Maximising the Log of Wealth

Last lesson left us with a paradox. A bet with a genuine edge — the 60/40 coin, where you win 60% of the time at even money — will still wipe you out if you bet your whole bankroll, and even betting most of it grinds you down over time. The arithmetic that says “positive expected value, so bet big” is answering a question you are not actually asking. So what is the right question? What single number should a compounding gambler try to make as large as possible?

The answer, discovered by John Kelly at Bell Labs in 1956 and weaponised by Ed Thorp at the blackjack tables a few years later, is deceptively small: maximise the expected logarithm of your wealth. Not your wealth. The log of your wealth. This lesson explains why that peculiar objective is the correct one for money that multiplies, then turns the crank and derives the Kelly fraction from it — a formula you can write on a napkin and that tells you exactly how much of your bankroll to stake.

Why logarithms: wealth multiplies, so growth adds

Here is the crux, and it is a fact about arithmetic, not psychology. When you bet a fraction of your bankroll, good and bad rounds multiply together, they do not add. Win the 60/40 coin at fraction $f$ and your wealth is multiplied by $(1+fb)$; lose and it is multiplied by $(1-f)$. After many rounds your wealth is a long product of these factors:

$W_n = W_0 \cdot (1+fb)^{\,\text{wins}} \cdot (1-f)^{\,\text{losses}}.$

Products are hard to reason about; sums are easy. The logarithm is the tool that turns a product into a sum — $\ln(xy)=\ln x+\ln y$ — so taking logs converts your multiplying bankroll into an adding one:

$\ln W_n = \ln W_0 + (\text{wins})\ln(1+fb) + (\text{losses})\ln(1-f).$

Now the long-run behaviour is a plain average. Divide by $n$ rounds and, by the law of large numbers, the fraction of wins tends to $p$ and losses to $q=1-p$. The per-round growth rate of log-wealth settles down to a fixed number we call $g(f)$:

$g(f) = p\,\ln(1+fb) + q\,\ln(1-f).$

This $g(f)$ is the time-average growth rate — the exponential rate at which a single bankroll, yours, actually compounds if you keep playing. Because $W_n \approx W_0\, e^{\,g(f)\,n}$, a positive $g$ means exponential growth, $g=0$ means you tread water forever, and $g<0$ means you decay to zero no matter how promising a single round looked.

The analogy. Think of two escalators. Plain expected wealth $E[W]$ is a lottery announcer shouting the average prize across a million ticket-holders — a number one jackpot winner can single-handedly inflate while everyone else goes home broke. The log-growth rate $g(f)$ is the escalator you personally are standing on: its slope is the rate your own money climbs, round after round, and no stranger’s jackpot changes the step under your feet.

The misconception to kill. “Log wealth is about being risk-averse — it’s a psychological utility function for cowards.” No. Even a perfectly risk-neutral robot that only cares about growing its money as fast as physically possible, with zero feelings about volatility, should maximise $E[\ln W]$ — because that is the growth rate. The log is not a fudge factor for nerves; it is the exact bookkeeping of a multiplying quantity.

When to use this lens

Reach for expected log wealth whenever outcomes multiply and you reinvest: bankrolls, portfolios, compounding returns, bet sizing over many independent rounds. If instead your payoffs simply add — a one-shot bet you will never repeat, a fixed stake you cannot re-wager — then plain expected value is the right tool and Kelly does not apply. Kelly is the mathematics of the repeated, reinvested, divisible wager.

