Quote from neke:
Let me give you a little quiz. Suppose your trade has two possible outcomes: a gain of +20% or a loss of -10% with equal chance. Would you trade this scenario? What percentage of your account will you commit to each trade? Give me you answer, and I will see how much of risk/reward analysis you understand
Thanks to all who took the time to reflect on this quiz. Let us assume each trade lasts exactly one week, and you can be in only one trade at a time. With 20% for each profitable trade, and -10% for each non-profitable trade,and with the chance of profitable trade being equal to that of an unprofitable trade, let's assume we use r as the leverage. The question becomes trying to determine the value of r that maximizes the return:
(1+0.2r)(1-0.1r).
Take the first derivative and equate to zero, the problem reduces to solving the equation: (0.2-0.1)=2*0.2*0.1r, which gives r = 2.5. This is the leverage that maximizes the return in the long run. (If the mathematics here looks fuzzy to you, just ignore the details and read the points I am trying to convey). Of course most brokerages will give you overnight leverage of only 2.0 for stock positions (which is my target instrument).
I have simulated returns with different leverage for a 52-week period. The returns are 20% for 26 weeks and -10% for 26 weeks, the actual weeks
being randomized. I have included details of a random distribution of weekly returns, starting from an initial 10K, and determined the returns at different leverage.
The summary of the returns over the 52-week period are as follows:
Leverage Total Returns Max Drawdown
2.500 2038% 92%
2.000 1804% 85%
1.000 640% 56%
0.500 214% 32%
0.125 37% 9%
Some folks (including my well-respected acerbits) will take on the last scenario (12.5% of account staked for a 37% annual return with 9% drawdown). My experience, and the response of many, tells me that most people are contented with measly return because of the scare of a bad drawdown. I completely understand this. We are all human. However, I believe we should each seek out whether we have any edge (in this regard your historical results over a long period will be the best indicator), determine our pain threshhold, and then set out to make the maximum advantage of that edge consistent with our pain tolerance. Even though 2.5 leverage is the optimizing level from the example above, I would not trade it. I am still human, and will be psychological distabilized by a 92% drawdown, especially if it happens near the start of the journey: We do not want anything that will make us doubt our edge or even ourselves. I believe I trade at a risk level I am comfortable with. If I suffer a 40% drawdown and still achieve my 550% target, so what? I have posted this primer on risk/reward just to answer those that think I am headed for disaster. Of course the outcomes in real-life will be more like a distribution rather than a two-state possibility. The stock market is a risk-based investment, and I do not understand folks who think others are nuts for taking measured risk, just because it does not conform to their risk tolerance profile.
Code:
Leverage 2.5 2 1 0.5 0.125
Week Return 10000 10000 10000 10000 10000
1 0.2 15000 14000 12000 11000 10250
2 0.2 22500 19600 14400 12100 10506
3 -0.1 16875 15680 12960 11495 10375
4 -0.1 12656 12544 11664 10920 10245
5 -0.1 9492 10035 10498 10374 10117
6 0.2 14238 14049 12597 11412 10370
7 0.2 21357 19669 15117 12553 10629
8 0.2 32036 27537 18140 13808 10895
9 -0.1 24027 22029 16326 13118 10759
10 0.2 36041 30841 19591 14429 11028
11 0.2 54061 43177 23509 15872 11304
12 0.2 81091 60448 28211 17460 11586
13 0.2 121637 84628 33853 19206 11876
14 -0.1 91228 67702 30468 18245 11727
15 -0.1 68421 54162 27421 17333 11581
16 -0.1 51316 43329 24679 16466 11436
17 0.2 76974 60661 29615 18113 11722
18 0.2 115460 84926 35538 19924 12015
19 0.2 173190 118896 42645 21917 12315
20 -0.1 129893 95117 38381 20821 12161
21 0.2 194839 133163 46057 22903 12465
22 0.2 292259 186429 55268 25193 12777 peak
23 -0.1 219194 149143 49742 23934 12617
24 -0.1 164396 119314 44767 22737 12460
25 -0.1 123297 95451 40291 21600 12304
26 0.2 184945 133632 48349 23760 12611
27 -0.1 138709 106906 43514 22572 12454
28 0.2 208063 149668 52217 24829 12765
29 -0.1 156047 119734 46995 23588 12606
30 -0.1 117036 95787 42296 22409 12448
31 -0.1 87777 76630 38066 21288 12292
32 -0.1 65833 61304 34259 20224 12139
33 0.2 98749 85826 41111 22246 12442
34 -0.1 74062 68660 37000 21134 12287
35 -0.1 55546 54928 33300 20077 12133
36 -0.1 41660 43943 29970 19073 11981
37 -0.1 31245 35154 26973 18120 11832
38 -0.1 23434 28123 24276 17214 11684 trough
39 0.2 35150 39373 29131 18935 11976
40 0.2 52725 55122 34957 20828 12275
41 0.2 79088 77170 41949 22911 12582
42 0.2 118632 108038 50338 25202 12897
43 -0.1 88974 86431 45305 23942 12736
44 -0.1 66731 69145 40774 22745 12576
45 0.2 100096 96802 48929 25020 12891
46 0.2 150144 135523 58715 27522 13213
47 -0.1 112608 108419 52843 26146 13048
48 0.2 168912 151786 63412 28760 13374
49 -0.1 126684 121429 57071 27322 13207
50 0.2 190026 170001 68485 30054 13537
51 0.2 285039 238001 82182 33060 13875
52 -0.1 213779 190401 73964 31407 13702
Drawdown 0.92 0.85 0.56 0.32 0.09
...How is this for a consolation?