# Capital Allocation for Portfolio of Multi-Strategy and Multi-Instrument

I would like to know if there is a way (or theory) to manage a multi-strategy, multi-instruments portfolio that would calculate the optimal weight to allocate capital for each combination of strategy and instrument (sometimes we may find one strategy works for many instrument or vice versa).

My first idea is that we can treat each combination of strategy and instrument as a imaginary instrument and introduce Markowitz's portfolio theory to find the optimal weight.

However, I also learned that the estimated return and covariance is very noisy in practice and deduce very different results from CAPM. May not be a ideal way. Another problem is that my strategies could be various across types and timeframe (from intra-day to month holding times). Estimating average return on daily basis could be misleading and underestimate return of strategies that works infrequently (for instance, strategies that take advantage on annual earning announcement or monthly FOMC meeting).

I checked Kelly formula and found the answer from it is exactly as Markowitz's theory. Thus, most issues on mean-variance theory (e.g. noise of estimation for mean and variance) applies here.

I wonder if anyone can share some thoughts on this issue. Any idea/example?

Many thanks.

Have a look at my paper http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2259133

I checked Kelly formula and found the answer from it is exactly as Markowitz's theory. >Thus, most issues on mean-variance theory (e.g. noise of estimation for mean and >variance) applies here.

Kelly is not exactly as Markowitz's theory but they are indeed closely related: as long as we [can] cut tails by assets' returns.

I also learned that the estimated return and covariance is very noisy in practice

Yes, they are! The rule of thumb is to decrease the estimation of expected return and increase the estimation of the volatility/correlation in order to avoid overbetting.

If you want to read more about practical application, have a look at my book: http://www.yetanotherquant.com

Hope it helps

Bayesian Odds Ratios can be used to compare models and allocate wealth to various models based on the relative probability that each particular model is "best."

You could begin to look into it more on the wiki site.

Others may have different views, but I've tried applying Kelly formula/fractional Kelly strategies to capital allocation, and find it rather unpractical and risky.

I would honestly suggest a three-tier optimization framework that I am myself adopting: Assuming you have $M$ number of models covering multiple instruments and strategies. Your goal is to pick a subset of them and assign a capital weighting to each selected model.

1) Universe Screening: Optimize the entire multi-strategy portfolio based on your objective metrics. For instance, you may consider the classical mean-variance framework. By now you should have a rough weighting scheme based on purely theoretical optimization.

2) Universe Refinement: based on stage one's weighting, pick the most favorable $N1 (N1<=M)$ models as your refined universe. Further refine the universe by

1. Analyzing their correlation structures and risk-profiles.
2. Analyzing which model is a diversification-contributor and which is diversification-consumer. Performing this analysis on a instrument, sector, asset-class level.
3. If you have fundamental views or bias based on a macro model,integrate it into your analysis by penalizing those models whose logic is not in line with the macros.

This step is rather conducive, as your logical analysis can complement the weakness of theoretical optimization. By then you have refined universe to $N2 (N2 <=N1)$ models.

3) Iteration: Perform 1) ans 2) iteratively. Step 1 is science driven as numerical optimization is quite standard but rather noisy. Step 2 is experience and logic driven, which is the more difficult part that is not well addressed by the modern portfolio theory.