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The word "benchmark" is often used in Finance, but in a rather fuzzy manner, there for a rough idea of what it is, and how it is 'defined'.

Can someone provide a rigorous and precise definition of what is it?

Benchmarks are often used in alternative investment funds to evaluate the performance of a portfolio manager. However, I find the choice of the benchmark of comparison somewhat arbitrary sometimes, and since fees are determined by how much you beat the benchmark, it would make sense to compare what is really comparable... and define precisely what is a financial benchmark, how to build it, or how to choose it?

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Some use the acronym SAMURAI: a benchmark should be

S - Specified in Advance

A - Appropriate

M - Measurable

U - Unambiguous

R - Reflective of Current Investment Thinking

A - Accountable

I - Investable

In principle it is the responsibility of the decision maker (the board of the pension fund) to set the benchmark, otherwise if the choice is left to the money manager he will of course be tempted to play games. Then the board advertises that it is looking for someone to manage XYZ amount in US equities according to an S&P 500 benchmark and solicits interested firms to compete for this job. That's the model.

In the retail space it is different and you are right that many mutual funds play games with benchmarks. For example one well known bond fund advertised that it beat its benchmark, which was US Treasury bonds, but the fund also included some mortgage and corporate bonds...

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One might define a benchmark as the status quo. If you simply bought all the stocks in the S&P500 for example, you would realize the returns on that index and it takes no active management from you, or a fancy portfolio manager. It is the default investment strategy, and although it is indeed arbitrary, the usual ones (S&P, Dow, etc.) are often used for equities since they are supposed to be a proxy for the market.

To see it another way, you'll often hear "underperform" and "outperform" in portfolio management circles, and we need a base case of what we're under or outperforming against. Since a benchmark is supposed to be the "no skills" strategy, it makes sense to gauge active portfolio managers' performance against such a strategy.

You may have also hear of "excess returns", sometimes defined as $\mu - r$ where $\mu$ is the return rate of an equity and $r$ is the return on a risk-free investment. The benchmark in this case is an asset with return rate $r$. Indeed, the rate $r$ is considered guaranteed, so you could always achieve this, and it makes more sense to gauge an equity's performance with respect to this default investment.

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