Modern Portfolio Theory: Optimizing Risk and Return

Modern portfolio theory says an investor should judge each holding by how it changes the risk and return of the whole portfolio, not by its own risk alone. The math can produce a "best" mix of assets, but that mix depends on forecasts that are hard to make, so the optimized answer is far less stable than the formulas suggest.

Key takeaways

Portfolio return is the weighted average of asset returns. In a long-only mix of two assets, portfolio volatility is below the weighted average of their volatilities whenever correlation is below 1.
In the hypothetical example below, a 50/50 mix has a 5.0% expected return and 8.85% volatility at a correlation of 0.2, and 10.11% volatility at 0.8.
Moving one expected-return input by a single percentage point shifts the example's tangency weight from 38.2% to 45.6%, which is the fragility in miniature.
A peer-reviewed study of 14 optimization models on seven datasets found none consistently better than equal weighting out of sample.
Constraints, covariance shrinkage, risk parity and equal weighting are different ways to reduce, not remove, estimation error.

What Markowitz actually proposed

Harry Markowitz published "Portfolio Selection" in the Journal of Finance (volume 7, issue 1, pages 77–91, March 1952). The Royal Swedish Academy of Sciences described the resulting theory in its 1990 Nobel Prize press release as an analysis of how wealth can be invested in assets that differ in expected return and risk, and how risk can thereby be reduced. Markowitz shared that prize with Merton Miller and William Sharpe.

The core move is to measure risk as the variance of the whole portfolio. As the Nobel release puts it, what matters is not each asset's risk in isolation but its contribution to the risk of the whole portfolio. That variance depends on three things for each pair of holdings: how volatile each is, how much of each you hold, and how correlated they are. A holding that looks risky alone can lower portfolio risk if it moves differently from everything else.

The two-asset formulas

Return:    Rp = wA*RA + wB*RB
Variance:  sp^2 = (wA*sA)^2 + (wB*sB)^2 + 2*wA*wB*rho*sA*sB
Min-var weight in A:  wA* = (sB^2 - rho*sA*sB) / (sA^2 + sB^2 - 2*rho*sA*sB)

Here wA and wB are portfolio weights (wB = 1 − wA), RA and RB are expected returns, sA and sB are volatilities (standard deviations), and rho is the correlation between the two assets. Volatility is the square root of variance. The third line is the weight that minimizes sp.

A two-asset example, step by step

Illustrative example. All inputs are hypothetical round numbers, not estimates for any real asset. Asset A has an expected return of 7% and volatility of 16%. Asset B has an expected return of 3% and volatility of 5%. The correlation is 0.2. All figures were computed with Python.

Weight in A	Expected return	Volatility (rho 0.2)	Volatility (rho 0.8)
0% (all B)	3.0%	5.00%	5.00%
10%	3.4%	5.07%	5.86%
20%	3.8%	5.60%	6.84%
30%	4.2%	6.48%	7.88%
40%	4.6%	7.59%	8.98%
50%	5.0%	8.85%	10.11%
60%	5.4%	10.19%	11.26%
80%	6.2%	13.04%	13.61%
100% (all A)	7.0%	16.00%	16.00%

For the 50/50 mix at a correlation of 0.2, the variance is (0.5 × 0.16)² + (0.5 × 0.05)² + 2 × 0.5 × 0.5 × 0.2 × 0.16 × 0.05 = 0.0064 + 0.000625 + 0.0008 = 0.007825. Its square root is 8.85%, well below the 10.5% weighted average of the two volatilities. At a correlation of 0.8 the same mix has 10.11% volatility, 1.26 percentage points higher, with the same 5.0% expected return.

The minimum-variance weight

At a correlation of 0.2, the formula gives a weight of 3.6% in A. That portfolio has a 3.14% expected return and 4.97% volatility, only slightly below the 5.00% of holding B alone. Adding a small amount of a much riskier asset barely lowers risk here because B is already low-volatility.

At a correlation of 0.8 the closed-form answer is −25.5% in A, meaning a short position, with volatility of 3.88%. A long-only investor cannot hold that, so the long-only minimum is 100% in B at 5.00%. The weight turns negative whenever rho exceeds sB/sA, which is 0.3125 in this example.

Frontier and capital market line

Repeating the calculation for every weight traces a curve of risk and expected return. The efficient frontier is the upper part of that curve: for each level of risk, the portfolio with the highest expected return. Portfolios below the minimum-variance point are dominated, since another mix offers more return for the same risk.

Add a risk-free asset that can be lent or borrowed at one rate, and the picture changes. The best risky mix is then the one with the highest Sharpe ratio, the expected return above the risk-free rate divided by volatility. That is the tangency portfolio, and combinations of it with the risk-free asset form a straight line.

The Nobel release describes this split in the CAPM: the investor's attitude to risk sets only how much goes into the risk-free asset or is borrowed, not what the risky portfolio contains. In the CAPM, where investors share the same information, that risky portfolio is the market portfolio and the line is called the capital market line. The release credits Sharpe as the leading figure among the 1960s researchers who built the CAPM on Markowitz's work.

Illustrative example, continued. With a hypothetical risk-free rate of 2% and a correlation of 0.2, the tangency portfolio holds 38.2% in A and 61.8% in B. It has a 4.53% expected return, 7.38% volatility and a Sharpe ratio of 0.343, compared with 0.313 for A alone and 0.200 for B alone. Raise A's expected return from 7% to 8% and the tangency weight moves to 45.6%. A one-point change in one forecast moved the answer by about seven percentage points.

