The analysis of risk of ruin has always been a very interesting topic. In this post I give the formulas for calculating ROR when the player initially chooses the betting unit as some Kelly fraction given the game parameters and his bankroll.

It is a common practice that a player forms blackjack bankroll, and then chooses appropriate Kelly fraction to bet. After choosing the Kelly fraction the player calculates the betting unit given the betting spread and other constraints. Denote the Kelly fraction by $p$, where $p=2$ for half Kelly, etc.

Suppose that the player plays a fixed unit size with Kelly fraction $p$. Thus, full Kelly, also so called logarithmic utility, corresponds to $p=1$. Kelly fraction $p$ corresponds to the so-called power utility, also called constant relative risk aversion utility, with risk aversion parameter $p$.

If the player finds the unit bet size as the corresponding Kelly fraction $p$ from the initial bankroll, and fixes the betting unit afterwards, then the risk of ruin can be calculated as
$$ROR = exp{-2 p}.$$

Note that, perhaps surprisingly, the ROR does NOT depend on the edge and risk of the game, nor on the bankroll. The reason for that is that the desirability of the game and the size of the bankroll are already reflected in the bet size. The only thing that matters is player?s risk aversion reflected by the Kelly fraction $p$.

For example, suppose that there are otherwise two identical games, except one has an edge of 1% and the other has an edge of 2%. The optimal bet size for the second game will be double the first game. But the ROR will be the same! Intuitively, one can understand this as saying that ?time runs twice that fast? for the second game. To obtain the same results in terms of risk and return, one just needs to play twice as many games with HALF the bet size in the first case.

Another interesting question is the probability of ruin before reaching a certain goal, i.e., before doubling the bankroll, after which point the player may for example want to increase the unit size.

Denote $B >= 1$ to be the desired multiple of bankroll. The corresponding risk of ruin before reaching the goal $B$ then equals
$$ROR(B) = (\exp{-2 p} ? exp{-2 B * p}) / (1- exp{-2 B * p} ).$$

Note that if $B$ is large (if $B$ converges to infinity) we get exactly the simpler formula above.

Here are some numbers:
Suppose full Kelly bettor ($p=1$). If the goal is doubling the bankroll, the player runs ROR(2) = 11.92%. The ROR(3) for $B=3$ (tripling the bankroll) is 13.32%, while it is 13.53% for $B = infinity$. $B = infinity$ means never hitting zero.

For half Kelly and $B = 2$ we get ROR(2) = 1.8%, while the ROR is 1.83% for $B = infinity$. This number changes very little ? if the initial bet size is chosen as half Kelly, it is indeed extremely unlikely to ever hit zero again, once the player doubles the bankroll.

For those mathematically inclined I will make another post on these pages shortly, showing how to derive the ROR formulas above.

Best regards,

Karel