M2: Fooled by Average
We are used to average things out for the ease of calculation. Let us see, how our calculations go for a toss in continuous stock market kind of games.
Imagine you have a special strategy, where each month either you make 60% profit OR 40% loss with equal probability. You play this game continually for five years (60 months). If you start with 100000, how will it turn out at the end of five year? Choose an option:
a. Will make BIG money
b. Will be a moderate winner
c. Will loose money
(If you are not a math person, you can continue reading overlooking at calculations, you will get the zest of story. Otherwise, solving before reading further will be a great value addition to one self)
Approach 1:
So, with the out come of +60% half the time & other half the time -40%, average gain per month is +10% (i. e. ½ x [(60%)+(-40%)])
If I make 10% every month, the amount will be:
End of 1st month : 1,00,000 x (1+.1)^1 = 1,10,000
End of 2nd month : 1,00,000 x (1+.1)^2 = 1,21,000
End of 3rd month : 1,00,000 x (1+.1)^3 = 1,33,100
End of 4th month : 1,00,000 x (1+.1)^4 = 1,46,410
…
End of 5yrs : 1,00,000 x (1+.1)^60 = 3,04,48,163
Woo!!! My 1Lac has turned into 3 Cr 4.5 Lacs at the end of the 5th year.
Approach 2:
Let us not take averaging, but calculate the values month by month:
End of 1st month:
1L will be 1.6L (1Lx1.6) or 60,000 (1Lx .6) (equal probability of 60% gain or 40% loss)
End of 2nd month:
If, 1.6L it will be 2.56L (1L x 1.6 x1.6) or 96,000 (1Lx1.6x.6)
If, 60,000 it will be 96,000(1Lx.6×1.6) or 36,000(1Lx.6x.6)
End of 3rd month:
If, 2.56L it will be 4.096L(1Lx1.6^3) or 1,53,600(1Lx1.6^2x.6)
If, 96,000 it will be 1,53,600(1Lx1.6^2x.6) or 57,600(1Lx1.6x.6^2)
If, 36,000 it will be 57,600(1Lx1.6x.6^2) or 21,600(1Lx.6^3)
…
OK, its complex tree for 60 months. I will take a short cut path of winning for 30 months and loosing for 30 months. Half the time out of 60 months, I am going to make 60% gain & other 30 months I am going to loose 40%. Using the formula, I will have: 1L*(1.6^30)x(.6^30), i. e., 29,385. This is the most likely outcome!!!
OOOPS!!! My 1Lac has turned roughly less than 30K at the end of 5th year. I HAVE LOST more than 70% WITH 60% GAIN/40% LOSS STRATEGY.
Which approach is correct?.
In Approach 1, If my pay off instead of +60% & -40%, is 20% & 0% OR +80% & -60%, I will still get the same answer!!! Don’t get fooled by average! The calculations are for the strategy that makes every moth 10% profit.
Also, Approach 1 works well in discreet games. If you invest 1L at the start of every month and take out 1.6L or 60K at the end of month. Again fresh investment of 1L at the start of next month. The calculation is fine with that approach.
Just cross check with other pay offs with approach 2:
| Pay Offs | Calculation | After 5 Yrs |
|---|---|---|
| +20% & 0% | 1Lx(1.2^30)x(1^30) | 2,37,37,631 |
| +40% & -20% | 1Lx(1.4^30)x(.8^30) | 29,95,992 |
| +60% & -40% | 1Lx(1.6^30)x(.6^30) | 29,385 |
| +80% & -60% | 1Lx(1.8^30)x(.4^30) | 5.24 |
| +100% & -80% | 1Lx(2^30)x(.2^30) | 0.00 |
If our strategy looses -60% or -80% half the time, for sure, we will end up with almost nothing.
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