3. Markowitz and the efficient frontier

Level 7: Linear algebra and portfolio optimization

In 1952 Harry Markowitz published an idea that today seems obvious, and that at the time was not obvious at all.

Investors were already doing it intuitively, but nobody had formalized it: it makes no sense to evaluate an asset in isolation, but rather how it fits into a complete portfolio.

From "diversify well" to a solvable problem

His contribution was to translate that intuition into a concrete, solvable mathematical problem.

The starting ingredients are three, and you already know all of them:

• A set of assets.
• Their expected returns.
• Their covariance matrix.

With that on the table, the question is: which combination of weights maximizes expected return for a given level of risk?

Or, equivalently, which combination minimizes risk for a given expected return? Both questions have the same answer. Only which variable you fix and which you leave free changes.

The cloud of all possible portfolios

The most intuitive way to understand it is geometric, without touching the algebraic machinery behind it.

Imagine a chart with risk, that is, portfolio volatility, on the horizontal axis and expected return on the vertical axis. Each point on that chart is a possible portfolio: a different combination of weights across your assets.

If you compute thousands of random weight combinations and plot them all, you get a cloud of points with a characteristic shape, resembling a crescent moon or the tip of a bullet pointing to the left.

The efficient frontier is the edge of that cloud

The upper-left edge of that cloud is the efficient frontier. That edge gathers exactly two things, which turn out to be the same set of portfolios:

• Those that, for each level of risk, offer the highest possible expected return.
• Those that, for each level of return, offer the lowest possible risk.

From that comes the important consequence. Any portfolio not on that edge is, by definition, improvable: there is another combination of weights that gives the same with less risk, or more with the same risk.

The leftmost point of them all

There is one point in the cloud that deserves a name of its own. It is the leftmost point of the whole cloud, where risk is as low as possible regardless of return, and it is the global minimum variance portfolio.

It has a peculiarity that will prove decisive later on. It is computed using only the covariance matrix, with no need to estimate expected returns.

That makes it, as you will see in a later module, more robust than the other points on the frontier against estimation error.

Why the cloud curves to the left

You already saw in the theory course why combining two assets with a correlation below 1 reduces risk below the simple weighted average of their individual risks.

Geometrically, that effect is precisely what curves the cloud of portfolios toward the left.

Plot only the combinations between two specific assets and check it. Instead of a straight line joining their two individual points, you get a curve that bends leftwards, the more so the lower the correlation between them.

With many assets at once the same thing happens, but multiplied across every possible pair in the covariance matrix, and that is what shapes the entire cloud and its frontier.

The tangency portfolio and the capital market line

There is one especially important point on the frontier, even though it is neither the lowest-risk one nor an arbitrary one: the tangency portfolio, the one that, combined with the risk-free asset (Treasury bills, in practice), maximizes the Sharpe ratio you already know from the theory course.

Draw a straight line from the risk-free asset's point to the point where that line is tangent to the efficient frontier, and you get what is called the capital market line.

Combining the risk-free asset with that tangency portfolio in different proportions moves you along that straight line, not along the curve. You obtain different levels of risk and return, but always keeping the same Sharpe ratio, the highest available in the whole asset universe.

It is an important result because it splits the problem into two independent decisions:

• Which combination of risky assets to hold, which is always the tangency portfolio, for any investor.
• How much weight to put in that combination versus cash or risk-free debt, which does depend on each investor's risk tolerance.

Minimum variance is not "the best" portfolio

It is important not to confuse the minimum variance portfolio with "the best" portfolio outright. It is the best only for someone who will not tolerate any additional risk in exchange for more return.

Most investors have different levels of risk tolerance. They will therefore choose other points on the frontier, further to the right, accepting more volatility in exchange for higher expected return.

And here is the final nuance about what all this really guarantees. Markowitz does not tell you which exact point is right for you.

It guarantees that, whatever risk you are willing to take on, there is a point on the frontier that gives you the highest possible expected return for that risk. And it guarantees that any portfolio off the frontier is, quite simply, a worse choice that could be improved without taking on more risk.