Let's say you have two trading strategies and all you're given is information about their average daily P&L and the Sharpe ratio of each strategy. Trading strategy A's daily average P&L is 10,000 USD and the Sharpe ratio is 5. Trading strategy B's daily average P&L is 5,000 USD and the Sharpe ratio is 10.
Given only this information (and nothing on correlation), how would you optimally allocate capital between the two trading strategies?
To be consistent with the average daily returns that you specified, your first strategy would need to have a daily standard deviation of 31,749 USD and the second a standard deviation of 7,937 USD.
How much weight you should assign to each strategy depends on your goal. You might want to maximize the daily profit, minimize the volatility, or maximize the information ratio. Each of these demands different allocations.
To maximize the daily profit, put all of your weight in the strategy with the highest expected return (strategy 1).
To minimize the volatility, you should weight each strategy according to the variance of the other strategy. This leads to putting 94.1% of the weight in the second strategy, and 5.9% in the first strategy. The exact ratio of the weights is 16:1, since the first strategy has 4x the volatility of the first (and hence 16x the variance).
To maximise the information ratio (equivalently the Sharpe ratio, assuming that the returns you mentioned are excess returns), the Kelly criterion then suggests that the allocation to each strategy should be proportional to the average return over the variance. Since the second strategy has half the expected return, and one-quarter the standard deviation, it should therefore receive 8x the capital allocation that the first strategy receives, since
$$ \frac{\frac{1}{2}}{\frac{1}{4}\times \frac{1}{4}} = 8 $$
If the strategies are somewhat positively correlated, you should allocated more to the second strategy (since it has higher Sharpe).
If you only have that type of information, then simply use:
w1 = 10000/(10000+5000)
w2 = 5000/(10000+5000)
Or replace the PNL by the Sharpe Ratio.
