Case study: portfolio returns and correlation as an angle

A portfolio is a vector of weights

Number the assets A, B and C as 1, 2 and 3. The weights $\mathbf{w} = (w_1, w_2, w_3)$ are the fractions of the money in each, as decimals: here $\mathbf{w} = (0.5, 0.3, 0.2)$. Lesson 0.2.4 worked out weights from dollar holdings.

Over a period, each asset earns a simple return (case study 1.4.7). Collect the three in a vector $\mathbf{r} = (r_1, r_2, r_3)$; here bold $\mathbf{r}$ holds simple returns, not log returns.

Case study 1.4.7 found that the portfolio's return is the weighted sum of these. That is the dot product of lesson 3.1.2: $R_p = \mathbf{w} \cdot \mathbf{r}$.

Suppose that in one year A returns 4%, B loses 10% and C gains 25%, so $\mathbf{r} = (0.04, -0.10, 0.25)$. Then $\mathbf{w} \cdot \mathbf{r} = 0.02 - 0.03 + 0.05 = 0.04$: the portfolio returns 4%.

The expected return

The same weighted sum of the assets' expected returns gives the portfolio's expected return. Write them as a vector $\boldsymbol{\mu} = (0.06, 0.08, 0.11)$, a bold mu.

With dollar holdings in place of weights, the dot product is in dollars. The holdings are $\mathbf{h} = 10{,}000{,}000\,\mathbf{w}$: $5 million, $3 million and $2 million. Then $\mathbf{h} \cdot \boldsymbol{\mu} = 760{,}000$ is the expected gain, and $\mathbf{h} \cdot \mathbf{r} = 400{,}000$ the gain in the year above.

The budget: $\mathbf{w} \cdot \mathbf{1} = 1$

Write $\mathbf{1} = (1, 1, 1)$ for the vector of ones. Its dot product with any vector adds up the components, so $\mathbf{w} \cdot \mathbf{1} = w_1 + w_2 + w_3$, the total weight. A fully invested portfolio has $\mathbf{w} \cdot \mathbf{1} = 1$: every dollar is placed somewhere.

Case study 1.2.4 put a weight $w$ in its first asset and $1 - w$ in its second. As a vector that is $(w, 1 - w)$, whose components add up to 1 for every $w$.

Its long-only portfolios run from $(1, 0)$ to $(0, 1)$. At $\rho = 0.8$ the least risky mix was $(1.2353, -0.2353)$, beyond that range: a short sale of the second asset.

Long and short weights

A negative weight is a short sale: the fund borrows an asset, sells it, and spends the cash on the others. A "130/30" portfolio is long 130% of its capital and short 30%. For example, $\mathbf{w} = (-0.3, 0.6, 0.7)$ is short A and long B and C.

Its weights still add up to 1, and its expected return is $\mathbf{w} \cdot \boldsymbol{\mu} = 0.107$. The sizes of the weights add up to 1.6: 160% of the capital is at work, which is also more risk.

Trades that keep the budget

Changing the weights by a vector $\mathbf{d}$ gives new weights $\mathbf{w} + \mathbf{d}$ (lesson 3.1.1). They still add up to 1 exactly when $\mathbf{d} \cdot \mathbf{1} = 0$, that is, when $\mathbf{d}$ is orthogonal to $\mathbf{1}$ (lesson 3.1.2). Whatever is sold pays for what is bought.

Moving 10% from A into C is $\mathbf{d} = (-0.1, 0, 0.1)$. The new weights are $(0.4, 0.3, 0.3)$, and the expected return rises by $\mathbf{d} \cdot \boldsymbol{\mu} = 0.005$, to 0.081.

A zero-cost portfolio has $\mathbf{w} \cdot \mathbf{1} = 0$, for example long $1 million of C and short $1 million of A. Nothing is invested, so a return per dollar invested would divide by 0. Desks measure it in dollars instead: in the year above, $\mathbf{h} \cdot \mathbf{r} = -40{,}000 + 250{,}000 = 210{,}000$.

Correlation as an angle

A portfolio's risk depends on how its assets move together. The usual measure is the correlation $\rho$ (rho), a number from $-1$ to 1 (case study 1.2.4). It is the cosine of an angle between two vectors.

