The efficient frontier

Chapter 4 produced a region of achievable portfolios. This chapter throws most of it away.

Dominance

Portfolio A dominates portfolio B if

$E(R_A) \ge E(R_B) \quad \text{and} \quad \sigma_A \le \sigma_B$

with at least one holding strictly. If A dominates B, nobody with any attitude to risk should hold B — they could have more return for no more risk, or the same return for less.

That is a remarkably strong claim and it costs almost nothing to make. It requires no utility function, no view about the investor, no assumption about risk appetite. It needs only that people prefer more return to less and less risk to more.

A portfolio nothing dominates is efficient. The set of efficient portfolios is the efficient frontier, which Markowitz describes as "the set of Pareto optimal expected return, variance of return combinations."

Where the frontier is

On a chart with risk across and return up, the frontier is the upper-left boundary of the achievable region.

Everything in the interior is dominated — move straight up to the boundary for more return at the same risk, or straight left for less risk at the same return.

And the lower half of the boundary is dominated too. This is the part people miss. Chapter 4's curve bends back on itself: below the minimum-variance point, adding more of the safe asset increases risk while reducing return. From chapter 4's numbers, the all-debt portfolio at 7.000% and 4.000% is beaten by the 2.53% equity portfolio at 7.101% and 3.985%.

So the minimum-variance portfolio is where the frontier starts. Everything to its lower right on the curve is inefficient, and the efficient frontier is the arc from that point upward.

The frontier is concave, bending down as it rises. Each additional unit of return costs more risk than the one before, which is chapter 4's accelerating third column stated as a property of the boundary.

What the frontier does not tell you

Three things, and each corrects a common over-reading.

It does not pick a portfolio. The frontier is a menu. Choosing among its points requires something the mathematics does not contain — how much risk this particular investor should take, which is the Risk subject's capacity-and-tolerance question, not a computation.

It does not say the inefficient portfolios are bad holdings for a person. Dominance compares two portfolios on two axes. It does not know that you might need the money in eighteen months, or that you will sell at the bottom if the portfolio halves. Chapter 12 returns to this.

It does not exist until someone estimates it. There is no observable efficient frontier. There is a frontier implied by a particular set of estimated means, variances and covariances, which is a different object.

Constraints bend it

The clean frontier assumes you can hold any weights, including negative ones.

Most individuals cannot. Shorting requires facilities and margin, and in India an individual cannot hold a short equity position beyond the trading day without using derivatives, with the lot sizes and margins the Option pricing subject sets out.

Imposing $w_i \ge 0$ shrinks the achievable region, so the constrained frontier lies below and to the right of the unconstrained one. Chapter 4's table showed one case directly: at a correlation of 0.8 the minimum-variance portfolio wanted an equity weight of −0.333. An investor who cannot short simply cannot reach that point, and their best available minimum-variance portfolio is worse.

Other constraints do the same. A cap on any single holding, a floor on liquid assets, a prohibition on certain sectors — each one removes portfolios from the region and can only move the frontier adversely.

The useful habit is to ask what a constraint costs, in points of return at the risk you want. Sometimes the answer is negligible and the constraint is free. That is a computation worth doing rather than an argument worth having.

The problem the frontier has

Here is the objection that matters most, and the one practitioners discovered the hard way.

Chapter 4 counted the inputs: 1,325 estimates for a 50-asset problem, 125,750 for 500. Every one is estimated from a finite sample and carries error.

An optimiser does not know which inputs are reliable. It finds the weights that look best given the numbers it was handed — and the numbers that look best are disproportionately the ones whose errors happened to be favourable. An asset whose return was overestimated and whose covariances were underestimated looks like a free lunch, so the optimiser loads up on it.

The consequences are specific and recognisable. Unconstrained mean-variance optimisation on historical inputs characteristically produces portfolios that are extremely concentrated, that change violently when the estimation window shifts by a month, and that perform worse out of sample than an equally weighted portfolio of the same assets.

The error is largest in the means. Variances and covariances can be estimated tolerably from a few years of data; expected returns cannot. The frontier's vertical position is the least reliable thing about it, which is unfortunate, because that is what the choice between its points depends on.

So the frontier should be read as a shape rather than a map. It tells you truthfully that risk and return trade off, that the trade is better at the left end, and that combinations beat components. It does not reliably tell you that a particular set of weights is optimal.

And the evidence for that caution is in the data this subject has been using. The Nifty 50 returned 6.38% a year with 13.82% volatility over five years, and 12.12% with 22.39% since inception. An optimiser fed the first pair would barely hold equities; fed the second, it would hold little else. Same asset, same country, two defensible windows, opposite answers — and nothing in the method warns you which window to use.

Working the problem

All debt: 7.000% return, 4.000% risk. The 2.53% equity portfolio: 7.101% and 3.985%.

Why all-debt is inefficient. The second portfolio has a higher expected return and a lower standard deviation. It dominates on both axes simultaneously, so by the definition above, all-debt is not on the efficient frontier. No investor who accepts the inputs should prefer it — the alternative is better on every dimension the framework measures.

What that does imply. Within this model, with these three estimates, holding no equity at all is strictly worse than holding a sliver. The conclusion is secure conditional on the inputs, and it does not depend on the investor's risk appetite — which is what makes dominance arguments powerful.

What it does not imply — four things.

Not that the gain is worth acting on. It is 0.101 points of return and 0.015 of risk. Rebalancing costs, as chapter 11 shows, are not zero, and a benefit this small can be entirely consumed by them.

Not that the inputs are right. The whole conclusion rests on the 0.2 correlation and the 11% equity expectation. At a correlation of 1.0 the result disappears. The dominance is a fact about the estimates, not about the world, and the estimates are the weak part.

Not that the person should hold equity at all. Someone who needs the money in a year, or who would sell after a fall, faces costs this model does not represent. The Risk subject's distinction between capacity and tolerance is exactly this gap, and chapter 12 argues it is the theory's deepest limitation rather than an application note.

And not that "inefficient" means "foolish". A portfolio can be inefficient in the mean-variance sense and perfectly sensible for its owner — an emergency fund held entirely in liquid instruments is dominated by some mixture containing a little equity, and should still be held entirely in liquid instruments, because its job is to be available on a bad day rather than to be efficient.

The honest summary. Dominance is the strongest argument in the subject, because it needs almost no assumptions about the investor. That strength is exactly why its weakness sits elsewhere — entirely in the inputs.

The point

A portfolio is dominated if another offers at least as much return with no more risk, and the undominated ones form the efficient frontier — a result that needs no assumption about the investor beyond preferring more return and less risk. The frontier runs from the minimum-variance portfolio upward; everything inside the region and everything below that point is dominated. Constraints that individuals actually face, such as being unable to short, shrink the achievable set and can only make the frontier worse. The real difficulty is that the frontier is estimated: the inputs grow with the square of the asset count, expected returns are the least reliable of them, and an optimiser systematically overweights whatever the estimation errors flattered — which is why the same index gives 6.38% with 13.82% over five years and 12.12% with 22.39% since inception, and why the frontier is better read as a shape than as a map.