Full position, half position, Sharpe ratio, geometric return

PortfolioAlert · 2026-02-11T11:53:16+00:00

The Sharpe ratio is used to measure the excess return obtained per unit of risk. By comparing the returns and volatility of full and half positions, analysis shows that in long-term investments, the geometric return of a half position outperforms that of a full position, revealing the impact of the "volatility tax." This phenomenon explains the efficiency advantage of holding a diversified asset portfolio.

PortfolioAlert

2026-02-11 11:53:16

Abstract generation in progress

Sharpe Ratio formula: (Rp - Rf) / σ, where Rp is the arithmetic return, Rf is the risk-free rate, and σ is the standard deviation of the asset.
The Sharpe Ratio essentially indicates how much excess return is earned per unit of total risk taken.
Let’s conduct a thought experiment: assume the benchmark for the stock market is the main index, and compare a fully invested position versus a half-position, tracking in real-time without considering wear and tear. In reality, the full position’s Sharpe Ratio is (Rp - Rf) / σ; for the half position, the arithmetic return is 0.5Rp + 0.5Rf, and the volatility is 0.5σ. Calculating this, the half position’s Sharpe Ratio actually equals that of the full position.
The Sharpe Ratio is about how, by combining risk-free assets and risky assets, one can adjust risk exposure while maintaining the same Sharpe Ratio.

But is that all? Many fund products favor the Sharpe Ratio, but in reality, it measures the trade-off between risk and return over a single period, not long-term performance.
In fact, for long-term investing—especially when involving compound interest—the focus is not on arithmetic returns but on geometric returns.
For geometric returns, an approximation is G = Rp - 0.5σ^2.
To simplify, set the risk-free rate to zero.
The geometric return for a full position is: Rp - 0.5σ^2
For a half position: 0.5Rp - 0.125σ^2
It’s easy to see that the geometric return of the half position is greater than half of the full position’s return.
This is what is called the “volatility tax”—volatility is the enemy of compound interest.
In fact, the earlier discussed Sharpe Ratio essentially states that, holding the asset constant, position control does not affect the Sharpe Ratio;
and the “volatility tax” in a sense suggests that, with the asset unchanged, smaller positions are more efficient.
Perhaps this explains why the so-called stock-bond balance offers a better investment experience?
Of course, this assumes that positions are adjusted in real-time and does not consider rebalancing effects.
Feel free to discuss.

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