There are different performance estimators that are industry-standard for assessing how a portfolio works. This is not an exhaustive list, but these are the four I believe are the most useful. All the measures presented below use historical data; they are referred to as ex-ante.
The Sharpe ratio is undeniably the most common performance ratio and is the industry standard. It is attributed to William Sharpe, who introduced it in 1996 (link). The Sharpe ratio often implies the Modern Portfolio Theory (MPT) developed by Markowitz (link).
The main idea of the Sharpe ratio is to express excess asset returns \(r_A\) over risk, adjusted by the risk-free rate \(r_f\). Sharpes above \(2.0\) are deemed good, as they mean that there are 2 units of return for 1 unit of risk. Market Sharpe usually sits around \(0.4\).\[Sharpe=\frac{\textbf{E}r_A - \textbf{E}r_f}{\sqrt{\textbf{Var}(r_A - r_f)}}\]
The Sharpe ratio has some downsides, mainly that it is difficult to manipulate when combining portfolios.
The Sortino ratio can be seen as an improvement of the Sharpe ratio. It has the same idea and a similar equation, except that only the downside deviation is used as a unit of risk. \[downside\ deviation = \sqrt{\textbf{Var}(r_A - r_f)\left|r_A < r_f\right.}\]
Because of that transformation, the Sortino ratio only penalises the downside risk. If the return distribution has a limited downside but a very large upside, the Sortino ratio will be much greater than the Sharpe ratio. \[Sortino=\frac{\textbf{E}r_A - \textbf{E}r_f}{downside\ deviation}\]
Sometimes, instead of \(r_f\), a Minimum Acceptable Return (MAR) is used instead.
Despite it being an intuitive improvement, it has not gained widespread adoption compared with the Sharpe ratio. That's probably because a known empirical feature of returns (part of the so-called stylised facts) is that they should be somewhat symmetrical over large periods of time. If they are not, it might be that the sample is too small, and the Sharpe ratio would capture these better.
The Treynor ratio is a bit of a hybrid between empirical results and the Capital Asset Pricing Model (CAPM; link). It uses a measured market beta \(\beta_A\)for the asset. \[Treynor = \frac{\textbf{E}r_A - \textbf{E}r_f}{\beta_A}\]
The idea is that instead of comparing asset excess return to risk, it is compared to the overall market risk.
One drawback of Treynor is that there is an error in the estimation of beta. However, it can be a good tool when comparing different portfolios that track the same market.
While it is uncomment to see it on financial reports, the information ratio \(IR\) is ubiquitous in quantitative finance and signal theory in general. The idea is instead of looking at the risk-free rate; the benchmark return \(r_b\) is used instead. \[IR = \frac{\textbf{E}r_A - \textbf{E}r_b}{\sqrt{\textbf{Var}(r_A - r_b)}}\]
The denominator in particular is called the tracking error. When a portfolio manager attempts to replicate the performance of an index for example, it is their main measure of success (the lower, the better).
The information ratio is often referred to as the "measure of skill" of an asset manager. However, it is only relevant for relative value strategies. It makes no sense for absolute return strategies.
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