A distance matrix is a square table of pairwise dissimilarities. For hierarchical risk parity (HRP), transform an asset correlation matrix R into D with Dᵢⱼ = √[½(1 − Rᵢⱼ)]. An asset’s distance from itself is zero; more highly correlated assets have smaller distances. section 11.2.4
HRP then builds a second matrix, D̄, whose entries are Euclidean distances between columns of D. D̄ measures whether two assets have similar correlation patterns relative to all other assets, rather than merely measuring their direct pairwise correlation. Small values identify candidates for early clustering. section 11.2.4
Eight-asset portfolio
Compute returns for VV, VB, AGG, VEA, BNDX, VWO, EMB, and DJP; derive correlations; then calculate D elementwise as ((1 − corr) / 2) ** .5. Iterate over asset pairs to populate D̄ with vector norms, producing the input used to find close clusters. section 11.2.4
Listing 11.5 Calculating the distance matrix
import itertools
tickers = ['VV', 'VB', 'AGG', 'VEA', 'BNDX', 'VWO', 'EMB', 'DJP']
rets = yf.download(tickers, start='2014-01-01', end='2021-12-31')['Adj Close']
rets = rets.pct_change().dropna(axis=0, how='any')[tickers]
cov, corr = rets.cov() * 252, rets.corr()
dist = ((1 - corr) / 2.) ** .5
d_bar = dist * 0
for i, j in itertools.permutations(dist.columns, 2):
d_bar[i][j] = np.linalg.norm(dist[i] - dist[j])
This implementation downloads asset returns, calculates correlation, transforms it into D, and computes D̄ with Euclidean norms. section 11.2.4
