asset to the portfolio if the asset has a lower sharpe ratio than the portfolio??

#2

Sharpe(Asset) > Sharp(Portfolio) * Corr(Asset,Portfolio)

#3

Add if:

Alternative Investment Sharpe Ratio > (Portfolio Sharpe Ratio x correlation with alternative investment)

#4

they got it

#5

Appreciate it if anyone can derive or explain the reasoning of Sharpe(Asset) > Sharp(Portfolio) * Corr(Asset,Portfolio)? To be honet, I just memorized the formula and applied it whenever applicable.

#6

add real estate

#7

If (Sharpe(Asset) > Sharp(Portfolio) * Corr(Asset,Portfolio))

{

Add it!

}

#8

tulkuu - When you add a new asset to the portfolio, it can increase Sharpe in 2 ways:

1 - it has its own Sharpe to be added to the portfolio.

2 - if the correlation is smaller than 1, then the risk of the portfolio will be smaller than the sum of the risks (which increases Sharpe, all other things equal).

The formula basically says 2 things:

1 - If new asset’s Sharpe is better, portfolio’s Sharpe will improve.

2 - Even f the new asset’s Sharpe isn’t better, the correlation effect can make the new portfolio have a better Sharpe than before.

I think it makes more sense to think in terms like Sharpe(asset)/Corr(asset, port) > Sharpe (port) , but the overall idea is the same.

Not the best explanation, but it was enough to make me understand the main point. Hope it helps.

#9

Hey Tulkuu - It would be something like this:

ac = Asset Class; p = Portfolio; Rfr = Risk Free Rate; SD = Std Dev; Corr(ac,p) = Cov(ac,p) / (SDac * SDp)

Sharpe AC = Sharpe P * Corr(AC,P) or

(Rac - Rfr) / SDac = [(Rp - Rfr) / SDp] * [Cov(ac,p) / (SDac * SDp)]

So the SDp’s on the right side cancel out and if you multiply both sides by SDac you get:

Rac - Rfr = Cov(ac,p) * (Rp - Rfr)

Now move the Rfr on the left to the right side of the equation:

Rac = Rfr + Cov(ac,p) * (Rp - Rfr)

Which is basically the CAPM.

so in summation: If the return of your asset class is greater than the expected return of your portfolio adjusted for the risk of the new asset class then you should add the new asset class to the portfolio.

#10

Thanks, crazy man and finninja…interesting to see it from different angles. The formula looks like an extended CAPM – regressing the new asset against the existing portfolio.

Or is it just an approximation since it assumes the weights hold the same? But the weights(sum to 1)will change if the new asset class is added in…

#11

I made a small mistake in my formula, SDp does not actually cancel out so the equation is actually:

Rac - Rfr = [(Rp - Rfr) / SDp] * [Cov(ac,p) / SDp]

or

Rac = Rfr + [Cov(ac,p) / SDp^2] * (Rp - Rfr)

But Cov(ac,p) / SDp^2] is the actual formula for Beta so this formula still reduces to the CAPM