Last year, before the crash of the emerging markets – pro articulum in general – Prof. Jeremy Siegel was featured in an advert played regularly on CNBC for Wisdom Tree, talking about the inherent "buy high, sell low" strategy embedded in cap. weighted indices.

The basic problem is that when the price of a subset of the index increases then their weight relative to the rest of the index also increases. The index tracking investor is then required to buy more of those components, at their new higher price. If their prices should subsequently decline, then the index tracking investor will be required to sell a little of the investment, for the same reasoning as before, at the new lower price.

Unfortunately, stocks do regularly go up and down relative to each other and so the logic embedded in the previous paragraph represents an embedded buy high – sell low strategy which is overlaid over the basic strategy represented by the index. This is one of the defects of cap. weighted indices and will lead a fund manager that attempts to track such an index to underperform through no fault of their own.

The Markowitz Portfolio is constructed to be Mean-Variance efficient and weights components so that the expected risk-adjusted profit from each position is equal. However, cap. weighting doesn't follow any utility driven formalism and it explicitly contradicts known facts about the market (it overweights large cap. stocks whereas academic reasarch by Fama and French indicates that small cap. stocks consistently outperform).

The adverts. caught my attention because I had just tackled a similar buy high – sell low defect in the basket I own to track the Compact Model Portfolio. The portfolio that tracks the CMP Index is equally weighted, meaning that we allocate the same fraction of the overall equity to each individual investment.

Now equal weighting also has an embedded strategy, but in this case it is reversion rather than momentum. With an equal weighted basket, every time returns occur we need to reduce the position in the stocks that outperformed and increase the position in the stocks that underperformed, in order that we maintain the equal weighting. This is an embedded sell high – buy low strategy.

I was aware of this, but as I watched my basket I realized that I kept repeating the opposite. On the daily rebalance, the strategy would buy some more of a stock that went up at the end of the day and then, then next day, if it lost money, it would sell at a loss. This was repeated again and again.

I finally realized that this was because I was rounding my position into round lots, of a given size. The conventional algorithm for rounding positive numbers is to add one half and then truncate to an integer. The number of lots to hold in a given company is the fraction of the capital allocated to that company divided by the product of the price and the lot size. Following conventional ½ rounding we tend to round up after we've made money and round down after we've lost money. This is an embedded buy high – sell low strategy.

I solved this by rounding against it. I round up on a losing day and round down on a winning day. i.e.

shares=lotsize×⌊capital/(lotsize×price)−½sign δprice⌋.

This seems to work.

n.b. The notation ⌊x⌋ means floor(x) which means the largest integer less than or equal to x.