The efficient frontier · RiskAtoms Academy

By the end you will be able to

1Describe the feasible set of portfolios for two assets and for three or more.
2Decide whether one portfolio dominates another, and pick out efficient and inefficient portfolios.
3Locate the minimum-variance point and explain why everything below it is inefficient.
4Explain the shape of the efficient frontier: a hyperbola with short sales, concave in every case.

Why it matters

A mix of 40% equities, 30% government bonds and 30% property expects 5.65% at 10.70% volatility. Another mix of the same three assets earns 5.84% at the same risk; another gives the same return with only 9.87%. Both hold almost no property. The first mix is not wrong, but it is not efficient.

Core explanation

The best you can do, for each level of risk

The feasible set

The feasible set is every point on the risk-return chart that some portfolio can reach. With two assets it is just the curve of L3.1.03: Two correlated assets. With three it becomes a region: mix two portfolios on one curve with a portfolio on another, and you land between them. The chart below shows 1,500 random long-only mixes of global equities, government bonds and property. Together they fill a region with a sharp left edge.

random long-only mixestwo-asset curvesefficient frontier

Illustrative universe. Each grey dot is one random mix of the three assets; the thin lines are the three two-asset curves. No dot lies above the blue line.

Efficient and inefficient

Investors like expected return and dislike volatility. So portfolio P dominates Q if it offers at least as much return for no more risk, and is strictly better on one of the two. A portfolio is efficient if nothing dominates it. Look at the cloud: from any dot you can move straight up (more return, same risk) or straight left (same return, less risk) until you hit the edge. Only the upper-left edge survives. That edge is the efficient frontier.

The whole left edge, top and bottom, is the minimum-variance frontier: for each expected return, the least volatile portfolio. Its leftmost point is the minimum-variance portfolio of L3.1.04: The minimum-variance portfolio. Everything on the lower half is inefficient: the point straight above it on the upper half has the same risk and more return. All-bonds is the clearest case: the minimum-variance mix has 5.41% volatility against 6%, and 4.01% expected return against 3.5%.

What the frontier does not say

It does not say which efficient portfolio to hold. Moving up the frontier buys return with risk, and the right trade-off depends on the investor's appetite for risk (L2.3.05: Risk appetite frameworks). It also depends entirely on the inputs: expected returns, volatilities and correlations that are estimates, not facts. L3.1.08: The risk-free asset and the capital market line adds a risk-free asset, which changes the answer sharply.

The shape

It bends. Near the minimum-variance point, a little more risk buys a lot more return. Higher up, each extra point of return costs more and more risk. The reason is diversification itself: mixing two frontier portfolios gives a volatility at most the weighted average of theirs (L3.1.01: Portfolio return and risk), so the frontier can never bulge to the right of a straight chord.
More assets can only push it out. Every two-asset mix is still available, so the three-asset frontier is never to the right of a two-asset curve. How far a new asset moves it depends on what it offers. Here property, with a modest return and a 0.6 correlation with equities, barely moves it: the frontier hugs the equities-and-bonds curve and never holds more than 3.5% property.
Constraints pull it in. Without short sales the frontier stops at the asset with the highest expected return, here global equities. With short sales it runs on forever, as a hyperbola, and lies to the left of the long-only one.

Sort itEfficient or not?

Long-only mixes of global equities, government bonds and property. Decide, then check.

All government bonds: 6.00% volatility, 3.50% expected return.
All global equities: 16.00% volatility, 7.00% expected return.
All property: 18.00% volatility, 6.00% expected return.
60% global equities, 40% bonds: 9.66% volatility, 5.60% expected return.
40% equities, 30% bonds, 30% property: 10.70% volatility, 5.65% expected return.
The long-only minimum-variance mix: 5.41% volatility, 4.01% expected return.

