Modern portfolio theory, worked through with 98 years of stock and bond returns

From 1928 to 2025, 10-year US Treasury bonds returned an average of 4.82% a year with a volatility of 7.90%. Moving 13.8% of a bond portfolio into the S&P 500 raised the average return to 5.79% and lowered the volatility to 7.36%. Stocks swung almost two and a half times as much as the bonds over those years, and the mix still swung less than the bonds on their own.

The two assets had a correlation of 0.02 between their annual returns over those 98 years. Their bad years rarely coincided, so part of each one's swings cancelled out. Harry Markowitz turned that observation into a method in 1952, and robo-advisers such as Wealthfront still build their recommended portfolios on it.

Modern portfolio theory (MPT), also called mean-variance analysis, is a mathematical method for choosing a mix of assets from three kinds of input: each asset's expected return, each asset's variance, and the covariance of every pair. It finds the portfolios with the highest expected return for each level of risk, where risk is the standard deviation of returns.

Expected return

The average return an asset or portfolio is expected to earn over the next period. A portfolio's expected return is the weighted average of its assets' expected returns.

Volatility

The standard deviation of returns, the square root of the variance. Modern portfolio theory uses it as the measure of risk.

Correlation

A number from -1 to +1 that measures how closely two assets' returns move together. The covariance is the correlation multiplied by the two volatilities.

Efficient frontier

The set of portfolios with the highest expected return for their volatility. Every portfolio below it is beaten by one on it with the same volatility.

Tangency portfolio

The mix of risky assets with the highest Sharpe ratio, found where a straight line drawn from the risk-free rate touches the efficient frontier.

Capital market line

That straight line. Each point on it combines the tangency portfolio with lending or borrowing at the risk-free rate.

How Harry Markowitz arrived at portfolio selection

Markowitz was a graduate student in economics at the University of Chicago when the idea came to him, one afternoon in the library, while reading John Burr Williams's Theory of Investment Value. Williams valued a stock at the present value of its future dividends. Markowitz saw what followed if investors cared only about expected value: each would put everything in the one security with the highest expected return. Real investors diversify, so they care about risk as well, and variance came to mind as the measure. Portfolio variance depends on how securities move together, which matched what diversification does.

"Portfolio Selection" ran in the March 1952 issue of the Journal of Finance, 15 pages long. It rejected the rule of maximizing expected return because that rule never implies that a diversified portfolio is better than an undiversified one, replaced it with the expected return and variance rule, and showed the efficient set geometrically for small numbers of securities. When Markowitz defended the dissertation built on the same work, Milton Friedman argued that portfolio theory was not economics and that the department could not give him an economics doctorate for it. He got the degree without long debate.

The rest of the theory arrived in pieces, some of them earlier than 1952:

• 1940. Bruno de Finetti solved a mean-variance problem for proportional reinsurance in an Italian actuarial journal. Markowitz later recognized de Finetti's priority in applying the mean-variance approach to finance, while arguing that de Finetti had not solved the general case of correlated risks.
• 1952. A. D. Roy published "Safety first and the holding of assets" in Econometrica in July, four months after Markowitz. It chose portfolios by their mean and variance to minimize the chance of falling below a disaster level.
• 1956 and 1959. Markowitz published a fast method for computing mean-variance frontiers, the critical line algorithm, then a book, Portfolio Selection: Efficient Diversification of Investments, written at the Cowles Foundation at Yale at James Tobin's invitation.
• 1958. Tobin added a riskless asset and showed that investors with different attitudes to risk can hold the same mix of risky assets, changing only how much cash they keep next to it.
• 1964. William Sharpe derived the capital asset pricing model by assuming that every investor follows Markowitz with the same inputs.
• 1990. Markowitz, Sharpe and Merton Miller shared the Nobel prize in economic sciences. Markowitz died on June 22, 2023, at 95.

Why portfolio risk depends on correlation

For two assets held in weights w₁ and w₂, the portfolio's expected return is the weighted average w₁E₁ + w₂E₂. Its variance has a third term:

σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂

Here σ is each asset's volatility and ρ the correlation between them. Only at a correlation of exactly +1 does the portfolio's volatility equal the weighted average of the two volatilities. Every lower correlation pulls it below that average. With the 1928 to 2025 volatilities of the S&P 500 (19.40%) and of 10-year Treasuries (7.90%), a 50/50 portfolio would have had these volatilities at different correlations:

At a correlation of -1, a portfolio with 28.9% in stocks and 71.1% in bonds would have had no volatility at all.

