First, a quick introduction to Modern Portfolio Theory (MPT). In the Markowitz world, risk is measured by the volatility of returns. If we plot the return-volatility results of all possible portfolio combinations, we get an area that looks like a hyperbola on the left side. The upper part of this edge is called the efficient frontier, because it represents the best possible returns attainable for a given amount of risk.
The line that starts at the risk-free rate and is tangent to the efficient frontier (the capital allocation line) has a slope equal to the best attainable Sharpe ratio by this universe of assets (it is easy to see why graphically). Thus whatever your risk appetite might be, you will hold the tangency portfolio, and use cash to adjust your leverage upwards or downwards for the portfolio to meet your risk appetite. Since the tangency portfolio offers the highest Sharpe ratio, this guarantees the best possible returns for the amount of risk you wish to take.
What I'm going to do is take the main ideas from MPT and apply them to a portfolio of trading strategies. The most important difference is that traders identify risk as several different things (such as maximum drawdown) beyond volatility.
For demonstration purposes I'll use three extremely basic EOD strats on SPY:
50/200 day MA cross long/short
RSI(2) 1-day mean reversal long/short
RSI(14) 50-day percentile long-only
As well as buying and holding SPY.
Some basic stats:
The equity curves:
Correlations:
Even these simplistic strategies, on the same instrument, two of which are even conceptually very similar, show very little correlation. The gains from good diversification in terms of risk will be gigantic. Note however that correlation does not tell the whole story. Assets/strategies with very low correlations over long periods may still be highly correlated during crash times. The analysis below results in an approach that implicitly takes into consideration correlations during such catastrophic periods; by minimizing maximum drawdown we implicitly take into account correlations during heavy drawdown periods.
Note that the absolute performance of the strategies is not what interests us in this case; what concerns us is the improvement we can achieve by managing a portfolio of strategies properly.
MPT optimizes for Sharpe ratio, but (at the cost of optimizational complexity) we can optimize for whatever we want. While a buy and hold investor with a 25 year horizon may not mind a 55% max drawdown, it's an impossible number for the trader. As such, risk for us is much more than simple volatility. We can define risk as a composite of several factors: max dd, volatility, downward volatility, average dd, average days to recover, etc.
Essentially it all depends on how much of each risk you're willing to trade for additional return. For example, you might say that 0.5% of additional max dd is acceptable if it will give you at least an additional 1% annual return.
To keep it simple I'll define risk in this case as maxdd*2 + daily return std/2. We'll call the ratio of annual returns over this risk measure the "return-to-risk ratio".
Plotting risk & return for various random portfolios we get this familiar-looking efficient frontier:
Let's go ahead and see what kind of values we get for various portfolios:
(the optimal portfolio weights were 21.29%, 1.71%, 77%)
By diversifying using "inferior" strategies, even with the simplistic equally weighted approach, we have a significant improvement in return-to-risk ratio.
Finally, let's do some walk-forward optimization with various lookback period lengths:
Somewhat surprisingly, walk-forward optimization does not manage to improve on the equally weighted portfolio. I would not expect this to be the case in general.
The nice thing about knowing the portfolio of strategies with the highest return-to-risk ratio is that if you want more or less risk (higher or lower returns), you can simply leverage this portfolio …