Betas

18 Sep, 2026

description: I build an intuition for betas from a geometric perspective using linear algebra. Then, I relate this approach to statistical moments and formulas commonly used in Finance

Motivation

Imagine two return series of a futures contract: the front contract and a deferred contract. These are reprsented as n-dimensional vectors, where n is the number of periods:

\begin{matrix} 𝐗 & : = returns on front contract \\ 𝐲 & : = returns on contract \end{matrix}

We want to know much of the front contract's movement "explains" the returns of the deferred contract.

The simplest approach is to assume a constant proportional relationship where multiplying the front contract by some scalar value approximates the deferred contract. The linear assumption means there will always be some residual between its estimate and the true value:

𝐲 = β 𝐗 + 𝐮

Notice the underlying assumption of the model here: $β$ is applied in equal measure to every single timestep in the vector. It naively assumes a stable relationship between the two securities over the entire observation window. In markets driven by complex, dynamic factors, we know this is rarely true!

Sidenote A more pedantic approach might be to represent this as a non-linear model or time-varying model such as: $𝐲_{t} = β_{t} 𝐗_{t} + 𝐮_{t}$ It is a conscious decision to avoid this complexity and stick to the simple, robust framework of OLS regression. To "bend" this linear assumption, we do it through feature engineering: for example, by applying an exponentially weighted moving average or log transforms.

Thus brings us back to our motivation: Beta ( $β$ ) is a scalar that expresses the linear relationship between two securities. The OLS model derives this by minimizing the unexplained variance, $𝐮$ . In practical terms, this ratio can be used for hedging! It is the scaling factor required to match the directional exposure of one asset using another.

Intuition

We can visualize this model from two geometric perspectives: the $(N \times K)$ observation space and the $(K \times N)$ feature space.

N x K

Imagine the respective returns represented as vectors in n-dimensional space. For each vector, their n observations act as coordinates that define a single point in n-dimensional space. Thus, $𝐗$ and $𝐲$ are just two vectors pointing somewhere in this space. If we assume there is no intercept, we simply mutiply f by our scalar $β$ .

Intuitively, this stretches the vector $𝐗$ along its own axes. We wish to find some value of $β$ that makes our estimated vector $β 𝐗$ "as close as possible" to $𝐲$ . Geometrically, imagine this as a triangle with edges $β f$ , $𝐲$ and the residual gap $𝐮$ . Our estimate $\hat{β} 𝐗$ is physically trapped on the 1D line defined by $𝐗$ .

To make our estimate "as close as possible" to d, we must find $β$ that scales $𝐗$ on that line that minimizes the length of $\hat{𝐮}$ . The shortest possible path from a point in space ( $𝐲$ ) to a line ( $𝐗$ ) is a perpendicular drop. Therefore, at the exact minimum distance, our residual vector $\hat{𝐮}$ must meet our regressor $𝐗$ at a perfect right angle. Recall that for orthogonal vectors, their dot product is zero. Conceptually, this means:

(β f)^{⊤} 𝐮 = 0

We factor out the constant ( $β$ ) and thus:

\begin{matrix} 𝐗^{⊤} 𝐮 & = 0 \\ 𝐗^{⊤} (𝐲 - β 𝐗) & = 0 \\ 𝐗^{⊤} 𝐲 - β (𝐗^{⊤} 𝐗) & = 0 \\ β & = \frac{𝐗^{⊤} 𝐲}{𝐗^{⊤} 𝐗} \end{matrix}

Sidenote: Correlation vs. Beta Let $θ$ be the angle between our estimate $β 𝐗$ and the target $𝐲$ . The geometric cosine of this angle ( $\cos θ$ ) measures how tightly the two vectors align in space. If our return vectors are perfectly mean-centered, this geometric angle is the exact equivalent of the statistical correlation, $ρ$ . While correlation measures alignment ( $\cos θ$ ), Beta measures scale—it is that alignment multiplied by the ratio of their lengths.

K x N

We can transpose this perspective. Instead of viewing $k$ variables as vectors in $n$ -dimensional space, we can view the $n$ observations as a cloud of points in a data space where $k$ features define the axes. Because we have one regressor ( $𝐗$ ) and one target ( $𝐲$ ), this is a simple 2D scatter plot.

Here, the beta is a k-1 hyperplane. Because our model has no intercept, the hyperplane is "pinned" to pivots about the origin. We seek to fit a plane that best fits this cloud of points.

Equivalence

We have derived beta geometrically as the ratio of two inner products:

β = \frac{𝐗^{⊤} 𝐲}{𝐗^{⊤} 𝐗}

Lets assume these return vectors are already mean-centered (see Aside: Constants). I now will show how this ratio is equivalent to standard statistical formulas for beta.

