Why Tangency Portfolio Weights Can Flip at High Correlation
Summary
The document presents a question about the two risky asset tangency portfolio formula. It explores how the calculated weight of one asset changes as the assets’ correlation rises, holding expected excess returns and volatilities fixed. The example reports a positive allocation at a lower correlation, a negative allocation at a higher one, and a large positive allocation when correlation reaches one; the author wonders how this squares with long-only constraints.
No answer or derivation is included, so the post does not resolve the apparent reversal. It does highlight that the unconstrained tangency solution can produce weights outside the long-only range, and that correlation affects the covariance terms in the formula. Applying a long-only constraint requires solving the constrained portfolio problem; simply interpreting an out-of-range unconstrained weight as a portfolio weight is not enough. The numerical illustration is limited to one set of input assumptions.
Key ideas
- The tangency weight formula depends on expected excess returns, asset variances, and covariance.
- The example shows that changing correlation can move the unconstrained weight below zero.
- At perfect correlation, the stated formula produces a large positive weight for the first asset.
- A long-only portfolio requires imposing constraints rather than directly accepting unconstrained formula weights.
- The document poses the issue but provides no answer or general resolution.
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Full text
# Problem with determining weights in tangency portfolio (2 risky assets)
# 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.