Shannon's Demon: Volatility Harvesting and the Loss Bias of Leverage

Imagine an asset that moves +10% or −10% each round with equal probability. On average it goes nowhere: the arithmetic mean return is zero. Yet if you hold it, you lose about half a percent per round.

That is the loss bias in its simplest form. A 10% loss needs an 11.11% gain to get back to even, so a +10%/−10% coin does not round-trip. The asymmetry compounds, and leverage makes it worse.

Now put half your money in the asset and half in cash, and rebalance after every round. The same losing asset loses less — and with a small upward edge, the rebalanced portfolio turns into a positive compounder. That is Shannon's demon. The two effects are the same arithmetic with opposite signs, and the same curve explains why a 2× or 3× ETF bleeds in a choppy market.

The Loss Bias in One Round

Take a fair coin: +10% with probability $1/2$, −10% with probability $1/2$. One up and one down gives

$1.10 \times 0.90 = 0.99.$

A 1% loss per pair, or about $0.5\%$ per round. The arithmetic mean is zero, but the geometric mean is

$\sqrt{0.99} \approx 0.99499,$

a compound growth rate of $-0.501\%$ per round. After 100 rounds the asset is at $0.99499^{100} \approx 0.605$ — down almost 40%.

This is not a market prediction. It is the arithmetic of percentages. The same asymmetry explains why a 50% loss needs a 100% gain to recover, and why a 10% loss needs 11.11%.

Leverage multiplies the daily move. A 2× fund goes up 20% or down 20%; a 3× fund goes up 30% or down 30%. The round-trip outcomes are:

The loss bias grows faster than leverage: roughly with the square of it. The 3× fund turns a fair coin into a near-total loss over 100 rounds.

The Demon

Now hold 50% in the asset and 50% in cash, rebalancing after each round. The portfolio moves +5% or −5%:

$1.05 \times 0.95 = 0.9975.$

Its geometric mean is $\sqrt{0.9975} \approx 0.99875$, or $-0.125\%$ per round — much better than the asset's $-0.501\%$, but still negative. Rebalancing harvested volatility, but there was no positive edge for it to harvest.

To get the classic positive demon, the asset must have a positive arithmetic edge. Give the coin a 52.5% chance of going up. Then the asset's expected log growth is exactly zero:

$p \ln(1.1) + (1-p)\ln(0.9) = 0 \quad\Longrightarrow\quad p \approx 0.525.$

The asset is volatile but goes nowhere. The 50/50 portfolio, rebalanced each round, has a log growth of about $+0.125\%$ per round, or a geometric $1.134\times$ after 100 rounds. The demon is real — but with ±10% moves it is small.

The same asset, four exposures: the 50/50 compounds, buy-and-hold is flat, and both leveraged products bleed. The demon and the loss bias are two sides of one coin.

The Formula

For a constant exposure $f$, the expected log growth per round is

$g(f) = p \ln(1 + 0.1f) + (1-p)\ln(1 - 0.1f).$

Differentiate and set it to zero. The growth-maximizing exposure is

$f^* = \frac{2p - 1}{0.1},$

clamped at zero. For a fair coin ($p=0.5$), $f^*=0$: cash is optimal. At $p=0.525$, $f^*=0.5$ — Shannon's 50/50. At $p=0.55$, $f^*=1$: hold the asset. At $p=0.6$, $f^*=2$: the growth-optimal exposure is 2× leverage.

This is the same Kelly formula as in the Kelly criterion article, specialised to a ±10% bet. The curve is a hill: the left side is the harvest, the peak is Kelly, and the right side is the loss bias.

For small moves, the same hill can be written as

$g(f) \approx f g + \frac{\sigma^2}{2}f(1-f),$

where $g$ is the asset's growth and $\sigma \approx 10\%$ is its volatility. The second term is the volatility harvest; it peaks at $f=1/2$ when $g=0$, and at $f=1/2+g/\sigma^2$ in general.

Leverage

A leveraged ETF is managed to a fixed daily multiple, not a fixed long-run multiple. A 2× fund on our ±10% coin goes up 20% or down 20% each round; a 3× fund goes up 30% or down 30%. The round-trip factors are $0.96$ and $0.91$, and after 100 rounds they are $0.96^{50}\approx 0.13$ and $0.91^{50}\approx 0.009$.

