Portfolio Volatility Constraints, Cash Weights, and Conic Optimization
Summary
The note considers a minimum-variance portfolio when weights need not sum to one and total variance must stay below a limit. It starts from the fully invested minimum-variance solution and asks whether scaling that portfolio can satisfy the volatility constraint. Scaling works as a feasible adjustment when the constraint is binding, but the answer interprets the unused weight as an implicit cash position.
This cash interpretation gives a theoretical basis for the scaling approach when the residual cash weight is positive, since adding cash lowers portfolio volatility linearly. It also reveals a limitation: the equivalence does not hold if the solution requires borrowing, represented by a negative cash position. For a general solution to the constrained problem, the answer recommends conic optimization. The exchange provides conceptual guidance rather than a worked optimization example or evidence about how commonly the shortcut is used.
Key ideas
- Scaling minimum-variance weights can meet a tighter variance constraint in the stated setup.
- The leftover portfolio weight can be interpreted as an implicit cash position.
- The cash interpretation works only when that cash position is nonnegative.
- Borrowing requirements expose a limitation of the simple scaling approach.
- Conic optimization is suggested for solving the constrained portfolio problem generally.
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# An ad hoc portfolio optimization scheme
# An ad hoc portfolio optimization scheme
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?
## Answer by Tim Wilding (score 2, accepted)
https://quant.stackexchange.com/a/38794
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).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.