The Kelly criterion and fractional Kelly

The last module said the position size is the one thing you truly control, and that you should size for the loss you can survive. That is the survival half of sizing. This module is the other half — the more ambitious question hiding underneath it: if you genuinely have an advantage, is there a mathematically correct amount to bet?

It turns out there is. In 1956 a scientist at Bell Labs named John Kelly worked out a formula that answers it exactly. Given your edge — how favourable a bet really is, how much you expect to win against how much you could lose and how often — the Kelly criterion tells you the single bet size that makes your money grow fastest over the long run. Not the size that feels bold, not the size that feels safe: the mathematically optimal one. It was later used by professional gamblers to beat blackjack and by investors like Ed Thorp to run remarkably successful funds.

So there is a right answer, and it is knowable. And here is the twist that makes this module worth reading: almost nobody sensible bets the full amount the formula gives. They deliberately bet a fraction of it. Understanding why the mathematically optimal bet is still too big to actually make is the most useful sizing lesson you will ever learn — because it is really a lesson about the gap between a clean formula and a messy world.

Start with the plain intuition, because the maths only dresses it up. Imagine two bets. In the first, you win slightly more often than you lose — a thin, real edge. In the second, you win far more often than you lose, and win big when you do — a fat edge. How much of your money should you put on each?

Obviously more on the second. A bigger, surer advantage deserves a bigger bet; a thin one deserves a small one. This is the whole idea of sizing by edge — commit more where your advantage is real and large, and less where it is slim, so your money flows toward your best opportunities. Kelly's contribution was to make "more" and "less" precise: he found the exact fraction that maximises long-run growth.

The plain-language version of the formula is easier than it looks. Your Kelly fraction rises when:

• you win more often (higher probability of being right), and
• you win more relative to what you risk (bigger reward-to-risk on each bet).

And it falls when the edge is thin or the payoff is poor. If you have no edge at all — a coin flip with even payoffs — Kelly says bet nothing, which is exactly right: with no advantage, betting only adds risk for no expected gain. That single feature already makes it wiser than most people's instinct, which is to bet on hunches that carry no real edge.

But notice the quiet danger built into all of this. Every input — how often you will really win, how much you will really make — is an estimate about the future. In a casino, the odds are fixed and known. In investing, you are guessing your own edge, and humans reliably guess it too high. The formula is exact; the numbers you feed it are hopeful. Hold that thought, because it is the entire reason for the fraction.

Here is Kelly in its simplest form, and you never need more than this. For a bet where you either win or lose:

Bet fraction = your edge ÷ the odds.

More precisely, if p is your chance of winning, q = 1 − p is your chance of losing, and b is how many rupees you win for each rupee risked, then the Kelly fraction is f = (p × b − q) ÷ b. The top of that fraction, p × b − q, is just your edge written out: your expected winnings minus your expected losses. If it is positive you have an edge; if it is zero or negative, Kelly says bet nothing.

A worked example makes it concrete. illustrative Suppose you judge a bet has a 60% chance of paying off, and when it pays it returns 1× what you risked (b = 1). Then f = (0.60 × 1 − 0.40) ÷ 1 = 0.20. Kelly says stake 20% of your capital. On a ₹10,00,000 portfolio, that is ₹2,00,000. Change nothing but the win chance — drop it to 55% — and the fraction falls to 10%, or ₹1,00,000. A smaller edge, a smaller bet. That is the formula doing exactly what intuition wanted.

Now the crucial picture — the one that explains the whole module. If you plot your long-run growth against how much you bet, you do not get a line that keeps rising. You get a hump. Growth climbs as you bet more, reaches a peak exactly at the full Kelly amount, and then — this is the vital part — turns and falls. Bet past Kelly and you grow more slowly, not faster. Bet at twice Kelly, and your long-run growth drops all the way back to zero. Bet beyond that, and you go broke with certainty, even with a real edge.

Study the shape of that hump, because it is the answer to the module's puzzle. The curve is not symmetric around its peak. Bet a little less than Kelly and you slide only slightly down the left side — you give up a small sliver of growth. Bet a little more than Kelly and you fall down a steeper right side, losing growth fast while your swings grow violent. Under-betting is a minor cost. Over-betting is a catastrophe. When your inputs are uncertain — and in investing they always are — you want to err firmly on the safe side of the peak.