The formal version

Write $\mu(w) = w'\mu$ and $\sigma(w)$ for the expected return and volatility of weights $w$ (the vector notation is built in L3.1.06: Many assets and the covariance matrix). The allowed weights satisfy $w'\mathbf 1 = 1$ , and $w \ge 0$ if short sales are banned.

\begin{aligned} \text{Feasible set:}\quad & \mathcal F = \{\,(\sigma(w),\ \mu(w)) : w \text{ allowed}\,\} \\ \text{Dominance:}\quad & P \succ Q \iff \mu_P \ge \mu_Q,\ \sigma_P \le \sigma_Q, \text{ at least one strict} \\ \text{Minimum-variance frontier:}\quad & \sigma_{\min}(m) = \min_w \{\,\sigma(w) : \mu(w) = m,\ w \text{ allowed}\,\} \\ \text{Efficient frontier:}\quad & \{\,(\sigma_{\min}(m),\ m) : m \ge m_{\text{mv}}\,\} \end{aligned}

where $m_{\text{mv}}$ is the expected return of the minimum-variance portfolio.

Why the frontier is concave. Take two efficient portfolios $a$ and $b$ and mix them, $t$ and $1-t$ . The mix is allowed (with or without short sales), and by L3.1.01: Portfolio return and risk:

\begin{aligned} \mu_{\text{mix}} &= t\,\mu_a + (1-t)\,\mu_b && \text{on the chord} \\ \sigma_{\text{mix}} &\le t\,\sigma_a + (1-t)\,\sigma_b && \text{on or left of the chord} \\ \sigma_{\min}(\mu_{\text{mix}}) &\le \sigma_{\text{mix}} && \text{the frontier is at least as far left} \end{aligned}

So the frontier always lies on or to the left of the chord between any two of its points: as a function $\sigma_{\min}(m)$ it is convex, and its upper half, read as return against risk, is concave.

The exact shape with short sales. L3.1.07: Mean-variance optimisation derives the frontier for any number of assets: $\sigma_{\min}^2(m) = (C m^2 - 2A m + B)/D$ , where $A, B, C$ and $D = BC - A^2 > 0$ are four numbers built from the expected returns and the covariance matrix. Completing the square shows the shape:

\begin{aligned} \sigma^2 &= \frac{C}{D}\Big(m^2 - \frac{2A}{C}m\Big) + \frac{B}{D} \\ &= \frac{C}{D}\Big(m - \frac{A}{C}\Big)^2 - \frac{C}{D}\cdot\frac{A^2}{C^2} + \frac{B}{D} = \frac{C}{D}\Big(m - \frac{A}{C}\Big)^2 + \frac{BC - A^2}{CD} \\ &= \frac{C}{D}\Big(m - \frac{A}{C}\Big)^2 + \frac{1}{C} \\ \Longleftrightarrow\quad & \frac{\sigma^2}{1/C} - \frac{(m - A/C)^2}{D/C^2} = 1 \end{aligned}

That is a hyperbola. Its vertex, the minimum-variance portfolio, is at $\sigma = 1/\sqrt C$ , $m = A/C$ . Far from the vertex it approaches two straight lines through $(0, A/C)$ with slopes $\pm\sqrt{D/C}$ . For our three assets:

A = 13.736, \quad B = 0.5901, \quad C = 342.68, \quad D = 13.536

\text{vertex: } \sigma = 5.40\%,\ m = 4.01\%; \qquad \text{asymptote slope } \sqrt{D/C} = 0.1987

The slope is a Sharpe-like ratio: far up the frontier, each extra point of volatility buys at most about 0.20 points of expected return. Without short sales the frontier is made of pieces of such hyperbolas, joined at corner portfolios where an asset enters or leaves.

Try it

Start at 60% equities and 40% bonds: it sits on the frontier. Now move 30 points into property and press “Same risk, more return”: see where the property goes. Then try all bonds and all property, and switch on the frontier with short sales to see how little a ban on short sales costs here.

Try itWhere does your mix sit?

Global equities60%

Property0%

Government bonds: 40% (the rest). No short sales.

Your mix

Efficient

Volatility9.66%

Expected return5.60%

No long-only mix of these three assets has a higher expected return for this volatility (within 0.03 points).

efficient frontier (long-only)lower, inefficient halfrandom mixesyour mix

Worked example

Two inefficient portfolios, and what beats them

All bonds against the minimum-variance mix (two assets)

Step 1 of 3

1
The minimum-variance mix
From L3.1.04: The minimum-variance portfolio: 14.65% in global equities, volatility 5.41%, expected return 4.01%.