The extra drag from leverage is approximately

$\frac{L(L-1)}{2}\sigma^2$

per round, where $L$ is the leverage multiple. At $\sigma \approx 10\%$, that is about 1% a round for a 2× fund and 3% a round for a 3× fund. The loss bias grows with the square of leverage.

A cleaner way to isolate the leverage effect is to make the index itself flat. Suppose the index goes up 10%, then down 9.09%, so it ends where it started. A 2× fund goes up 20%, then down 18.18%, and ends at $1.20 \times 0.81818 \approx 0.982$ — a 1.8% loss. A 3× fund ends at $1.30 \times 0.72727 \approx 0.945$, a 5.5% loss. Repeat ten times and the index is still flat, while the 2× fund is down about 17% and the 3× fund about 43%.

The Playground

The playground below uses the ±10% coin. Choose the probability of an up round and the number of rounds; the four strategies are the 50/50 portfolio, buy-and-hold, and 2× and 3× daily-reset ETFs.

Things to try:

• Fair coin (50%): every strategy loses, including the 50/50. The demon reduces the damage but cannot create a profit without an edge.
• Goes nowhere (52.5%): the asset is flat, the 50/50 compounds at about 0.13% a round, and the 2× and 3× funds lose. This is Shannon's demon in its purest form.
• 55% (Kelly 1×): the asset grows, and the 2× fund is exactly twice Kelly — its growth is essentially zero. This is the same "twice Kelly stops growing" point from the Kelly article.
• 60% (Kelly 2×): the asset's log growth is 1.5% a round, the 2× fund's is 2.0%, and the 3× fund is past the peak. Leverage is not always the villain; it depends on the edge relative to the variance.
• Set rounds to 500 and watch the 3× median collapse. The range chart shows the 10th–90th percentile spread; the 3× fund has a few spectacular winners and a very low median.

The Catch

The demon needs an edge. The fair ±10% coin has no edge; the best the demon can do is lose less. The 50/50 harvest is about 0.125% per round, while the asset's own negative drift costs about 0.25%; the net is still negative. The positive demon requires the asset's arithmetic mean to exceed its volatility drag.

Trends hurt. Rebalancing sells winners. If the asset rises 10% twice, buy-and-hold turns $1 into $1.105 while the 50/50 turns it into $1.1025 — a small shortfall. In a strong trend, the shortfall compounds. The demon is a bet on oscillation and mean reversion, not on direction.

Costs matter. The harvest at ±10% is only about 0.125% per round. A spread or commission of a few basis points on each rebalance eats a meaningful fraction of that. Rebalance in bands, not on every tick.

The edge must be real. The difference between a fair coin and the zero-growth coin is only 2.5 percentage points. If you estimate $p$ with error, the "optimal" exposure can be on the wrong side of the peak. Use a fraction of Kelly, not the full theoretical value.

Leveraged ETFs add their own risks. Daily resetting, financing costs, fees, tracking error, and the possibility of a catastrophic daily move. The loss bias is the price of constant leverage; it can be worth paying for a deliberate, short-term trade in a trending market, but it is a poor deal for a long-term holding.

What to Take Away

The ±10% coin makes the arithmetic transparent. A fair coin loses about 0.5% a round even though its average return is zero. Rebalancing half against cash cuts that to 0.125%. A 2.5-point edge turns the demon positive; leverage turns it sharply negative.

The same curve describes all of it:

$g(f) = p \ln(1 + 0.1f) + (1-p)\ln(1 - 0.1f).$

The peak is Kelly, the left side is the harvest, and the right side is the loss bias. Before adding leverage, ask where you are on the curve — and how confident you are in the edge that put you there.

Further Reading

• William Poundstone, Fortune's Formula — the history of Shannon's idea, Kelly, and Thorp
• J. L. Kelly, "A New Interpretation of Information Rate" (1956) — the original growth-optimal sizing paper
• David Luenberger, Investment Science — the "volatility pumping" view of rebalanced portfolios
• E. Robert Fernholz, Stochastic Portfolio Theory — the "excess growth rate" of diversified portfolios
• David Booth & Eugene Fama, "Diversification Returns and Asset Contributions" (1992) — the empirical rebalancing premium
• SEC, Leveraged and Inverse ETFs: Specialized Products with Extra Risks for Buy-and-Hold Investors