Why Sharpe Ratio, Not Return-to-Volatility, Ranks Risky Portfolios
Summary
The document examines why the optimal risky portfolio can appear to differ when optimizing with or without access to lending and borrowing at a risk-free rate. Its central correction is that maximizing expected return divided by volatility is not equivalent to maximizing the Sharpe ratio, which subtracts the risk-free rate from expected return before dividing by volatility. Although subtracting a constant from returns alone would preserve rankings, the risk-free adjustment to the ratio depends on each portfolio’s volatility.
The explanation uses the capital market line: adding or removing lending or borrowing changes an investor’s position along that line, while the tangency portfolio is identified by its Sharpe ratio. It provides the algebraic relationship between excess return, volatility, and the line’s gradient. The discussion is conceptual and does not provide the original optimization setup or numerical evidence, so it does not diagnose every possible source of discrepancies in a particular implementation.
Key ideas
- The tangency portfolio is ranked by its Sharpe ratio, which uses excess return per unit of volatility.
- Return divided by volatility is not generally equivalent to the Sharpe ratio.
- The risk-free rate adjustment varies across portfolios because it is divided by each portfolio’s volatility.
- Lending or borrowing at the risk-free rate moves an investor along the capital market line.
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# Portfolio optimization with Scipy in Python
# Portfolio optimization with Scipy in Python
I performed Scipy portfolio optimization in two scenarios: 1) when I cannot lend or borrow at the risk-free rate; 2) when I can lend and borrow at rf=1.5%. Now, optimal risky portfolio weights anyway should not change because in the scenario where the risk-free asset is not present, we are maximizing the return to volatility ratio. The portfolio with the highest return to volatility ratio should also be the one with the higher Sharpe ratio (we are just subtracting a constant from the numerator). Instead, I have small discrepancies that you can see in the image I uploaded. Is this due to Python numerical storage or should I look for other reasons?
## Answer by KaiSqDist (score 0, accepted)
https://quant.stackexchange.com/a/78974
Strictly speaking, when you borrow (lend) at the riskless rate for the Sharpe-maximized-portfolio you move up (down) the capital market line (CML) as shown below. The return-vol ratio you used above is not exactly the same as the gradient (which is known as Shape and portfolio allocation), which is the thing that stays constant with leverage.
\begin{equation} Gradient_{TP} = \frac{ER_{TP}-ER_{i}}{Vol_{TP}-Vol_{i}} = \frac{ER_{i}-r_f}{Vol_{TP}-0} = Sharpe_{TP} \neq Return{\text-}Vol = \frac{ER}{Vol} \end{equation}
where the $TP$ refers to the tangency portfolio (or ideal market portfolio in the diagram below) and the difference between the $ER_{TP}$ and $ER_{i}$ are infinitesimal (same for the $Vol$s).
The correct way to perform the ranking would be to use the Sharpe and not your return-vol ratio.
--- EDIT ---
Decreasing the return-vol ratio by the riskless rate is not adjusting it by a constant, what you are decreasing it by is actually the riskless rate scaled by a different vol everytime:
\begin{equation} \frac{r_f}{Vol_1} \neq \frac{r_f}{Vol_2} \neq \frac{r_f}{Vol_3} \neq ... \end{equation}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.