We're talking about betting a fixed percentage x% of our bankroll and winning n times and losing m times after (n+m) bets.

• When we win, we win a certain fraction of our bet, namely W₁, W₂, ... W_n.
• When we lose we lose a fraction of our bet, namely L₁, L₂, ... L_m.
• If our bankroll is B and we win, our bankroll increases to B + x*B*W_k
  hence our bankroll changes by a factor (1+x*W_k).
• If our bankroll is B and we lose, our bankroll decreases to B - x*B*L_j
  hence our bankroll changes by a factor (1-x*W_j).
• After (n+m) trades (with n wins and m losses) our initial bankroll has changed by the factor:
  G(x) = (1+x*W₁)(1+x*W₂)...(1+x*W_n) (1-x*L₁)(1-x*L₂)...(1-x*L_m)
• Taking logarithms (to the base e) we get:
  ***     log[G(x)] = Σlog(1+x*W_k) + Σlog(1-x*L_j)   where   Σ   means "add them all up".
• Since log(1+z) = z - z²/2 + z³/3 - + ... when z is small
  (and we can get log(1-z) by replacing z by -z), we expand and get:
  log[G(x)] = Σ (x*W_k - x²*W_k²/2 + - ...) - Σ (x*L_j + x²*L_j²/2 + + ...)

>Wait! log(1+z) = z - z2/2 etc. etc. What's that about? We'll approximate log(1+z) by z - z2/2 which is pretty good when z is small ... as we can see in Figure 1 >And is z small? For us, z is one of those things like xWk and both x and Wk are fractions less than "1", for example we might have x = 0.25 (meaning we bet 25% of our bankroll) and Wk = 0.05 (meaning our WIN is 5% of our bet) in which case         z = x*W = (0.25)(0.05) = 0.0125 (meaning 1.25%) so it's normally going to be pretty small so we can ... >Okay! Okay! Please continue.

Figure 1

• So far we have:   log[G(x)] = Σ (x*W_k - x²*W_k²/2 + - ...) - Σ (x*L_j + x²*L_j²/2 + + ...)
• Retaining just the first two terms, we get:

>Aha! I can see that the average of the wins & losses are going to make their appearance ... AND the average of their squares! Isn't the average square related to the volatility?

Yes indeed! In fact, (1/n)ΣW_k² - { (1/n)ΣW_k }² = (Standard Deviation of the W_k)² which we'll call S²(W).
See SD stuff
Further, we'll let (1/n)ΣW_k = W, the Average of the W_k.
That mean we can use:   ΣW_k² = nS²(W)+nW².
We can then rewrite Equation (1) as:

(2)       log[G(x)] = {nW - mL}x - (x²/2){nS²(W)+nW² + mS²(L)+mL²}

Note that this has the form: G(x) = A x - (1/2)B x2 which describes a parabola and we might expect it to have a maximum ... at x = A/B, as indicated in Figure 2A and ... >Hold on! You replace log(1+z) by z - z2/2 and pick out a maximum, but log(1+z) doesn't HAVE a maximum! Very good point! Let's look at log(G(x)) for an actual stock (involving several hundred weekly returns). Our original equation ***, above, can be rewritten:     log[G(x)] = Σlog(1+x*Rk) where R is an actual weekly return, either a positive Wk or a negative Lj. Let's compare this to the approximation where we replace ln(1+z) by z - z2/2 and use the average of the postive and negative weekly returns, W and L ... as in Equation (2). We get Figure 2B. >They're not the same! Aaah, but the location of the maximum, that's pretty good, eh what?

Figure 2A Figure 2B

Before we continue, note that if G is the Gain Factor over (n+m) trades, then the Gain Factor per trade is G^1/(n+m)
and log[G^1/(n+m)] = (1/(n+m)) log[G] ... so we'll divide both sides of Equation (2) by (n+m) and put
p = n/(n+m) and q = m/(n+m) and rewrite Equation ...

>So just give me the final answer, okay?
The maximum Gain Factor per trade occurs for:

best x = { pW - qL} / {pS2(W)+pW2 + qS2(L)+qL2} where     W = average of the Winning returns     L = average of the Losing returns     S(W) = Standard deviation of the Winning returns     S(L) = Standard deviation of the Losing returns     p = probability of a win     q = 1 - p = probability of a loss

>So, is it any good?
Here's a few, where we trade every week (as explained in Part II) and we invest a percentage x% of our bankroll and we compare the actual stock perforkmance (meaning x% = 100%, so we invest everything with each trade) versus investing various Kelly-fractions as given in Part I, Part II and above:

Kelly #1:     x = p - q/(W/L) = (p*W - q*L)/W
Kelly #2:     x = (p*W - q*L)/(W*L)
Kelly #3:     x = (p*W - q*L) / {p*S²(W)+p*W² + q*S²(L)+q*L²}

>Can you just give us the pictures?

>I like Kelly #2 best ... the green guy! It's always at the top of the curve, eh?
Uh ... you're looking at the right-hand curve.
But if you use the actual weekly stock prices, and invest using Kelly #3, you get a bigger final portfolio.
That's the blue guy, in the left-hand charts.
Note that it (usually) gives lower volatility.
Note, too, that Kelly #1 = Kelly #2 if L = 1 (meaning that you lose all of your investment when you lose).

>Like ... if you're betting on a horse race, eh?
Yes.

>So who uses Kelly #1?
How many guesses do I get?

To download a spreadsheet^*, go to Part II and RIGHT-click on the picture of the spreadsheet and Save Target.

*   The spreadsheet keeps changin' so don't expect it to look like the one shown in Part II

One other thingy:
There's another spreadsheet, explained like this, which does Monte Carlo simulations.
To download a .ZIPd version, RIGHT-click here and Save Target.