Portfolio Optimization Methods Beyond Exhaustive Weight Searches

# Portfolio optimisation - Non brute force solutions to optimisation problems


# Portfolio optimisation - Non brute force solutions to optimisation problems












Recently I wrote a program in Python which extracts stock data for a designated period and frequency of the predetermined stocks and then optimises the portfolio using the Sharpe ratio. In order to generate the different kinds of weights, I wrote a generator that gives me all the possible weights (all weights are non-negative and have to sum to 100).

Now the problem with this is that while I can still confidently simulate a 5 stock portfolio (which takes about 5 minutes), simulating a 6 stock portfolio would take up to 10 hours on my machine, given all the possible distributions of weights (if I recall correctly it should be around 96 millions). I would want to push this as far as possible, optimally to a point where the gains from un-systematic risk reduction are minimal as a consequence of adding additional stocks.

Now my question is what are the "smart ways" to do this (obviously the way I am using is neat since you get all the possible outcomes, but cant be considered seriously given the resource and time intensity)? What kind of optimisation techniques would one use in order to make this process faster?

I am not searching for any single solution, but rather for an elaboration as to what optimization techniques exist out there, that could be used in order to approach this problem and what are their advantages or/and disadvantages.

Thank you for any suggestions!

## Answer by Chris Taylor (score 2, accepted)

https://quant.stackexchange.com/a/35162

There are a few commonly used solutions

- Analytic solutions. These can be particularly effective when the objective function is linear/quadratic, and the constraints are all linear equality constraints (e.g. self-financing, zero beta, sector neutrality etc)

- Quadratic optimization. This works well with linear/quadratic objective function, and linear equality or inequality constraints (all the above, plus turnover constraints, position limits, sector exposure limits etc)

- Conic optimization. This extends the applicability of the above solutions to objective functions that can be non-linear/quadratic and constraints that can be non-linear, as long as the objective function and constraints can be transformed to either linear/quadratic or conic.

- Non-linear optimization. Generic non-linear optimizers (e.g. interior point methods or Nelder-Mead) can sometimes give satisfactory solutions on a wide range of objective functions and constraints -- the only requirements generally being that the objective function/constraints are sufficiently smooth (no discontinuities). It helps if you can calculate an analytic gradient.

To optimize for the Sharpe ratio in cases 1 and 2, you generally have to consider a family of solutions $w_\lambda$ that maximise

$$\alpha^T w_\lambda - \lambda w_\lambda^T\Sigma w_\lambda$$

and chose the value of $\lambda$ that gives the highest Sharpe Ratio. In cases 3 and 4 you can optimize for the Sharpe ratio directly.

## Answer by RRG (score 2)

https://quant.stackexchange.com/a/35160

The maximum Sharpe portfolio has a closed form solution.

Let $w_i$ be the weight of asset $i$ so that $\sum_i w_i = 1$. The weights for the maximum Sharpe portfolio is then $$ \bf{w} = \frac{\bf{\Sigma}^{-1}\bf{\mu}}{\bf{1}^T\bf{\Sigma}^{-1}\bf{\mu}}, $$ where $\bf{\Sigma}$ is the return covariance matrix, $\mu$ is the excess return expectations, and $\bf{1}$ is a vector of ones.