Cont. from above...... Now, even above solution is not correct because just looking at say year 2004 trade list of multiple systems and computing the correlation table from that would be just one instance of the correlation table. As trades in individual systems can come in different sequences (the whole premise behind doing Monte Carlo), therefore we do not know the true correlation among systems by just looking at 1 year realized trades. Finally, I think an "easier and practical solution" would be to attack this problem in a 3-pronged way: A) Classify all the systems into 3 categories - best of the breed (say PF > 2.8, Calmar > 10 etc.), very good (say PF > 2, Calmar > 7) and definitely tradeable (say PF > 1.5, Calmar > 4 etc.). Construct a portfolio including all the best of the breed systems with weight 1 given to all of them. Then, deciding whether to include other systems proceed to B) and C). B) Just look at the "long-term (not 6 months, at least computed over 3 years)" correlation of all the systems against some commonalities, instead of constructing the whole system-system correlation table. I assume these long term correlations would be more stable compared to system-system correlations computed every 6 months or 1 year. Examples would be: Against level of VIX or ATR of the particular market. Forecast what kind of volatile market we are in - Look at VIX or ATR. Then based on historical correlation of system with VIX/ATR, decide which systems need to be included in the portfolio, given the market regime we are in. C) After building the portfolio in B), apply optimization on this multi-system portfolio to compute system weights. Objective function can be maximize Calmar, or Sharpe etc. On an on-going basis, repeat the process every half year to every year (again using long term correlations of systems against market regimes and checking that individual system edges have not deteriorated). I will DEFINITELY do modifications to my approach as I start implementing it but this is the most practical and mathematically complete that I have been able to think through till now. I started with saying optimization should not be done on portfolio level, rather Monte Carlo should be done. I have ended in step C) above saying optimize portfolio. It is for sure a hard problem
Sounds interesting, but can you explain the sampling method? If you have already a joint distribution, how would you use the correlation matrix, f.i. 10x10 in your 10 systems example, to sample the data for Monte Carlo?
Exactly my thoughts, too. You're way ahead of me in terms of programming and mathematics. Some simple thoughts: 1) Why not calculate portfolio drawdown using actual back-test results, per-strategy, per-time frame. Each result produces an equity curve, and then amalgamate those equity curves, to yield the result. 10 strategies over 100 simulations each = 1,000 equity curves? More to the point, 10 strategies, over say, 30 instruments, over 15 time frames each = 4,500 instances, or individual equity curves. Add or amalgamate the performance of those curves together, and the result would net the total equity curve had those strategies been run concurrently over the backtested period. This rough result would give an idea of what your maximum portfolio drawdown would look like under vanilla conditions. You'd have to assume an average risk per trade (say around 1/30th of 1% of net capital, but whatever it is, the results should be more or less the same). A good programmer could whip something up in excel, that would allow 'tinkering' with position sizing amongst various strategies, to yield a more realistic looking total equity curve, later on, when you know which strategies are more profitable than others. Some other questions: Why use monte carlo, other than to confirm the edge? Strategies, if profitable, will yield a positive expectancy when historically back and forward-tested. Monte Carlo is sorta overkill and not necessary, imo. But each their own. As for ranking, yes, I agree. It's not just return or raw performance. Other factors play a role, and it really comes down to personal preference to pick superior qualities, and which weighted blend of factors are more important than others. Every trader is different, but for me, the top two are 1) performance and 2) drawdown. That's it. What little I know about the Sharp Ratio is it measures consistency, which in itself is a reflection of drawdown, but gives undue importance to minimal drawdown at the sake of returns. So I'd be looking for something a bit more custom. Maybe even going over performance tables manually and examining return versus DD. For me, I'd much rather take a strategy that returns 100% pa with a 10% DD, than one that yields 30% pa with a 3% DD. Expectancy doesn't incorporate drawdown either. So in my mind, it's a bit weak. If we go back to the idea of 'Tiers', maybe group strategies according to drawdown first (Tier 1 = less than 10% drawdown, Tier 2 = less than 20% drawdown, Tier 3 = less than 30% drawdown etc), then rank each strategy within each Tier according to their return. For me, that would be ideal. This catches strategy correlation, indirectly. Not the best solution, but gives a rough idea. Once that was done, you could toy around with your excel spreadsheet or whatever, and test various position sizing for the different strategies (higher for higher ranked strategies, lower for lower ranked strategies), which, again, yields a result of your total portfolio draw down over the historical back test period. To me, that backtest period should be pretty big. 5 years minimum. 10 year, even better. But there's all sorts of tweaks and averages that can be thrown in. Real-time management could be monitored by net profit/loss and historical performance. This way, the squeaky wheel gets the grease and anything that falls outside expected historical performance gets singled out for review. Losers can be taken offline. Max loss parameters can be set for any given strategy (no more than 1% total capital loss, per day, per strategy etc).
Good point, it's true that, it's just that it's fully automated and my options trading modules have only been tested for the very basic functionalities. I'll get to that in a few months.
Or, as an alternative, you can assign capital allocation based on market risk factors (that's how I do it). In short, you figure out market factor sensitivity for each strategy and allocate new trades to keep the whole portfolio more or less factor neutral. The two main improvements over any equity-curve based method is that you can use it for convex instruments (e.g. credit or volatility) and that you can fine-tune the risk preferences.
The process goes like this - you generate 'independent' standard normal random numbers and then transform them into 'correlated' standard normal random numbers - using Cholesky decomposition. This transformation is done using the historical correlation matrix between say 10 systems. These different randomly generated series of numbers (after transformation) are multiple trades from different signals. This is the way to sample from a joint distribution. Once you have the correlated random numbers, simulation is standard stuff. To put it simply, you are using the historical correlation matrix to construct the joint distribution. Main assumptions are: 1) Trade distribution is normal (where in finance you don't have that assumption ) 2) Number of trades per system >> Number of systems 3) correlation matrix is positive definite. This could be one of the trickiest assumptions to meet - because otherwise the computation fails. Some of my strategies have both profit targets and stop losses. So, they are really not normal distribution rather binomial distribution. I guess I would be able to apply this technique for them as well. But I need to confirm.
