Modern Portfolio Theory Calculator

1. Pick 2–6 assets and enter expected return and volatility for each.
2. Enter correlations between assets; use values from research or history.
3. Add your intended weights; the tool normalizes to 100% automatically.
4. Set a risk‑free rate and simulation count for suggestions.
5. Submit to view your metrics plus optimized comparisons.
6. Download CSV for spreadsheets, or PDF for sharing.

Inputs that drive the model

Each asset uses an annual expected return, annual volatility, and pairwise correlation. A four‑asset example can produce 6 unique correlations. Entering consistent ranges matters: 5% return and 15% volatility are typical for diversified equity, while high‑grade bonds often show 3–6% volatility.

From correlation to covariance

The calculator converts correlation into covariance using Σ_ij = σ_iσ_jρ_ij. If Asset A has 18% volatility and Asset B has 6% volatility, with −0.10 correlation, the covariance becomes −0.00108. This term reduces portfolio risk when combined with positive‑covariance pairs.

Portfolio return and risk

Expected portfolio return is the weighted sum Σw_iμ_i. Risk is computed by wᵀΣw and reported as standard deviation. A 60/40 mix can lower risk even when each component is volatile, provided correlations are well below 1.0.

Sharpe ratio and comparisons

Sharpe uses (E[R_p]−R_f)/σ_p. With a 2.5% risk‑free rate, a 7.0% portfolio return and 10.0% risk yields a Sharpe of 0.45. Raising return by 1% without increasing risk lifts Sharpe materially, which is why the max‑Sharpe search is useful.

Efficient frontier sampling

The tool generates thousands of random portfolios and retains points that set the best observed return for a given risk level. If you run 6,000 simulations, you typically see a dense scatter cloud plus a smooth frontier curve. More simulations improves the frontier resolution and helps locate better weight combinations. To stress‑test assumptions, adjust one input at a time. A 0.10 drop in equity‑equity correlation can cut portfolio risk by more than one percentage point in many mixes. Likewise, increasing bond volatility from 6% to 8% raises covariance terms and shifts min‑variance weights toward cash‑like assets. Re‑run the chart to see how the frontier bends; tighter clouds suggest more stable estimates. If Sharpe rankings flip across small changes, treat the output as exploratory and widen your safety margins.

Interpreting optimized weights

Max‑Sharpe weights emphasize return per unit of risk, while min‑variance weights emphasize stability. In long‑only mode, weights stay non‑negative and sum to 100%. If shorting is enabled, leverage is capped by Σ|w|, helping prevent extreme exposures that can distort practical allocation decisions.

1) Are expected returns required to be realistic?

No, but unrealistic inputs can create misleading “optimal” weights. Use defensible assumptions from research, historical estimates, or your own scenario analysis.

2) What if my weights do not sum to 100%?

The calculator automatically normalizes your weights to sum to 100%. This keeps comparisons consistent across your portfolio and the suggested portfolios.

3) Why does correlation matter so much?

Correlation changes covariance, which drives portfolio variance. Lower or negative correlations can reduce risk without reducing expected return, improving risk‑adjusted performance.

4) How accurate is the max‑Sharpe result?

It is an approximation from randomized search. Increasing simulation count usually improves results and stability, especially when assets have similar risk‑return profiles.

5) When should I enable shorting?

Enable it only if you understand borrowing costs, margin rules, and downside risks. Use the leverage cap to keep exposures within a controlled range.

6) Does the chart guarantee future performance?

No. The chart summarizes inputs and assumptions, not forecasts. Use it to compare scenarios and diversify thoughtfully, not to predict returns.