Monte Carlo Portfolio Simulator
This tool generates thousands of randomly weighted portfolios and plots each one on a risk–return scatterplot. Each dot represents a portfolio formed by assigning different weights to the selected assets. Together, the cloud reveals the investment opportunity set. Dot color indicates the Sharpe ratio — redder dots have a higher Sharpe ratio and deliver more return per unit of risk taken.
Assets
| ✓ | Name | E(r) % | σ % |
|---|
Correlations
Edit upper-triangle cells; lower triangle mirrors automatically.
Simulation Settings
Portfolios to simulate
5,000
Risk-free rate (%)
4.50%
Show risk-free asset & CAL
How to read this chart
Each dot is a possible portfolio.
The simulator assigns random weights to the assets and records each portfolio's expected return
and volatility. With enough dots the cloud fills out the feasible set — every portfolio
constructible from these assets. Portfolios toward the upper left are generally
preferable: more return for less risk.
Dominated portfolios and the efficient frontier.
A portfolio is dominated if another portfolio offers higher expected return for the
same risk, or equal return with less risk. The upper-left boundary of the cloud is the
efficient frontier — no dominated portfolio lies on it. Rational investors
should only hold efficient portfolios.
Diversification and correlations.
When assets do not move in perfect lockstep (correlation < 1), combining them pushes the
frontier to the left: the same expected return becomes achievable at lower risk. The lower the
pairwise correlations, the greater the benefit. Try setting some correlations to negative
values and re-running the simulation to see the frontier expand.
The tangency portfolio and the Capital Allocation Line (CAL).
The tangency portfolio is the risky portfolio with the highest Sharpe ratio — it sits
where a ray from the risk-free rate just touches the efficient frontier. Any investor who can
borrow or lend freely at the risk-free rate should hold a combination of (1) the risk-free
asset and (2) the tangency portfolio. The CAL traces all such combinations;
it dominates the risky frontier everywhere except at the tangency point itself.
Key formulas
Portfolio expected return — the weighted average of the individual asset returns:
E(rp) = w1·E(r1) + w2·E(r2) + … + wN·E(rN)
Portfolio variance — accounts for each asset's own volatility and every pairwise correlation:
σ²p = Σi Σj wi · wj · ρij · σi · σj
When correlations are below 1, portfolio variance is less than the weighted average of individual
variances — this is the mathematical foundation of diversification.
Sharpe ratio — excess return earned per unit of volatility:
Sharpe = (E(rp) − rf) / σp
A Sharpe of 0.80 means the portfolio earns 0.80% of extra return for every 1% of risk taken.
The tangency portfolio maximises this ratio among all risky portfolios.