Selecting a Fixed-Size Asset Basket by Sharpe Ratio
Summary
The document explains how to choose a fixed-size, equally weighted basket from a larger asset universe using estimated returns and a covariance matrix. It formulates the selection as a binary optimization problem: indicators represent whether assets are included, while the objective measures expected portfolio return relative to risk. Since this objective is not convex, the proposed route is to trace the efficient frontier instead.
For a series of target returns, solve a mixed-integer convex quadratic program that minimizes variance while meeting the return target and holding the basket size fixed. Then compare the Sharpe ratios of the resulting frontier portfolios. The highest feasible target return can be bounded using the assets with the largest expected returns. A second answer recommends optimization heuristics, which can handle other objectives such as drawdown and may work at larger scales, though they can require custom programming. The document provides no empirical comparison of these approaches; results depend on the quality of the return and covariance estimates and on the solver or heuristic used.
Key ideas
- Represent each asset choice with a binary indicator and constrain the basket to a fixed size.
- The maximum Sharpe portfolio must lie on the return-variance efficient frontier.
- Trace that frontier by minimizing variance subject to different minimum-return targets.
- Heuristics offer flexibility for alternative portfolio objectives but may require implementation work.
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Full text
# how to choose top n assets?
# how to choose top n assets?
I have m assets, and have estimated their future returns and covariance matrix.
I would like to invest in an evenly weighted n product basket from this universe, where 0<n<m.
How do i find the basket with the highest sharpe, without having to calculate the sharpe for every possible portfolio?
## Answer by David Nehme (score 9)
https://quant.stackexchange.com/a/9794
If $Q$ is your covariance matrix, and $r$ is a vector of your expected returns, then the maximum Sharpe ratio is given by the following math program. $${\rm maximize} \frac{r^t x}{\sqrt{0.5 x^t Q x}}$$ subject to $$ 1^t x = m$$ $$ x \in \{0,1\}^n$$ Where $x$ is a vector of indicators of which of the $n$ assets are part of the $m$ selected assets. While the objective is not convex, its solution lies of the efficient frontier of $$\rm{maximize}\ r^t x, -0.5 x^t Q x.$$ You can compute the efficient frontier of the return/variance by solving the following mixed-integer convex-quadratic program for multiple values of $r^*$. $$ {\rm minimize} \frac{1}{2} x^t Q x - \epsilon r^t x$$ subject to $$ r^t x \ge r^*$$ $$ 1^t x = m$$ $$ x \in \{0,1\}^n$$
where $r$ is a vector of expected returns, and $r^*$ is a target expected return. The optimal value for any any value of $r^*$ that is feasible will yield a point on the efficient frontier. The portfolio with the best Sharpe ratio will be on the efficient frontier, so by iteratively solving the above problem for multiple values of $r^*$, you can find the value of $r^*$ that yields the best Sharpe ratio. The highest feasible value for $r^*$ can be computed by taking the top $m$ expected returns.
## Answer by Enrico Schumann (score 3)
https://quant.stackexchange.com/a/9822
For such a problem ("selecting n out of m") you can use optimisation heuristics. These algorithms work well even for large n and m, and they are flexible: you may as well select a portfolio that minimises some other function, for instance, the portfolio's drawdown. The downside is that you may have to do some programming yourself.
An example very similar to your problem is described in "Heuristic Methods in Finance" ( http://ssrn.com/abstract=1794290 ) The R code for the example is here http://cran.r-project.org/web/packages/NMOF/vignettes/LSselect.pdfShown 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.