Here are ten months of returns for two stocks, X and Y. They are invented: each month both follow a common market move plus a move of their own. The moves are drawn at random with a fixed seed and rounded to 0.1%.

Each stock's returns form a vector in $\mathbb{R}^{10}$, $\mathbf{x} = (x_1, \ldots, x_{10})$ and $\mathbf{y}$, one component per month.

The means are $\bar{x} = 1.37\%$ and $\bar{y} = 1.19\%$ ($\bar{x}$ is read "x bar"). Centring subtracts each mean from every month:

This is a scaling and a subtraction from lesson 3.1.1, with $\mathbf{1}$ now ten ones. $\tilde{\mathbf{x}}$ is read "x tilde", and $\tilde{\mathbf{y}}$ is made the same way. A centred vector's components add up to 0, so $\tilde{\mathbf{x}} \cdot \mathbf{1} = 0$.

The subscript $t$ numbers the months. In 8 of the 10, X and Y sit on the same side of their means, both above or both below. Each of those months adds a positive product $\tilde{x}_t\tilde{y}_t$ to the dot product $\tilde{\mathbf{x}} \cdot \tilde{\mathbf{y}}$; months 5 and 9 subtract a little.

Here $\tilde{\mathbf{x}} \cdot \tilde{\mathbf{y}} = 0.0061107$, $\lVert \tilde{\mathbf{x}} \rVert = 0.1509$ and $\lVert \tilde{\mathbf{y}} \rVert = 0.0663$, in decimals. So $\rho = 0.6113$, and the angle is 52.32°. NumPy's np.corrcoef gives the same 0.6113.

Two vectors in $\mathbb{R}^{10}$ still lie in one flat plane through the origin, so the angle between them can be drawn.

Divide $\tilde{\mathbf{x}} \cdot \tilde{\mathbf{y}}$ by $n - 1 = 9$, with $n$ the number of months, and you get the sample covariance, 0.000679. Divide each length by $\sqrt{9} = 3$ and you get the sample standard deviations, X's being 0.050284; lesson 5.5.2 defines both. The factors cancel in the cosine, and case study 3.2.5 gathers covariances into a matrix.

The level's correlations as angles

The three assets A, B and C have correlations $\rho_{AB} = 0.3$, $\rho_{AC} = 0.1$ and $\rho_{BC} = 0.5$; case study 3.2.5 uses them. As angles:

Around 90°, where $\rho$ is near 0, the cosine changes fastest: a small turn moves $\rho$ the most. Near 0° and 180° it changes slowest: from 0° to 25.84°, $\rho$ only falls from 1 to 0.9.

Correlations of 1, 0 and −1

The extreme correlations are the extreme angles of lesson 3.1.2.

• $\rho = 1$: 0°. A fund holding X with double exposure, less a fee of 0.5% a month, returns $\mathbf{y} = 2\mathbf{x} - 0.005\,\mathbf{1}$. Centring removes the fee, and scaling by 2 keeps the direction (lesson 3.1.1). So $\tilde{\mathbf{y}} = 2\tilde{\mathbf{x}}$, the angle is 0° and $\rho = 1$.
• $\rho = -1$: 180°. A fund that returns $-\mathbf{x}$ every month has $\tilde{\mathbf{y}} = -\tilde{\mathbf{x}}$, pointing the opposite way.
• $\rho = 0$: 90°. The centred vectors are orthogonal.

A short position turns returns round, so it turns the vector round too. $-\mathbf{y}$ has correlation $-0.6113$ with $\mathbf{x}$, an angle of $127.68° = 180° - 52.32°$. Writing returns in percent scales both vectors by 100 and leaves $\rho$ at 0.6113.

Zero correlation, and why centring matters

Take four months with returns $(1, 2, 3, 4)$ for one asset and $(4, 2, 2, 4)$ for another, in percent. Centred, they are $(-1.5, -0.5, 0.5, 1.5)$ and $(1, -1, -1, 1)$, and their dot product is $-1.5 + 0.5 - 0.5 + 1.5 = 0$. So $\rho = 0$.

Yet the second asset depends on the first: it is high exactly when the first is extreme. Correlation measures only how closely two series follow a straight-line relation, not every link between them.

