Trouble Understanding Optimal F Example

Context

I am going through Mathematics of Money Management by Ralph Vince and am struggling to understand an example that is provided on Optimal F. Was hoping to get some clarification here on the example on right side of this page

I am trying to reproduce the plot in the page with some simple python code to really solidify my understanding. But fundamentally I don't think I understand this example

The author refers to a game being offered in which a player has 50% chance of winning $2 and a 50% chance of losing $1. A natural question is if you have some amount of capital/bankroll how many times do you play this game (# of bets)?

The metric of Terminal Wealth Relative (TWR) is used to answer this question by maximizing it. TWR is simply just final capital / initial capital.

Then we consider a hypothetical situation where the game has been played 40 times and the outcome is 40 sequences of +2, -1. The optimal # of bets is apparently $1 for every $2.50 you have and your TWR would be 10.55 meaning you made 10.55 times your initial capital.

Question

I don't really understand what optimal # of bets means in the context of the example or plot. Since the plot shows for 20 sequences of +2, -1 and we bet for $1 for every $2.50 we have, does that mean that we have to have an initial capital of $160? I don't think this is the case but I'm really unsure on how to interpret the 40 bets (20 sequences) and how that leads to a whopping TWR of 10.55?

If anyone understands this, some help would be greatly appreciated

Also I think this follows the rules of sub, but mods feel free to take down if it doesn't.

Submitted September 26, 2020 at 03:16PM by whiteboy2471
via https://ift.tt/2G5XNSP

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