Deriving the Kelly fraction

We have the objective $g(f)$. Now we find the fraction $f$ that maximises it — and this is where a napkin and one line of calculus produce a famous formula. A smooth hump is maximised where its slope is zero, so we differentiate $g$ with respect to $f$ and set the result to $0$:

$g'(f) = \frac{p\,b}{1+fb} - \frac{q}{1-f} = 0.$

Cross-multiply the two fractions:

$p\,b\,(1-f) = q\,(1+fb).$

Expand both sides — $pb - pbf = q + qbf$ — and gather every term with $f$ on one side:

$pb - q = pbf + qbf = bf\,(p+q) = bf,$

using $p+q=1$. Divide by $b$ and you have the Kelly fraction:

$\boxed{\,f^{*} = \frac{pb - q}{b} = p - \frac{q}{b}\,}$

Read it as a sentence: bet your edge divided by the odds. The numerator $pb-q$ is your edge — expected profit per unit staked — and dividing by the payoff odds $b$ scales it into a fraction of bankroll. Bigger edge, bet more; longer odds (bigger $b$), bet a smaller slice, because each win already pays handsomely.

For an even-money bet ($b=1$) the formula collapses to something you can do in your head:

$f^{*} = 2p - 1 = p - q,$

which is simply your edge. A 60% coin at even money: $f^*=2(0.6)-1=0.2$, bet a fifth. A 55% coin: bet a tenth. A 50% coin: bet nothing — no edge, no wager.

Worked example A: the 60/40 even-money coin

Take the canonical bet: win probability $p=0.6$, lose $q=0.4$, even money $b=1$. The formula gives $f^{*}=0.6-0.4=\mathbf{0.20}$ — stake 20% of your bankroll every round. Let us verify it really is the peak by tabulating the growth rate $g(f)$ across fractions. (The middle column is $g$ in log units; the right column converts it to an effective compounded return per round, $e^{g}-1$.)

The peak sits exactly at $f^*=20\%$, earning about +2%/round of compounding — modest per round, but $e^{0.02n}$ over hundreds of rounds is a fortune. Two features to file away for the next lesson: half-Kelly (10%) still banks about 1.5%/round, roughly three-quarters of the peak growth for half the stake; and double-Kelly (40%) has already fallen back through zero — a positive-edge bet delivering negative long-run growth. Betting everything ($f=1$) means a single loss multiplies you by $(1-1)=0$: instantaneous, permanent ruin, which repeated play guarantees.

Worked example B: favourable odds (2-to-1)

Now change the payoff. Suppose a fair-ish coin, $p=0.5$, but a win pays 2-to-1 ($b=2$) — you risk one unit to win two. Is this worth betting on, and how much? First the edge:

$\text{edge} = pb - q = (0.5)(2) - 0.5 = 1.0 - 0.5 = 0.5 > 0,$

so yes, strongly favourable. The Kelly fraction is edge over odds:

$f^{*} = \frac{pb-q}{b} = \frac{0.5}{2} = 0.25,$

and the alternative form agrees: $f^{*}=p-\dfrac{q}{b}=0.5-\dfrac{0.5}{2}=0.5-0.25=0.25$. Stake 25%. Notice the discipline here: a bet you win only half the time still deserves a quarter of your bankroll because the winning rounds pay double — but not more than a quarter, because the losing half will show up just as often and you must survive it.

Worked example C: a thin real edge

Real edges are rarely 20%. When Ed Thorp took card-counting to the blackjack tables, his advantage over the house at a favourable count was on the order of 1% at roughly even money. Kelly then says stake about $f^{*}\approx 1\%$ of your bankroll — and serious advantage players bet a fraction of even that, for reasons the next two lessons make painfully clear. The lineage matters: Kelly’s 1956 paper grew out of Claude Shannon’s information theory (its title is literally about information rate), Thorp turned it into a blackjack and then a Wall Street fortune at the hedge fund Princeton–Newport, and the same formula now sizes positions for professional gamblers and quantitative traders alike.

The lesson of example C is sobering: a 1% edge is real money over time — about a 1%/round growth engine — but it is also a whisper. Overestimate it even slightly, bet as if it were 3%, and you have shoved yourself over the peak toward the ruin side of a hump you cannot see. Precisely how dangerous that is — and why “err low” is the only safe direction — is the subject of the next lesson.