Is there an optimal sizing?

Short answer: Yes.

Longer answer:

If sizing too large destroys a +EV sizing, and sizing too small yields too little returns, then we can conclude that there should be an optimal middle ground value that yields the highest +EV return.

Just bet more when EV is higher?

Let's do some thought experiment.

Should we trade a larger size if our EV is higher? Instinctively, I would say yes.

Assuming we do, let's say our EV is a 20% return, maybe we risk 5% of our capital.

The higher the EV, the more we should bet right? What if the EV is a 200% return? Let's bet more, we'll bet 10 times more than before since the EV is 10 times higher. We risk 50% of our capital.

What if the EV is a 400% return instead, do we bet 2 times more than before? If so, we risk 100% of our capital. This means that if we are wrong, we lose everything.

Example

Let's assume our 400% EV equation looks like this:

EV = (1 + W) ^ P(W) * (1 - L) ^ P(L)

= (1 + 25) ^ 0.5 * (1 - 0.05) ^ 0.5 (we win 2500% and lose 5%)

= approx 5 = 400%

Side note: This EV equations looks absolutely amazing. If you find a strategy like this, you are set for life.

As mentioned, we aim to lose 100% of our capital if we are wrong. The EV equation indicates that we lose 5% if we are wrong. Hence, we need to lever up 20X to lose 100% (5% loss * 20 times =100% loss).

In this equation, we are right half the time and wrong the other half.

If we start with $10,000, we bet $200,000 ($190,000 is borrowed). If we win the first time, we end up with $200,000 * 25 - $190,000 = $4,810,000.

That is a incredibly massive profit. Then we play again by betting 20X of $4.81 million = $96.2 million.

Then we lose, we end up with a loss of $96.2 * 0.05 = $4.31M. i.e. we lose everything.

This is bad for us. We're bankrupt with no hope of recovery.

Even if our trades look like this WWWWL, i.e. we win the first 4 times and lose on the 5th trade, we still end up with $0.

Conclusion

Thus, we conclude that we don't always blindly bet more when the EV is higher.

Why does overbetting turn +EV to -EV?

Assuming this is our EV inputs:

• P(W) = 0.5
• W = 0.8
• P(L) = 0.5
• L = 0.4
• Initial capital = $1,000
  

If we bet our entire capital and lose, we lose 40% of our capital. i.e. $1,000 to $600.

If we bet our entire capital and win, we win 80% of our capital. i.e. $1,000 to $1,800.

Thus, since P(W) and P(L) are 0.5, if we play 2 times, win once and lose once, we get $1,000 * 1.8 * 0.6 = $1,080. We profit $80 every 2 trades, or $40 per trade on average. This is +EV.

Double the sizing?

Let's say that we double the sizing.

If we bet our entire capital and lose, we lose 80% of our capital. i.e. $1,000 to $200.

If we bet our entire capital and win, we win 160% of our capital. i.e. $1,000 to $2,600.

We play 2 times, win once and lose once, we get $1,000 * 2.6 * 0.2 = $520. We lose $480 every 2 trades, or $240 per trade on average. This is -EV.

Explanation

This happens because of the "if we lose 50%, we need to win 100% to breakeven" effect.

This means that if we lose 50% and go from $1,000 to $500, our new initial capital is now $500. We need to win 100% in our next trade to go from $500 to $1,000.

The chart above shows how much you need to make if you are down by X amount.

Let's go back to our previous example.

L = 0.4. We are down 40%. Thus, we need 66.67% to breakeven.

If we double our sizing, L becomes 0.8. We are down 80%. Now we need 400% to breakeven. Doubling our sizing doubles our W. But a jump of 66.67% to 400% is a jump of 6 times! Just a doubling of our W is not enough.

Hence, overbetting turns a +EV strategy to -EV because the sizing multiplier does not increase our W enough to match the increase in L. This is especially prevalent as L gets larger.

Instructor: Lucas Liew