Position sizing under fat tails

M3.1 to M3.5 were diagnosis. This is the treatment: given that returns have kurtosis of 20, correlations converge in crises, and every risk estimate is unreliable, how much should you actually bet?

It is the stage of M2.1's pipeline that gets the least attention and destroys the most people. A mediocre signal sized sensibly survives. An excellent signal sized aggressively does not — and the fat tails are precisely why.

Kelly, and what it recommends

The Kelly criterion maximises the long-run growth rate of wealth. For continuous returns it has a famously simple form:

   f*  =  μ / σ²        the growth-optimal fraction of capital

   S&P 500, 2016–2026:   μ = 12.7%,  σ = 18.1%

   f*  =  0.127 / 0.181²  =  0.127 / 0.0329  =  3.87×  LEVERAGE

Kelly says: borrow to hold nearly four times your capital in equities. That should immediately feel wrong, and it is worth understanding precisely how it is wrong, because the failure is instructive rather than a flaw in the mathematics.

Run it through the actual path

Take those same 2,513 real daily returns and apply constant leverage L, rebalanced daily:

   leverage    final wealth    max drawdown    worst day
   ──────────────────────────────────────────────────────────
      1.00          3.56×          33.9%        −12.0%
      1.94          8.65×          57.9%        −23.2%   ← half Kelly
      3.00         16.50×          76.7%        −36.0%
      3.87         20.73×          86.8%        −46.4%   ← full Kelly
      5.00         17.97×          94.5%        −59.9%

Two rows deserve attention.

Full Kelly (3.87×) does produce the highest wealth — 20.7× — and an 86.8% drawdown to get it. That is not a risk of ruin; it is near-certain career death. No allocator, fund, or human tolerates losing 87% en route to a good outcome. Kelly maximises terminal growth and is entirely indifferent to the path, which is a mismatch with every real constraint anyone operates under.

At 5× leverage, wealth actually falls — 17.97× against 20.73× — while drawdown climbs to 94.5%. This is the clearest possible demonstration that leverage is not monotone in return. Past the growth optimum, variance drag dominates: compounding is multiplicative, and a −50% day requires a +100% day to undo. The growth rate under leverage L is approximately

   g(L)  ≈  L·μ  −  ½·L²σ²

   linear gain, QUADRATIC penalty — so there is always a peak,
   and beyond it more risk buys less money

Anyone who has levered a good strategy into a bad outcome has met this curve.

Three reasons to bet less than Kelly says

1. You don't know μ. Kelly takes μ and σ as known. M2.1 established that a Sharpe estimate carries a standard error around 0.45 on five years of daily data — so f* = μ/σ² is a ratio of estimated quantities, and the numerator is the badly estimated one. Overestimate μ by a factor of two and you bet twice Kelly, which the table above shows is on the wrong side of the peak. Since the growth curve is flat near its maximum but falls steeply beyond it, the asymmetry is decisive: underbetting costs you a little, overbetting costs you everything. Half-Kelly captures roughly 75% of the growth for a fraction of the drawdown, which is why it is the practitioner default.

2. Kelly's derivation assumes continuous rebalancing. It assumes you can adjust as the price moves. Real markets gap — overnight, on news, on 2020-03-16. At 3.87× leverage a −12% gap is a −46% loss with no opportunity to reduce on the way. And M0.7's margin mechanics mean that loss arrives as a cash call the following morning. Fat tails plus leverage plus daily settlement is precisely the combination that ends funds.

3. It assumes you know the distribution. With α ≈ 3 (M3.2), the moments Kelly needs are estimated from a sample dominated by a handful of days. Sizing off σ when σ itself is unstable is building on the least reliable statistic you have.

Volatility targeting — the practical alternative

The insight from M3.3 was that volatility is forecastable even when returns aren't. That is exactly what a position sizer needs:

                        σ_target
   position_t  =  ─────────────────────
                    σ̂_t   (forecast)

   calm market   → σ̂ low  → size UP
   turbulent     → σ̂ high → size DOWN, automatically

This works because it uses the part of the distribution that is predictable, and it side-steps the estimation problem that wrecks Kelly — you never need μ, only σ, which is the quantity you can actually forecast. It delivers roughly constant risk through time rather than constant exposure, it de-levers into crises without anyone deciding to, and empirically it improves the Sharpe of most strategies while sharply reducing drawdowns.

It is not free. Vol targeting is procyclical — it sells after volatility rises, which is after prices have fallen — so at scale it can amplify the very selloff it responds to. This is a live criticism of risk-parity and vol-target funds as a class, and it is another instance of M3.5's point that the mechanism, not the statistics, transmits stress.

Drawdown is the real constraint

The deepest reframing here is what you are optimising at all. Variance is the textbook risk measure because it is analytically convenient. Nobody is fired for variance.

   what the model minimises      what actually ends things
   ────────────────────────────────────────────────────────────
   variance                      DRAWDOWN — investors redeem,
                                 risk limits bind, you are
                                 stopped out at the bottom
   expected loss                 a single day you cannot fund
   long-run growth rate          a path you do not survive to
                                 the end of

Which surfaces the distinction that makes all this coherent. Ensemble average is not time average. Averaged across many parallel traders, a high-leverage strategy might have a wonderful expected return, carried by the few who avoided the bad path. But you only get one path. The quantity that governs a single realisation is the time-average growth rate, and it is systematically lower than the expected return whenever returns are multiplicative — dramatically so under leverage and fat tails.

This is the ergodicity argument, and it is the rigorous version of an old piece of trading-desk wisdom:

Survival is a precondition, not a preference. A strategy that compounds at 8% for twenty years beats one that compounds at 40% for four and then goes to zero — and the second one looks far better right up until it doesn't.

Source: Ed Thorp, A Man for All Markets, for Kelly from someone who used it in both casinos and markets. On fractional Kelly and estimation error, MacLean, Thorp & Ziemba, The Kelly Capital Growth Investment Criterion. Ole Peters, “The ergodicity problem in economics” (Nature Physics, 2019) is short, well argued, and the clearest statement of the time-average point — a physics paper about finance, which is the theme of this whole module.