Relative Danger of Trading Account Sizes

Most everyone intuitively knows that it’s harder to bankrupt bigger accounts than it is to bankrupt smaller ones (assuming all else approximately equal). But, for any two account sizes, just how much more or less danger is there? Twice as much? Ten times? Can you put a number on it? In this article, I’ll show how you can numerically judge the relative danger of bankrupting accounts, and then show how you can adjust risk to trade a small account with the same loss profile as bigger accounts.

Definitions

First, lets all get on the same page about what I’m calling “danger.” In a previous article, I talked about the probability that someone using a trading system will sustain enough consecutive losses to empty the trading account. I considered an account to be empty when it drops below the $25,000 freeze point for pattern day traders in the USA. I called the chances that an account would end up empty the “Probability of Financial Ruin.” If that’s not a scary enough name, maybe pick “Probability of Having to Go Back to Your Day Job.”

In this article, I use the same concept, but I drop the probability and trading system, and think purely in terms of the number of losses an account can sustain before reaching the freeze point. This is one way of characterizing the relative danger of trading two account sizes, if you assume most other factors are about equal. For instance, if account A can sustain 40 losses, and account B can sustain 80, then I’ll say account B is twice as “safe” as account A. I’ll also say account A is twice as “scary” as account B. I use “scary” because terms like “risky” have too many meanings already.

We’ll use three example accounts throughout: a $50k account, a $200k account, and a $1Million account.

Risking a Fixed Value per Trade

One approach to trading is to risk a fixed dollar amount per trade. The dollar amount chosen is typically a percentage of the initial trading stake. I don’t recommend this approach for any account size, and the numbers below will show you why.

The number of losses an account can sustain, under this approach is:

Where “n” is the number of consecutive losses an account can sustain, “stake” is the amount of money a trader starts with, and “pct” is the percent of the initial stake that a trader chooses to use as the amount to risk per trade.

Here’s how our accounts fare under this approach, assuming they risk 2% of their initial stake on each trade:
$50k account can lose 25 consecutive times
$200k account can lose 43 consecutive times.
$1Mil account can lose 48 consecutive times.

So, from this perspective, a $50k account is 1.72 times and 1.95 times more scary than the respective larger accounts. Notice that the larger accounts look a lot closer, with the $200k only 1.11 times as scary as the $1Mil account. This makes sense, because if it weren’t for the $25,000 freeze point, all accounts would have a worst losing streak of 50 trades, at 2% each. Even a $2Billion account could only lose 50 times in a row this way. Surely there is something better?

Risking Fixed Pct of Current Equity per Trade

This is the approach I recommend, and the approach most traders I know use. Instead of risking a fixed value, you risk a fixed percentage of whatever your current account value is. The advantages of this approach, such as risking smaller amounts during losing streaks, have been discussed in detail elsewhere. Here, I’m only concerned with the practical effects related to financial ruin.

With this approach, the number of losses an account can sustain is:

Again, “n” is the number of sustainable consecutive losses, “stake” is the amount of money the trader starts with, and this time “pct” is the percent of current account value risked per trade.

Here’s how our accounts fare under this approach, assuming 2% risk per trade:
$50k account can lose 34 consecutive times
$200k account can lose 103 consecutive times.
$1Mil account can lose 182 consecutive times.

First, note that this is a less scary picture all around than the previous approach. Also, this approach scales better with account size. As such, the $50k is 3.2 times and 5.6 times more scary than the respective bigger accounts. The $1Mil account owner should be happy to spot that the $200k account is 1.77 times more scary with this approach. At least he or she gets something for all that wealth this time!

As it seems better in every way, this is the trading approach assumed for the rest of this article.

Reducing Small Account Danger

So, now that we have a numerical way to judge the relative danger level of different account sizes, we can also numerically even out the playing field by adjusting the percent risked. In plainer english, we know that a small account will always be scarier than a large account when they both risk the same percent of current equity. We want to know how much less a small account should risk than a larger account, if it wants to be just as safe as the larger account.

This rather ugly equation will tell you just that:

Here, “pct” and “stake” have the same meanings as in the previous section. They are subscripted “little” and “big” to differentiate the values for the two account sizes in question.

This tells us that if a $200k account wants to be as safe as a $1Mil 2% account, it should only risk 1.13%. To double check, we can use the formula from the previous section to calculate that at 1.13%, it can lose 182 consecutive times. That is exactly as safe as the $1Mil account at 2%, so this equation works.

As you might expect, our smaller account must cut risk much more sharply to be as safe as the bigger players. For the $50k account to be as safe as the $200k 2% account, it should only risk 0.67%. To be as safe as the $1Mil 2% account, it should only risk 0.38%. So, initially, that’d be a risk of $335 and $190 per trade, respectively.

A Slightly Different Spin

Instead of comparing account sizes, we can also just ask, “what percent can I risk if I want to be able to sustain n consecutive losses?” This equation will tell us that:

Summary

We’ve seen a way to judge the relative safety of different accounts and risk amounts, in terms of the number of losses it would take to freeze the account. We’ve also seen a couple of ways to tailor the percent risked so that an account will have the desired level of safety.

As always with these articles, I hope there was some food for thought for you in here. There’s a lot of general discussion about percent risked, but I don’t see anyone spelling the consequences out numerically. And, for some reason, it seems like nobody takes the $25,000 pattern day trader threshold into account, even though it’s a very important number for stock traders.