single-index model

The performance of an investment can be modeled as

where the 'market return' is that of some sufficiently broad index such as the S&P 500, and $\epsilon$ is a zero-mean residual term. This equation underlies the terms 'alpha' and 'beta' as they are used in finance:

Alpha: by what amount did an investment outperform the market during a given time period?

Beta: how correlated is an asset with the market?

Typical assets have beta between zero and one, indicating that their performance is some average of market conditions and their own idiosyncratic factors. But assets exist with $\beta < 0$ (synthetically constructed via short positions, or naturally, e.g., gold, which tends to go up when the market goes down) and with $\beta > 1$ (implying some amount of direct or indirect leverage).

Beta is also known as the 'hedge ratio' because it tells you how to hedge your position against the broader market. If some asset has a beta of 2.0, then to hedge $1k of this asset you'd short $2k of the overall market. This hedged position is now uncorrelated with the market, in that a $1 increase in the asset will tend to correspond to a $1 decrease in your market position, and vice versa.

Technically these returns are relative to the risk-free rate $r_\text{rf}$, so the full equation is written $(r_\text{i} - r_\text{rf}) = \alpha_i + \beta_i (r_\text{m} - r_\text{rf}) + \epsilon_i$. Thus, the presence of any beta in your portfolio implies some amount of risk.

References: https://en.wikipedia.org/wiki/Single-index_model