Sorry, I am confused here - what do you mean by "monte-carlo", re-sampling or actual generation of price paths? // actually, you seem to have replied above, still trying to undestand the methodology I usually look at the 2-dimensionally re-sampled multi-strategy paths - positional re-sampling for the joined equity curve, where the joined equity curve is generated by boolean re-sampling. Gives fairly conservative results, but thatâs exactly what I need to understand the leverage and reserve parameters.
This is what most of us do. The problem is that your risk-attribution/optimization process (how you "figure out factor sensitivities") ends up loading you up on hidden or unaccounted-for risk factors -- the "alpha alignment problem" that Axioma and other firms have been going on about lately.
Oh no! Johann must be scared shitless now! So now you are going to stalk him on ET until you drive him away? Great idea Bill! Good way to get yourself banned.
Thanks, I know a bit of maths and finance, but I am average in programming Its straightforward to calculate portfolio DD using actual backtest results on multiple strategies. Multicharts portfolio tester and Tradestation Portfolio maestro will do it for you. As you said, you can take out multiple backtests and combine them yourself in excel. But it is an incomplete result. Here is why: Say you have 3 strategies: S1, S2 and S3 and you have 8 trades in each. 5 wins in all cases. Lets say av win:av loss is 1:1. Let us say you get following trade series for all these strategies. Sign of the historically realized trades in S1: +,+,-,-,+,+,-,+ Sign of the historically realized trades in S2: +,-,+,+,-,+,-,+ Sign of the historically realized trades in S3: +,-,+,-,+,+,-,+ So, Historical Portfolio trades: 3+,1-,1+,1-,1+,3+,3-,3+ So, Historical Portfolio equity curve: 3+,2+,3+,2+,3+,6+,3+,6+ ...(A) If there is no serial dependency, it means trades in a strategy are independent. Then, there is no reason sometime in future, you can't have 3 (-)s together in any strategy. Situation will be really bad when all these consecutive losses happen in all the strategies together. Following are two possibilities: Sign of the future realized trades in S1: +,+,-,-,-,+,+,+ Sign of the future realized trades in S2: +,-,-,-,+,+,+,+ Sign of the future realized trades in S3: +,-,-,-,+,+,+,+ So, future Portfolio trades: 3+,1-,3-,3-,1+,3+,3+,3+ So, future Portfolio equity curve: 3+,2+,1-,4-,3-,0,3+,6+ ...(B) Sign of the future realized trades in S1: -,+,-,-,+,+,+,+ Sign of the future realized trades in S2: +,-,-,-,+,+,+,+ Sign of the future realized trades in S3: -,-,+,-,+,+,+,+ So, future Portfolio trades: 1-,1-,1-,3-,3+,3+,3+,3+ So, future Portfolio equity curve: 1-,2-,3-,6-,3-,0,3+,6+ ...(C) Can you see the massive difference in portfolio equity curves in historical and hypothetical future case - even when all the systems have maintained their respective edges. This is an iceberg type problem - something not clearly visible and appears infrequently. All the individual strategies maintaining their edges is like calm seas, and under that calmness, the portfolio having an unexpectedly bad DD is like an iceberg. Its very rare, but it can and does happen. It will be instructive to put all these numbers in excel and graph the future equity curve (last line) in (A), (B) and (C) - you will see that even though individual strategies have maintained their edges, portfolio equity curve in (B) and (C) are horrible. Only properly done Monte Carlo can help shed light on this iceberg problem. Monte Carlo is not used to confirm the edge of an individual strategy. Rather, Monte Carlo is used to look at possible trade sequences in future. So, that you can determine the proper position size model. Even if the edge persists, there could be a large DD in future simply because lots of losses happen in sequence or close sequence. Following doesn't pertain to portfolio construction, but I will write it. I use following 3 guidelines/test to qualify robustness of my edges: 1) The idea should make sense based on my understanding of markets like trend following, breakouts, mean reversions etc. It should be based on some long lasting market character or markets participants behavior. 2) The entry should have an edge. You can generate random entries, same holding period as your strategy and see if the profitability persists. 3) The strategy should remain profitable over a large value of parameters. Also, as few parameters as possible for robustness. Agree with you on everything. I also don't want 30% return with 3% DD. I will rather have 200% return with 30% DD Sharpe is inappropriate - primarily because it hates upside volatility equally as downside volatility. Sortino or Calmar ratios are much better measures of performance. I suggest you read their definitions and track them for your various strategies. Agreed on a long backtest period. However, I think position sizing on each individual strategy should be decided before constructing the strategy portfolio. So, you could decide to use Fixed fractional with 3% risk on S1 and fixed fractional with 1.5% risk on S2 and 2.5% risk on S3. After you have decided on position sizing for each strategy, then only strategy portfolio construction process should start. After successful portfolio construction, you should be able to allocate weights w1, w2 and w3 to your strategies. So, your w1 could be 0.7, w2 could be 0.8 and w3 could be 1. Which means for 1 contract for S3, you will trade 0.7 contracts of S1 and 0.8 contracts of S2. Ofcourse, you will continue to use the 3%, 1.5% and 2.5% fixed fractions on S1, S2 and S3. This is the way at least I would do it. Disclaimer: I have just started attacking this problem of strategy portfolio construction - after concentrating for last 2 years on developing edges. So, a lot of my ideas are still in developmental stage and I am bound to revise them as I develop tools to implement these ideas.