We start by relating lens of linear algebra and statistics. By working with the inner products and vector lengths, we can map vector geometry directly to various statistical moments:

\begin{matrix} Var (𝐗) & = \frac{𝐗^{⊤} 𝐗}{n - 1} & Cov (𝐗, 𝐲) & = \frac{𝐗^{⊤} 𝐲}{n - 1} \\ σ_{X} & = \frac{‖ 𝐗 ‖}{\sqrt{n - 1}} & ρ & = \frac{𝐗^{⊤} 𝐲}{‖ 𝐗 ‖ ‖ 𝐲 ‖} = \cos θ \end{matrix}

Clockwise from top left

Variance ( $σ^{2}$ ): The normalized inner product of a vector with itself.
Covariance ( $Cov$ ): The normalized inner product of two different vectors.
Correlation ( $ρ$ ): The geometric cosine of the angle $θ$ between the two vectors.
Standard Deviation ( $σ$ ): The normalized Euclidean length (norm) of the vector.

This gives us a lookup to translate a geometric beta into common statistical operations:

1. Beta = Covariance over Variance If we take our geometric ratio and divide both the numerator and denominator by $n - 1$ , the scaling factors perfectly cancel out. This immediately yields the textbook OLS formula: $β = \frac{\frac{1}{n - 1} (𝐗^{⊤} 𝐲)}{\frac{1}{n - 1} (𝐗^{⊤} 𝐗)} = \frac{Cov (𝐗, 𝐲)}{Var (𝐗)}$

2. Beta = Correlation and Volatility Ratio ( $ρ \frac{σ_{y}}{σ_{X}}$ ) We can expand the inner products using their pure geometric definitions:

the inner product of two vectors is the product of their lengths and the cosine of the angle between them.
The inner product of a vector with itself is its length squared.

Substituting these definitions and referencing the lookup, we get:

\begin{matrix} β = \frac{𝐗^{⊤} 𝐲}{𝐗^{⊤} 𝐗} & = \frac{‖ 𝐗 ‖ ‖ 𝐲 ‖ \cos θ}{‖ 𝐗 ‖^{2}} \\ = \cos θ \frac{‖ 𝐲 ‖}{‖ 𝐗 ‖} \\ = ρ \frac{‖ 𝐲 ‖ / \sqrt{n - 1}}{‖ 𝐗 ‖ / \sqrt{n - 1}} \\ = ρ \frac{σ_{y}}{σ_{X}} \end{matrix}

This gives us the practical risk-translation formula for beta:

β = ρ \frac{σ_{y}}{σ_{X}}

Why does this matter? In a standard OLS regression, the correlation and the volatilities are rigidly bound to the exact same dataset and lookback window. However, in dynamic financial markets, the structural alignment of two assets and their individual volatilities can behave very differently. By breaking beta into its modular components of alignment ( $ρ$ ) and scale ( $\frac{σ_{y}}{σ_{X}}$ ), we can mix and match estimators to build a "better beta":

Mismatched Time Horizons: Correlation is noisy and often requires a long, stable lookback window (e.g., 6 to 12 months) to capture the true structural relationship between two contracts. Volatility, on the other hand, clusters and reacts violently to recent events. Risk managers may prefer a much shorter or exponentially weighted window (like an EWMA) for the standard deviations to immediately capture the current market regime.
Structural Alignment vs. Reactive Volatility: A researcher might measure the correlation ( $ρ$ ) using a long, unweighted historical window (e.g., 6 months of daily returns) to capture the true, stable structural relationship between the front and deferred contracts. However, they could calculate the standard deviations ( $σ_{y}$ and $σ_{X}$ ) using a short-term Exponentially Weighted Moving Average (EWMA). This creates a hybrid beta that anchors to a reliable baseline correlation but scales the immediate hedge ratio based on the localized, present-day volatility of the futures curve.

Aside: Constants

In OLS, we must decide whether to include a constant (an intercept, $α$ ).

In the $k \times n$ matrix space, adding a constant introduces a translation operation. It removes the constraint that the line must pass through the origin, allowing the hyperplane to shift up or down the vertical axis to better center itself through the point cloud.

In the $n \times k$ -dimensional vector space, adding a constant introduces a new dimension: a vector of ones, $ι$ . We are now projecting $𝐲$ onto a subspace defined by both $𝐗$ and $ι$ . Intuiively, we can see that this is now a 2-dimensional subspace.

To isolate just the beta multiplier, we use the Frisch-Waugh-Lovell (FWL) Theorem. Geometrically, the FWL theorem proves we can isolate beta by first projecting both $𝐲$ and $𝐗$ onto the orthogonal complement of the constant vector $ι$ , and then solving for those projected vectors.

(Potentially, FWL will be explored in a future post)

If we regress these new, mean-centered vectors against each other, we obtain the exact same beta as the full model.