`portfolio-theory`¶

Pack: 100xOS shared skills

Category: modeling

Field: economics

License: private (curator-owned)

Updated: 2026-05-20

Stages: formal-modeling

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Portfolio Theory¶

Overview¶

Portfolio theory studies how investors allocate wealth across assets to optimize the risk-return tradeoff. From Markowitz's mean-variance framework to modern factor-based and risk-parity approaches, the field provides both normative guidance (how should you invest?) and positive tools (how do we measure portfolio performance?).

Mean-Variance Optimization (Markowitz, 1952)¶

Setup¶

Given N assets with: - Expected returns: mu (N x 1 vector) - Covariance matrix: Sigma (N x N, positive definite) - Portfolio weights: w (N x 1, sum to 1)

Portfolio return: E[R_p] = w' * mu Portfolio variance: Var(R_p) = w' * Sigma * w

Optimization Problem¶

Minimize w' * Sigma * w subject to w' * mu = mu_target, w' * 1 = 1

Efficient frontier: the set of portfolios with minimum variance for each target return.

Global minimum variance (GMV) portfolio: w_gmv = (Sigma^{-1} * 1) / (1' * Sigma^{-1} * 1)

Tangency (maximum Sharpe ratio) portfolio: w_tan = (Sigma^{-1} * (mu - r_f * 1)) / (1' * Sigma^{-1} * (mu - r_f * 1))

Python

import numpy as np
from scipy.optimize import minimize

def max_sharpe_portfolio(mu, Sigma, rf=0):
    """Maximum Sharpe ratio portfolio (tangency portfolio)."""
    n = len(mu)
    excess = mu - rf

    def neg_sharpe(w):
        ret = w @ excess
        vol = np.sqrt(w @ Sigma @ w)
        return -ret / vol

    constraints = [{'type': 'eq', 'fun': lambda w: np.sum(w) - 1}]
    bounds = [(0, 1)] * n  # long-only
    w0 = np.ones(n) / n
    result = minimize(neg_sharpe, w0, bounds=bounds, constraints=constraints)
    return result.x

def min_variance_portfolio(Sigma):
    """Global minimum variance portfolio."""
    n = Sigma.shape[0]
    ones = np.ones(n)
    Sigma_inv = np.linalg.inv(Sigma)
    w = Sigma_inv @ ones / (ones @ Sigma_inv @ ones)
    return w

Estimation Error Problem¶

Mean-variance optimization is extremely sensitive to estimation error in mu and Sigma: - Small errors in expected returns produce large weight swings. - Estimated optimal portfolios often perform poorly out of sample. - The "error maximization" critique (Michaud, 1989).

Remedies: - Shrinkage estimators for Sigma (Ledoit-Wolf). - Bayesian approaches (Black-Litterman). - Constraints (no short-selling, position limits). - Resampled efficient frontier (Michaud, 1998). - Focus on minimum variance (which only needs Sigma, not mu).

Black-Litterman Model (1992)¶

Combines market equilibrium (CAPM implied returns) with investor views.

Steps¶

Equilibrium returns: Start from market-cap-weighted implied excess returns: pi = delta * Sigma * w_mkt where delta = (E[R_m] - r_f) / sigma_m^2.
Investor views: Express K views as P * mu = q + epsilon, epsilon ~ N(0, Omega).
P: K x N pick matrix (e.g., "Asset A outperforms Asset B by 2%").
q: K x 1 view returns.
Omega: K x K confidence matrix (diagonal: less confident → higher variance).
Posterior returns: mu_BL = [(tauSigma)^{-1} + P'Omega^{-1}P]^{-1} * [(tauSigma)^{-1}pi + P'Omega^{-1}*q]
Optimize: Use mu_BL in standard mean-variance optimization.

tau ~ 0.025 (a scalar controlling prior uncertainty relative to data).

Advantages¶

Starts from a reasonable equilibrium (not zero expected returns).
Views are blended with equilibrium, not imposed.
Produces stable, diversified portfolios.
Investor only needs to express relative or absolute views on specific assets.

