Lesson 7.1: The Covariance Matrix & The Geometry of Portfolio Risk

A Masterclass Edition lesson on the cornerstone of portfolio theory.

Welcome. Today, we will embark on a journey to answer a single question: What is the true risk of a portfolio?

Along the way, we will discover that a concept you learned in basic statistics—variance—blossoms into one of the most beautiful and powerful structures in all of finance: the **Covariance Matrix**. And we will see how the language of linear algebra allows us to express a deeply complex idea with breathtaking simplicity.

The Soul of Variance

What are we really measuring?

Let's start with an idea. What is risk? In finance, we often define risk as **volatility**, or the degree to which an asset's returns swing around its average.

Let `R` be a single return for an asset, and `μ` be its average return. The deviation for that day is `(R - μ)`. Some deviations are positive, some are negative. To get a sense of the average "magnitude" of deviation, we square it, making everything positive: `(R - μ)²`.

**Variance**, `σ²`, is simply the *expected value* (the long-term average) of this squared deviation.

σ2=E[(R−μ)2]\sigma^2 = E[(R - \mu)^2]

This is the bedrock. Everything we do will be built on this single, intuitive idea.

The Two-Asset Portfolio - A Tale of Interaction

Deriving Portfolio Variance

The return of our portfolio, `R_p`, is a simple weighted sum: `R_p = w₁R₁ + w₂R₂`

The average return, `μ_p`, is also a weighted sum: `μ_p = w₁μ₁ + w₂μ₂`

Now for the big question: What is the variance of this portfolio, `σ²_p`? Let's use our fundamental definition: `σ²_p = E[(R_p - μ_p)²]`.

Substitute our portfolio formulas: `R_p - μ_p = (w₁R₁ + w₂R₂) - (w₁μ₁ + w₂μ₂) = w₁(R₁ - μ₁) + w₂(R₂ - μ₂)`.

Now we square it, using `(a + b)² = a² + 2ab + b²`:

(Rp−μp)2=w12(R1−μ1)2+w22(R2−μ2)2+2w1w2(R1−μ1)(R2−μ2)(R_p - \mu_p)^2 = w_1^2(R_1 - \mu_1)^2 + w_2^2(R_2 - \mu_2)^2 + 2w_1w_2(R_1 - \mu_1)(R_2 - \mu_2)

We need the expected value of this entire expression. Since `E[]` distributes across sums and weights are constants:

σp2=w12E[(R1−μ1)2]+w22E[(R2−μ2)2]+2w1w2E[(R1−μ1)(R2−μ2)]\sigma^2_p = w_1^2 E[(R_1 - \mu_1)^2] + w_2^2 E[(R_2 - \mu_2)^2] + 2w_1w_2 E[(R_1 - \mu_1)(R_2 - \mu_2)]

We have just re-discovered the definitions of variance and covariance!

  • E[(R1−μ1)2]=σ12E[(R_1 - \mu_1)^2] = \sigma_1^2 (Variance of Asset 1)
  • E[(R2−μ2)2]=σ22E[(R_2 - \mu_2)^2] = \sigma_2^2 (Variance of Asset 2)
  • E[(R1−μ1)(R2−μ2)]=σ12E[(R_1 - \mu_1)(R_2 - \mu_2)] = \sigma_{12} (Covariance of Asset 1 and 2)

Substituting these in gives the classic two-asset formula:

σp2=w12σ12+w22σ22+2w1w2σ12\sigma^2_p = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_{12}

The Leap into Linear Algebra

From Summation to Matrices
The two-asset formula is hideous. What if we have 1000 assets? This does not scale. Let's rewrite it in a more structured way.

Let's organize the risks into a **Covariance Matrix, Σ**, and our weights into a vector `w`.

Σ=[σ12σ12σ21σ22]\Sigma = \begin{bmatrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{21} & \sigma_2^2 \end{bmatrix}
w=[w1w2]w = \begin{bmatrix} w_1 \\ w_2 \end{bmatrix}

Let's propose a structure: `wᵀΣw` and see what happens.

Step 1: Compute `Σw`

Σw=[σ12σ12σ21σ22][w1w2]=[σ12w1+σ12w2σ21w1+σ22w2]\Sigma w = \begin{bmatrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{21} & \sigma_2^2 \end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} = \begin{bmatrix} \sigma_1^2 w_1 + \sigma_{12} w_2 \\ \sigma_{21} w_1 + \sigma_2^2 w_2 \end{bmatrix}

Step 2: Compute `wᵀ(Σw)`

wT(Σw)=[w1,w2][σ12w1+σ12w2σ21w1+σ22w2]w^T(\Sigma w) = [w_1, w_2] \begin{bmatrix} \sigma_1^2 w_1 + \sigma_{12} w_2 \\ \sigma_{21} w_1 + \sigma_2^2 w_2 \end{bmatrix}

Multiplying this out gives:

w1(σ12w1+σ12w2)+w2(σ21w1+σ22w2)=w12σ12+w1w2σ12+w2w1σ21+w22σ22w_1(\sigma_1^2 w_1 + \sigma_{12} w_2) + w_2(\sigma_{21} w_1 + \sigma_2^2 w_2) = w_1^2\sigma_1^2 + w_1w_2\sigma_{12} + w_2w_1\sigma_{21} + w_2^2\sigma_2^2

Since `σ₁₂ = σ₂₁`, this simplifies to the exact same formula as before:

σp2=w12σ12+w22σ22+2w1w2σ12\sigma^2_p = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_{12}

This is the miracle. The chaotic, expanding sum of terms is perfectly and compactly organized by the structure of matrix multiplication `wᵀΣw`.

Summary: The Triumph of Structure
  • Portfolio risk comes from individual volatilities (**variances**) and their interactive movements (**covariances**).
  • A simple summation formula becomes impossibly complex as assets grow.
  • By organizing weights into a vector `w` and risks into a **Covariance Matrix `Σ`**, we express total portfolio variance with the simple and scalable formula: **`σ²_p = wᵀΣw`**.
  • This formula works because the structure of matrix multiplication is the perfect mirror for the structure of portfolio risk.
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Lesson 6.3: Advanced SVD Applications
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Lesson 7.2: Portfolio Optimization & The Efficient Frontier