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Key Takeaways
- Modern Portfolio Theory compares expected return with portfolio variance, whose square root is return volatility.
- For fixed positive weights, correlation below one lowers volatility relative to the weighted average of the two standalone volatilities.
- The efficient frontier is the set of portfolios that deliver the highest expected return for each level of risk.
- A capital allocation line combines a risky portfolio with a risk-free asset. The maximum-slope feasible line identifies the tangency portfolio for the selected inputs and constraints.
- Specify expected returns, covariances, position limits and financing terms, then compare the resulting feasible portfolios.
What is Modern Portfolio Theory (MPT)?
Modern Portfolio Theory (MPT) is Harry Markowitz’s framework for building portfolios from the expected returns, variances and correlations of their assets. It measures portfolio risk as return variance, shows how holding imperfectly correlated assets lowers volatility below the weighted average of their volatilities, and identifies the portfolios that offer the highest expected return for each level of risk.
How MPT calculates portfolio return and risk
Expected portfolio return is the weighted sum of asset expected returns. Portfolio variance includes each asset’s variance and the covariance between holdings.
Choose the assets, holding period and position limits before calculating the feasible set.
Formula
E(Rp) = w1 × E(R1) + w2 × E(R2); variance(Rp) = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2- w1 and w2 are the portfolio weights of the two assets and sum to 1.
- σ1 and σ2 are the standard deviations of each asset return; a variance is a squared standard deviation.
- For two positive weights, lowering correlation reduces the cross term and variance. If one weight is negative, the effect reverses. The example uses long-only weights.
MPT example: a 60/
A trader blends two assets. Stock A has an 11% expected return and 20% volatility. Bond B has a 4% expected return and 6% volatility. Their return correlation is 0.10. The trader holds 60% in A and 40% in B.
- Expected return: 0.60 × 11% + 0.40 × 4% = 6.6% + 1.6% = 8.2%
- Variance: 0.60² × 0.20² + 0.40² × 0.06² + 2 × 0.60 × 0.40 × 0.10 × 0.20 × 0.06 = 0.0144 + 0.000576 + 0.000576 = 0.015552
- Volatility: square root of 0.015552 = 12.47%
- Weighted-average volatility (no diversification benefit): 0.60 × 20% + 0.40 × 6% = 14.4%
The 60/40 mix has 8.2% expected return and 12.47% volatility. With the same weights and standalone volatilities but perfect positive correlation, volatility would be 14.4%. The stated 0.10 correlation lowers it by 1.93 percentage points.
The efficient frontier in MPT
For the long-only two-asset example, the minimum-variance mix has 5.83% A and 94.17% B. It has 4.41% expected return and 5.88% volatility.
The efficient branch runs from that mix toward higher expected-return allocations. The 60/40 example appears on the same computed curve.
The risk-free asset and the tangency portfolio
For the chart, introduce a 2% risk-free rate. The maximum-Sharpe risky mix holds 30.24% A and 69.76% B, producing 6.12% expected return and 7.69% volatility.
Its capital allocation line has a slope of 0.54. Combine that risky mix with the risk-free asset within the chosen borrowing and lending constraints. Different forecasts, financing terms or portfolio constraints move the tangency mix.
Correlation and diversification in MPT
For positive weights, lowering correlation reduces the variance cross term. With a long and a short weight, that relationship reverses.
Compare the new portfolio with the correct reference: an original allocation, a weighted-volatility baseline or another feasible mix. Adding a low-correlation asset and changing weights is a complete portfolio calculation. Review expected return, volatility, concentration, drawdown and scenario results together.
Limitations of Modern Portfolio Theory
- The output is only as good as the inputs. Expected returns are forecasts, and small changes in them or in covariances are enough to move the optimal mix substantially, so rerun the optimization across a range of inputs before you commit to the weights.
- Variance is the only risk measure, and it scores gains and losses alike. Check fat tails and drawdowns separately before you size the allocation.
- Volatilities and correlations change over time. When correlations rise, a long-only portfolio built on historical estimates carries more volatility than planned, so re-estimate the inputs on a schedule.
- The tangency line prices borrowing and lending at the 2% risk-free rate. Real borrowing costs more and position limits cap leverage, so price the line at your own financing terms.
- Diversification lowers volatility below the weighted average: the example’s 12.47% sits under the 14.4% average and still above Bond B’s 6%. Judge a mix against the right reference before you call it safer.
Bottom line
Modern Portfolio Theory shows how holding imperfectly correlated assets lowers volatility below the weighted average of their volatilities and how to choose the best trade-off between expected return and risk. Use it with carefully estimated inputs and realistic constraints, and check drawdown and scenario results alongside variance.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- CFA Institute: Portfolio Risk and Return: Part I
- CFA Institute: Portfolio Risk and Return: Part II
- Fidelity: Why Diversify
How entries are sourced, calculated and updated: editorial methodology.