Risk Parity¶

Allocate portfolio such that each asset contributes equally to total portfolio risk.

Equal Risk Contribution (ERC)¶

For asset i, its risk contribution: RC_i = w_i * (Sigma * w)_i / sqrt(w' * Sigma * w)

Target: RC_i = RC_j for all i, j.

This means: w_i * (Sigma * w)_i = w_j * (Sigma * w)_j.

Python

def risk_parity(Sigma):
    """Equal risk contribution portfolio."""
    n = Sigma.shape[0]

    def risk_contribution(w):
        port_vol = np.sqrt(w @ Sigma @ w)
        marginal = Sigma @ w / port_vol
        rc = w * marginal
        target = port_vol / n
        return np.sum((rc - target) ** 2)

    constraints = [{'type': 'eq', 'fun': lambda w: np.sum(w) - 1}]
    bounds = [(0.01, 1)] * n
    w0 = np.ones(n) / n
    result = minimize(risk_contribution, w0, bounds=bounds, constraints=constraints)
    return result.x

Properties¶

Does not require expected return estimates (only Sigma).
Produces more diversified portfolios than mean-variance.
In practice, often combined with leverage (borrow at risk-free to target a volatility level).
Bridgewater's "All Weather" strategy is risk-parity-inspired.

Covariance Estimation¶

Sample Covariance¶

Sigma_hat = (1/(T-1)) * X' * X (where X is demeaned returns matrix)

Unbiased but noisy, especially when N is large relative to T.
Eigenvalues are spread out (largest too large, smallest too small).
If N > T, the matrix is singular.

Ledoit-Wolf Shrinkage (2003)¶

Sigma_shrunk = delta * F + (1 - delta) * Sigma_hat

F: structured target (e.g., diagonal matrix, single-factor model covariance, constant correlation).
delta: shrinkage intensity (estimated to minimize expected loss).
Optimal delta has a closed-form formula.

Python

from sklearn.covariance import LedoitWolf

cov_estimator = LedoitWolf().fit(returns)
Sigma_shrunk = cov_estimator.covariance_
print(f"Shrinkage coefficient: {cov_estimator.shrinkage_:.3f}")

Factor Model Covariance¶

Sigma = B * Sigma_f * B' + D

B: factor loadings (N x K).
Sigma_f: factor covariance (K x K).
D: diagonal idiosyncratic variance (N x N).
With K << N factors, this is a low-rank plus diagonal structure.
Better conditioned than the sample covariance when N is large.

Performance Evaluation¶

Return Metrics¶

Cumulative return: Product(1 + R_t) - 1. Annualized return: (1 + cumulative)^(252/T) - 1 for daily data. Annualized volatility: std(R_t) * sqrt(252) for daily data.

Risk-Adjusted Return Metrics¶

Sharpe ratio (Sharpe, 1966): SR = (E[R_p] - R_f) / sigma_p

The standard risk-adjusted performance metric.
Annualize: SR_annual = SR_daily * sqrt(252) (approximately).
Bootstrapped confidence intervals recommended (Lo, 2002).
Beware: SR assumes normal returns. Strategies with negative skewness or fat tails can have high SR but large tail risk.

Sortino ratio: Sortino = (E[R_p] - R_f) / sigma_downside

Uses only downside deviation (volatility of negative returns).
Better for asymmetric return distributions.

Information ratio: IR = alpha / sigma_epsilon

alpha and sigma_epsilon from a regression of portfolio returns on benchmark.
Measures active return per unit of active risk (tracking error).

Calmar ratio: Annualized return / Max drawdown.

Drawdown Analysis¶

Maximum drawdown: Largest peak-to-trough decline.

MDD = max_{t} (max_{s<=t} V_s - V_t) / max_{s<=t} V_s

Python

def max_drawdown(returns):
    """Compute maximum drawdown from a return series."""
    cumulative = (1 + returns).cumprod()
    running_max = cumulative.cummax()
    drawdowns = (cumulative - running_max) / running_max
    return drawdowns.min()

Drawdown duration: Time from peak to recovery (or ongoing if not recovered).

