Question

It is possible to form a two-stock portfolio that is risk free if the correlation coefficient of the returns of the two stocks is?

Answer 1

Yes, it is possible to form a two-stock portfolio that is risk free when the two assets are perfectly negatively correlated.

1] The SD of a two asset portfolio is given by:

SD [Volatility] of a two asset portfolio [asset 'a' and asset 'b'] is given by the formula, sdp = [sda^2*wa^2+sdb^2*wb^2+2*sda*sdb*wa*wb*Cor(a,b)]^0.5
Where,
sda and sdb are the standard deviations, wa and wb are the weights of the two assets in the portfolio and Cor(a,b) the correlation of the returns of the two assets.

2] When correlation between the two assests is -1 [perfectly negatively correlated], the above equation becomes:

SD of portfolio = wa*sda-wb*sdb

3] For being riskless, the sum of the above equation for portfolio should be set as 0:

wa*sda-wb*sdb = 0

Now, wb = 1-wa, the above equality becomes:

wa*sda-(1-wa)*sdb = 0

wa*sda = sdb-wa*sdb

wa*(sda+sdb) =sdb

wa = sdb/(sda+sdb) and wb = sda/(sda+sdb)---These are the weights of the two risky assets to give an portfolio SD of 0 [riskless portfolio]