Full position, half position, Sharpe ratio, geometric return

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 is the stock market index, and compare a full position in the index versus a half position, tracking in real-time without considering slippage. 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 the full position’s.
The Sharpe Ratio is about how, by combining risk-free assets and risky assets, you can adjust risk exposure while maintaining the same Sharpe Ratio.

But is that all? Many funds 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, assume the risk-free rate is 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 growth.
In fact, the earlier discussed Sharpe Ratio essentially states that, holding the underlying asset constant, position sizing does not affect the Sharpe Ratio;
and the “volatility tax” in some sense suggests that, with the underlying unchanged, smaller positions are more efficient.
Perhaps this explains why the so-called balanced stock-bond portfolio often provides a better investment experience?
Of course, this assumes real-time position adjustments and does not consider rebalancing effects.
Feel free to discuss.