Risk-Budgeted Portfolios: Scale Invariance and Long-Short Constraints

Summary

The document asks how a common equal-risk-contribution (ERC) objective compares with a formulation that maximizes the sum of log weights under a unit variance constraint. The first method computes each asset’s share of portfolio variance, then minimizes squared deviations from target shares. The alternative is presented as a way to avoid a fully invested constraint, which can be unsuitable for long-short futures portfolios because net capital allocation does not describe their exposure in the same way it does for a long-only cash portfolio.

The discussion highlights a key modeling issue: risk contributions are unchanged when all weights are scaled together, so the share-based objective alone does not determine portfolio scale. A normalization or other constraint is needed for a well-defined solution. The log-weight formulation also presumes positive weights, so it is not directly a general long-short solution. The document poses these questions but does not provide an answer, derivation, or empirical comparison; equivalence depends on assumptions and constraints beyond what is included here.

Key ideas

• ERC targets allocate specified shares of total portfolio variance across assets.
• A fully invested constraint fixes net weights, which may not represent exposure in a long-short futures portfolio.
• Risk-share objectives are scale-invariant and need a separate normalization to determine portfolio size.
• A log-weight objective requires positive weights and does not directly cover arbitrary short positions.
• The document raises the equivalence question but does not resolve it.

Tags

# Equivalent constructions of risk budgeted portfolios?


# Equivalent constructions of risk budgeted portfolios?












When building an ERC portfolio given covariance $S$, I am familiar with the approach in optimalPortfolio. It takes the risks of a given weight vector $(w_i * (S w)_i)$, divides it by total risk to get the vector of current budget $Rb = (w_i * (S w)_i) / w^T S w$ and it minimizes the objective $\sum (Rb - b)^2$ where $b$ is your target budget (e.g. the vector of $1/n$ if you want ERC)

That is very intuitive, however, when reading this article Equal Risk Contribution Portfolios it mentions on page 4 near the bottom:

> Finally, the investor must think carefully about her various and possibly conflicting goals when deciding on a portfolio layer. It’s typical to see both the MV and ERC portfolios written with a “fully invested” constraint, i.e.∑iwi= 1.This requirement may or may not make sense; for a long-short futures portfolio, for example, it is not appropriate.

And subsequently it says the portfolio objective for the ERC becomes

$$\text{max}_w \sum \log w_i \text{ subject to } w^T S w \leq 1$$

Why might it be saying "fully invested" for long short futures portfolio not appropriate? Why the need for this second formulation (is it equivalent?), what is the limitation with the former specification?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.