A vector is the most basic object in quantitative finance. When you hold a portfolio of three assets in proportions 50 percent, 30 percent, and 20 percent, you are really holding the vector . Almost everything that follows in this course is a manipulation of vectors and the matrices that act on them, so we start here.
Formally, the space is the set of all ordered lists of real numbers. A typical element is written
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Lesson close · Linear Algebra & Optimization for Quants
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Sequence
Continue to Matrices & Matrix Multiplication, the next prerequisite step in this course.
We think of as an arrow from the origin to the point with those coordinates, or simply as a column of numbers. The number is the -th component. In finance might be the number of assets, the number of risk factors, or the number of dates in a return series.
Two vectors of the same length add component by component, and a scalar scales every component:
These two operations make a vector space. Combining them gives a linear combination , the workhorse expression of the whole subject. A portfolio return is exactly such a combination: if assets have returns and weights , the portfolio return is .
The dot product (or inner product) of two vectors in is the single number
It is symmetric, , and linear in each argument. The portfolio return above is precisely the dot product , which is why dot products are everywhere in finance: they collapse a weighted bet across many assets into one expected payoff.
The Euclidean norm (length) of a vector is defined through its own dot product:
For this is just the Pythagorean theorem. A unit vector has norm ; we normalize any nonzero by dividing by its norm, .
The dot product and the norm are tied together by the geometric identity
where is the angle between the two vectors. Solving for the angle gives the cosine formula
Apply the law of cosines to the triangle whose two sides are and and whose third side is the difference :
Now expand the left side algebraically using the definition of the norm:
Setting the two expressions equal, the and terms cancel, leaving . Dividing by yields , exactly the identity above.
When degrees, , so . Two vectors are orthogonal precisely when their dot product is zero. In finance, orthogonal return streams are uncorrelated bets that diversify each other, which is the mathematical heart of risk reduction.
Let and . First the dot product:
The norms:
The cosine of the angle:
So degrees. The vectors point in nearly the same direction, consistent with a high cosine. As a sanity check, gives , so and are orthogonal.
A portfolio is a weight vector , expected returns form a vector , and the expected portfolio return is the dot product . The cosine formula reappears as the correlation between two return series, and orthogonality is the formal statement that two strategies are uncorrelated. Norms measure the size of a position or the magnitude of a return, and they underlie the risk and tracking-error metrics you will meet later. Mastering these three operations, addition, the dot product, and the norm, gives you the language in which every later model is written.
Adapted from MIT OpenCourseWare (18.06 Linear Algebra / 6.255 Optimization), CC BY-NC-SA.
Knowledge check
Q1. For vectors x and y in R-n, the dot product x dot y equals:
Q2. Two nonzero vectors are orthogonal exactly when:
Q3. The Euclidean norm of a vector x is defined as:
Q4. If a portfolio has weights w and assets have expected returns mu, the expected portfolio return is:
Problems
Let x = (1, 2, 2) and y = (2, 0, 1). Compute cos(theta), the cosine of the angle between them. Give the answer as a decimal.
For a portfolio with weight vector w = (w1, w2) and asset return vector r = (r1, r2), write the portfolio return as a dot product and state the general formula for n assets.
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