The factor model's implied correlation¶

What the factor model says returns share.

Requires

the Loading of each instrument on each factor, each factor's variance over the return's window, each instrument's own variance.

Provides

the correlation matrix of the instruments' returns.

Law

C = L S L' + E, normalised by its diagonal: returns co-move exactly as far as they load on the same factors.

Source

Elton, Gruber, Brown & Goetzmann, Modern Portfolio Theory and Investment Analysis, ch. 8.

Proof

invariant, limit.

No Trait form: a tool, not a step law: a matrix over every pair of instruments, L S L' + E normalised, read off the factor model; it runs where a game prepares its data, never in a step, so it has no Trait form.

finance.implied_correlation(loadings, factor_variances, idiosyncratic_variances)

In [1]:

import os, sys
root = os.getcwd()
while not os.path.isfile(os.path.join(root, 'docsite.json')):
    root = os.path.dirname(root)
sys.path.insert(0, os.path.join(root, 'tools'))
import numpy as np
import matplotlib.pyplot as plt
from lusine import dmath as D, library
from lusine.kernels.finance.implied_correlation import implied_correlation

import os, sys root = os.getcwd() while not os.path.isfile(os.path.join(root, 'docsite.json')): root = os.path.dirname(root) sys.path.insert(0, os.path.join(root, 'tools')) import numpy as np import matplotlib.pyplot as plt from lusine import dmath as D, library from lusine.kernels.finance.implied_correlation import implied_correlation

The reference¶

The kernel's reference, in numpy: the body under the declaration above, and what every section below runs. It stays as the kernel's test beside its Trait form.

In [2]:

library.reference(implied_correlation)

library.reference(implied_correlation)

Out[2]:

the reference, implied_correlation.py

def implied_correlation(loadings, factor_variances, idiosyncratic_variances):
    """THE IMPLIED CORRELATION: what the factor model says returns share.

    Requires: the `Loading` of each instrument on each factor, each factor's
    variance over the return's window, each instrument's own variance.
    Provides: the correlation matrix of the instruments' returns.
    Law: C = L S L' + E, normalised by its diagonal: returns co-move exactly as
    far as they load on the same factors.
    Source: Elton, Gruber, Brown & Goetzmann, Modern Portfolio Theory and
    Investment Analysis, ch. 8.
    Proof: invariant, limit."""
    L = np.asarray(loadings, dtype=np.float64)
    C = L @ np.diag(np.asarray(factor_variances, dtype=np.float64)) @ L.T + np.diag(np.asarray(idiosyncratic_variances, dtype=np.float64))
    s = np.sqrt(np.diag(C))
    return C / np.outer(s, s)

Invariant: a correlation matrix¶

Unit diagonal, symmetric, positive semi-definite, every entry within [-1, 1], for random loadings and variances.

In [3]:

rng = np.random.default_rng(6)
for _ in range(50):
    L = rng.normal(size=(8, 5))
    C = implied_correlation(L, rng.uniform(1e-6, 1e-3, 5), rng.uniform(1e-7, 1e-4, 8))
    assert np.allclose(np.diag(C), 1) and np.allclose(C, C.T) and np.min(np.linalg.eigvalsh(C)) > -1e-12 and np.all(np.abs(C) <= 1 + 1e-12)
print('a correlation matrix every time')

rng = np.random.default_rng(6) for _ in range(50): L = rng.normal(size=(8, 5)) C = implied_correlation(L, rng.uniform(1e-6, 1e-3, 5), rng.uniform(1e-7, 1e-4, 8)) assert np.allclose(np.diag(C), 1) and np.allclose(C, C.T) and np.min(np.linalg.eigvalsh(C)) > -1e-12 and np.all(np.abs(C) <= 1 + 1e-12) print('a correlation matrix every time')

a correlation matrix every time

Limit: what nothing is shared is uncorrelated; one factor alone is one¶

Two instruments loading on different factors are uncorrelated; two on one factor with no noise of their own move as one.

In [4]:

C = implied_correlation(np.array([[1.0, 0.0], [0.0, 1.0]]), np.array([1e-4, 1e-4]), np.array([1e-5, 1e-5]))
assert abs(C[0, 1]) < 1e-15
C = implied_correlation(np.array([[1.0], [2.0]]), np.array([1e-4]), np.array([0.0, 0.0]))
assert abs(C[0, 1] - 1) < 1e-12
noise = np.logspace(-8, -3, 30)
rho = [implied_correlation(np.array([[1.0], [1.0]]), np.array([1e-4]), np.array([n, n]))[0, 1] for n in noise]
assert np.all(np.diff(rho) < 0)
fig, ax = plt.subplots(figsize=(7, 3))
ax.semilogx(noise, rho)
ax.set_xlabel('own variance')
ax.set_ylabel('correlation')
plt.show()

C = implied_correlation(np.array([[1.0, 0.0], [0.0, 1.0]]), np.array([1e-4, 1e-4]), np.array([1e-5, 1e-5])) assert abs(C[0, 1]) < 1e-15 C = implied_correlation(np.array([[1.0], [2.0]]), np.array([1e-4]), np.array([0.0, 0.0])) assert abs(C[0, 1] - 1) < 1e-12 noise = np.logspace(-8, -3, 30) rho = [implied_correlation(np.array([[1.0], [1.0]]), np.array([1e-4]), np.array([n, n]))[0, 1] for n in noise] assert np.all(np.diff(rho) < 0) fig, ax = plt.subplots(figsize=(7, 3)) ax.semilogx(noise, rho) ax.set_xlabel('own variance') ax.set_ylabel('correlation') plt.show()