Efficient frontier — Markowitz, the gentle intro

The map of all possible portfolios

Here's a thought experiment. Imagine all the ways you could combine N assets in different proportions. Each combination has a particular expected return and a particular σ. Plot every possible combination on a chart with σ on the x-axis and expected return on the y-axis.

You don't get a line. You get a cloud of dots — every possible portfolio. Some have high return for low σ (great); some have low return for high σ (terrible); most fall somewhere in between.

The efficient frontier is the upper-left edge of that cloud. It's the set of portfolios where, for any given σ, you can't get a higher expected return without taking more σ. Or equivalently: for any given expected return, you can't get less σ without giving up some return.

Anything not on the frontier is dominated by something on the frontier (same σ, more return; or same return, less σ). So if you're going to invest, you'd want to be on the frontier.

Why this exists at all — the diversification math

If correlation between assets were always +1, the cloud would be a straight line: combinations of two assets give returns and σ's that are weighted averages of the components. No frontier shape, no improvement possible by mixing.

But correlations are usually below +1. So when you mix two assets, σ goes down faster than the weighted average suggests (the math from the last two lessons). The cloud bulges to the upper-left. The bulge is the diversification benefit visualized. The frontier is the bulge's edge.

Add more assets, and the bulge grows. With enough assets, the frontier curves smoothly — the curve famously named after Markowitz, who formalized this in 1952 (and won a Nobel for it in 1990).

What "minimum-variance portfolio" means

One special point on the frontier: the minimum-variance portfolio. The single combination with the lowest σ across all possible weight choices. It's the leftmost point on the frontier — least shake possible from these assets.

Below the minimum-variance point, the frontier curls back on itself (you can have less return at the same σ — but that's not "efficient," it's "stupid"). So we usually trim the frontier to start at the minimum-variance point and go up-right from there.

The whole upper section is the efficient frontier. The minimum-variance point is where it starts.

What the frontier doesn't tell you

The frontier shows you all efficient combinations. It doesn't tell you which efficient combination is right for you. That depends on:

• Your risk tolerance — how much σ you can stomach
• Your time horizon — long-term investors can tolerate more σ
• Your goals — capital preservation vs. growth vs. income
• Whether you can also hold cash or borrow at a risk-free rate (Track 9 expands this)

Two investors with different risk tolerances will pick different points on the same frontier. Both are right for their situation. The frontier just tells you what's possible.

The takeaway

Plot every possible portfolio's expected return vs. σ. You get a cloud. The upper-left edge is the efficient frontier — portfolios where you can't get more return without more σ, or less σ without less return. It exists because correlations between assets are usually less than +1, which lets diversification reduce σ. The frontier's leftmost point is the minimum-variance portfolio. Track 9 expands this with risk-free assets and CAPM. Right now, just see the picture.