Position_Concentration_Risk

Notes on diversification and portfolio risk

Intuition

Think about card counting in blackjack. You might have a small edge — say a 51% chance of winning. If you bet your whole bankroll on a single hand, the expected outcome is in your favor, but the variance is huge. You could easily lose everything.

If instead you spread the same bankroll across many hands, each independent of the others, the variance drops. The expected return stays the same, but your risk of losing everything gets much smaller.

The lesson: independent bets make your edge safer to realize.

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Simulating Blackjack

We can model this with probability simulations.

• One hand: results are binary (win or lose), variance is very high.
• 100 hands: the expected win rate (51%) shows up more clearly, variance shrinks, but it’s still noisy.
• 10,000 hands: the distribution tightens around the expected edge, making the outcome much more reliable.

This shows why you don’t want to risk everything on a concentrated bet! Variance dominates in the short run, and only diversification lets your edge shine through.

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Extending to Portfolios

The same principle applies to investing. Returns are uncertain, but if you spread your bets across many assets, risk falls.

• Case 1: Single Asset Your portfolio is entirely exposed to the volatility of that one asset.

• Case 2: Correlated Assets Adding assets that move together doesn’t help. If they all rise and fall at the same time, you’ve basically made the same bet multiple times.

• Case 3: Uncorrelated Assets Holding independent assets smooths out shocks. When one drops, another might rise. Portfolio volatility falls well below the volatility of any single component.

This is the real benefit of diversification: not changing expected returns, but lowering the variance of outcomes.

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Portfolio Math

In finance, risk is usually measured as volatility (standard deviation).

For a two-asset portfolio, variance is:

$$ \sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho_{12}\sigma_1\sigma_2 $$

• $w$ = portfolio weights
• $\sigma$ = asset volatilities
• $\rho$ = correlation between assets

The last term shows the impact of correlation:

• If correlation = 1, no diversification benefit.
• If correlation = 0, variance is much lower than the weighted average of risks.

With more than two assets, the same principle extends: the lower the correlations, the bigger the risk reduction.

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Practical Limits

In theory, holding many uncorrelated assets reduces risk further and further. In practice, constraints like transaction costs and minimum capital sizes limit how many positions you can hold.

Fortunately, you don’t need hundreds of assets. The biggest benefits are captured within the first ~20 independent positions.

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Final Point

Diversification is risk management. It doesn’t increase expected return, but it protects you from the danger of any single position dominating your portfolio.

Spread your bets across as many uncorrelated assets as possible. This way, your edge has the best chance to work without being wiped out by variance.