# How to determine the optimal start capital for a strategy?

Suppose my strategy generates a stream of daily profits distributed like 𝒩[μ=1€, σ=10€].

Intuitively, if I trade with 10€ start capital:

• I could very well be ruined on the first day, if the first day profit is below -10€, which could happen with 13.6% probability (given μ=1€, σ=10€)
• However if, by lucky chance, I don’t face any nasty drawdown, I can build a fortune starting with only 10€ capital, i.e. stellar ROI

On the other hand, if I trade with 10'000€ start capital:

• I will be able to withstand the ugliest drawdowns, I will never stop trading!
• but my ROI will be terribly worse, because the initial investment is 1000 times higher than the previous case

So:

• 10€ start capital: stellar ROI, but too risky
• 10'000€ start capital: no risk of ruin, but terrible ROI

How do determine the start capital, with the best risk-reward trade off?

• The main question is: why do you care about ROI ? Is the strategy designed to manage your own money or client’s money ? That makes a big difference when it comes to which metric are important. – Ezy Jan 1 at 12:48
• In addition in practice a strategy return is never normally distributed so is your question purely academic ? – Ezy Jan 1 at 12:51
• @Ezy, the strategy is for my own money, but I still have the risk-reward dilemma, and I'm looking for an analytical approach to solve it 😋 – elemolotiv Jan 1 at 12:54
• If this is your own money you should concern yourself more with your tails than with optimizing ROI based on the bulk of the returns distribution. ROI is a metric you show to clients, what matters for your own money is the pnl itself. The other aspect is: if you do not allocate to this strat what else would you do with this money ? – Ezy Jan 1 at 12:56
• @Ezy - it seems a longer story than what I thought, do you know some good book where all this is comprehensively discussed? thanks 🙇‍♂️ – elemolotiv Jan 1 at 13:03

From an academic standpoint there is not an objective best risk-reward tradeoff. Here the ROI is a direct function of your aversion to the risk of ruin. It is an entirely subjective parameter.

• thanks for the comment! I appreciate my ROI itself is a subjective utility function. If I don't get better advices, I'll mark your answer as "accepted" – elemolotiv Jan 1 at 13:51
• @elemolotiv no problem. Again things would be a little different if you were providing more info like tail distribution of losses, max drawdown in backtest etc... then you could infer more realistically things like “what is the min capital i need allocate to ensure absence of ruin before my exit at 95 percentile” but still at the end of the day your choice of ruin scenario would still be subjective. That being said for long only type of strategies and if you invest in securities instead of derivatives (like futures) then capital is needed to setup the portfolio itself. – Ezy Jan 1 at 14:55

The optimal starting point should be between minimum and maximum capital available.

The minimum capital required for a strategy is you will be buy the required quantity of securities as per your strategy. For example, if your strategy trades in 5 different securities and you are required to buy 1 quantity of all 5 of them and the sum of the prices of 5 instruments in $$1000$$ then the minimum capital required is $1000. The maximum capital on the other hand is with that capital you will not overwhelm the order book or move the market against you by placing order. For example, if you only trade in a ill liquid security where only 5 quantities are traded in a minute and price of the security is $$1000$$ then the maximum capital you should have is $5000.

In real life, the maximum can be determined through the average daily traded volume of a security. Your strategy requirement for a security shouldn't exceed the average daily volume traded for a security.

Optimal capital should be between your minimum and maximum possible. Generally, you can take average of of the two to determine the capital allocation to a strategy.

The answer depends on personal risk-aversion. Two ideas came to my mind.

The first is based on statistics (eg. value at risk, stopping time, distribution of the minimum) which is easy to measure.

The second is prospect theory.

Kelly criterion answers a more general question specifically what is the optimal size of a series of bets in order to maximise wealth.

• thanks @Jonas for the ideas and useful links! – elemolotiv Jan 3 at 14:36