### Evaluating Leveraged Trading Systems

Trading systems generate some return on every trade. When comparing trading systems to determine which one to use, it is important to properly evaluate the expected returns and risk. This is particularly important when evaluating leveraged trading systems. This is a description of the main quantities I use to analyze the value of a trading system with an example of applying it to a new system.

The basic assumption is that the distribution of returns is constant. If this is not true, then the trading system doesn't work. It's just lucky sometimes. The other basic assumption is that there is no serial correlation in the trade results. If there is serial correlation, you should account for it in your system until there isn't (for example, if the wins and losses come in streaks, you should only take a trade when the previous one would have been profitable). The only assumption about the distribution of returns is that the returns are all small enough that no one trade will wipe you out.

Using the formula for log expectancy, you can easily derive the Kelly ratio for optimal leverage: l=mu/sig^2. Furthermore, you can determine that the long-run average trade value is mu-sig^2/2. The central limit theorem says that this formula will be valid when you average over at least 30 trades, regardless of the distribution (if it is close to normal, the formula is valid for fewer trades). When you use fixed leverage l, the average trade return becomes l*mu-l^2*sig^2/2. With optimal leverage, this is mu^2/sig^2-mu^2/sig^2/2=(mu/sig)^2/2. So with fixed leverage, the best return you can get will be from the system that has the highest Sharpe ratio. The Sharpe ratio should be calculated based on however often you readjust your leverage. If you adjust it daily, you should look at the daily Sharpe ratio. If you adjust it once per trade, you should use the per-trade Sharpe ratio.

At the moment, my trading account is fairly small. Since I am planning on trading e-mini futures, the amount of additional money required to add another contract is a large percentage of my account balance. When trading a fixed number of contracts, the optimal performance criteria is different. At that point, the main question is how small of an account can you safely trade with. After a reasonable number of trades, the mean value of the trades will have a normal distribution with mean n*mu and standard deviation sqrt(n)*sig. That means that an m standard deviation drawdown will be n*mu-m*sqrt(n)*sig. Differentiating with respect to n shows that the max drawdown will be at (m*sig/(2*mu))^2 trades. Putting this into the maximum drawdown formula, the max drawdown is m^2*sig^2/(4*mu)-m^2*sig^2/(2*mu)=-(m/2)^2*(sig^2/mu). Which means that you want to minimize sig^2/mu, which is the same as maximizing mu/sig^2 (which is also the optimal leverage).

To determine the minimum amount of money to trade a leveraged system safely, you multiply the expected drawdown by the contract value: MIN=(m^2*sig^2/4/mu)*VAL. For 2 standard deviations, this is VAL/l, where l is the optimal leverage. For 3 standard deviations, it is 9/4 of that value (a little more than twice as high).

You should also check what the value of n comes out to. If it is very low, then the max expected drawdown is actually higher because you can't assume that the distribution of the means will be normal.

For example, a trading system I have been working on for trading the NASDAQ 100 futures has mu=0.837% and sig=2.327%. The Sharpe ratio is 0.35, so the maximum return per trade (with leverage) is 6.47%. The optimal leverage is 15.47 (the maximum leverage due to current margin requirements is about 5.4). The maximum expected drawdown value for 2 standard deviations is $1391 and for 3 standard deviations it is $3130. However, for 2 standard deviations, n=7.7 trades and for 3 standard deviations, n=17.4 trades, so those estimates are probably low. For 4 standard deviations, n=30.9 trades, which is enough for the formula to be valid. The maximum expected drawdown amount for 4 standard deviations is $5565. Since the margin requirement is $4000, this system should not be traded in an account with less than about $9565 in it if you want to be really safe, although it is probably ok to trade with as little as $7170.

I haven't actually worked out the math for how differently skewed trade results change all of this. Based on actual simulations, there is a 1% chance of a 12% drawdown ($2507) and a 2% chance of a 10% drawdown ($2125). The simulated result with a probability corresponding to 4 standard deviations is a drawdown of 23% ($4970). That suggests that this system can be safely traded with as little as $9000, or $6500 if you don't mind a 1% chance of dropping below the margin requirements. A second contract should be safe to add once you are above $10K, although it isn't entirely safe until you are above $14K. Above $12k, it is probably best to trade this system with maximum allowable leverage (about 5.4). This will produce an expected logarithmic return of about 3.7% per trade (about 19 trades to double). With about 1.5 trades per month, this is an expected doubling time of about 1 year.

