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What Are Portfolio Optimisation Models?

A column on how mean-variance optimisation and Black-Litterman approach the question of allocation weights

Building a portfolio, the question after what to hold is at what weights. Optimisation models are what appear when you want to think about those weights structurally rather than only by feel. Here we set out the two best-known ones — mean-variance optimisation and Black-Litterman — in terms an individual investor can read.

What you'll learn
  • Optimisation models are tools for comparing which weights are efficient, given the same products.
  • Mean-variance optimisation is the starting point of modern portfolio theory and draws the efficient frontier.
  • But it is sensitive to its assumptions and prone to producing extreme allocations.
  • Black-Litterman is a development that contains that extremity and makes more realistic allocations easier to reach.

What mean-variance optimisation is doing

Mean-variance optimisation compares allocation weights in pursuit of a higher return at the same risk and lower risk at the same return. It lines up candidates — what proportion of equity A and equity B, how to combine equities and bonds — and looks for the efficient combinations among them.

What emerges from this is the efficient frontier: the line joining, among the achievable allocations, the combinations with the highest expected return at each level of risk. For the main text of Chapter 4, grasping how to ask where on this line do I sit is enough.

Mean-variance optimisation — searching for efficient allocations
Among many possible allocations, it finds the combination with the highest return at a given level of risk (the efficient frontier). Grey dots are candidate allocations; the blue line is the frontier.
At the same risk, aim for a higher returnAllocations on the efficient frontier are, in theory, the most efficientRisk (size of movement) →Expected return →Efficient frontierThe most efficient mix at this level of riskSame risk, buta lower return

Why it is hard to use as-is

Mean-variance optimisation is elegant theory, but a slight change in the assumed expected returns or correlations can move the result substantially. So it can produce answers that are hard to adopt in practice, such as placing an extremely large weight in one holding.

Real markets also do not move as calmly as the theory. In falling markets, correlations can rise sharply and moves can be larger than assumed. So while mean-variance optimisation is useful as an elegant starting point, care is needed before treating it as a final answer.

Mean-variance optimisation is sensitive to assumptions
A slight change to your expected-return estimate can produce a mix concentrated in one holding. That is why, correct as it is in theory, it is hard to use as-is in practice.
A small change in assumptions moves the result a great dealChange the expected return by just 0.5% and the mix can tilt to an extremeOriginal assumptionsAs estimated35%30%25%Equity A35%Equity B30%Bonds25%Cash10%A slight changeA slight changeReturn adjusted by 0.5%70%20%Equity A70%Equity B5%Bonds20%Cash5%This is mean-variance optimisation's weakness: it is too sensitive to its assumptions

What Black-Litterman adds

Black-Litterman is a model designed to contain that tendency towards extreme allocations. The idea is to place the market's own allocation as the base, then reflect your own outlook only slightly on top. Because you do not decide everything from your own forecast from the outset, the allocation is less prone to jumping.

Black-Litterman is widely used by institutional investors and robo-advisers because this shape — adjust gradually from a base — fits practice. It is considered easier than mean-variance optimisation for arriving at a sensible allocation.

Black-Litterman builds an allocation from a base plus fine-tuning
Start from the whole market's allocation, then add a little of your own outlook. Because you do not decide everything by forecast from the outset, extreme tilts are less likely.
Black-Litterman — adding a little outlook on top of the market baseBecause you do not decide everything from your own forecast, the mix is less likely to jump1Start from the market allocationMarket-cap weights= the market's own viewAn unbiased starting point2Add your own outlook“US equities look a little strong”“Bonds may be weak”Reflected modestly3An adjusted allocationNo extreme tiltsRealistic weightsBase + fine-tuning = stabilityThis stability is why it is widely used by institutional investors and robo-advisers

How should an individual investor read this?

What an individual investor wants first is not memorising the equations. It is enough to sort them as mean-variance optimisation is the basic form for thinking about efficient weights and Black-Litterman is the development that contains its weakness. From there, when deciding an actual allocation, the natural flow is returning to the main text of Chapter 4 and asking which point on the efficient frontier matches your risk tolerance.

What the tool compares matters more than the model's name

Neither mean-variance optimisation nor Black-Litterman is a mechanism for predicting the future. They are easier to understand read as tools for asking which weights make more sense, given the same products.

Returning to the main text

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The terms appearing across the learning chapters, organised so you can look them up by chapter and category

On this page

What mean-variance optimisation is doingWhy it is hard to use as-isWhat Black-Litterman addsHow should an individual investor read this?Returning to the main text