Modern portfolio theory
Based on Wikipedia: Modern portfolio theory
The Mathematical Proof That You Shouldn't Put All Your Eggs in One Basket
Your grandmother was right about diversification. But in 1952, a young economist named Harry Markowitz proved why she was right—and in doing so, transformed how the entire financial world thinks about risk.
The core insight is deceptively simple: you shouldn't evaluate an investment by itself. What matters is how it behaves in combination with everything else you own.
This idea earned Markowitz a Nobel Prize and spawned what we now call Modern Portfolio Theory, or MPT. But beyond the academic accolades lies a genuinely useful framework for thinking about the trade-off every investor faces: how much risk are you willing to accept in pursuit of higher returns?
The Problem Everyone Had Been Ignoring
Before Markowitz, the conventional wisdom was straightforward: pick good investments. Find stocks that will go up. Avoid the ones that will go down. Simple.
Except it wasn't simple at all. Even the best analysts couldn't reliably predict which individual investments would outperform. And focusing solely on picking winners missed something crucial about how portfolios actually behave.
Markowitz asked a different question. Instead of "which investments are best?" he asked "how do investments interact with each other?"
The answer changed everything.
The Free Lunch of Finance
Economists love to say there's no such thing as a free lunch. But diversification comes remarkably close.
Here's the key insight: when you combine investments that don't move in perfect lockstep, something almost magical happens. The risk of the combined portfolio is less than what you'd expect from simply averaging the risks of the individual pieces.
Let me make this concrete. Imagine you own two stocks. Each one, on its own, swings up and down by about 20 percent in a typical year. You might assume that owning both would give you a portfolio that also swings by 20 percent.
But if these stocks don't move together perfectly—if sometimes one zigs while the other zags—then the combined portfolio will actually swing by less than 20 percent. The individual movements partially cancel each other out.
This is the free lunch: you can reduce risk without necessarily giving up any expected return. You just need investments that aren't perfectly correlated.
Correlation: The Hidden Variable
Correlation is the mathematical measure of how much two things move together. It ranges from negative one to positive one.
A correlation of positive one means perfect lockstep. When one goes up, the other always goes up by a proportional amount. Owning both is essentially like owning more of one thing—you get no diversification benefit at all.
A correlation of zero means complete independence. The movements of one tell you nothing about the movements of the other. This is where diversification really starts to work.
A correlation of negative one means perfect opposition. When one goes up, the other always goes down. This is the diversifier's dream, though it's rare to find in practice.
Most real investments fall somewhere between zero and one. Stocks in the same industry tend to be highly correlated. Stocks in different countries less so. Stocks and bonds even less. Finding genuinely uncorrelated assets is one of the great quests of portfolio management.
The Efficient Frontier: A Map of Possibilities
Markowitz's framework lets you visualize all possible portfolios on a two-dimensional map. The horizontal axis shows risk, typically measured by standard deviation—how much the portfolio's value tends to bounce around. The vertical axis shows expected return.
Plot every possible combination of investments, and you get a cloud of points. But here's what's interesting: the cloud has a distinct boundary on its upper-left edge.
This boundary is called the efficient frontier, and it represents the best possible trade-offs. Every portfolio on this line offers either the highest expected return for its level of risk, or the lowest risk for its level of return. You literally cannot do better without accepting more risk or less return.
Portfolios inside the cloud are inefficient—you're taking on risk without being compensated for it, or leaving returns on the table. A rational investor would never choose one of these when a better option exists on the frontier.
The shape of this frontier is worth noting: it curves like the edge of a bullet. Investment professionals sometimes call it the "Markowitz bullet." At the leftmost point sits the Global Minimum Variance Portfolio—the combination with the absolute lowest risk possible from any mix of the available assets.
The Two Fund Theorem
One of Markowitz's most elegant results is the two fund theorem, sometimes called the separation theorem. It states something remarkable: you can construct any portfolio on the efficient frontier by combining just two other portfolios that also lie on the frontier.
Think about what this means. You don't need to hold dozens of carefully calibrated positions. You could achieve any efficient allocation by mixing just two well-chosen funds in different proportions.
This has profound practical implications. It's part of why index funds and simple two-fund portfolios can be surprisingly effective. The complexity of optimization can often be reduced to a much simpler choice.
Adding a Risk-Free Asset Changes Everything
The original efficient frontier assumes you're choosing among risky assets. But what if you can also hold cash—or more precisely, something like a government treasury bill that offers a modest but guaranteed return?
This addition transforms the picture entirely. Instead of a curved frontier, you now get a straight line called the Capital Allocation Line. This line starts at the risk-free rate on the vertical axis and extends upward, tangent to the original curved frontier.
The point where the line touches the curve is special. It represents the optimal portfolio of risky assets—the one that, when combined with the risk-free asset, offers the best possible risk-return trade-off at any risk level.
Every investor, regardless of their risk tolerance, should hold this same optimal risky portfolio. The only difference is how much they put in the risky portfolio versus the safe asset. Conservative investors hold mostly cash. Aggressive investors hold mostly the risky portfolio—or even borrow to hold more than 100 percent of their wealth in it.
This is a startling conclusion. It says that everyone should own the same thing, just in different amounts. The work of portfolio selection collapses into a single choice: how much risk do you want?
What Counts as Return?
When Markowitz and his successors talk about return, they mean total return—everything you receive from holding an investment, not just price appreciation.
For stocks, this includes dividends. A stock that pays steady dividends might have a similar total return to one that pays nothing but appreciates more in price.