The basic assumption is that the distribution of returns is constant. If this is not true, then the trading system doesn't work. It's just lucky sometimes. The other basic assumption is that there is no serial correlation in the trade results. If there is serial correlation, you should account for it in your system until there isn't (for example, if the wins and losses come in streaks, you should only take a trade when the previous one would have been profitable). The only assumption about the distribution of returns is that the returns are all small enough that no one trade will wipe you out.

Using the formula for log expectancy, you can easily derive the Kelly ratio for optimal leverage: l=mu/sig^2. Furthermore, you can determine that the long-run average trade value is mu-sig^2/2. The central limit theorem says that this formula will be valid when you average over at least 30 trades, regardless of the distribution (if it is close to normal, the formula is valid for fewer trades). When you use fixed leverage l, the average trade return becomes l*mu-l^2*sig^2/2. With optimal leverage, this is mu^2/sig^2-mu^2/sig^2/2=(mu/sig)^2/2. So with fixed leverage, the best return you can get will be from the system that has the highest Sharpe ratio. The Sharpe ratio should be calculated based on however often you readjust your leverage. If you adjust it daily, you should look at the daily Sharpe ratio. If you adjust it once per trade, you should use the per-trade Sharpe ratio.

At the moment, my trading account is fairly small. Since I am planning on trading e-mini futures, the amount of additional money required to add another contract is a large percentage of my account balance. When trading a fixed number of contracts, the optimal performance criteria is different. At that point, the main question is how small of an account can you safely trade with. After a reasonable number of trades, the mean value of the trades will have a normal distribution with mean n*mu and standard deviation sqrt(n)*sig. That means that an m standard deviation drawdown will be n*mu-m*sqrt(n)*sig. Differentiating with respect to n shows that the max drawdown will be at (m*sig/(2*mu))^2 trades. Putting this into the maximum drawdown formula, the max drawdown is m^2*sig^2/(4*mu)-m^2*sig^2/(2*mu)=-(m/2)^2*(sig^2/mu). Which means that you want to minimize sig^2/mu, which is the same as maximizing mu/sig^2 (which is also the optimal leverage).

To determine the minimum amount of money to trade a leveraged system safely, you multiply the expected drawdown by the contract value: MIN=(m^2*sig^2/4/mu)*VAL. For 2 standard deviations, this is VAL/l, where l is the optimal leverage. For 3 standard deviations, it is 9/4 of that value (a little more than twice as high).

You should also check what the value of n comes out to. If it is very low, then the max expected drawdown is actually higher because you can't assume that the distribution of the means will be normal.

For example, a trading system I have been working on for trading the NASDAQ 100 futures has mu=0.837% and sig=2.327%. The Sharpe ratio is 0.35, so the maximum return per trade (with leverage) is 6.47%. The optimal leverage is 15.47 (the maximum leverage due to current margin requirements is about 5.4). The maximum expected drawdown value for 2 standard deviations is $1391 and for 3 standard deviations it is $3130. However, for 2 standard deviations, n=7.7 trades and for 3 standard deviations, n=17.4 trades, so those estimates are probably low. For 4 standard deviations, n=30.9 trades, which is enough for the formula to be valid. The maximum expected drawdown amount for 4 standard deviations is $5565. Since the margin requirement is $4000, this system should not be traded in an account with less than about $9565 in it if you want to be really safe, although it is probably ok to trade with as little as $7170.

I haven't actually worked out the math for how differently skewed trade results change all of this. Based on actual simulations, there is a 1% chance of a 12% drawdown ($2507) and a 2% chance of a 10% drawdown ($2125). The simulated result with a probability corresponding to 4 standard deviations is a drawdown of 23% ($4970). That suggests that this system can be safely traded with as little as $9000, or $6500 if you don't mind a 1% chance of dropping below the margin requirements. A second contract should be safe to add once you are above $10K, although it isn't entirely safe until you are above $14K. Above $12k, it is probably best to trade this system with maximum allowable leverage (about 5.4). This will produce an expected logarithmic return of about 3.7% per trade (about 19 trades to double). With about 1.5 trades per month, this is an expected doubling time of about 1 year.

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