Hierarchical risk parity (HRP)

Идея простыми словами

Optimisers that invert the covariance matrix become unstable when tickers are highly correlated. HRP never inverts it. It first clusters the tickers (indices with indices, currencies with currencies), orders them so that similar ones sit together, and then divides the money between the two halves of the tree by their risk, again and again down to single tickers.

Как считается

1) Distance between tickers dᵢⱼ = √((1 − ρᵢⱼ) ÷ 2); single-linkage clustering builds the tree. 2) The tree gives an order with similar tickers side by side. 3) Recursive bisection: split the ordered list in two, measure each half's variance with inverse-variance weights inside it, give each half money in inverse proportion to its variance, repeat inside each half.

dᵢⱼ = √((1 − ρᵢⱼ) ÷ 2)
V(half) = w̃ᵀ Σ_half w̃,   w̃ ∝ 1 ÷ diag(Σ_half)
α = 1 − V_left ÷ (V_left + V_right);  left × α,  right × (1 − α)

Как использовать

• With many tickers, or groups of similar ones (several indices, several currency pairs).
• When other models give unstable weights from one study to the next.
• It is long only and needs no expected returns.

Выводы

• Tickers that cluster together share one budget, so adding a third stock index takes money from the other indices, not from gold.
• The order in the study's notes shows how it grouped the tickers.

На что обратить внимание

• Results depend on how the tree splits; with few tickers it can be lopsided.
• It ignores expected returns entirely.

Немного истории

Marcos López de Prado, "Building Diversified Portfolios that Outperform Out of Sample" (Journal of Portfolio Management, 2016). It brought machine-learning style clustering into portfolio construction.