__Martingale:__The gambler doubles his bet after every loss...

Martingale vs. Non Martingale (Simplified RoR vs Profit in 3 trade runs with all possibilities worked out)

Lets take a very small run and calculate all possibilities. We will start with 4 units to risk over 3 trades with 8 possible total outcomes all worked out to provide an over all EV of 3 trades. The system has a 50% chance of winning and wins 1.1x risk when it wins. Lets define Risk of Ruin as losing 75% of the 4 unit start account. Of course we don't normally only make 3 trades and risk such a high proportion of our risk capitol but I will show later how the conclusions hold true to some degree when we add trades or increase our edge or reduce risk or reduce the martingale multiplier.

__Standard risk 0.43 unit on each trade NON martingale:__

sum L L L

-1.29 -0.43 -0.43 -0.43

W L L

-0.39 0.47 -0.43 -0.43

L W L

-0.39 -0.43 0.47 -0.43

W W L

0.52 0.47 0.47 -0.43

L L W

-0.39 -0.43 -0.43 0.47

W L W

0.52 0.47 -0.43 0.47

L W W

0.52 -0.43 0.47 0.47

W W W

1.42 0.47 0.47 0.47

EV of 8 runs0.52

__Martingale (2x) start risk 0.43 unit:__

sum L L L

-3.01 -0.43 -0.86 -1.72

W L L

-0.82 0.47 -0.43 -0.86

L W L

0.09 -0.43 0.95 -0.43

W W L

0.52 0.47 0.47 -0.43

L L W

0.60 -0.43 -0.86 1.89

W L W

0.99 0.47 -0.43 0.95

L W W

0.99 -0.43 0.95 0.47

W W W

1.42 0.47 0.47 0.47

EV of 8 runs0.77

Hey the Martingale really is more profitable! Well, of course it is, we have taken on more risk so it's not a fair comparison. Notice in the martingale we have 1 run that is -3 units. -3 units from our start balance of 4 is a ruin. 1 ruin in 8 is a 12.5% RoR. Lets boost our risk on the NON Martingale run so it too has a 12.5% RoR.

__Standard Risk 1 NON Martingale__

sum L L L

-3.00 -1.00 -1.00 -1.00

W L L

-0.90 1.10 -1.00 -1.00

L W L

-0.90 -1.00 1.10 -1.00

W W L

1.20 1.10 1.10 -1.00

L L W

-0.90 -1.00 -1.00 1.10

W L W

1.20 1.10 -1.00 1.10

L W W

1.20 -1.00 1.10 1.10

W W W

3.30 1.10 1.10 1.10

EV of 8 runs1.20

OK, there we have it, NON Martingale is miles ahead of Martingale in EV when we compare strategies with the same RoR. What if we reduce the Martingale Multiplier? Apparently, 1.5 multiplier and a 0.63 start size gives us a 12.5% RoR so we can compare....

__Martingale (1.5x) start risk 0.63__

sum L L L

-2.99 -0.63 -0.95 -1.42

W L L

-0.88 0.69 -0.63 -0.95

L W L

-0.22 -0.63 1.04 -0.63

W W L

0.76 0.69 0.69 -0.63

L L W

-0.02 -0.63 -0.95 1.56

W L W

1.10 0.69 -0.63 1.04

L W W

1.10 -0.63 1.04 0.69

W W W

2.08 0.69 0.69 0.69

EV of 8 runs 0.93

Still no dice for Marty. The best profitability to RoR ratio is NON martingale. What if we increase the edge of the actual system? Lets bring it from 1.1 to 1.5...

__Martingale (1.5x) start risk 0.63__

sum L L L

-2.99 -0.63 -0.95 -1.42

W L L

-0.63 0.95 -0.63 -0.95

L W L

0.16 -0.63 1.42 -0.63

W W L

1.26 0.95 0.95 -0.63

L L W

0.55 -0.63 -0.95 2.13

W L W

1.73 0.95 -0.63 1.42

L W W

1.73 -0.63 1.42 0.95

W W W

2.84 0.95 0.95 0.95

EV of 8 runs4.65

__Standard Risk 1 NON Martingale__

sum L L L

-3.00 -1.00 -1.00 -1.00

W L L

-0.50 1.50 -1.00 -1.00

L W L

-0.50 -1.00 1.50 -1.00

W W L

2.00 1.50 1.50 -1.00

L L W

-0.50 -1.00 -1.00 1.50

W L W

2.00 1.50 -1.00 1.50

L W W

2.00 -1.00 1.50 1.50

W W W

4.50 1.50 1.50 1.50

EV of 8 runs6.00

Again the best EV comes from NON Martingale. Can we do even better if we reduce risk after a loss instead of increasing it? Lets try a simple percentage risk of current equity strategy. Apparently we can achieve a RoR of 12.5% by risking 37% of current equity on each trade...

__Percentage Risk 37% (NON Martingale)__

sum L first trade balL2nd balLfin bal

-3.00 -1.48 2.52 -0.93 1.59 -0.59 1.00

WLL

-1.53 2.22 6.22 -2.30 3.92 -1.45 2.47

LWL

-1.53 -1.48 2.52 1.40 3.92 -1.45 2.47

WWL

2.09 2.22 6.22 3.45 9.67 -3.58 6.09

sum L first trade bal L 2nd bal W fin bal

-1.53 -1.48 2.52 -0.93 1.59 0.88 2.47

WLW

2.09 2.22 6.22 -2.30 3.92 2.17 6.09

LWW

2.09 -1.48 2.52 1.40 3.92 2.17 6.09

WWW

11.04 2.22 6.22 3.45 9.67 5.37 15.04

EV of 8 runs9.73

OK so the percent risk strategy has double the EV as a 1.5x Martingale strategy of the same RoR. OK well, what if we redefine RoR to be only -0.25(6.25% dd)? That should give us more realistic risk compared to start capitol.

__Percentage Risk 2.1% (NON Martingale)__

sum L first trade bal L 2nd bal L fin bal

-0.25 -0.08 3.92 -0.08 3.83 -0.08 3.75

WLL

-0.05 0.13 4.13 -0.09 4.04 -0.08 3.95

LWL

-0.05 -0.08 3.92 0.12 4.04 -0.08 3.95

WWL

0.17 0.13 4.13 0.13 4.26 -0.09 4.17

sum Lfirst trade balL2nd balWfin bal

-0.05 -0.08 3.92 -0.08 3.83 0.12 3.95

WLW

0.17 0.13 4.13 -0.09 4.04 0.13 4.17

LWW

0.17 -0.08 3.92 0.12 4.04 0.13 4.17

WWW

0.39 0.13 4.13 0.13 4.26 0.13 4.39

EV of 8 runs0.51

__Martingale (1.5x) start risk 0.052__

sum LLL

-0.25 -0.05 -0.08 -0.12

W L L

-0.05 0.08 -0.05 -0.08

L W L

0.01 -0.05 0.12 -0.05

WWL

0.10 0.08 0.08 -0.05

LLW

0.05 -0.05 -0.08 0.18

WLW

0.14 0.08 -0.05 0.12

LWW

0.14 -0.05 0.12 0.08

WWW

0.23 0.08 0.08 0.08

EV of 8 runs0.38

Martingale loses again but here we can see the EV differences starting to get more subtle.

**We now know that if we want to increase profits by accepting more risk, we are always better off taking additional risk by increasing our percent risk of equity than we are switching to a Martingale.**

May all your fits be loose.