Specific risk, market risk and the number of stocks

The same arithmetic extends to many assets. In an equal-weighted portfolio of N stocks that each have volatility σ and the same correlation ρ with each other, the variance is σ²/N + (1 - 1/N)ρσ². The first term falls toward zero as N grows. The second converges to ρσ², the risk the stocks share, however many are added.

Take σ at 35% and ρ at 0.25, round numbers chosen for the illustration. One stock has a volatility of 35%, 10 stocks 19.95%, 30 stocks 18.35% and 100 stocks 17.76%. The floor is 17.5%, which is 35% times the square root of 0.25. The part that diversification removes is called specific, idiosyncratic or unsystematic risk. The part that stays is systematic, or market, risk.

Real stocks differ in volatility and correlation, and a portfolio picked at random can end up concentrated by chance. Allowing for that, a randomly chosen portfolio needs at least 30 stocks to be well diversified for an investor who borrows and 40 for one who lends. The figure widely accepted before that 1987 result was about 10 stocks.

A two-asset efficient frontier from 98 years of US returns

The inputs are the averages and volatilities of annual total returns from 1928 to 2025 in the NYU Stern historical returns dataset. They are arithmetic averages of nominal returns, before fees and taxes:

• S&P 500, dividends included: average 11.85% a year, volatility 19.40%
• 10-year US Treasury bond: average 4.82%, volatility 7.90%
• 3-month Treasury bill: average 3.41%
• Correlation between the stock and bond returns: 0.02

The Sharpe ratio column uses the 3.41% bill return as the risk-free rate. Our guide to the Sharpe ratio covers the formula and its limits.

Red: the efficient frontier, from the minimum-variance portfolio to 100% stocks. Dashed: the inefficient lower branch, from the minimum-variance portfolio down to 100% bonds. Black line: the capital market line from the average Treasury bill return through the tangency portfolio. Annual returns, 1928 to 2025.

Every portfolio between 100% bonds and the minimum-variance point at 13.8% stocks sits on the dashed lower branch. Each is beaten by a portfolio on the upper branch with the same volatility. The all-bond portfolio's 7.90% volatility, for instance, matches a mix with about 27.5% in stocks that averaged 6.76% a year, almost two points more.

The red upper branch, from 13.8% stocks to 100%, is the efficient frontier for these two assets. Which point on it suits an investor depends on how much volatility that investor accepts. With risky assets alone, the theory leaves that choice open.

The tangency portfolio and the capital market line

A Treasury bill held to maturity has a known nominal return, so modern portfolio theory treats it as the riskless asset. Holding a share k of any risky portfolio P and the rest in bills gives an expected return of R_f + k(E_P - R_f) and a volatility of kσ_P, a straight line in the chart. The steepest such line touches the frontier at the portfolio with the highest Sharpe ratio, the tangency portfolio.

With two risky assets, the tangency weight in stocks has a closed form. With e_s and e_b the excess returns of stocks and bonds over bills:

w_s = (e_sσ_b² - e_bρσ_sσ_b) ÷ (e_sσ_b² + e_bσ_s² - (e_s + e_b)ρσ_sσ_b)

With e_s at 8.44%, e_b at 1.41% and the volatilities and correlation above, w_s comes to 50.8%. That portfolio averaged 8.40% a year with a volatility of 10.67%, a Sharpe ratio of 0.467, against 0.435 for stocks alone and 0.178 for bonds alone. A plain 60/40 mix scored 0.464 over the same years. Our article on the 60/40 portfolio covers where that split came from and how it has fared.

The capital market line in the chart runs from the bill return through the tangency portfolio, and its slope is that Sharpe ratio of 0.467. It touches the frontier only at the tangency point and lies above it everywhere else, so at any volatility a mix of the tangency portfolio and bills had a higher average return than any stock and bond mix on its own.

Tobin's separation: one risky portfolio, different amounts of cash

Once bills are available, an investor who wants less risk than the tangency portfolio holds it alongside bills, and one who wants more borrows to hold more of it. The mix of risky assets stays the same for both. Two cases on the 1928 to 2025 numbers, assuming borrowing at the bill rate:

A target volatility of 8%

75% in the tangency portfolio, which means 38.1% in stocks and 36.9% in bonds, plus 25% in bills, averaged 7.15% a year. The best stock and bond mix with the same 8% volatility, about 29% stocks, averaged 6.85%.

A target volatility of 15%

140.6% of capital in the tangency portfolio, with 40.6% borrowed, averaged 10.42%. The unlevered mix with 15% volatility, about 77% stocks, averaged 10.20%.

