The math, shown.

Markowitz^[1] casts allocation as a trade-off between return and variance. Tracing the minimum-variance weight for each attainable return level sweeps out the efficient frontier. The long-only, fully-invested program above is a convex quadratic program; Argmin solves it with CVXPY (Clarabel/OSQP).

The tangency (max-Sharpe) portfolio is the frontier point that maximizes the reward-to-risk ratio relative to the risk-free asset:

The sample covariance $S$ is unbiased but ill-conditioned when the number of assets is comparable to the sample length: its extreme eigenvalues are systematically distorted, which mean-variance amplifies into unstable weights. Ledoit and Wolf^[2] shrink $S$ toward a structured target $F$ (here a constant-correlation matrix) by an intensity $\delta \in [0,1]$ chosen to minimize expected Frobenius loss in closed form.

On this window the optimal intensity is δ = 0.0064. A small $\delta$ means the sample matrix is already well-conditioned over 2007-2024; the estimator still removes the noisiest off-diagonal extremes, as the side-by-side below shows.

Risk parity targets equal risk contributions rather than equal capital. The marginal risk of asset $i$ is $(\Sigma w{)}_i$, so its risk contribution is ${\text{RC}}_i=w_i(\Sigma w{)}_i$. The equal-risk-contribution portfolio satisfies

Argmin solves this through Spinu's^[3] convex log-barrier formulation above: the unique interior minimizer has equal risk contributions after normalization, and the log term acts as a smooth barrier keeping every weight strictly positive.

HRP^[6] avoids inverting $\Sigma$ entirely - the step that makes mean-variance fragile. It proceeds in three stages: (1) convert correlations to the distance metric above and build a hierarchical clustering tree; (2) quasi-diagonalize by reordering assets so the most-correlated sit adjacent; (3) allocate top-down by recursive bisection, splitting each cluster's budget in inverse proportion to its variance.

The dendrogram below is the actual linkage tree estimated on this universe; leaf order is the quasi-diagonal sequence the bisection then walks.

The diversification ratio^[5] is the weighted average of asset volatilities divided by the portfolio volatility. It equals one only when assets are perfectly correlated and grows as the portfolio harvests diversification, so maximizing it,

produces the “most diversified portfolio.” The objective is scale-invariant; Argmin solves the equivalent convex problem of minimizing $w^{\top }\Sigma w$ subject to $w^{\top }\sigma =1$ and then renormalizing.

Black-Litterman^[4] starts from market-implied equilibrium returns obtained by reverse optimization,

where $\delta$ is the market risk-aversion and $w_{\text{mkt}}$ the market-cap weights. Investor views are encoded as $P\mu =Q$ with uncertainty $\Omega$, and the posterior above is the precision-weighted blend of the prior $\pi$ and the views $Q$. With no views it collapses to ${\mu }_{\text{BL}}=\pi$, recovering the equilibrium portfolio - which is the regularized prior that tames mean-variance's sensitivity to noisy $\mu$.

Note: Argmin uses an equal-weight market proxy (no live market-cap feed), so $\pi$ reflects a 1/N reference rather than true float weights. This is stated again in the limitations.

A Sharpe ratio is itself an estimate with sampling error, and that error is larger for skewed, fat-tailed returns. The probabilistic Sharpe ratio^[7] gives the probability that the true Sharpe exceeds a benchmark $S^{\ast }$, adjusting for skewness ${\hat{\gamma }}_3$ and kurtosis ${\hat{\gamma }}_4$ of the returns.

When many strategies are tried, the best in-sample Sharpe is upward-biased. The deflated Sharpe ratio corrects for this multiple-testing by raising the benchmark to the expected maximum of $N$ trials:

Here $\gamma$ is the Euler-Mascheroni constant and $N$ the number of trials. Argmin deflates over 7 candidate strategies. We also report a block-bootstrap confidence interval (block size 21 days, 1,000 resamples) to preserve autocorrelation in the resampled returns.