Calculating Portfolio Sharpe and Sortino Ratios from Returns
Summary
The document considers how to calculate Sharpe and Sortino ratios for a portfolio whose asset weights can change over time. It describes forming each period’s portfolio return from the weighted returns of its holdings, then calculating the desired risk-adjusted ratio across that portfolio return series.
The answer recommends computing the ratios directly from portfolio returns. It briefly claims that weighting individual assets’ ratios can be equivalent, but gives no derivation or conditions for that claim. In general, Sharpe and Sortino ratios are nonlinear statistics, so a weighted average of asset ratios is not generally equal to the ratio of portfolio returns; portfolio covariance and downside behavior matter. The practical takeaway is to construct the portfolio return series first, then compute its ratios, while treating the stated equivalence as unsupported.
Key ideas
- Form a return series for the portfolio using its asset weights at each timestep.
- Calculate portfolio Sharpe and Sortino ratios from that portfolio return series.
- Individual asset ratios do not generally combine linearly because portfolio risk depends on joint returns.
- The answer recommends direct calculation but does not substantiate its equivalence claim.
Tags
Full text
# Portfolio vs individual security Sharpe and Sortino ratios # Portfolio vs individual security Sharpe and Sortino ratios For an individual security calculating it's Sharpe and Sortino ratios is straightforward. What I'm curious about is the following: Let's say I have a portfolio of several securities, which is a distribution of my total capital: for example Asset A has 25%, Asset B has 50%, and Asset C has 25%. At every timestep `t`, let's assume that I can adjust these percentages to maximize my profits, and that the total distribution always has to add up to 100%. So at each timestep `t` my portfolio has a return of `r_t`, which is the dot product of the distribution vector (`a`) for each asset at time `t` with the vector of the change in price for each asset since time `t-1`. If I want to calculate the Sharpe and Sortino for the portfolio, would I: - Calculate the Sharpe and Sortino ratios for each individual security at time `t` and again take a dot product between my distribution vector `a` and the vector of each sharpe/sortino ratio for each security - Directly calculate the Sharpe and Sortino ratios of the portfolio using the returns of the portfolio (`r_t`) across all timesteps `t`. Another good question would be: are both of these approaches fundamentally the same? Thanks in advance for your help! ## Answer by develarist (score 1, accepted) https://quant.stackexchange.com/a/57573 If you want to calculate the Sharpe and Sortino ratios for the portfolio, you should - directly calculate them using the returns of the portfolio Even if the individual sharpe ratios for each of the $N$ assets being dot-multiplied by the portfolio weights is equivalent to the above approach, you would be calculating $N$ number of Sharpe/Sortino ratios when you could have just calculated the one that you want: the portfolio's ratio
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.