# Problem with determining weights in tangency portfolio (2 risky assets)

I use the following well known formula in order to determine the weight of asset i in the tangency portfolio (in the case of two risky assets):

$w_{i,T}=\frac{\sigma[r_2]^2E[R_1]-\sigma[r_1,r_2]E[R_2]}{\sigma[r_2]^2E[R_1]-\sigma[r_1,r_2]E[R_2]+\sigma[r_1]^2E[R_2]-\sigma[r_1,r_2]E[R_1]}$

where $E[R_i]=r_i-r_f$ is the excess return on asset i (in excess of the riskless rate).

Whilst I think I understand the underlying rational and derivation of this formula, it leads to some weird behavior which I don't understand.

For instance, let me choose as input $E[R_1]=0,05$, $E[R_2]=0,1$, $\sigma_1=0,12$, $\sigma_2=0,20$ and let me play around with the correlation coefficient $\rho_{1,2}$ (where $\sigma_{1,2}=\rho_{1,2}\sigma_1\sigma_2$). The higher the correlation, the lower the weight of asset 1. For instance, in the case of $\rho_{1,2}=0,8$ the weight of asset 1 turns out to be 14,29%. In the case of $\rho_{1,2}=0,9$, the weight of asset 1 is -80%. In the case of a long-only restriction, I’d assume that asset 1 gets a weight of 0% and asset 2 a weight of 100% - which makes intuitively sense. However, if the correlation is $\rho_{1,2}=1,0$, the weight is 250% - i.e. again assuming a long-only constraint, the weights in the tangency portfolio would be now the other way around. This behavior is not limited to the specific input parameters. Why is that? Obviously there is something about this formula and tangency portfolio concept which I don’t fully understand yet. I would appreciate any help. Thank you.

• try checking the expected return of the minimal variance portfolio, if this is below the risk-free rate, everything breaks. – Mark Joshi May 30 '16 at 23:37
• Thanks for your comment. Indeed - given my other input parameters, for correlation coefficients >0.95 the expected return of the portfolio becomes negative, i.e. definitively lower than the risk-free rate.... I see the results but I don't quite understand yet what that actually means. – tk79 Jun 1 '16 at 12:38
• well the tangent point ends up being on the lower half of the hyperbola instead of the upper half, so the portfolio is optimally inefficient. See my "introduction to mathematical portfolio theory" – Mark Joshi Jun 2 '16 at 0:05