Overview
Trend Diversification
The standard mathematical treatment of diversification given above ignores the trends developing within each asset or asset class. Theoretically, it is possible for two assets to trend in the direction while the day to day returns are uncorrelated, or even anticorrelated. From this perspective, the most effective way to diversify portfolio is to find assets that do not form down trends at the same time that the portfolio is trending down.
Note that we do not seek assets that trend in oppositive directions. We dont necessarily want our diversifying assets to trend down while our portfolio is trending up.
As an example, consider a manager trying to diversify an equity portfolio. Equities trend up most of the time (say 90%). If the manager found an asset that trended in the opposite direction to equities, they would get an asset that trends down 90% of the time. (A true anti equity trending asset would be short equities, but then you would end up with the risk free rate).
Down Trend Overlap
Down trend overlap takes two assets (prices series) and measures the percentage of the time that
{% overlap = \frac{time \, both \, trending \, down}{time \, at \, least \, one \, trending \, down} %}
Calculation
Calculating down trend overlap is relatively easy. If the data includes a column indicating the direction of trend1 and the direction of trend2 (that is, trend1 is positive if the trend is positive, and negative if the trend is negative and zero if there is no trend)
let downtrend = data.filter(p=>p.trend1<0 || p.trend2<0);
let overlapping = downtrend.(p=>p.trend1*p.trend2>0);
let overlap = overlapping.length/downtrend.length;