## Tuesday, March 31, 2009

### kelly's criterion

Kelly criterion

There is high influence of mathematics in arbitrage, trading ,hedge fund in short the whole stock market. Today I am trying to explain formulae which are usually used by fund managers to find out how much they can get as average return on their investment; however these are mere on paper calculation sometimes overly optimistic assumptions which should be taken care very cautiously. [http://www.cisiova.com/betsizing.asp]

Avg returns= (1 + W1*F)^(P1) * (1 + W2*F)^(P2)

Where Wi = increase in percentage value of stock you are expecting
Pi = chance of happening this rise based on the backtest
F = fraction u put in the wager

When you play with stocks there is however a funny but simplified formula given by epchan.blogspot.com it says that if you are expecting your stock to increase by 1% or decrease by 1% with 50-50 odds what would you do ?… many will jump on hold the stock as it is going to remain flat…. But formula says otherwise [reference]

Y = m-s^2/2

where S = standard deviation =1%                                                                                                                m = expected return value =1%                                                                F = 0.5% which means we are going to take loss

Kelly criterion is very and some times overly optimistic, I have shown 2 different flavors which are currently used in the market. For more info check out William Poundstone’s Fortune’s Formula (amazon)

How ever there is another complicated formula for this, we can find out how much fraction can we bet to optimize profit for our bank roll….[refence]

y = [(1+v*f)^p]*[(1-f)^(1-p)] = Avg return

f = Fraction optimized for gain
f = [p*(v+1)-1]/v (derivative of y wrt f)
p = Probability of gain
v = Odds of gain

so if odds of winning are v = 2:1 and backtest probability is 50-50 the f = ¼ optimized fraction pluggin into to equation y value would provide us 6.1% avg retrun on our bet.

My interest is purely in the fun of the mathematics and not the application so let me throw a disclaimer of using these formulae’s at your own risk.We have seen effects of these modeling in past when modelers failed to model the financial melt down but in their defense I should also say that a Quant is not astrologer he models based on back testing data given to him . So he cannot predict any event never ever happened before in the history of stock market.