The Caladan Weekly | The Arithmetic Trap: Why Most Investors Optimize for the Wrong Number

The Arithmetic Trap: Why Most Investors Optimize for the Wrong Number

I might be preaching to the choir here but: walk into any investment pitch and you’ll hear the same obsession. Hedge funds promise 20% alphas. ETFs advertise 2x or 3x exposure. Everyone’s chasing the highest arithmetic return they can find, competing on who can project the biggest numbers.

But they might be optimizing for the wrong variable.

There’s a mathematical trap that destroys this entire framework. It wipes out leveraged portfolios, ruins option sellers, and turns strategies with impressive backtests into slow-motion disasters.

The math itself is straightforward, but it gets ignored in practice more often than you’d expect.

The Trap Hiding in Plain Sight

Start with a thought experiment.

Here’s a simple coin flip game: heads doubles your money, tails loses 60%. Fair coin, so the arithmetic expected return is +20% per flip. Most investors would call this attractive. Some would leverage it.

Run this 1,000 times across 25,000 simulated paths and the median outcome approaches zero.

Most paths end in ruin despite the positive expected value. The arithmetic mean keeps climbing (pulled up by lucky outliers), while the geometric mean (what you actually get) declines toward zero.

This is volatility drag, and the formula is straightforward:

Volatility Drag ≈ σ² / 2, where σ is annual volatility.

For a 30% volatility asset, that’s a 4.5% annual drag. Your 10% arithmetic return becomes 5.5% geometric.

At 70% volatility, you need over 25% arithmetic returns just to break even geometrically. This shows up across real markets, and it gets worse when leverage enters the equation.

How Leverage Amplifies the Problem

Volatility drag scales with variance, not volatility. This matters enormously because it means doubling your leverage quadruples your drag.

Take a 30% volatility asset. Unleveraged drag is 4.5%. At 2x leverage, volatility becomes 60%, but drag jumps to 18%. That’s 4x worse, not 2x.

Real world example: MicroStrategy stock delivered 87% annualized returns over five years. A 2x leveraged long position? Only 44% annualized. The 2x leveraged short position lost 98%.

Daily rebalancing in leveraged products creates path dependency. During volatile periods, the rebalancing mechanism systematically destroys value that can’t be recovered.

So if leverage multiplies the drag problem, does that mean we should avoid it entirely? Not necessarily. There’s a way out, but it requires thinking differently about what we’re actually leveraging.

The Counterintuitive Solution

The solution isn’t to avoid leverage. It’s to leverage a diversified portfolio instead of a single asset.

Compare these approaches:

The diversified portfolio delivers 5x the compound growth despite half the arithmetic return. Lowering volatility through diversification creates exponentially larger benefits than chasing higher returns.

Rebalancing adds another layer. When you periodically restore target weights, you’re mechanically selling high and buying low across return streams. This works best with low-correlation assets.

But if volatility creates this mathematical headwind, how much leverage should anyone actually use? The Kelly formula provides a framework for thinking about this.

The Optimal Sizing Problem

The Kelly formula gives optimal leverage as:

f = μ / σ²*

Where μ is expected return and σ² is variance. Notice it’s inverse to variance, not volatility. This relationship explains why high volatility investments can warrant position sizes below available capital.

Same expected return, but 2x the volatility means you should use less than 1x leverage. High volatility investments often warrant underweighting relative to available capital.

Most practitioners use fractional Kelly (1/2 or 1/4) to reduce wealth volatility while capturing most of the growth benefit.

Understanding optimal sizing opens up another angle. If volatility creates drag for long positions, there might be an edge in being on the other side of the trade.

The Premium for Taking the Other Side

Implied volatility consistently exceeds realized volatility. The VIX typically overpredicts S&P 500 volatility by several percentage points. This gap can be harvested by selling volatility.

The catch: you’re collecting small premiums while exposed to catastrophic tail risk. “Picking up nickels in front of a steamroller” captures the profile.

Historical backtest data (2004-2013):

The VRP-targeted approach switches between long and short volatility ETNs based on the spread between VIX and 10-day realized volatility. When the 5-day moving average of (VIX – realized vol) exceeds zero, go short volatility. Otherwise, go long.

High returns, but drawdowns exceed 50%. Position sizing matters critically. Keep allocations in low single digits.

Selling volatility harvests a premium. But there’s a flip side that reveals something even more counterintuitive about how returns compound.

When Losing Money Improves Performance

Three ways to add protection to a 100% equity portfolio:

1. Store-of-Value (T-bills): 2% real return, zero crash correlation
2. Alpha (CTAs, gold): 20% in crashes, 10% in moderate drawdowns, 5% otherwise
3. Insurance (tail hedging): 900% in crashes, -100% all other years

Simulated 20-year results with multiple crash scenarios:

The insurance approach has 0% average return but delivers the best geometric growth. A 3% allocation returning 0% on average outperforms like it’s returning 30% annually. The extreme convexity during crashes more than compensates for the steady bleed.

The math: preventing severe drawdowns maintains compounding ability. A 50% loss requires a 100% gain to break even. Avoiding that hole matters more than the premium cost.

All of this diversification, rebalancing, and optimal sizing depends on one critical assumption. When that assumption breaks, everything changes.

The Diversification Illusion

Diversification depends on correlation. Below 0.5 helps significantly. Above 0.8 offers little benefit.

Problem: correlations shift during crises, exactly when you need diversification most.

Traditional bonds maintain stable diversification. High yield bonds act more like equity during stress. Managed futures show modest negative correlation that holds up reasonably well across regimes.

This correlation instability explains why portfolios that look diversified in backtests can collapse during actual market stress. The protection can disappear when it matters most.

What the Math Actually Tells Us

Most investors optimize for arithmetic returns because they’re easier to calculate and more impressive to pitch. But portfolios compound geometrically, not arithmetically. The gap between them scales with the square of volatility, creating a systematic bias in how people think about returns.

This isn’t necessarily about being conservative or avoiding risk. It’s about understanding what drives long-term wealth accumulation. Sometimes the path to higher compound returns runs through lower arithmetic returns and less volatility, not higher returns and more risk.

The math doesn’t care about intuition, but it’s worth paying attention to.

Commentary by Chris Park, Director of BitGo Korea

“The math doesn’t punish optimism, it just exposes it. Compounding is ruthless to variance, indifferent to narrative, and perfectly fair over time. Once you see that, alpha becomes a volatility management problem, not a return forecasting one.

Most investors intuitively chase arithmetic returns, but long-term wealth is geometric. The subtlety is that variance, not bad timing or lack of conviction, is often what kills compounding. Managing volatility is managing survival.”