Let’s start with a simple thought experiment.
Suppose I offer you a bet: flip a coin, heads you win $2, tails you lose $1. On average, that’s a profit of $0.50 per flip. Most people would happily take it.
Now scale it up: heads you win $2,000, tails you lose $1,000. The expected gain is now $500. Still sounds good… but you might hesitate.
Go even bigger: heads you win $2 million, tails you lose $1 million. Mathematically, the expected gain is $500,000. But most people would refuse — because a $1 million loss could ruin them.
So what’s going on?
The problem is that expected wealth treats all pounds as equally valuable, no matter how rich you are or what losing them would mean for your life.
A slight reformulation
Consider a bet where you put down $x. If the coin lands heads, you end up with 3x in total (ie a 2x profit). If it lands tails, you lose the x you put down. How much would you choose for x?
This is essentially the same question as before — but phrased in terms of how much you’re willing to risk. Too small a bet and you are leaving some of that juicy profit on the table. Too large and you risk financial ruin.
Why Utility Makes More Sense
People tend to think in percentages, not absolute amounts. The same dollar gain means different things depending on where you start:
- A $1k profit is exciting if you have $100k in total wealth.
- It’s barely noticeable if you have $1m.
This idea — that the value of money depends on how much you already have — is captured by utility functions. Instead of maximising wealth directly, we maximise a function \(u(W)\) that reflects how much “happiness” or “satisfaction” we get from having wealth \(W\).
Log Utility Captures Percentage Thinking
If people really do think in percentage gains and losses, then our utility function should treat a 10% increase in wealth the same way, no matter our starting point.
For example:
- A person seeing their wealth going from 100 to 110 sees a 10% gain.
- A person seeing their wealth going from 200 to 220 also sees a 10% gain.
If we use the utility function
$$u(W) = \log(W)$$
then these two changes give the same increase in utility: $$log(110)−log(100)=log(220)−log(200)$$
A 10% gain always adds the same amount to \(\log(W)\), regardless of whether you start with 100 or 1,000,000.
This makes log utility a natural way to model percentage-based thinking. It values proportional changes equally and automatically adjusts for diminishing marginal value of money — the very problem expected wealth ignores.
With this in mind, we can now look at situations where wealth changes happen repeatedly — like making the same bet many times — and see how log utility guides the optimal decision.
The Formal Setup (Kelly Betting)
Let’s say:
- You start with wealth \(W_0\)
- You bet a fraction \(f\) of your wealth on each round
- You win with probability \(p\), and lose with probability \(q = 1 – p\)
- If you win, your payout is \(b \cdot f \cdot W_0\)
- If you lose, you lose \(f \cdot W_0\)
So after one bet:
- Win: \(W_1 = W_0 (1 + fb)\)
- Lose: \(W_1 = W_0 (1 – f)\)
After N bets, assume we get to a wealth of \( W_N \). Writing it as a product of individual gains from each bet:
$$ \frac{W_N}{W_0} = \frac{W_N}{W_{N-1}} \cdot \frac{W_{N-1}}{W_{N-2}} \cdots \frac{W_1}{W_0} $$
Each term \(W_i / W_{i-1}\) represents the “wealth multiple” from a single bet, and we assume these are independent and identically distributed (i.i.d.).
We want to maximise the long term utility \(\log(W_N)\) which is the same as maximising the long term log growth \(\log(W_N / W_0)\):
$$ \log\left(\frac{W_N}{W_0}\right) = \sum_{i=1}^N \log\left(\frac{W_i}{W_{i-1}}\right) $$
Since the terms in the sum are i.i.d, our objective becomes maximising the expected log return (growth) per bet, or equivalently just the first bet:
$$G(f) = \mathbb{E}[\log(W_1)/ \log(W_0) = p \log(1 + fb) + q \log(1 – f) $$
This is the Kelly objective function which we want to maximise.
At this point, it might look like we took logs just to make the maths easier by turning a product into a sum. That’s true — it does make the maths nicer — but that’s not the real reason. The motivation goes back to what we discussed earlier: log-utility captures percentage thinking and avoids the “all-in” behaviour that comes from maximising raw expected wealth. It’s about aligning the maths with how people actually value gains and losses.
Differentiate and solve:
$$G'(f) = \frac{pb}{1 + fb} – \frac{q}{1 – f} = 0$$
Solve for \(f\) to find the optimal bet size that maximises long-term compounded wealth.
$$f = p – \frac{q}{b}$$
Conclusion
The Kelly formula emerges naturally when we maximise long-term log-utility, which we motivated from everyday percentage thinking. It’s not just a neat trick for gamblers — it’s the mathematical formalisation of “risk an amount that’s meaningful as a percentage of your wealth, but not so much that you blow up on a bad streak.”
In practice, many investors choose to bet less than the Kelly fraction (e.g., half-Kelly) to reduce volatility, but the principle remains: it’s about growing wealth at the fastest possible rate without risking ruin.
In theory, the Kelly fraction \(f\) maximises the long-term growth rate of your wealth. In practice, however, full Kelly can be very aggressive — especially if your estimates for \(p\) or \(b\) aren’t perfect.
Because of this, many investors and gamblers use Half-Kelly:
- You compute \(f_{\text{Kelly}}\) from the formula.
- Then you bet half that fraction.
In other words, you sacrifice a small amount of growth for a big drop in risk. This trade-off can be worth it for those who value smoother returns or who worry their inputs for \(p\) and \(b\) might be off.