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Say at each time $t$ I have a covariance matrix for the next period. Call this $\Sigma_{t+1}$. If I choose portfolio weights $w$ to minimize the variance, subject to the constraint that $\sum_i w_i = 1$, then the weight vector is $$ w^* = \frac{\Sigma^{-1} 1}{1^t\Sigma^{-1}1}. $$

If I relax the assumption that the weights sum to $1$, and instead I constrain them by forcing the sum to be less than or equal to $1$, and I constrain the overall variance $w^T \Sigma w \le $c, is there some quick adjustment of $w^*$, or do I have to learn about different procedures besides Lagrange multipliers?

I can see that multiplying the optimal weights $w^*$ by $\sqrt{\frac{c}{w^{*T}\Sigma w^*}}$ enforces the constraint if $c < w^{*T}\Sigma w^*$. Is this commonly done in practice? Is there theoretical justification for this?

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There is a theoretical justification for your use of $w^*$ constrained to less than 1. I am not sure how often this is done in practice, but this kind of approach is used for optimization. Adding an implicit cash position to your vector of weights would generate an identical solution to the one you outlined above for the constrained volatility problem. Adding cash to the portfolio reduces the volatility in a linear fashion. This solution is only identical if the implicit cash position is positive, and this shows some of the limitations of this approach. For example, what if we require borrowing - a negative cash position?

Generally speaking, if you want to solve the problem properly for $w$, you should use some form of conic programming (https://en.wikipedia.org/wiki/Second-order_cone_programming).

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  • $\begingroup$ Could you explain that first part. If you add a risk less asset to the collection your covariance matrix becomes singular and there’s no inverse for it. $\endgroup$ – Taylor Mar 15 '18 at 15:30
  • $\begingroup$ The cash is implicit and there is no need to invert the full, expanded matrix. Your maths remains the same and the solution only relies on the covariance matrix of the risky assets. $\endgroup$ – Tim Wilding Mar 16 '18 at 15:46

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