Modern Portfolio Theory and the Efficient Frontier

Why this matters

The first module of this track built the parts: expected return, standard deviation, covariance, and the ratios that combine them. Modern portfolio theory puts them to work on the question an investor needs answered: given these investments, how should I combine them? It's the standard framework for discussing how investments should be combined. Understanding it explains why diversified portfolios are built the way they are, and understanding its weak point tells you how much to trust a portfolio that came out of a model.

The idea

Harry Markowitz's contribution was to show that, under certain conditions, choosing a portfolio comes down to balancing its expected return against its variance, and that the variance depends on the covariances between every pair of assets as well as on each asset's own.¹ Two things follow.

• An investment should be judged by what it adds to the portfolio. A volatile asset that moves differently from everything else can make the whole steadier.
• Some portfolios are simply better than others. If two portfolios have the same risk and one has a higher expected return, nobody should want the other.

The portfolios that survive that second test, the ones with the highest expected return for each level of risk, form the efficient frontier.

Tracing a frontier

With two assets, every possible portfolio is a mix of the two, and the frontier can be laid out in a table.

Calculated for this lesson from Aswath Damodaran's yearly data, with 3-month Treasury bills as the risk-free asset, before fees, taxes, and inflation.² These are arithmetic averages of past returns. They describe what happened, and they aren't forecasts.

Read down the standard deviation column. It falls before it rises.

• The lowest-risk mix wasn't all bonds. Risk bottomed out at about 14% in shares, with a standard deviation of 7.4% against 7.9% for bonds alone. Adding a volatile asset to a steady one made the portfolio steadier, because the two had almost no correlation.
• An all-bond portfolio was inefficient. A mix with about 28% in shares had the same 7.9% standard deviation as bonds alone and an average return of 6.8%, two points higher. Anyone holding only bonds for safety was accepting a lower return for no reduction in risk.
• Above the lowest-risk point, everything is a tradeoff. From 14% in shares upward, each step adds return and adds risk. Those mixes are the efficient frontier. The theory doesn't say which one to hold. That depends on how much risk the investor can take, which is the subject of Time Horizon and Risk Tolerance.

You can try other mixes, and other periods, with the tool from Asset Allocation.

Historical US returns for the S&P 500, 10-year Treasury bonds, and 3-month Treasury bills, from Damodaran's dataset. Rebalanced yearly, before fees and taxes, and not adjusted for inflation. This is a way to explore history, not a forecast or a recommendation of any mix.

Adding a risk-free asset

So far the choice has been among risky portfolios. Add an asset with no risk, such as Treasury bills, and something changes. An investor can now hold any blend of bills and one risky portfolio. The blends of bills with a given risky portfolio lie on a straight line: more in the risky portfolio means proportionally more risk and more excess return.

The steepest such line belongs to the risky portfolio with the highest Sharpe ratio. In the table, that's about half in shares, with a Sharpe ratio of 0.47. That line is called the capital market line, and in the theory it replaces the curved frontier. Every investor should hold the same risky portfolio and set their own risk by how much they keep in bills. A cautious investor holds mostly bills. An aggressive one borrows at the risk-free rate to hold more than 100% in the risky portfolio.

That last step is where theory and practice separate. Ordinary investors can't borrow at the Treasury bill rate, and borrowing to invest has the dangers set out in Margin and Short Selling. So an investor who wants more return than the best risky portfolio offers moves along the frontier and holds more in shares.

Look again at the Sharpe ratio column. From 40% in shares to 70%, it barely moves: 0.46, 0.47, 0.46, 0.46. The "best" portfolio isn't a sharp peak. It's a broad plateau, and any mix on it was about as efficient as any other.

Why optimizers give fragile answers

With many assets, nobody traces the frontier by hand. A program called an optimizer takes an expected return and a standard deviation for each asset, and a correlation for each pair, and returns the efficient portfolios. The mathematics is exact. The inputs aren't.

Expected Return and Standard Deviation showed that 98 years of data leave the expected return on shares uncertain by several points. For bonds the standard error is 7.9% ÷ √98 = 0.8%. Here's what happens to the highest-Sharpe portfolio when the inputs move by amounts that small.

Expected return on shares	Expected return on bonds	Share of the best portfolio in shares
11.9%	4.8%	51%
11.9%	4.0%	72%
11.9%	5.8%	37%
9.9%	4.8%	44%
13.9%	4.8%	56%

The risks and the correlation are unchanged throughout. Moving the expected return on bonds by about one standard error either way moves the answer from 37% in shares to 72%. The data can't tell those inputs apart, and the optimizer reports each answer with the same confidence.

With two assets the damage is limited. With twenty, it isn't. An optimizer looks for assets that appear to offer high return, low risk, and low correlation with everything else, and the assets that look best on those counts tend to include the ones whose figures are wrong in a flattering direction. So it can put the most money where the estimates are least reliable. Left alone, an optimizer can produce an extreme portfolio: most of the money in a few assets, and none in many others.

The usual remedies are limits on how much any asset can hold, expected returns pulled toward a common figure, and a simple fixed mix used as a check on anything the model proposes. What These Models Miss returns to this.

What to keep

• Think about the portfolio. The question for any new holding is what it does to the whole.
• Diversify across things that behave differently. It's the one way to lower risk without giving up expected return.
• Avoid inefficient portfolios. Concentrating in one asset "for safety" or "for return" can mean accepting more risk than necessary for what's earned.
• Don't chase the exact optimum. The plateau is wide and the inputs are soft. A sensible mix held steadily is a better bet than a precise one that changes every time the estimates do.

Key takeaways

• The efficient frontier is the set of portfolios with the highest expected return for each level of risk.
• In US data from 1928 through 2025, the lowest-risk mix of shares and bonds held about 14% in shares, and all-bond portfolios were inefficient.
• With a risk-free asset, the best risky portfolio is the one with the highest Sharpe ratio, and risk is set by how much is held in it.
• The top of the frontier is a plateau: mixes from 40% to 70% in shares had nearly the same Sharpe ratio.
• Optimizers magnify errors in their inputs, so their precise answers deserve little trust.

Check your understanding

Portfolio X has an expected return of 6% and a standard deviation of 9%. Portfolio Y has an expected return of 7% and a standard deviation of 9%. Both are built from the same assets. What does the theory say about X?

1. It's the safer choice, since a lower return means lower risk.
2. It's inefficient, since Y offers more return for the same risk.
3. It's on the frontier, since every mix of the assets is on it.

An optimizer recommends 80% in one asset class out of ten. When the expected returns are re-estimated from a different ten-year period, it recommends 5%. What's the right conclusion?

1. The second answer is right, since it uses the more recent data.
2. The average of the two answers, about 42%, is the true optimum.
3. Neither is reliable, since the answer follows noise in the inputs.

What's next

The capital asset pricing model says one kind of risk, market risk, explains returns. Factor Exposure looks at the evidence that there are others, and at what's inside funds built to capture them.