The gains are about 0.3 and 0.22 percentage points a year, before costs. A margin loan or futures financing costs more than the bill rate, and a spread of 0.53 points on the 40.6% borrowed erases the second gain entirely. Leverage multiplies the excess return and the volatility by the same factor, so the Sharpe ratio stays the same until financing costs come in.

From the capital market line to the CAPM

If every investor uses the same inputs and borrows and lends at the same rate, all of them hold the same tangency portfolio. In equilibrium it has to be the market portfolio, every risky asset held in proportion to its market value. The line from the risk-free rate through that market portfolio is the capital market line in the strict sense. Drawn from one investor's own estimates, as in the chart above, the same construction is called a capital allocation line.

Sharpe's 1964 paper carried the equilibrium one step further. An asset's expected return then depends only on its beta, its covariance with the market divided by the market's variance:

E(R_i) = R_f + β_i(E(R_m) - R_f)

Specific risk can be diversified away at almost no cost, so in this equilibrium it earns no premium, and a holder of a few stocks carries risk that pays nothing extra. In the data, low-beta stocks have earned more than the model predicts and high-beta stocks less. That evidence, and the factor models built to explain it, are covered in our article on factor investing.

Mean-variance optimization with many assets

With N assets, the weights form a vector w, the expected returns a vector μ and the covariances a matrix Σ. The portfolio's expected return is w'μ and its variance w'Σw. Each point on the frontier solves one quadratic program: minimize w'Σw, subject to w'μ equal to a target return and the weights summing to 1. Without other constraints, Lagrange multipliers give the answer in closed form. A ban on short sales or a cap on each weight calls for a numerical solver, the job Markowitz's critical line algorithm did in 1956.

Two results carry over from the two-asset case. The frontier of risky assets alone has the curved shape sometimes called the Markowitz bullet, with the global minimum-variance portfolio at its leftmost point. Any portfolio on that frontier is a combination of two other frontier portfolios, the two-fund theorem, and adding the riskless asset reduces this to one fund, the tangency portfolio.

The number of inputs grows quickly. N assets need N expected returns, N variances and N(N - 1)/2 covariances, a total of N(N + 3)/2:

• 10 asset classes: 65 inputs
• 100 stocks: 5,150 inputs
• 500 stocks: 125,750 inputs

Sharpe's 1963 single-index model ties each stock to one market index, which cuts the count to 3N + 2, or 1,502 inputs for 500 stocks.

Estimation error in expected returns and covariances

The optimizer treats its inputs as known facts. Expected returns in particular are measured with large errors. With a volatility of 19.40%, the standard error of a 98-year average of annual stock returns is 1.96 percentage points, 19.40% divided by the square root of 98. The 95% confidence interval around the 11.85% average runs from about 8.0% to 15.7%.

Feeding different windows of the same two series into the tangency formula shows what that uncertainty does to the weights:

The 1966 to 1995 window asks for a 2.9% short position in bonds. Over 2016 to 2025, bonds returned 1.22% a year on average against 2.25% for bills, and an optimizer barred from short sales puts everything in stocks.

The optimizer leans hardest on the assets whose expected return was overestimated, whose variance was underestimated or whose correlation with the rest came out too low. Richard Michaud called mean-variance optimizers "estimation-error maximizers" in 1989 for that reason. In simulations, errors in expected returns were over ten times as damaging as errors in variances and over twenty times as damaging as errors in covariances, and their relative weight grew with the investor's risk tolerance.

Across seven datasets, none of 14 mean-variance models and refinements beat an equal-weighted portfolio consistently on Sharpe ratio, certainty-equivalent return or turnover, once tested on data they had not been fitted to. For sample-based mean-variance weights to beat equal weighting, a portfolio of 25 assets needs an estimation window of about 3,000 months, which is 250 years. A portfolio of 50 assets needs about 6,000 months.

How practitioners reduce estimation error

• Weight constraints. A ban on short sales and a cap on each weight stop the optimizer from building extreme long and short positions on small differences in estimated returns.
• Covariance shrinkage. The sample covariance matrix is pulled toward a simpler target, such as the same correlation for every pair, which removes part of its noise. The Ledoit and Wolf method of 2004 sets the shrinkage intensity from the data.
• Equilibrium expected returns. The Black-Litterman model, published in 1992 by Fischer Black and Robert Litterman of Goldman Sachs in the Financial Analysts Journal, starts from the expected returns implied by market-capitalization weights and moves them toward the investor's own views in proportion to the confidence attached to each. Wealthfront uses it to blend CAPM returns with its own forecasts.
• Leaving expected returns out. A minimum-variance portfolio needs only the covariance matrix. Risk parity needs only volatilities and correlations. Equal weighting needs no estimates at all.

