We're playing a game: poker, blackjack, the horses, the stock market, the ...

>Okay, okay! I get it ... a game! So?

We start with some bankroll $B₀ and, each time we play, we bet a fraction of the bankroll, say x (where x = 0.12 means 12%). We play a hundred times and win n times and lose m times.

>So n + m = 100.
Pay attention.
When we bet x B and win, our bankroll increases by a certain percentage of our bet, say W (where W = 0.25 means our winnings are 25% of our bet).

>If I double my money, then W = 2, eh?
No, W = 1, meaning my winnings equal my bet. If I bet $20 and get back $40, then my winnings are $20 so W = 1. On the other hand, if I bet $20 and get back $25, then my winnings are $5 which is 5/20 = 0.25 so W = 0.25 meaning 25% of my bet (which was $20). On the other hand, if I bet $20 and got back $45 my bet was $20 and my winnings were $25 so my winnings were 25/20 = 1.25 and that's W. If my bet was ...

>Wait! In Part I you said W was the amount we won, not the fraction of our bet!
Well, imagine that our bankroll is $1.00 so that if we win $0.65 that makes the fraction AND the amount: W = 0.65, see?

>I think it's ridiculous, but please continue.
Okay. We're talking about winning.
Starting with $B₀, we win n times, each time betting a fraction x of our bankroll and increasing our bankroll by a fraction W of our bet. If our current bankroll were B, we'd bet x*B and win W times this bet.

>If our bankroll is B, then we bet x*B and win W*x*B, right?
Yes, so our bankroll increases from B to B + W*x*B = B(1+x*W).

The Gain Factor for a WIN is:   1 + x*W

When we lose, we lose (perhaps) a different fraction of our bet, say L.

>Wouldn't you lose ALL of your bet ... so L = 1?
Yes, if you were playing blackjack or you were betting on a horse race, but if you put your bet on some stock then sold at a loss, you might only lose 10% of your bet so L = 0.10 and ...

>Okay, I get it.
Good.
If our bankroll is B and we bet x*B, we'd lose a fraction L of that bet, meaning we'd lose L*x*B and our new, diminished bankroll, is B - L*x*B = B(1-x*L).

The Gain Factor for a LOSS is:   1 - x*L

Remember, each time we win we multiply our bankroll by (1 + x*W) and each time we lose we multiply by (1 - x*L).

The Gain Factor for n Wins and m Losses is:   G(x) = (1 + x*W)n (1 - x*L)m and The Gain Factor per trade is:   G(x)1/(n+m) = g(x) = (1 + x*W)p (1 - x*L)q where p = n/(n+m) = the probability of a WIN   and   q = m/(n+m) = 1 - p = the probability of a LOSS.

>Don't tell me! You want the largest value of (1 + x*W)p (1 - x*L)q Yes, by choosing the appropriate value of x. For example, look at Figure 1 where we start with $1000 and bet 100 times and look at our final bankroll for various values of x. It's a graph of your final bankroll, namely:     B(x) = 1000 (1+0.20x)45 (1-0.15x)55 where B0 = $1000. >Aha! W=0.20 and L = 0.15 and you win 45 times and lose 55 times and you should bet x = 25% of your bankroll to get the maximum gain after 100 bets. Very good! You win 20% and lose 15% of your bet ... and you win 45% of the time.

Figure 1

>And you'll get the same answer for the "best" x ... as you got in Part I, eh?
Let's see. First we'll ...

>Wait! How about some examples, applied to stocks!
Okay, let's consider the daily returns of some stock, say GNSS, over the past few years.
We get positive returns about 49% of the time with an average return of 4.70% (so we'll take W = 0.0470) and an average negative return of 4.22% (so we'll take L = 0.0422). The "best" investment is about 57% of our bankroll:

>Is that a spreadsheet?
Well, yes, but it's not finished yet ...

>It says you win 48.6 times out of 100? Are you kidding? I mean ...
The n-value really gives the probability of winning. For the above example, it's 48.6% of the time.

>What about GE?
Okay, but remember that the best x-value may be more than 100% or even negative! GE is such an example. We plot the final portfolio for x between 0% and 100% and get a maximum, in this range, of 0%. (So we don't bet anything on this stock!) The Kelly percent is negative and we should plot the final portfolio for negative x-values as well:

>Choose x = -240%? What does that mean?
Beats me ... but the Math doesn't know that you can't bet -240% of your bankroll.

>Maybe it means sell short, eh?
Who knows. Anyway, continuing ...
>And if x = 150%? What then?
I guess you borrow the money ... or buy on margin  

Okay, we want to maximize g(x) = (1 + x*W)^p (1 - x*L)^q     the Gain Factor per trade.

First we take logarithms (to the base e) of each side, getting:
        log(g) = p log(1+x*W) + q log(1-x*L)
Then we differentiate with respect to x:
        g'/g = p*W/(1+x*W) - q*L/(1-x*L)
Then we set g' (that's dg/dx) = 0, so:
        p*W/(1+x*W) = q*L/(1-x*L)
Then we solve for x:
        best x-value = { p*W - q*L }/(L*W)
Remember: p = the probability of winning and q = 1 - p = the probability of losing:

Note 2: If, when we lose, we lose ALL of our bet, then L = 1 and we get:
        best x-value = { p*W - q }/W = p - q/W

You may want to play with this Kelly Criterion:

Probability of a Win:   p = %     Percentage of your bet (when you win):   W = %     Expressed as a percentage Percentage of your bet (when you lose):   L = %     Expressed as a percentage best x-value = { p*W - (1-p)*L }/(W*L) = %

If the amount to bet is large, like 50%, you may want to divide the number by 10 or 20 or whatever. That's like choosing a different denominator for the Kelly Ratio ... and some people do!

>So, is Kelly any good?
Let's check it out for a particular stock, starting with a $10K bankroll.
This is what we'll do, after determining the values of p, W and L from historical data.

1. At the start of each week we buy the stock at the opening price for that week, investing x% of our bankroll.
2. At the end of the week we sell the stock at the closing price.
3. We repeat steps 1 and 2, for a jillion weeks.
4. We change the value of x and repeat steps 1, 2 and 3.
5. We determine the best value of x and compare with the Kelly Ratio.

>How do you calculate p and W and L?
We look at the past few hundred weeks (for the particular stock we're investigating) and:

• Calculate the percentage of times that the stock increased in price over the week. That's our p.
• We look at all the positive gains (for each week) and find their average. That's our W.
• We look at all the negative gains (for each week) and find their average. That's our L.

>That's confusing. I mean ...
If, for week #k, the return is 1.23% then that gives our Winnings for that week, as a percentage of our investment,
so W(k) = 0.0123, and if the return were -2.34%, then L(k) = .0234, and our W and L values are the average
over all k-values of all the W(k) and L(k), see?

>Shouldn't L be negative, 'cause it's a loss and ...?
No. The negative sign is already in the formula: best x-value = { p*W - (1-p)*L }/(W*L)

>So, where are your examples?
Here's a few where we invest, each week, various percentages of our current bankroll.
For x% = 100%, the chart gives the actual performance of the stock.

>BUT WHAT'S THE KELLY VALUE !?
All are greater than 100%
But you'll have noticed that the formula we obtained above is different than the formula we gave in Part I
... and I don't think much of either!

Let's do it differently.
>Do we have to?
Why not? Besides, would you believe anything which says:

Calculate your portfolio growth with a constant x% by assuming
constant winnings and constant losses and constant winning probability.

for Part III