Combining Uncorrelated Strategies to Improve Sharpe Ratio
Summary
The document explains how combining two uncorrelated return streams can raise a portfolio’s Sharpe ratio. It presents a regression argument: when the streams are uncorrelated, the first strategy’s beta relative to the second is zero, so its excess return is treated as alpha and its residual risk equals its own volatility. Its appraisal ratio therefore matches its standalone Sharpe ratio.
For optimal combination, the squared Sharpe ratios add, giving the square root of their sum as the combined Sharpe ratio. With the stated inputs of two and five, the example gives approximately 5.38. The document also mentions a numerical proof elsewhere but does not reproduce it. The result assumes uncorrelated returns and an optimal allocation under the stated framework; it does not discuss estimation error, changing correlations, costs, or implementation constraints.
Key ideas
- For uncorrelated strategies, the beta of one strategy relative to the other is zero.
- The first strategy’s appraisal ratio equals its Sharpe ratio under the stated regression setup.
- The optimal combined Sharpe ratio is the square root of the sum of the individual squared Sharpe ratios.
- The numerical example combines Sharpe ratios of two and five to obtain about 5.38.
- The discussion does not address estimation uncertainty or trading costs.
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Full text
# sharpe ratio of 2 uncorrelated strategies
# sharpe ratio of 2 uncorrelated strategies
I was asked this question the other day:
By having two uncorrelated portfolios, one with sharpe ratio 2 and the other with sharpe ratio 5, what is the max sharpe ratio we can achieve.
I tried computing the sharpe ratio of the combined portfolio, where portfolio 1 was assigned a weight and portfolio 2 -> 1- weight and then compute the derivative with respect to w to get the optimal weight. Unfortunately, I cannot seem to be able to get a numeric result. If anyone knows how to solve this, I'd really appreciate it!
Thank you in advance!
## Answer by KaiSqDist (score 1)
https://quant.stackexchange.com/a/81104
Here is the numeric proof that a combination of the two assets will always get the same maximum Sharpe ratio (combined). I don't have the mathematical proof though:
On a side note, here is the mathematical formula: Optimise the Sharpe ratio of a portfolio of uncorrelated assets
## Answer by phdstudent (score 1)
https://quant.stackexchange.com/a/82308
The simple way to see this is the following.
Asset 1 has a Sharpe Ratio of 2. Asset 2 has a Sharpe Ratio of 5.
If you run a regression of the returns of asset 1 into asset 2 you get:
$$r_{1,t}-r_f = \alpha_1 + \beta_1 (r_{2,t} -r_f) + \epsilon_{1,t} $$
Now because assets 1 and 2 are uncorrelated that implies $\beta_1$ = 0. This further implies that $E[r_1,t] - r_f = \alpha_1$ and $\sigma_1^2 = \sigma_{\epsilon_1}^2$ (because $\sigma_1^2 = \beta_1 \sigma_2^2 + \sigma_{\epsilon_1}^2$).
This implies that the appraisal ratio of asset one vis-a-vis asset 2 is given by: $$\frac{\alpha_1}{\epsilon_1} = \frac{E[r_1,t] - r_f}{\sigma_1} $$
It is also known that the Sharpe ratio of your new portfolio which optimally combines asset 1 and asset 2 is given by:
$$SR_{new} = \sqrt{SR_2^2 + \bigg ( \frac{\alpha_1}{\epsilon_1} \bigg )^2} = \sqrt{SR_2^2 + SR_1^2} = \sqrt{5^2+2^2} = 5.38$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.