Explain the Kelly criterion and why full Kelly sizing is too aggressive for real investing.
Marcus is deciding how much of their stock money to put into Larkspur (an invented company). A friend of theirs who reads a lot about trading has passed on a formula that is said to give the mathematically perfect position size for the fastest growth, once you know your edge and the odds. When Marcus puts Larkspur's numbers into it, the answer is to hold about half of their stock money in that one company. The answer is clearly wrong for Marcus, and you need to understand why before you rely on the formula yourself.
The formula is the Kelly criterion: the bet size that maximises the long-run growth rate of your money when you have an edge. John Kelly, a researcher at Bell Labs, published it in 1956. The mathematician Edward Thorp later used it for blackjack and afterwards for investing.
A simple bet needs only the chances and the payout. Start with the probability of winning. Divide the probability of losing by the odds paid. Subtract that second number from the first, and what remains is the fraction of your money to stake. For real use, halve that fraction or go lower.
Say a bet wins 55% of the time, loses 45% of the time and pays even money (made-up figures). At even money you win what you stake, so the odds are 1. The working is 0.55 - 0.45 / 1, which comes to 10%, so full Kelly says to stake 10% of your money on each bet. Half Kelly stakes half the full fraction, 5% here. Quarter Kelly stakes a quarter of it.
Run that 55% bet 100 times. Staking 10% each time, a typical result turns S$1 into about S$1.65. Staking 5% each time, S$1 grows to about S$1.46. Half Kelly risks half as much per bet and still keeps about three quarters of full Kelly's growth rate.
Growth increases as the stake rises, up to the Kelly fraction. Beyond that point growth drops, and once the stake passes twice Kelly it goes negative. At 20% per bet, which is twice Kelly for this bet, S$1 finishes 100 bets at about S$0.99, roughly where it started. Stake more than twice Kelly and you lose money even with a genuine edge. The two kinds of mistake don't cost the same: staking too little gives up some growth, while staking too much gives up far more and brings drawdowns that can ruin you.
Kelly gives the best size only when the edge is known exactly. Casino odds can be worked out precisely. Investment odds can only be estimated, and estimates usually lean optimistic.
Go back to the same bet and suppose it actually wins 50% of the time. There is no edge, but you size it as if the win rate were 55%, five points too high. Staking 10% a bet for 100 bets, S$1 falls to about S$0.61, a 39% loss. Staking 5%, it falls to about S$0.88, a 12% loss. With no real edge, every position is too big, and the smaller one loses less. The same pattern holds in general: underestimating your edge gives up some growth, and overestimating it loses capital.
Knowing the edge exactly still leaves the problem of how far your money falls along the way. Full Kelly produces drawdowns most people can't sit through. Standard continuous-time maths gives a full Kelly bettor a 50% chance that their money halves at some point, and a half Kelly bettor about a one in eight chance. Lesson 2.2, "Drawdown and time under water are the risks you feel", showed that deep, frequent drawdowns are the ones that lead investors to abandon their plans. Professionals who use Kelly at all generally bet half of it or less. You never know your edge for certain, so lean towards the smaller size.
For a stock, the Kelly fraction is roughly the expected return above cash divided by the variance. Variance is volatility squared. Estimate the expected return above cash, square the volatility to get the variance, divide the first by the second for full Kelly, and then take a fraction such as half or a quarter.
Marcus's DCF puts Larkspur's value at about S$1.80, and the stock trades at S$1.60. From that gap they estimate an expected return above cash of 6% a year. Volatility is 35%, which gives a variance of 0.1225. Dividing 0.06 by 0.1225 gives about 49% of their stock money as full Kelly, which is where the friend's formula got its answer. Half Kelly is about 24% and quarter Kelly about 12%.
All of this rests on the S$1.80 valuation. In Lesson 8.6, "Sensitivity tables and scenarios instead of one target price", Larkspur's value ranged from about S$1.05 to S$2.40, so the real edge could be below 6% or zero. With a 3% edge, the Kelly fraction falls to about 24%. If the true edge is half your estimate, the correct Kelly fraction is also half, so a half Kelly position built on an inflated estimate is actually a full Kelly position.
The formula also sizes each bet on its own, and real holdings affect each other. Marcus owns a chip designer as well, and both that company and Larkspur depend on the chip cycle. Sizing each one by Kelly as if they were independent counts the shared chip-cycle risk twice. Lesson 11.5, "Correlation clusters: hidden bets across your holdings", covers how to handle positions like these.
Marcus decides to use Kelly only as an upper limit. Their rule is that no single stock goes above quarter Kelly on their own edge estimate, or above their position limit, whichever is lower. To apply it, work out quarter Kelly from your own edge estimate, compare it with your position limit, and use the lower of the two as your maximum size.
For Larkspur, quarter Kelly comes to about 12%, well above the position limit Marcus set in Lesson 11.3, "Position and sector limits, and concentration risk". The position limit therefore sets the size. The Kelly ceiling only becomes the lower number when the estimated edge is small, and that is when you want it to limit you.
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%.
Work out full Kelly and half Kelly for a made-up bet at even odds with a 55% win rate, then write down what happens if your real win rate is 50%.
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