6. Asset weighting: from 1/N to risk parity

Level 7: Linear algebra and portfolio optimization

Splitting the money into equal parts, estimating absolutely nothing, turns out to be surprisingly hard to beat.

Given what the previous module showed, it is fair to ask whether it is worth solving, in practice, the full Markowitz problem with estimated expected returns. Or whether, on the contrary, there are simpler alternatives that are more robust to estimation error.

The answer, surprisingly, is not obvious, and it has generated one of the most persistent debates in modern quantitative management.

The equal-weighted portfolio

The simplest possible starting point is the equal-weighted portfolio, known as 1/N. The rule could not be more direct:

• If you have 10 assets, each weighs 10%.
• If you have 500, each weighs 0.2%.

It requires estimating absolutely nothing, neither expected returns nor a covariance matrix.

That makes it immune, by construction, to the estimation error problem described in the previous module.

The uncomfortable result of the DeMiguel study

There is an influential study by Victor DeMiguel and several co-authors which compared 1/N against various versions of optimized portfolios across multiple markets and historical periods.

Its result was uncomfortable for the defenders of pure optimization. Out of sample, that is, evaluating the outcome on data later than the data used to compute the weights, 1/N matched or beat the optimized portfolios on measures such as the Sharpe ratio, in most of the cases analyzed.

The most widely accepted explanation is not that Markowitz is mathematically ill-posed. It is that the estimation error in its inputs, especially in the expected returns, destroys in practice much of the theoretical advantage the model should offer over such a simple rule.

Risk parity: equalizing risk, not capital

At the other end of the complexity spectrum sits risk parity.

Its central idea differs from 1/N. Instead of equalizing the capital invested in each asset, it seeks to equalize each asset's contribution to total risk.

This forces you to use each asset's volatility, which, unlike expected returns, can be estimated with relative reliability, as you saw in the previous module. But it deliberately uses no estimate of expected return.

Concretely, imagine that one asset has four times the volatility of another. In the simplest version of the scheme, risk parity will assign it roughly a quarter of the capital weight of the less volatile asset.

The aim of that split is that both end up contributing a similar amount of risk to the portfolio total, even though the capital invested in each is very different.

Why risk parity almost always uses leverage

This has a practical consequence worth anticipating, namely that risk parity systematically overweights lower-volatility assets, typically bonds over equities.

As a result of that bias, the expected return of the resulting portfolio, without any further adjustment, tends to fall below what many investors need to meet their objectives.

That is why, in real implementations, risk parity strategies almost always incorporate leverage: money is borrowed to increase the portfolio's total exposure, keeping the same relative proportions among assets.

The absolute expected return thus returns to competitive levels without giving up the balanced distribution of risk.

But that introduces, in turn, a new risk that the logic of risk parity does not itself capture: leverage risk and margin calls during episodes of market stress, something that became especially visible during March 2020 and other liquidity crisis episodes.

The map of the three approaches

Comparing these three approaches, 1/N, risk parity and full Markowitz, leaves a useful map of what is gained and lost with each.

1/N uses no information at all about risk or expected return. That makes it extremely robust to estimation error, but also blind to real and known differences between assets: treating a high-volatility stock the same as a low-volatility one, with no adjustment whatsoever, does not always make economic sense.

Risk parity uses risk information, that is, volatilities and, in fuller versions, correlations too, but it deliberately gives up any view on which asset will perform better.

That makes it robust to the most severe problem from the previous module, in exchange for not trying to maximize return in any sense, only distributing risk in a balanced way.

Full Markowitz uses all the available information, both risk and expected return, and is therefore, in theory, the optimal approach. But it is also the most fragile in the face of errors in those inputs.

The intermediate points on the map

Between those extremes there are intermediate approaches that occupy points of their own on this same map of trade-offs between robustness and use of information:

• The minimum variance portfolio, which uses only the covariance matrix, with no expected returns, but does actively seek to minimize risk rather than merely distribute it.
• The maximum diversification portfolio.