You will be able to explain the Kelly criterion and why full Kelly sizing is too aggressive in practice.
A friend of Marcus who reads a lot about trading told him there was a formula that gives the mathematically perfect position size. Put in your edge and your odds, and it tells you exactly how much to bet to grow your money fastest. Marcus tried it on Larkspur with his own estimates and the formula said to put about half his stock money into it. That answer was the most useful thing the formula taught him, because it was so obviously wrong for him.
This lesson explains the Kelly criterion, why the people who use it professionally bet a fraction of what it says, and how to use it as a ceiling rather than a target, with made-up figures throughout.
In 1956 John Kelly, a researcher at Bell Labs, published a formula for the bet size that maximises the long-run growth rate of your money when you have an edge. The mathematician Edward Thorp later applied it to blackjack and then to investing. For a simple bet, the Kelly criterion says to stake a fraction of your money equal to your chance of winning, minus your chance of losing divided by the odds you're paid.
Take a made-up bet that pays even money, so you win what you stake, and that you win 55% of the time. Kelly says stake 0.55 minus 0.45 divided by 1: 10% of your money on each bet. Half Kelly is 5%.
Compound the typical result over 100 such bets and full Kelly turns S$1 into about S$1.65, while half Kelly turns it into about S$1.46, so halving the bet keeps about three quarters of the growth rate. Twice Kelly, staking 20% a bet, ends at about S$0.99: all that extra risk bought nothing, and beyond twice Kelly you lose money even with a real edge.
Kelly gives the best answer only if you know your edge exactly. In a casino game the odds can be calculated. In investing they can only be estimated, and estimates are usually too optimistic.
Run the same bet with a true win rate of 50% instead of 55%, so there's no edge at all, while you keep betting as if there were. At 10% a bet, S$1 shrinks to about S$0.61 over 100 bets. At 5% a bet, it shrinks to about S$0.88. Overestimating your edge by five points turned full Kelly into a 39% loss and half Kelly into a 12% loss. The two errors are lopsided, since underestimating an edge costs you some growth while overestimating it costs you capital.
Now apply the investment version to Larkspur. For a stock, the Kelly fraction is roughly the expected return above cash divided by the variance, which is volatility squared. Marcus's made-up estimates were an expected return of 6% a year above cash, from his DCF value of about S$1.80 against a price of S$1.60, and a volatility of 35%. The formula gives 0.06 divided by 0.1225, about 49% of his money. Half Kelly is about 24%, quarter Kelly about 12%.
Look at what sits inside that 6%. It comes from a valuation that lesson 8.6, Sensitivity tables and scenarios instead of one target price, showed could reasonably range from about S$1.05 to S$2.40. If the real edge is 3% rather than 6%, the right Kelly fraction halves to about 24%, so Marcus's "half Kelly" would in fact be full Kelly. If the edge is zero, any position is overbetting.
Even when the edge is known, full Kelly produces swings most people couldn't live with. In the standard continuous-time version of the maths, a full Kelly bettor has a 50% chance of seeing their money halve at some point, and at half Kelly the chance falls to about one in eight. That's the price of the fastest growth: deep, frequent drawdowns that lesson 2.2, Drawdown and time under water are the risks you feel, showed are the ones investors abandon plans over.
There's one more gap between the formula and a real portfolio. Kelly sizes one bet in isolation. Your positions aren't isolated. Larkspur and the chip designer both depend on the chip cycle, so sizing each by Kelly as if it stood alone would bet on that cycle twice. Lesson 11.5, Correlation clusters: hidden bets across your holdings, deals with that.
For all these reasons, professional investors who use Kelly at all tend to bet half of it or less.
What stays useful from the formula is the shape of its curve. Growth rises as you bet more, up to the Kelly fraction, then falls, and past twice Kelly it turns negative, so betting too little costs some growth while betting too much costs far more and adds ruinous drawdowns. When you're unsure of your edge, which is always, err small.
That makes Kelly useful as an upper limit on size. Marcus wrote: "No single stock above a quarter of the Kelly fraction on my own estimate of its edge, and never above my position limit, whichever is lower." For Larkspur, quarter Kelly of about 12% sits far above the limit he sets in lesson 11.3, Position and sector limits, and concentration risk, so the limit decides. The Kelly ceiling only bites when his estimated edge is small, which is exactly when it should.
For the activity, you'll work out full and half Kelly for a made-up bet that wins 55% of the time at even odds, then see what each does if the true win rate is 50%.
Calculate full and half Kelly for a made-up bet with a 55% win rate at even odds and write what happens if your true win rate is 50%.
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