Visualizers
Efficient Frontier Explorer
Mix two assets and watch the risk-return frontier bend as you change their correlation, the free lunch of diversification, made visual.
Efficient frontier
Mixing two investments that don't rise and fall in perfect lockstep lets you lower your overall risk without giving up much return, that is why the curve bends to the left. The further the correlation sits from +1, the more the curve bows out and the bigger this “free lunch” becomes; at +1 the two assets move together, there is no benefit, and the frontier collapses to a straight line. The single leftmost point is the minimum-variance mix, the blend with the smallest possible swings. The solid stretch is “efficient” (the most return for each level of risk); on the dashed stretch you could earn more for the same risk, so no one should sit there.
Illustrative two-asset model. Educational only, not investment advice.
Learn how it works
Five worked examples. Read a couple before you dive in, try to answer first, then reveal the solution.
Perfect correlation, no free lunch
Give two assets different risk and return, then drag the correlation ρ all the way to +1. Look at the shape of the curve connecting them.
Show solution
The 'curve' collapses into a straight line between the two assets.
When ρ = +1 the assets move in lockstep, they rise and fall together, always. Mixing them just blends their risk and return in a plain, proportional way.
- There's no bend, no shortcut, no benefit to diversifying.
- A 50/50 blend has exactly the middling risk you'd guess.
This is the one case where diversification does nothing, and it's the baseline everything else improves on.
The bow to the left
Now set correlation ρ = 0.2 and watch the line. It stops being straight.
Show solution
The curve bows to the left, toward less risk.
Because the assets no longer move perfectly together, when one zigs the other sometimes zags, and those offsetting moves cancel out some of the bumps.
- Pick a point on the bowed curve and compare it to the straight line: same return, less risk.
- That saved risk, gained just by blending, is the closest thing investing has to a free lunch.
This leftward bow is the entire reason people hold more than one thing.
Lower correlation bends it more
Slowly drag ρ down from +1 toward 0 and into negative territory. Watch how far the curve bends and where its leftmost tip lands.
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The lower the correlation, the more the curve bends and the further left its tip reaches.
- High ρ: nearly straight, little benefit.
- Low or negative ρ: deep bow, and the lowest-risk blend gets safer and safer.
Correlation is the dial that controls how much diversification helps. Assets that don't move together are worth far more as a pair than their individual risks would suggest, that's the whole magic.
Perfect hedge at ρ = −1
Push correlation ρ all the way to −1 and look for the point where the curve nearly touches the zero-risk edge.
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At ρ = −1 the two assets are mirror images, whenever one goes up by a step, the other goes down. With the right mix, their swings cancel almost completely.
- The curve reaches nearly to a vertical, near-zero-risk line.
- You can build a blend whose ups and downs almost perfectly offset, leaving a smooth ride.
This is the idealized perfect hedge. Real assets never hit exactly −1, but it shows why negatively-related holdings are so prized.
Finding the calmest blend
On any bowed curve, find the minimum-variance point, the leftmost tip that the tool marks for you.
Show solution
The leftmost point is the blend with the lowest possible risk you can build from these two assets.
- Everything on the curve to the right of it takes on more risk.
- It usually isn't 100% of the safer asset, a dash of the riskier one, moving differently, can actually lower total risk.
That counterintuitive result is diversification in one picture: adding a risky asset can make the whole portfolio calmer, as long as it doesn't move in step with what you already hold.
What you'll learn
Why combining imperfectly-correlated assets lowers risk, where the minimum-variance mix sits, and what Markowitz's efficient frontier really shows.