Methodology & Concepts

Transparent documentation of the mathematics, academic foundations, and data sources powering our optimizer.

Modern Portfolio Theory (MPT)

Portfolio Optimizer

Our optimization engine implements Mean-Variance Optimization (MVO), first introduced by Harry Markowitz in his seminal 1952 paper(Markowitz, 1952). This mathematical framework constructs portfolios by quantifying the trade-off between expected return and risk (variance), enabling investors to maximize returns for a given level of risk tolerance.

Core Calculations

Expected Portfolio Return

E(R_p) = \sum_{i=1}^{n} w_i \cdot E(R_i)

E(R_p)

= Expected portfolio return

w_i

= Weight of asset i

E(R_i)

= Expected return of asset i

The portfolio's expected return is the weighted sum of individual asset expected returns. Each weight represents the proportion of capital allocated to that asset.

Portfolio Variance

\sigma_p^2 = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \sigma_{ij}

\sigma_p^2

= Portfolio variance

\sigma_{ij}

= Covariance between assets i and j

Total portfolio risk accounts for both individual asset variances and the covariances between all asset pairs—the key insight of diversification.

The Efficient Frontier

Portfolio Optimizer

Efficient Frontier Chart

The Efficient Frontier is the set of optimal portfolios offering the highest expected return for each level of risk. Portfolios below this curve are dominated—they provide inferior returns for equivalent risk. Rational investors should only hold portfolios on or above this frontier (Markowitz, 1959).

We construct the efficient frontier by solving a series of constrained optimization problems (SLSQP method)—minimizing volatility for each of 200 target return levels. The feasible set visualization uses 2,000 Monte Carlo-sampled random portfolios.

Optimization Objectives

Portfolio Optimizer

Maximum Sharpe Ratio

Finds the portfolio with the highest risk-adjusted return—the tangency portfolio where the Capital Market Line touches the efficient frontier.

\max \frac{E(R_p) - R_f}{\sigma_p}

Minimum Variance

Constructs the lowest-risk portfolio possible given the asset universe, regardless of expected returns.

\min \sigma_p^2 = \mathbf{w}^T \Sigma \mathbf{w}

Risk Parity

Allocates weights such that each asset contributes equally to total portfolio risk.

RC_i = w_i \cdot \frac{(\Sigma \mathbf{w})_i}{\sigma_p} = \frac{\sigma_p}{n}

Modern Portfolio Theory (MPT)

Portfolio Optimizer

Core Calculations

Expected Portfolio Return

E(R_p) = \sum_{i=1}^{n} w_i \cdot E(R_i)

E(R_p)

= Expected portfolio return

w_i

= Weight of asset i

E(R_i)

= Expected return of asset i

The portfolio's expected return is the weighted sum of individual asset expected returns. Each weight represents the proportion of capital allocated to that asset.

Portfolio Variance

\sigma_p^2 = \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \sigma_{ij}

\sigma_p^2

= Portfolio variance

\sigma_{ij}

= Covariance between assets i and j

Total portfolio risk accounts for both individual asset variances and the covariances between all asset pairs—the key insight of diversification.

The Efficient Frontier

Portfolio Optimizer

Efficient Frontier Chart

Optimization Objectives

Portfolio Optimizer

Maximum Sharpe Ratio

Finds the portfolio with the highest risk-adjusted return—the tangency portfolio where the Capital Market Line touches the efficient frontier.

\max \frac{E(R_p) - R_f}{\sigma_p}

Minimum Variance

Constructs the lowest-risk portfolio possible given the asset universe, regardless of expected returns.

\min \sigma_p^2 = \mathbf{w}^T \Sigma \mathbf{w}

Risk Parity

Allocates weights such that each asset contributes equally to total portfolio risk.

RC_i = w_i \cdot \frac{(\Sigma \mathbf{w})_i}{\sigma_p} = \frac{\sigma_p}{n}