In my last post I showed that there are far more that a googol permutations of portfolio of 100 assets with (positive, non-zero) weights in increments of 10 basis points, or 0.1%. That number can be expressed as C(999,99), or C(999,900) or 999!/(99!*900!), or ~6.385*10138. Out of sheer audacity, I will call this number Balhiser’s first constant (Kβ1). [Wouldn’t it be ironic and embarrassing if my math was incorrect?]
In the spirit of Alan Turing’s 100th birthday today and David Hilbert’s 23 unsolved problems of 1900, I propose the creation of an initial set of financial problems to rate the general effectiveness of various portfolio-optimization algorithms. These problems would be of a similar form: each having a search space of Kβ1. There would be 23 initial problems P1…P23. Each would have a series of 37 monthly absolute returns. Each security will have an expected annualized 3-year return (some based on the historic 37-month returns, others independent). The challenge for any algorithm A to score the best average score on these problems.
I propose the following scoring measures: 1) S”(A) (S double prime) which simply computes the least average semi-variance portfolio independent of expected return. 2) S'(A) which computes the best average semi-variance and expected return efficient frontier versus a baseline frontier. 3) S(A) which computes the best average semi-variance, variance, and expected return efficient frontier surface versus a baseline surface. Any algorithm would be disqualified if any single test took longer than 10 minutes. Similarly any algorithm would be disqualified if it failed to produce a “sufficient solution density and breadth” for S’ and S” on any test. Obviously, a standard benchmark computer would be required. Any OS, supporting software, etc could be used for purposes of benchmarking.
The benchmark computer would likely be a well-equipped multi-core system such as a 32 GB Intel i7-3770 system. There could be separate benchmarks for parallel computing, where the algorithm + hardware was tested as holistic system.
I propose these initial portfolio benchmarks for a variety of reasons. 1) Similar standardized benchmarks have been very helpful in evaluating and improving algorithms in other fields such as electrical engineering. 2) Providing a standard that helps separate statistically significant from anecdotal inference. 3) Illustrate both the challenge and the opportunity for financial algorithms to solve important investing problems. 4) Lowering barriers to entry for financial algorithm developers (and thus lowering the cost of high-quality algorithms to financial businesses). 5) I believe HAL0 can provide superior results.