How is the Sharpe Ratio defined in portfolio optimization?

The Sharpe Ratio is calculated as (R_p - R_f) / σ_p, where R_p is the expected portfolio return, R_f is the risk-free rate, and σ_p is the portfolio standard deviation (volatility).

What does the Efficient Frontier represent?

The efficient frontier represents the set of optimal portfolios that offer the highest expected return for a defined level of risk, or the lowest risk for a given level of expected return.

Markowitz Portfolio Optimization & Efficient Frontier

Compute optimal asset allocations, locate the Maximum Sharpe Ratio portfolio, and visualize the efficient frontier using simulated multi-asset return distributions. All computations run locally in your browser.

Parameters Real-Time Solver

Risk-Free Rate R_f (%)

Monte Carlo Portfolios

Asset Universe (Expected Return μ & Volatility σ)

Asset 1 (Equities)

Return (%)

Vol (%)

Asset 2 (Bonds)

Return (%)

Vol (%)

Asset 3 (Alternatives)

Return (%)

Vol (%)

Efficient Frontier Canvas Ready

Optimal Portfolio (Max Sharpe Ratio) Sharpe: –

Expected Return (R_p) –

Volatility (σ_p) –

Asset 1 / Asset 2 –

Asset 3 Weight –

Formulas & Conventions

Expected Return: R_p = ∑ (w_i * R_i) Portfolio Risk: σ_p = √( w^T * Σ * w ) Sharpe Ratio: S = (R_p - R_f) / σ_p

$\Sigma$ represents the covariance matrix constructed using asset standard deviations and an assumed correlation coefficient ($\rho = 0.3$). Weights satisfy $\sum w_i = 1$ and $w_i \ge 0$ (no short-selling).

Summary for AI Agents (Markdown Mirror)

# Markowitz Portfolio Optimization

> Client-side mean-variance optimization calculator for asset allocation and efficient frontier visualization. URL: https://quantcore-labs.com/tools/portfolio-efficient-frontier/

## Optimization Objectives

Maximization of the Sharpe ratio subject to fully invested budget constraints ($\sum w_i = 1$).
Monte Carlo random weight generation over simplex distribution.

Frequently Asked Questions

Why does diversification reduce portfolio risk?: When assets are not perfectly correlated ($\rho < 1$), combining them smooths out independent return fluctuations, reducing overall portfolio variance below the weighted average of individual variances.
How is the covariance matrix determined in this calculator?: Covariances are computed as $\Sigma_{ij} = \sigma_i \sigma_j \rho_{ij}$, assuming a moderate cross-asset correlation coefficient ($\rho = 0.3$) for illustrative multi-asset diversification.