Combining systems and diversification
Combine two trading systems, calculate portfolio volatility with correlation, and compare equal-capital and inverse-volatility weights.
What you will learn
- Why combining systems with low correlation can improve return per unit of risk
- How to calculate the volatility of a two-system portfolio
- How equal-capital and inverse-volatility weights differ
- Practical rules for choosing, weighting and checking a set of systems
Explanation
Correlation and diversification between assets are covered in the ETF and risk management courses. Here you apply the same mathematics to return streams of systems: the monthly returns that each system produces.
What makes systems different
Two systems diversify each other when their returns do not move together. Sources of difference are:
- Style: a trend-following system profits from long moves; a mean-reversion system profits when prices return to an average after a short move.
- Market: share indices, bonds, currencies, commodities.
- Timeframe: a system with weekly signals and one with daily signals react to different moves.
Running the same idea with slightly different parameters gives highly correlated systems and little diversification.
Portfolio volatility
Volatility here means the annualised standard deviation of returns. For two systems with weights wA and wB, volatilities σA and σB and correlation ρ (rho, from −1 to +1):
σP = √(wA²σA² + wB²σB² + 2·wA·wB·ρ·σA·σB)
The portfolio return is simply the weighted average: rP = wA·rA + wB·rB. Return is averaged, but volatility is reduced when ρ is below 1. That is the benefit of diversification.
To compare, use the return-to-volatility ratio (return ÷ volatility). It is the Sharpe ratio from lesson 5 with the risk-free rate set to zero for simplicity.
Weighting methods
- Equal capital: each system gets the same euro amount. The more volatile system then dominates the risk.
- Inverse volatility: weight each system by 1 ÷ its volatility, so each contributes similar risk. With two systems: wA = (1/σA) ÷ (1/σA + 1/σB).
Netting signals
If two systems trade the same instrument, one may buy while the other sells. You can run them in separate sub-accounts, or net the signals (trade only the difference), which saves costs but must be tested as a combined system.
Rules of thumb
- Calculate correlations on monthly returns over the full test, and also over the worst periods. Correlations measured in the past can change, and they often rise during market stress.
- Check whether drawdowns overlap in time, not only correlation numbers.
- Prefer a few genuinely different systems over many similar ones.
- Rebalance weights on a fixed schedule (for example quarterly), written as a rule.
Worked example
Hypothetical example: System A (trend-following) has an expected annual return of 8% and volatility of 16%. System B (mean-reversion) has 5% return and 8% volatility. Their correlation is 0.1. Assumptions: returns and volatilities stay constant, annual rebalancing, costs and taxes excluded.
Stand-alone ratios: A = 8 ÷ 16 = 0.50; B = 5 ÷ 8 ≈ 0.63.
Equal capital (50/50):
Return = 0.5 × 8% + 0.5 × 5% = 6.5%.
Variance = 0.25 × 256 + 0.25 × 64 + 2 × 0.5 × 0.5 × 0.1 × 16 × 8 = 64 + 16 + 6.4 = 86.4 (in %²).
Volatility = √86.4 ≈ 9.3%. Ratio = 6.5 ÷ 9.3 ≈ 0.70.
Inverse volatility: 1/16 = 0.0625 and 1/8 = 0.125, so wA = 0.0625 ÷ 0.1875 = 1/3 and wB = 2/3.
Return = 1/3 × 8% + 2/3 × 5% = 6.0%. Volatility ≈ 7.9%. Ratio ≈ 0.76.
| Portfolio | Return | Volatility (ρ = 0.1) | Ratio | Volatility (ρ = 0.6) | Ratio |
|---|---|---|---|---|---|
| A only | 8.0% | 16.0% | 0.50 | 16.0% | 0.50 |
| B only | 5.0% | 8.0% | 0.63 | 8.0% | 0.63 |
| 50/50 | 6.5% | 9.3% | 0.70 | 10.9% | 0.60 |
| Inverse volatility | 6.0% | 7.9% | 0.76 | 9.5% | 0.63 |
Both combinations beat each system alone when ρ = 0.1. If correlation rises to 0.6, most of the benefit disappears. The value of combining depends on correlation staying low.
Common mistake
Practice activity
On paper: System A has 10% return and 20% volatility; System B has 6% return and 12% volatility. Use 50/50 weights. (a) Calculate portfolio return. (b) Calculate portfolio volatility and the ratio for ρ = 0 and for ρ = 0.8. (c) Compare with the stand-alone ratios.
Show solution
(a) 0.5 × 10% + 0.5 × 6% = 8%.
(b) ρ = 0: variance = 0.25 × 400 + 0.25 × 144 = 136, volatility ≈ 11.7%, ratio ≈ 0.69. ρ = 0.8: variance = 136 + 2 × 0.25 × 0.8 × 20 × 12 = 136 + 96 = 232, volatility ≈ 15.2%, ratio ≈ 0.53.
(c) Each system alone has a ratio of 0.50. With ρ = 0 the mix is clearly better; with ρ = 0.8 it is only slightly better.
Limitations: real returns, volatilities and correlations change over time, and past values are not forecasts.
Key takeaways
- Combining systems averages returns but can reduce volatility when correlation is low.
- Use the two-asset volatility formula to quantify the effect.
- Inverse-volatility weights balance risk contributions better than equal capital.
- Check correlations and overlapping drawdowns in stress periods, not only on average.
Next step
In the next lesson you will move a tested system into live trading and define monitoring rules and stop criteria in advance.
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