Free lesson · Linear algebra

Your portfolio is a vector; your risk is a matrix

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Start with the idea

Vectors let you keep the sign and size of every position; matrices retain the dependence between every pair. The off-diagonal terms matter because assets can move together even if their individual volatilities look modest.

Symbols, units & horizon

wᵢ: signed capital weight of asset i
rᵢ: decimal return of asset i
N: asset count
r_p: portfolio return
⊤: transpose
Σ: covariance matrix
σᵢⱼ: covariance of returns i and j
ρᵢⱼ: return correlation
σᵢ: asset volatility
σ_p²: portfolio variance
ΣᵢΣⱼ: sum over every asset pair

When and why to use this

Use weighted returns in daily P&L attribution and the covariance quadratic form in risk forecasts, optimisation and hedge sizing.

A single number is a scalar: a price, a return, a weight. A list of numbers is a vector: the returns of five assets today, or the weights of five positions. A grid is a matrix: the returns of five assets over 250 days (250 × 5), or how each pair of assets co-moves (5 × 5).

Portfolio return is a weighted sum, which is a dot product:

r_{p} = 𝐰^{⊤} 𝐫 = \sum_{i = 1}^{N} w_{i} r_{i}

Algebra and arithmetic

Expand the portfolio dot product

Capital allocated to asset i is $w_{i} V_{0}$ . Its gain is $w_{i} V_{0} r_{i}$ .
Sum gains and divide by V₀: $r_{p} = \sum_{i} w_{i} r_{i} = w^{T} r$ .

Work it by hand

Weights .6,.4 and returns 2%,−1% give .6(.02)+.4(−.01)=.008, or 0.8%.

That is the easy half. The hard half is variance, because variance of a sum is not the sum of variances. It picks up every pairwise covariance:

σ_{p}^{2} = 𝐰^{⊤} Σ 𝐰 = \sum_{i} \sum_{j} w_{i} w_{j} σ_{i j}, σ_{i j} = ρ_{i j} σ_{i} σ_{j}

Algebra and arithmetic

Expand the covariance quadratic form

Write the centred portfolio return as $\sum_{i} w_{i} (r_{i} - μ_{i})$ . Square it and take expectations: $\sum_{i} \sum_{j} w_{i} w_{j} E [(r_{i} - μ_{i}) (r_{j} - μ_{j})]$ .
The expectation is covariance σᵢⱼ. Substitute $σ_{i j} = ρ_{i j} σ_{i} σ_{j}$ . With two assets, variance is $w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} ρ σ_{1} σ_{2}$ .

Work it by hand

Equal weights, vols .1,.2, correlation .25: variance=.0025+.01+.0025=.015; volatility=√.015=12.247%.

$Σ$ is the covariance matrix: diagonal entries are variances, off-diagonal entries are covariances. For two assets the expression expands to $w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} ρ σ_{1} σ_{2}$ ; for 50 assets it has 2,500 terms, and the matrix form is the only sane way to write it.

Two assets, each 20% vol, 50/50 weights, correlation 0. Portfolio vol?

$σ_{p}^{2} = 0.25 (0.04) + 0.25 (0.04) + 0 = 0.02$ , so $σ_{p} = 0.141$ . Two uncorrelated 20% assets make a 14% portfolio. With ρ = 1 you would get exactly 20%.

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.

import numpy as np  # dependency: numpy

def portfolio(weights, returns, covariance):
    w, r, cov = map(np.asarray, (weights, returns, covariance))
    variance = float(w @ cov @ w)
    return float(w @ r), variance, np.sqrt(max(0, variance))

def covariance_from_correlation(volatilities, correlations):
    vol = np.asarray(volatilities)
    return np.outer(vol, vol)*np.asarray(correlations)

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

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