7.1 The coordinated max-median rule for portfolio selection  (Page 2/6)

The coordinated max-median rule

Introduction

We now consider a strategy which allows us to implicitly capture the joint performance of securities as part of our selection criteria. Our goal is to pick, from the universe of investible stocks, a meaningful handful on which to equally allocate a given investment quantity on a yearly basis. As a first step, we consider the S&P 500 constituents to be our universe of stocks from which can select a critical few (a number which we have set initially, and somewhat arbitrarily, to 20. This quantity seemed both appealing and reasonable in terms of being financially manageable and computationally feasible). It is also worthwhile noting, that we regard limiting an investor to select from the S&P 500 (or any other well-known index) as both a reasonable and soundly restricted starting point. Furthermore, we also know that stocks listed in the S&P 500 are representative of various market sectors (inherently diversified) as well as of various reasonable company sizes (in terms of market capitalization). Additionally, other filtering criteria inherent in a reasonable-size index (in terms of the number of constituents), seem to provide a good baseline both as a benchmark (to outperform) and as a sensible constraint to the universe of all potentially-considered stocks.

Preliminary setup

Our first step is to select a subset of stocks from a given index in which to allocate a given investment at any given point in time. Here, and in general, we can start by considering a subset of n stocks from a given index I with K constituents. Our evaluations considered I =S&P 500 (for which $K=500$ ) and assembling baskets of $n=20$ each time. Based on this setup, there is a total of $C_{20}^{500}\approx 2.667\times {10}^{35}$ unique baskets of randomly selected securities that we could potentially consider. Clearly, if we require evaluating some optimal objective function over all possible combinations, this becomes computationally infeasible.

Instead, we proceed by selecting stocks according to some plausible robust criterion that can be applied to any randomly assembled basket (the most appealing, both in terms of interpretation and prior results, being the median of the portfolio daily-returns). We also note, and quite emphatically so, that to both evaluate a meaningful amount of portfolios as well as to assess procedure repeatability, we clearly need to parallelize this effort.

Algorithm

Consider the following algorithm:

1. Pick n -stocks (e.g. $n=20$ ) from the S&P 500 Index at random.
2. Form Portfolio j (start with $j=1$ ) at time $t=0$ , i.e. $P_j(t=0)$ , by equal-weight investment in these n -stocks.
3. On a day-to-day basis (and for T trading days in any given year) compute the daily-returns for Portfolio j :
   $$r_j\left(t\right):=\frac{P_j\left(t\right)-P_j(t-1)}{P_j(t-1)};t=1,2,...,T$$
4. Sort these for the years trading days.
5. Calculate the median daily-return for Portfolio j , let $\widetilde{P_j}:=\mathtt{median}\left(P_j\right)$ .
6. Repeat Steps (1-5) above for $j=1,2,...,J$ (e.g. $J=10,000$ ) additional randomly selected portfolios.
7. Pick the portfolio with the highest median, i.e. $P_{j^{*}}$ s.t. $j^{*}=\underset{j\in \{1,2,...,J\}}{argmax}\left[\widetilde{P_j}\right]$ .
8. Invest equally in $P_{j^{*}}$ .
9. Hold for one year, then liquidate.
10. Repeat Steps (1-9), yearly, over the time-frame of interest.