Strategy selection as function of correlation / sharpe ratio

[AlphaTango]

Greetings to all,

If I were to construct a portfolio and choose two of three strategies say S1 (sharpe ratio 2.7); S2 (S.R. 1.6) & S3 (S.R. 2.2). Equally weighted S1&S2 has S.R. 3.2 while S1&S3 has 3.6

Would you ever chose S2(uncorrelated) over S3(small -ve correl) to go with S1 in a portfolio? If so, under what circumstances and what's thought process?

inter-correlations: s1,s2 0.0 ; s1,s3 -0.1 ; s2,s3 0.8
correl vs. SPX total returns : s1,spxTR -0.4 ; s2,spxTR -0.3 ; s3,spxTR 0.0

PS: S2 & S3 are very similar and hence highly correlated

-many thanks in advance

 

[sle]

The "right" answer, obviously, is to allocate some risk to all strategies, unless it's somehow difficult from labour or technology perspective.

Correlation to the market, VIX or anything else is a good guide to understanding the alpha/beta in your strategy on a standalone basis. Unless you're living in a benchmarked world, you should only use the correlation between the strategies for portfolio formation.

The theoretical approach (no liquidity constraints, log-normal prices, unlimited and reliable history etc) is to construct a mean-variance portfolio and pick your spot on the efficient frontier. If you have a sub-set of strategies that are highly correlated, the efficient frontier will down-weigh them. For two strategies, you can do it analytically.

In practice, with noisy histories, leverage constraints and other issues, it becomes more complex.

 

[maler]

If you put some confidence intervals on your correlation estimates I suspect the difference
between a 0.0 and -0.1 is meaningless. This, together with the high correlation between S2 and S3
suggests that they try to capture the same juice, (perhaps two implementations of the same idea?).
Merge the best of both and then allocate between the new strat and S1.

 

[truetype]

If I were to construct a portfolio and choose two of three strategies say S1 (sharpe ratio 2.7); S2 (S.R. 1.6) & S3 (S.R. 2.2). Equally weighted S1&S2 has S.R. 3.2 while S1&S3 has 3.6

Would you ever chose S2(uncorrelated) over S3(small -ve correl) to go with S1 in a portfolio? If so, under what circumstances and what's thought process?

inter-correlations: s1,s2 0.0 ; s1,s3 -0.1 ; s2,s3 0.8
correl vs. SPX total returns : s1,spxTR -0.4 ; s2,spxTR -0.3 ; s3,spxTR 0.0

PS: S2 & S3 are very similar and hence highly correlated

This is standard mean-variance optimization, treating each strategy as a security. There are a million free tools on the Web. But be careful with highly-correlated strategies. The optimizer will want to levered-buy the one with a higher Sharpe and levered-short the lower.

 

[drm7]

Mathematically speaking, if the Sharpe Ratio of the new asset is greater than the Sharpe Ratio of the existing portfolio times the correlation of the new asset with the existing portfolio, then it is efficient to add the new asset.

(Rnew-Rrf)/STDEVnew > (Rexisting-Rrf)/STDEVexisting * Corr(new vs. existing)

 

[sle]

Mathematically speaking, if the Sharpe Ratio of the new asset is greater than the Sharpe Ratio of the existing portfolio times the correlation of the new asset with the existing portfolio, then it is efficient to add the new asset.

(Rnew-Rrf)/STDEVnew > (Rexisting-Rrf)/STDEVexisting * Corr(new vs. existing)

Hmm, I am not sure this heuristic really works for all possible corner cases. For example, if you have a current strategy set with a relatively high Sharpe and a new strategy with a low Sharpe but somewhat negative correlation.

 

[drm7]

Hmm, I am not sure this heuristic really works for all possible corner cases. For example, if you have a current strategy set with a relatively high Sharpe and a new strategy with a low Sharpe but somewhat negative correlation.

I'm not sure of all the math behind it, but I'm pretty sure it's a closed-form problem. It's in pretty much every portfolio management textbook and is on all of the CFA tests. If you google "when to add an asset to a portfolio" you may get a source that will go a little deeper and answer your question.

As an aside, this equation is often used to explain why gold miner stocks have had historically poor risk-adjusted returns (other than crappy management and poor capital allocation, lol). Since gold miners have a pretty low (sometimes negative) correlation with the broad stock market, investors are willing to accept lower returns due to their status as "diversifiers." It's a little too "ivory tower" for my taste, but there's probably a grain of truth to it.

 

[sle]

I'm not sure of all the math behind it, but I'm pretty sure it's a closed-form problem. It's in pretty much every portfolio management textbook and is on all of the CFA tests.

If it's a variance frontier approximation, then I corrected it for you; it's one of those tiny nuance things

Mathematically speaking, if the Sharpe Ratio of the new asset is greater than the Sharpe Ratio of the existing portfolio times the correlation of the new asset with the existing portfolio, then it is efficient to add the new asset.

Mathematically speaking, if the Sharpe Ratio of the portfolio inclusive of the new asset is greater than the Sharpe Ratio of the existing portfolio times the correlation of the new asset with the existing portfolio, then it is efficient to add the new asset.

So the math now actually makes sense:
O = periodic returns for the old portfolio
N = periodic returns for the asset to be added
W = portfolio weight to be allocated to the new asset
mean(W*N + (1-W)*O)/std(W*N + (1-W)*O) > cor(O, N) * mean(O)/std(O)

PS. In general, in the world of trading strategies it's assumed that your strategies are self-financing and thus risk free rate can be omitted from the Sharpe ratio calculation

 

[AlphaTango]

Mathematically speaking, if the Sharpe Ratio of the new asset is greater than the Sharpe Ratio of the existing portfolio times the correlation of the new asset with the existing portfolio, then it is efficient to add the new asset.

(Rnew-Rrf)/STDEVnew > (Rexisting-Rrf)/STDEVexisting * Corr(new vs. existing)

Does "Corr(new vs. existing)" here include the sign (rather than the absolute value) i.e. any negatively correlated new strategy will always get added to the portfolio?

Also, what should be the optimal weights for the new portfolio? (I've so far looked at but I'm not too sure about brute-force back test optimization of the weights to maximize sharpe ratio)

Thanks everyone for the discussion so far!

 

[drm7]

Does "Corr(new vs. existing)" here include the sign (rather than the absolute value) i.e. any negatively correlated new strategy will always get added to the portfolio?

Also, what should be the optimal weights for the new portfolio? (I've so far looked at but I'm not too sure about brute-force back test optimization of the weights to maximize sharpe ratio)

Thanks everyone for the discussion so far!

1. Yes, the sign matters, so negatively correlated assets/strategies help a lot!
2. See secretsanta's correction to my post above. I don't want to provide incorrect info!
3. Optimizing weights requires a lot of computing power above 5 assets, as your correlation matrix grows exponentially with each asset.

With all that said, I wouldn't get too precise around optimizing around sharpe ratios, as a) past performance will change (i.e. sharpe's won't persist); and b) correlations between assets and strategies can change a lot, which will trash your careful optimization.