For bonds, total return includes coupon payments—the periodic interest the bond pays—as well as any change in the bond's price. Bond pricing gets complicated because of something called accrued interest. When you buy a bond between coupon payments, you pay the seller for the interest that's accumulated since the last payment. The price including this accrued interest is called the dirty price, while the quoted price without it is the clean price.
Serious portfolio analysis also accounts for transaction costs: brokerage commissions, exchange fees, taxes, and custody fees. These frictions reduce your actual returns and can meaningfully affect which portfolios are truly optimal.
The Math Behind the Magic
For those comfortable with mathematics, Modern Portfolio Theory is elegant in its formulation. The expected return of a portfolio is simply the weighted average of the expected returns of its components. If you put 60 percent in stocks expecting 8 percent returns and 40 percent in bonds expecting 4 percent, your portfolio's expected return is 0.6 times 8 plus 0.4 times 4, or 6.4 percent.
Risk is where it gets interesting. The variance of a portfolio isn't just the weighted average of variances. It depends critically on the covariances between assets.
Covariance measures how two assets move together, combining information about their correlation and their individual volatilities. The formula for portfolio variance includes a term for each asset's own variance, weighted by its squared portfolio weight. But it also includes cross-terms for every pair of assets, weighted by the product of their weights.
When correlations are low, these cross-terms contribute less, and the portfolio variance ends up lower than you'd expect from the individual variances alone. This is the mathematical machinery behind diversification's free lunch.
Finding the Optimal Portfolio
In practice, constructing an efficient portfolio means solving an optimization problem. You want to minimize portfolio variance for a given target return—or equivalently, maximize expected return for a given risk tolerance.
Markowitz himself developed an algorithm called the critical line method to solve this problem. It handles the real-world constraints that make simple calculus insufficient: requirements that portfolio weights sum to one, bounds on how much you can hold in any single asset, and the mathematical complications that arise when some assets are highly correlated.
Today, standard software packages like MATLAB, Excel, and R include optimization routines that can solve these problems. The mathematics is well-understood. The challenge lies elsewhere.
The Estimation Problem
Here's the dirty secret of Modern Portfolio Theory: it tells you exactly what to do once you know the expected returns, variances, and correlations. But it doesn't tell you how to know these things.
The typical approach is to look backward. Calculate average historical returns. Measure how much prices have fluctuated in the past. Compute correlations from historical data. Then assume the future will resemble the past.
This assumption is problematic. Markets change. Relationships between assets shift. An asset class that was uncorrelated with stocks for decades might suddenly start moving in lockstep. A sector that delivered strong returns historically might face structural decline.
Small errors in these inputs can lead to dramatically different optimal portfolios. The mathematics amplifies estimation errors rather than dampening them. An asset that looks slightly better than it actually is might get a huge portfolio weight, while one that looks slightly worse gets ignored entirely.
This sensitivity to inputs has spawned an entire field of robust portfolio optimization—techniques designed to produce sensible portfolios even when the inputs are imprecise.
Assumptions and Their Limits
Modern Portfolio Theory rests on several assumptions that don't perfectly match reality.
It assumes investors care only about expected return and variance. But real investors might also care about the chance of catastrophic losses, or about how their portfolio performs relative to peers, or about various psychological factors that don't fit neatly into a mean-variance framework.
It assumes returns follow a normal distribution—the familiar bell curve. But actual market returns have "fat tails," meaning extreme events happen more often than the bell curve predicts. The stock market crash of 1987, when the Dow Jones Industrial Average fell 22 percent in a single day, would be virtually impossible under normal distribution assumptions. Yet it happened.
It assumes correlations are stable. But correlations tend to increase during market crises, precisely when diversification would be most valuable. Assets that seemed uncorrelated in calm times suddenly start moving together when panic sets in.
It assumes markets are efficient and investors are rational. But behavioral economics has documented numerous ways in which actual investors deviate from rationality, and these deviations can affect prices.
A Curious Footnote: The Italian Precursor
Markowitz published his groundbreaking paper in 1952, and for decades, he was credited as the sole originator of mean-variance analysis. But in 2006, financial historians discovered that an Italian mathematician named Bruno de Finetti had published essentially the same ideas in 1940—twelve years earlier.
De Finetti worked in the context of proportional reinsurance, not stock portfolios, and his paper was published in Italian in an obscure actuarial journal. It remained unknown to English-speaking economists for over six decades. Academic recognition sometimes depends as much on language and venue as on ideas.
This doesn't diminish Markowitz's contribution. He developed the theory independently, worked out many extensions, and—crucially—brought it to the attention of the financial world. But de Finetti's priority is now acknowledged in the historical record.
The Lasting Impact
Despite its limitations, Modern Portfolio Theory fundamentally changed how people think about investing. Before Markowitz, diversification was folk wisdom—sensible but vague advice without rigorous foundation. After Markowitz, it became a mathematically precise concept with clear implications.
The theory provides a language for discussing risk and return trade-offs. It explains why and how diversification works. It offers a framework for thinking about what an optimal portfolio might look like, even if the precise optimum is unknowable.
Most importantly, it shifted attention from individual securities to portfolios as a whole. The relevant question is no longer "is this a good investment?" but rather "how does this investment interact with everything else I own?"
This perspective has permeated financial practice. Portfolio managers routinely analyze correlations and optimize allocations. Financial advisors frame recommendations in terms of risk tolerance and efficient frontiers. Even everyday investors implicitly apply Markowitz's ideas when they spread their retirement savings across different asset classes.
Your grandmother's advice about eggs and baskets was always sound. Harry Markowitz just told us exactly how many baskets we need, and how many eggs should go in each.