A property-heavy mix against the three-asset frontier

Step 1 of 4

1
Where it sits
40% global equities, 30% bonds, 30% property: $\sigma = 10.70\%$ , $\mu = 5.65\%$ . Deep inside the cloud of the first chart.

Portfolio	Volatility	Expected return	Efficient?
All government bonds	6.00%	3.50%	No
All global equities	16.00%	7.00%	Yes
All property	18.00%	6.00%	No
60% global equities, 40% bonds	9.66%	5.60%	Yes
40% equities, 30% bonds, 30% property	10.70%	5.65%	No
The long-only minimum-variance mix	5.41%	4.01%	Yes

The long-only minimum-variance mix holds 14.7% global equities, 85.3% government bonds, 0.0% property: property adds nothing at the lowest risk.

Regulatory lens · looking ahead

No rule requires a firm to sit on the frontier. In practice, insurers and pension funds draw frontiers when they set a strategic asset allocation, but measure risk against their liabilities (the surplus) and often in terms of the regulatory capital each mix uses. That asset-liability view comes in M3.6: Interest rate risk in the banking book and ALM, and how Solvency II capital depends on the assets held comes in M4.6: Solvency II and insurance regulation.

Spark Atoms

Spark AtomTwo discoveries in one year

In 1952, the same year as Markowitz's “Portfolio Selection”, A. D. Roy published “Safety First and the Holding of Assets” in Econometrica, which also traded off the mean against the variance of a portfolio, with correlations. Markowitz later wrote that Roy deserved a share of the credit for the idea.

Spark AtomEfficient is not the same as good

Every point on the frontier is efficient, including a portfolio that is 100% global equities. Efficiency only rules portfolios out; it never rules one in. Choosing among efficient portfolios needs a view on how much risk to take.

Key takeaways

The feasible set is every (volatility, return) point some allowed portfolio reaches: a curve for two assets, a region for three or more.
P dominates Q if it has at least the return for no more risk, and is better on one. Efficient portfolios are those nothing dominates.
The minimum-variance frontier is the left edge of the feasible set; its upper half, from the minimum-variance point up, is the efficient frontier.
The frontier is concave because mixing never adds risk beyond the weighted average. With short sales it is a hyperbola with vertex (1/√C, A/C).
With these inputs property barely earns a place: a 40/30/30 mix with 30% property is beaten by 5.84% at the same 10.70% risk.

Reaction

Check what stuck

Glossary

Feasible set: Every combination of volatility and expected return that some allowed portfolio reaches.
Dominance: Portfolio P dominates Q if P has at least Q's expected return and at most Q's volatility, and is strictly better on one of the two.
Efficient portfolio: A portfolio that no other allowed portfolio dominates.
Efficient frontier: The set of efficient portfolios: the upper-left edge of the feasible set, from the minimum-variance point upwards.
Minimum-variance frontier: For each expected return, the portfolio with the lowest volatility. Its upper half is the efficient frontier.
Corner portfolio: A point on a constrained frontier where an asset enters or leaves the portfolio. Between corners the weights change in straight lines.

Deep Dives & Origins

Deep Dives

H. M. Markowitz, Portfolio Selection: Efficient Diversification of Investments, Wiley, 1959 (Cowles Foundation Monograph 16). Advanced.
Z. Bodie, A. Kane and A. J. Marcus, Investments, McGraw-Hill, the chapter on optimal risky portfolios. Intermediate.

Origins

H. Markowitz, “Portfolio Selection”, Journal of Finance, 7(1), 1952, pp. 77–91.
A. D. Roy, “Safety First and the Holding of Assets”, Econometrica, 20(3), 1952, pp. 431–449.
H. M. Markowitz, “The Early History of Portfolio Theory: 1600–1960”, Financial Analysts Journal, 55(4), 1999.
R. C. Merton, “An Analytic Derivation of the Efficient Portfolio Frontier”, Journal of Financial and Quantitative Analysis, 7(4), 1972.

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