Free lesson · Math & notation

Vectors, matrices and transpose notation

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Start with the idea

Write a two-asset example explicitly. Once the dimensions and signs work there, vector notation lets the same operations scale to a larger book.

Symbols, units & horizon

w: ordered vector of signed capital weights
r: matching decimal return vector
T: transpose, not elapsed time here
Σ: covariance matrix in squared return units
Σᵢⱼ: covariance of assets i and j
Σ with limits: summation rather than the matrix name
i, j: asset indices

When and why to use this

Use a dot product for return and a quadratic form for variance. This is the bridge between arithmetic and portfolio models.

A vector is an ordered list. A portfolio’s weights form one vector and the assets’ returns form another. A matrix is a rectangular table; a covariance matrix records a relationship for each pair of assets. Superscript T or ⊤ means transpose: swap rows and columns.

w^{T} r = \sum_{i} w_{i} r_{i}, w^{T} Σ w = \sum_{i} \sum_{j} w_{i} w_{j} Σ_{i j}

Algebra and arithmetic

Multiply two-vector and two-matrix expressions

For w=(w₁,w₂), r=(r₁,r₂), $w^{T} r = w_{1} r_{1} + w_{2} r_{2}$ .
First calculate Σw, then take the dot product with w. Expanding a symmetric two-by-two matrix gives $w_{1}^{2} Σ_{11} + 2 w_{1} w_{2} Σ_{12} + w_{2}^{2} Σ_{22}$ .

Work it by hand

w=(.5,.5), r=(.02,−.01) gives return .005. With diagonal covariance (.01,.04) and zero covariance, variance=.0125.

Capital Σ in this context is a matrix, while Σ with limits can be a summation symbol. The entry Σᵢⱼ is row i, column j. Matrix multiplication is ordered: multiply matching coordinates and add. The superscript −1 on a matrix means its inverse, if one exists, not taking each entry’s reciprocal.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

# Dependency: numpy. Asset ordering must match in all arrays.
import numpy as np

def portfolio_moments(weights, returns, covariance):
    w, r, cov = map(np.asarray, (weights, returns, covariance))
    return float(w @ r), float(w @ cov @ w)

print(portfolio_moments([.5, .5], [.02, -.01], [[.01, 0], [0, .04]]))

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