Skip the centring, and the cosine of the raw vectors is 0.866, an angle of 30°. Both series are positive every month, so the raw arrows point roughly the same way. For X and Y the raw cosine is 0.6472 rather than 0.6113, so always subtract the means first.

import numpy as np


def _sample():
    """The lesson's ten invented months for stocks X and Y."""
    rng = np.random.default_rng(449)
    market = rng.normal(0.008, 0.04, 10)
    x = np.round(market + rng.normal(0.002, 0.025, 10), 3)
    y = np.round(0.6 * market + rng.normal(0.003, 0.02, 10), 3)
    return x, y


def _raises_value_error(f, *args):
    try:
        f(*args)
    except ValueError:
        return True
    return False


def test_portfolio_return():
    """Weights (0.5, 0.3, 0.2) and returns (4%, −10%, 25%) give 4%"""
    expect_close(portfolio_return(np.array([0.5, 0.3, 0.2]), np.array([0.04, -0.10, 0.25])), 0.04, what="w · r")


def test_expected_return():
    """With expected returns (6%, 8%, 11%) the portfolio expects 7.6%"""
    expect_close(portfolio_return(np.array([0.5, 0.3, 0.2]), np.array([0.06, 0.08, 0.11])), 0.076, what="w · μ")


def test_short_weights():
    """A 130/30 portfolio, (−0.3, 0.6, 0.7), expects 10.7%: negative weights are allowed"""
    expect_close(portfolio_return([-0.3, 0.6, 0.7], [0.06, 0.08, 0.11]), 0.107, what="w · μ for (−0.3, 0.6, 0.7)")


def test_weights_must_add_to_one():
    """Weights adding to 0.999, dollar holdings or percentages: ValueError"""
    mu = np.array([0.06, 0.08, 0.11])
    assert _raises_value_error(portfolio_return, np.array([0.333, 0.333, 0.333]), mu), "expected ValueError for weights adding to 0.999"
    assert _raises_value_error(portfolio_return, np.array([5e6, 3e6, 2e6]), mu), "expected ValueError for dollar holdings"
    assert _raises_value_error(portfolio_return, np.array([50, 30, 20]), mu), "expected ValueError for weights in percent"


def test_sizes_must_match():
    """Three weights and two returns: ValueError"""
    assert _raises_value_error(portfolio_return, np.array([0.5, 0.3, 0.2]), np.array([0.06, 0.08])), "expected ValueError for sizes 3 and 2"


def test_correlation_sample():
    """The ten months of X and Y have correlation about 0.6113, as np.corrcoef says"""
    x, y = _sample()
    expect_close(correlation(x, y), np.corrcoef(x, y)[0, 1], what="correlation of X and Y")


def test_centring():
    """(1, 2, 3, 4) and (4, 2, 2, 4) have correlation 0: centre before taking the cosine"""
    expect_close(correlation([1, 2, 3, 4], [4, 2, 2, 4]), 0.0, what="correlation of (1, 2, 3, 4) and (4, 2, 2, 4)", abs_tol=1e-12)


def test_scaling():
    """Percent units and double exposure leave the correlation alone; a short position flips its sign"""
    x, y = _sample()
    rho = np.corrcoef(x, y)[0, 1]
    expect_close(correlation(100 * x, 100 * y), rho, what="correlation with returns in percent")
    expect_close(correlation(x, 2 * x - 0.005), 1.0, what="correlation of X with 2X − 0.005")
    expect_close(correlation(x, -y), -rho, what="correlation of X with −Y")
    expect_close(correlation(x, -x), -1.0, what="correlation of X with −X")


def test_constant_series():
    """A constant series has no correlation: ValueError, even when centring leaves rounding dust"""
    x, _ = _sample()
    assert _raises_value_error(correlation, x, np.full(10, 0.003)), "expected ValueError for a series that is 0.003 every month"
    assert _raises_value_error(correlation, x, np.full(10, 0.01)), (
        "expected ValueError for a series that is 0.01 every month: centred, it is about 1e-18, not exactly 0"
    )


def test_bad_lengths():
    """Series of different lengths, or a single month: ValueError"""
    x, y = _sample()
    assert _raises_value_error(correlation, x, y[:-1]), "expected ValueError for 10 and 9 months"
    assert _raises_value_error(correlation, [0.01], [0.02]), "expected ValueError for a single month"