Normal returns, fat tails and unstable correlations

The model assumes a single holding period, investors who agree on the inputs and care only about mean and variance, no taxes or trading costs, and unlimited borrowing and lending at the riskless rate. Mean and variance describe a distribution completely only when returns are normally distributed, or when the investor's utility is quadratic.

Stock returns have fatter tails than the normal distribution allows. The S&P 500 moved more than five standard deviations in a day on 102 days from 1928 to 2025, where a normal distribution predicts 0.014 such days, as detailed in our article on fat tails. On October 19, 1987, the index fell 20.47%, a move of about 17 standard deviations.

Variance also counts a large gain as risk in the same way as a large loss, while investors feel losses more than gains of the same size, the pattern known as loss aversion. Markowitz devoted chapter IX of his 1959 book to the semi-variance, which counts only returns below a target. Versions of the method built on downside risk, grouped under the name post-modern portfolio theory, replace variance with semi-variance or with downside deviation, the measure in the denominator of the Sortino ratio.

Stock and bond correlation by decade

The 0.02 correlation between stocks and bonds over 98 years averages out decades that looked nothing alike. Measured over 10-year windows in the same data:

• 1956 to 1965: -0.60
• 1984 to 1993: +0.66
• 2000 to 2009: -0.86
• 2016 to 2025: +0.66

In 2022 the S&P 500 returned -18.04% and the 10-year Treasury -17.83%, and a 50/50 mix of the two lost 17.93%. Correlations also rise when markets fall. Between international equity markets, correlation increases in bear markets but not in bull markets, so diversification across countries weakens in the years when prices drop.

How Markowitz invested his own retirement money

When Markowitz joined the TIAA-CREF retirement plan in the 1950s, shortly after publishing "Portfolio Selection", he split his contributions 50/50 between bonds and stocks to limit his future regret if the market moved sharply either way, and later said he should have computed the historical covariances and drawn an efficient frontier. By 2009 he was dividing his money among asset classes in the proportions of efficient portfolios he had seen.

The 1928 to 1951 returns, the history he could have measured at the time, would have given a very different answer. Bonds then had a volatility of 2.71% and stocks 25.65%, and the tangency formula puts 4.1% in stocks. The full 1928 to 2025 record puts the tangency portfolio at 50.8%, close to the split he chose to limit regret.

This article is general information, not investment advice.

Common questions about modern portfolio theory

What is modern portfolio theory in simple terms?

Modern portfolio theory is a method for choosing a mix of investments by looking at how their returns move together as well as at their own risk and return. Assets whose bad years rarely coincide offset part of each other's swings. From 1928 to 2025, a mix of 13.8% S&P 500 stocks and 86.2% 10-year Treasury bonds averaged 5.79% a year with a volatility of 7.36%, while the bonds alone averaged 4.82% with a volatility of 7.90%.

What is the difference between modern portfolio theory and the CAPM?

Modern portfolio theory tells one investor how to combine assets, given estimates of their expected returns, volatilities and correlations. The capital asset pricing model, published by William Sharpe in 1964, assumes that every investor follows modern portfolio theory with the same estimates and can borrow and lend at the same riskless rate. All of them then hold the market portfolio, and each asset's expected return depends only on its beta: E(Ri) = Rf + beta times (E(Rm) - Rf).

What is the formula for modern portfolio theory?

A portfolio's expected return is the weighted average of its assets' expected returns. For two assets its variance is w1 squared times sigma1 squared, plus w2 squared times sigma2 squared, plus 2 times w1 times w2 times rho times sigma1 times sigma2, where w are the weights, sigma the volatilities and rho the correlation. With many assets the variance is w'Σw, and each point of the efficient frontier is the set of weights that minimizes that variance for a given expected return.

Is modern portfolio theory still used?

Yes. Wealthfront, for example, builds its recommended portfolios with mean-variance optimization and sets expected returns with the Black-Litterman model, because optimization on naively estimated inputs produces concentrated portfolios. Out of sample, none of 14 mean-variance models and refinements tested across seven datasets beat an equal-weighted portfolio consistently.

Who created modern portfolio theory?

Harry Markowitz, in his paper Portfolio Selection, published in the Journal of Finance in March 1952. James Tobin added the riskless asset in 1958 and William Sharpe derived the capital asset pricing model from the theory in 1964. Markowitz and Sharpe shared the 1990 Nobel prize in economic sciences with Merton Miller. Bruno de Finetti had solved a similar problem for reinsurance in 1940, and A. D. Roy published a related approach in July 1952.