Why optimizers are fragile

Expected returns are the weak input

The formulas need an expected return for every asset, plus a volatility and a correlation for every pair. Volatility and correlation can be estimated from past prices, though with error. Expected returns are much harder to pin down, and the optimizer treats whatever number it receives as certain. The one-point example above shows how a small error in one forecast moves the output.

Equal weighting is hard to beat

DeMiguel, Garlappi and Uppal compared the out-of-sample performance of mean-variance optimization, and variants designed to reduce estimation error, against the naive 1/N rule that splits money equally. As the abstract of their 2009 Review of Financial Studies paper (volume 22, issue 5, pages 1915–1953) reports, none of 14 models was consistently better than 1/N across seven empirical datasets on Sharpe ratio, certainty-equivalent return or turnover. The authors attribute this to estimation error offsetting the gains from optimizing. Calibrated to US equity market parameters, they estimate the data window needed for these strategies to beat 1/N at about 3,000 months for 25 assets and about 6,000 months for 50.

Both numbers come from the paper's calibration, not from a general law. They show the size of the data problem under one set of assumptions, not that optimization can never help.

Correlations and the normal-distribution assumption

The two-asset table treats correlation as a fixed number and returns as roughly bell-shaped. Longin and Solnik tested this on international equity markets and reported, in the abstract of their 2001 Journal of Finance paper (volume 56, issue 2, pages 649–676), that they reject multivariate normality for the negative tail but not the positive tail. They also found that correlation relates to the market trend rather than to volatility itself: it increases in bear markets but not in bull markets.

In the example, that is the move from the 0.2 column to the 0.8 column, at the moment losses arrive. A frontier built on calm-period correlations can overstate the protection a mix offers in a downturn.

Practical fixes and alternatives

Each approach below addresses estimation error in a different way, and each has its own cost.

Weight constraints. Long-only rules and caps per asset stop the optimizer from taking the extreme positions it favors, such as the short position in the correlation-0.8 case above. The cost is that the result is no longer the unconstrained optimum.
Covariance shrinkage. Ledoit and Wolf argued in "Honey, I Shrunk the Sample Covariance Matrix" (Journal of Portfolio Management, volume 30, issue 4, 2004) that the sample covariance matrix should not be used for optimization, and proposed pulling its most extreme coefficients toward more central values. It still relies on past data and does not address expected-return error.
Risk parity. Weights are set so each asset contributes about the same amount of portfolio risk. It needs no expected-return forecasts, but it still needs volatility and correlation estimates. With two assets, equal risk contributions mean weights in inverse proportion to volatility: 23.8% in A and 76.2% in B in the example, for a 3.95% expected return and 5.90% volatility. Reaching the 50/50 mix's 5.0% while keeping risk contributions equal would take borrowing.
Equal weighting (1/N). It needs no estimates at all. It ignores differences in risk between assets: in the example's 50/50 mix at a correlation of 0.2, asset A supplies about 87% of the portfolio's variance.

None of these makes the future knowable. They trade a small amount of theoretical optimality for lower sensitivity to bad inputs.

Frequently asked questions

Is modern portfolio theory the same thing as the Capital Asset Pricing Model?

No. Markowitz's theory describes how an individual investor can combine assets given their expected returns, risks and correlations. According to the Nobel press release, researchers in the 1960s, with Sharpe the leading figure, used it as the basis for the CAPM, a theory of how financial assets are priced.

What does a correlation of 1 do to portfolio risk?

At a correlation of 1, a long-only mix's volatility equals the weighted average of the two volatilities. In the example, a 50/50 mix would have 10.5% volatility, so diversification would reduce no risk. Below 1, the combined figure falls under that average, as the 8.85% and 10.11% results show.

Can the minimum-variance weight be negative?

Yes. In the two-asset formula the weight in the more volatile asset (A in the example) is negative whenever the correlation exceeds the ratio of the lower volatility to the higher one, 0.3125 in the example. A negative weight means shorting that asset, which many investors cannot or will not do, so long-only versions add a constraint.

Is the tangency portfolio the same as the maximum-Sharpe portfolio?

Yes, in the standard setup with a risk-free asset. It is the mix of risky assets with the highest ratio of excess return to volatility. In the example it is 38.2% A and 61.8% B at a Sharpe ratio of 0.343, against 0.313 for A alone.

How is volatility different from variance?

Volatility is the standard deviation, the square root of variance. Asset A's 16% volatility corresponds to a variance of 0.0256. The formulas add variances and covariances, which is why the 50/50 calculation above produces 0.007825 before taking the square root.

A checklist for using the theory

Treat the output as a range, not a single answer: rerun the calculation with expected returns one or two points higher and lower and note how far the weights move.
Test volatility at a higher correlation than the historical figure, as the 0.8 column does, to see how the mix behaves in a downturn.
Set practical limits first, such as no shorting and a cap per holding, so the result is something you could hold.
Compare the result against an equal-weight mix. If the optimized mix is not clearly better under stressed inputs, the simpler one is easier to maintain.
Once target weights are set, check drift on a schedule. The rebalancing calculator shows how far current holdings are from target and what would restore them.