the Kelly Ratio

In 1956, John Kelly* at AT&T's Bell Labs did research on telephone transmission in the presence of noise and ...

>Huh?
Pay attention. We'll just talk about the results of his analysis as it has come to be applied to the stock market (called, among other things, Kelly Ratio/Value/Criterion), in particular, what it says about how much money to put into a single trade, given the historical evolution of the stock: the percentage of times that you'd win, the average winnings per trade compared to the average loss per trade and ...

>Can you just give me the answer, please?
Okay.
Play.

Probability of a Win:   p = %     It may be 30% or 60% for a stock, but 50% for a coin toss!
Average amount of winnings (when you win):   W =     Only the ratio W/L is important!
Average amount of losses (when you lose):   L =
the Kelly Ratio = p - (1-p)/{ W/L } = %

Kelly% = the percentage of your capital to be put into a single trade

Note: For a p = 50% probability of winning (like tossing a coin),
and equal winnings as losses (so W = L), Kelly says
"Forget it!

... but if you expect to win p = 60% of the time, then Kelly says
"Put 20% of your capital into your next trade

>Yeah, so are you going to explain why it ...
Well, Kelly's original paper is quite mathematical ...
>Okay, forget it!
... but we can explain the formula like so:

  • When you make a winning trade, you average $500.
  • Your losses, per trade, average $350.
  • The probability of winning (and making, on average, $500) is 0.60 ... the Winning Probability.
  • Out of 1000 trades, you'd expect to win 0.60 *1000 = 600 times and lose (1-0.60) * 1000 = 400 times.
  • The wins provide (600)*500 = $300,000 and the total expected losses are (400)*350 = $140,000
    ... for 1000 trades.
  • Hence, the expected gain is $300,000 - $140,000 = $160,000
    ... for 1000 trades.
  • The expected gain per trade is $160 ... dividing by 1000.
  • Then you expect to make this amount per trade ... on average.
  • Then these expected winnings of $160 is just 160/500 = 0.32 or 32% of your winning trades.
  • That's the Kelly Ratio!
In general:
  • When you make a winning trade, you average $W.
  • Your losses, per trade, average $L.
  • The probability of winning (and making, on average, $W) is p ... the Winning Probability.
  • Out of N trades, you'd expect to win p *N times and lose (1-p) * N times.
  • The wins provide $(p*N)*W and the total expected losses are $[(1-p)*N]*L ... for N trades.
  • Hence, the expected gain is (p*N)*W - [(1-p)*N]*L ... for N trades.
  • The expected gain per trade is (p)*W - (1-p)*L ... dividing by N.
  • As a fraction of W, we get:     Kelly Ratio = { p*W - (1-p)*L } / W

>Is that the same formula as you got above?
Yes.

>But it's interpreted as the percentage of your capital to invest in each trade. Why?
If the expected Gain per trade is p*W - (1-p)*L and you'd like to make your winning gain, namely $W, then you'd have to make W/{ p*W - (1-p)*L } trades so you have to have enough money to make these trades so if this number was 4 then you'd invest just 1/4 or 25% of your capital on each trade so you'd have enough money to make four trades so you should only invest a fraction Kelly = { p*W - (1-p)*L }/W on each trade or you may just want to average $W over the long haul or maybe ...

>zzzZZZ
My sentiments exactly ... but not everybuddy uses the same formula for their Kelly Criterion, although the expression
p*W - (1-p)*L seems to be in everybuddy's Kelly Criterion. That's your expected winnings per bet.
For application to the stock market, and the use of the above formula, see this.

But, let's continue with another derivation:

for Part II

* In 1961, Kelly was involved in making a computer sing "A Bicycle Built for Two". Arthur C. Clark heard the computer-synthesized song when he visited the labs and had Hal the computer sing it in "2001: A Space Odyssey"
... when Hal was being disconnected.