Weight Shrinkage for Portfolios of Nearly Identical Assets
Summary
The document examines how portfolio methods allocate between two assets with almost identical risk characteristics and slightly different estimated returns. It contrasts mean-variance optimization, Black-Litterman, and the Resampled Efficient Frontier, asking how to favor a near-equal split when estimation and model error make a concentrated allocation undesirable. The accepted reply points to shrinkage applied directly to portfolio weights, rather than to expected returns, as a way to avoid the specific issue described for Black-Litterman. Another reply notes that truly indistinguishable assets should receive symmetric weights, since swapping their labels should not change the result.
The example and discussion illustrate sensitivity to small return differences and the importance of symmetry and estimation error in portfolio construction. They do not provide a full derivation, comparative empirical results, or a universally preferred allocation rule. The comments also distinguish algorithm behavior from investor preference: a near-equal allocation may reflect a robustness objective, while a method's output depends on its assumptions, constraints, and numerical implementation.
Key ideas
- Small differences in estimated returns can produce concentrated allocations between highly correlated assets.
- Shrinking portfolio weights directly can address an issue that arises when Black-Litterman shrinks expected returns.
- An allocation method should behave symmetrically when otherwise identical assets have their labels exchanged.
- The Resampled Efficient Frontier is described as tending toward an approximately equal split in this example, with other noted side effects.
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Full text
# Portfolio construction for almost identical assets
# Portfolio construction for almost identical assets
The problem I am looking at concerns the treatment of almost identical assets in portfolio construction.
Let us assume that we have two assets, both with a standard deviation $\sigma=0.2$ and a correlation matrix $$ C = \begin{pmatrix} 1 & 0.98 \\ 0.98 & 1 \end{pmatrix}. $$ One asset has a return of $\mu=0.101$ and the other has a return of $\mu=0.1$.
An argument usually employed in justifying portfolio construction models is that as the two assets are almost identical, the allocation outcome does not matter and for example putting the full (or a large) allocation into the asset with the slightly higher return is perfectly reasonable.
However, setting aside any additional costs that might arrive from investing in two assets instead of one, in terms of guarding against estimation and model error it might be preferable to invest half the total allocation into each asset.
For the example above, I am showing below the results for Black-Litterman 1, 2 and the Resampled Efficient Frontier by Michaud 3, with the Mean-Variance solution as a reference (labelled historical) on the left-hand for both charts.
Black-Litterman
For the prior I have used the Equal-Risk-Contribution portfolio, which I understand is a deviation from the conventionally used market-equilibrium portfolio. The ERC weights are naturally 50%-50% for the assets. However, as Black-Litterman shrinks to the returns implied by ERC and not to the weights, the outcome is not a 50%-50% split at each level of risk.
Resampled Efficient Frontier
Due to the nature of the sampling, the resampled efficient frontier will produce weights that will always split approximately 50%-50%. However, unfortunately the REF has some other undesirable side-effects 6, among them increasing the weight of high-volatility assets when using a long-only constraint and reducing to the Mean-Variance solution when allowing shorting.
My question is: Is there a way to arrive at the behavior of the Resampled Efficient Frontier in the special case of two almost identical assets with either Black-Litterman or some other portfolio construction methodology? Is there a fundamental point that I am overlooking in this?
1 The Intuition Behind Black-Litterman Model Portfolios 2 A Step-By-Step Guide to the Black-Litterman Model Incorporating User-specified Confidence Levels 3 Estimation Error and Portfolio Optimization: A Resampling Solution 6 Resampled Efficiency and Portfolio Choice
## Answer by Hans-Peter Schrei (score 1, accepted)
https://quant.stackexchange.com/a/75390
The approach that I eventually adopted to meet this criterion is, as I subsequently found out, close in spirit to the approach in Turnover Minimization: A Versatile Shrinkage Portfolio Estimator.
As the shrinkage is done on weights and not on returns as in Black-Litterman, the problem does not arise in this approach.
## Answer by André Bittencourt (score 0)
https://quant.stackexchange.com/a/73788
If you have only two assets, with the same volatility and return, regardless the correlation, they should split 50%/50% no matter the methodology used (may deviate due to numerical procedure).
Call the first asset A and the second B, your algorithm gives you 70%/30% weights. As the assets are (for this algorithm) indistinguishable, you can rename then B and A. Now, should your algo gives the opposite weights just because you rename them?
I might get you wrong, but it seems your problem is ill-proposed, doesn't it?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.