Now answer the doubt this module exists for, the one every beginner asks: if the maths says bet 20%, and the maths is proven, why on earth would I bet less? illustrative

Three reasons, and together they are decisive.

First, your edge is a guess, and betting too much is punished far harder than betting too little. Kelly assumes you know your edge exactly. You do not — you estimated a 60% win chance, but the true figure might be 55%, or 50%. Look again at the hump: if you bet the full 20% Kelly told you, but your real edge only justified 10%, you are now betting at twice the correct fraction — the point where long-run growth collapses toward zero. Because humans reliably over-estimate their own edge, full Kelly on a flattered estimate is a fast road to the right-hand cliff. Betting a fraction is the cushion for being wrong about how right you are.

Second, full Kelly is savagely volatile even when your edge is exactly correct. Full Kelly is mathematically optimal for growth, but it produces gut-wrenching swings — drawdowns of 50% or more are routine along the way. Almost no human can hold a strategy through that without panicking and abandoning it at the worst moment, which destroys the very growth the maths promised. A sizing rule you cannot emotionally survive is not optimal in any real sense.

Third — and this is the beautiful part — cutting the bet costs you far less than you would expect. Because the hump is nearly flat near its top, betting half of full Kelly captures roughly three-quarters of the long-run growth while cutting the size of your swings by about half. You give up a quarter of the growth and buy back half the pain. That is not a close call. That trade — a little less growth for a much smoother, more survivable ride — is why serious investors almost universally use fractional Kelly: they compute the full figure and then deliberately bet a half, a third, or a quarter of it.

So the honest answer to "the maths says bet big — why bet small?" is this: the maths says bet big only if your inputs are perfect and your nerves are infinite. Neither is true. Fractional Kelly keeps almost all of the reward, removes most of the danger, and protects you from the one error the formula cannot — over-estimating yourself. It is the same survival instinct as the last module, now dressed in arithmetic.

Kelly is a sharp tool, and like every sharp tool it is dangerous when used where it does not fit.

It cannot know your real edge — and it trusts whatever you tell it. The formula's entire output rests on inputs you supplied: your win probability, your payoff. Feed it an honest edge and it guides you well; feed it a flattered one and it will confidently size you into ruin. Kelly does not check your inputs. Fractional Kelly exists precisely because you cannot fully trust them.

It assumes bets you can repeat many times. Kelly optimises long-run growth over many independent bets. A single, once-in-a-lifetime, all-or-nothing wager is not what it was built for, and applying it there can justify a recklessly large stake on something you only get to do once.

It ignores correlation between your bets. Classic Kelly sizes one bet in isolation. If you run it separately on five positions that are really the same bet — the five-banks problem again — you can end up with five "correct" Kelly sizes that together are a wildly oversized single wager. Kelly must be paired with the correlation thinking from earlier in this part.

It is about growth, not about sleep. Even used perfectly, full Kelly optimises how fast money compounds, not how comfortable the ride is. If the swings would make you abandon the plan, the mathematically optimal size is the wrong size for you. The best rule is the one you can actually keep.

Kelly is famous, which means it is misused in famous ways. Watch for these.

1. Betting full Kelly on a guessed edge. The formula's precision seduces people into trusting inputs that are really hopes. Because over-betting is punished so much more than under-betting, a flattered edge run at full size is one of the fastest ways to blow up with a real advantage.

2. Forgetting the hump turns down. Many assume "more edge, more bet, always more growth." Past the peak, betting more lowers growth and, eventually, guarantees ruin. There is such a thing as too big even when you are genuinely right.

3. Applying it to a one-shot bet. Kelly is about compounding over many repetitions. Used to justify a huge stake on a single irreversible decision, it is being asked to do a job it was never designed for.

4. Running it position-by-position and ignoring correlation. Five separate "optimal" bets on the same underlying risk stack into one enormous, un-optimal bet. Size the portfolio's real bets, not each name as if it stood alone.

5. Mistaking discomfort for sub-optimality. Cutting to half or quarter Kelly feels like leaving money on the table. It is not — it is buying survivability cheaply, keeping most of the growth while removing most of the ruin. The discomfort is the price of a ride you can actually stay on.