Attribution¶

Brinson-Fachler decomposition: For a benchmark-relative portfolio: - Allocation effect: being over/underweight in asset classes that performed well/poorly. - Selection effect: picking better/worse securities within each asset class. - Interaction effect: the cross-term.

Factor attribution: Decompose returns into factor exposures * factor returns + alpha.

R_p = alpha + Sum(beta_k * F_k) + epsilon

Report: what fraction of return came from market, size, value, momentum, etc.

Portfolio Construction in Practice¶

Constraints¶

Long-only: w_i >= 0. Dramatically changes the efficient frontier.
Position limits: w_i <= w_max (e.g., 5% per stock).
Sector limits: sum of weights in a sector <= threshold.
Turnover limits: constrain sum of |Delta_w_i| to control transaction costs.
Tracking error: sigma(R_p - R_b) <= TE_max.

Transaction Costs¶

Proportional costs: c * |Delta_w_i| * Portfolio_value.
Market impact: increases with trade size (square-root model: impact ~ sqrt(Volume)).
Net-of-cost optimization: subtract estimated costs from expected returns.

Rebalancing¶

Calendar rebalancing: monthly, quarterly, annually.
Threshold rebalancing: rebalance when weights drift beyond bands.
Tradeoff: more frequent rebalancing keeps weights closer to optimal but incurs higher costs.

Practical Checklist¶

Inputs: Use Ledoit-Wolf or factor model for covariance. For expected returns, use Black-Litterman or skip returns entirely (minimum variance / risk parity).
Constraints: Always impose long-only and position limits in practice. Unconstrained MVO is unstable.
Out-of-sample testing: Rolling window or expanding window optimization. Report out-of-sample Sharpe, turnover, and max drawdown.
Performance metrics: Report annualized return, volatility, Sharpe, max drawdown, Calmar ratio. Include benchmark comparison.
Transaction costs: Estimate and subtract from returns. Report gross and net performance.
Factor attribution: Decompose returns into factor contributions and alpha.
Robustness: Test sensitivity to estimation window, rebalancing frequency, and constraint parameters.
Statistical significance: Bootstrap Sharpe ratios (Lo 2002). Difference in Sharpe ratios: Ledoit-Wolf (2008) test.
Economic significance: Report dollar returns, capacity, and scalability.
Visualization: Efficient frontier, cumulative return plots, drawdown plots, rolling Sharpe.

Key References¶

Markowitz, H. (1952). Portfolio selection. Journal of Finance.
Black, F. and Litterman, R. (1992). Global portfolio optimization. Financial Analysts Journal.
Ledoit, O. and Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance.
Maillard, S., Roncalli, T., and Teiletche, J. (2010). The properties of equally weighted risk contribution portfolios. Journal of Portfolio Management.
DeMiguel, V., Garlappi, L., and Uppal, R. (2009). Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? Review of Financial Studies.
Lo, A.W. (2002). The statistics of Sharpe ratios. Financial Analysts Journal.
Michaud, R.O. (1989). The Markowitz optimization enigma: Is 'optimized' optimal? Financial Analysts Journal.
Brinson, G.P., Hood, L.R., and Beebower, G.L. (1986). Determinants of portfolio performance. Financial Analysts Journal.

portfolio-theory¶

Portfolio Theory¶

Overview¶

Mean-Variance Optimization (Markowitz, 1952)¶

Setup¶

Optimization Problem¶

Estimation Error Problem¶

Black-Litterman Model (1992)¶

Steps¶

Advantages¶

Risk Parity¶

Equal Risk Contribution (ERC)¶

Properties¶

Covariance Estimation¶

Sample Covariance¶

Ledoit-Wolf Shrinkage (2003)¶

Factor Model Covariance¶

Performance Evaluation¶

Return Metrics¶

Risk-Adjusted Return Metrics¶

Drawdown Analysis¶

Attribution¶

Portfolio Construction in Practice¶

Constraints¶

Transaction Costs¶

Rebalancing¶

Practical Checklist¶

Key References¶

`portfolio-theory`¶