## Abstract

A portfolio-resampling procedure invented by Richard and Robert Michaud is a subject of highly controversial discussion and big scientific dispute. It has been evaluated in many empirical studies and Monte Carlo experiments. Apart from the contradictory findings, the Michaud approach still lacks a theoretical foundation. I prove that portfolio resampling has a strong foundation in the classic theory of rational behavior. Every noise trader could do better by applying the Michaud procedure. By contrast, a signal trader who has enough prediction power and risk-management skills should refrain from portfolio resampling. The key note is that in most simulation studies, investors are considered as noise traders. This explains why portfolio resampling performs well in simulation studies, but could be mediocre in real life.

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## Notes

- 1.
The Michaud approach is a US patented procedure with patent number 6,003,018.

- 2.
In the following, I use both terms “investor” and “trader” synonymously.

- 3.
Here it is implicitly assumed that the search costs for the information \(\mathcal {I}\) are negligible.

- 4.
Here \(\bar{w}\) denotes the

*theoretical*expectation of \(\tilde{w}\) but not some corresponding approximation or estimate. - 5.
Here the mean portfolio \(\bar{w}_t\) is a function of time, since the distribution of \(\tilde{w}_t\) might depend on \(t\in \mathbb {Z}\,\).

- 6.
In Sect. 4.2 I discuss the apparent link between the random-walk and the efficient-market hypothesis.

- 7.
In most publications, \(\Sigma \) typically denotes the covariance matrix \(Var (R)\) but not the uncentered second-moment matrix of \(R\). For short-term asset returns, the difference between \(Var (R)\) and \(E (RR')\) is negligible.

- 8.
Here \(\Sigma \) refers to the

*covariance matrix*of \(R\) (Michaud 1998). - 9.
Hence, \(\mathcal {C}\) belongs to the \(N\)-dimensional Euclidean simplex.

- 10.
This point has been suggested by Christoph Memmel in a personal communication.

- 11.
In the random-walk model (see Example 4), \(\theta \) is the parameter of \(F\), i.e., the joint cumulative distribution function of the asset returns.

- 12.
For example, it is typically assumed that the random-walk model is true, \(\Sigma \) is known, and \(\mathcal {C}=\mathbb {R}^N\). Then the desired portfolio corresponds to \(w^*=\Sigma ^{-1}\mu /\alpha \), which is a linear function of \(\mu \,\).

- 13.
I would like to thank Winfried Pohlmeier for this hint.

- 14.
- 15.
More details on this subject can be found in Frahm (2013).

- 16.
This means \(R\) is stochastically independent of \(\tilde{w}_t\sim \tilde{w}\) for \(t=1,2,\ldots ,n\,\).

- 17.
Here I do not distinguish between theories where a rational individual act

*as if*he maximizes the expected utility based on his subjective probabilities and other theories implying that he actually*does*. - 18.
This is also reflected by Markowitz’ two-stage portfolio optimization approach discussed in Sect. 2.

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## Author information

## Additional information

Many thanks belong to Uwe Jaekel, Bob Korkie, Christoph Memmel, Gert Mittring, Winfried Pohlmeier, Rainer Schüssler, and Niklas Wagner for their fruitful comments and suggestions. Moreover, I thank all participants of the workshop on portfolio optimization in Konstanz (2012) for their contributions.

## Proofs

### Proofs

In the following, it is implicitly assumed that the given expectations, variances, and covariances exist and are finite. Moreover, it is assumed that \(\Sigma \) is positive definite so that the inverse \(\Sigma ^{-1}\) exists.

###
**Proposition 1**

Let \(f\) be an increasing and strictly concave function from \(\mathbb {R}\) to \(\mathbb {R}\,\). Further, let \(g\) be a concave function from \(\mathbb {R}^N\) to \(\mathbb {R}\,\). Then \(f\circ g\) is strictly concave, too.

###
*Proof*

Since \(f\) is strictly concave, it holds that

for every \(\pi \in ]0,1[\,\). Further, since \(g\) is concave, we have that

Since \(f\) is increasing, it turns out that

for every \(\pi \in ]0,1[\,\), which means that \(f\circ g\) is strictly concave.

### Theorem 1

We have that

and, since \(\tilde{w}\) is independent of \(R\,\), it follows that \(p(w,r)=p(w)p(r)\,\). This means

Since \(u\) is increasing and strictly concave in \(W\,(1+r+w'r)\), which is linear and thus concave in \(w\), Proposition 1 implies that \(u\big (W\,(1+r+w'r)\big )\) is strictly concave in \(w\,\). Further, since \(\tilde{w}\) is mixed, its probability distribution is not degenerated and thus

by Jensen’s inequality. Hence, it follows that

Since \(\mathcal {C}\) is convex and \(\tilde{w}\in \mathcal {C}\), we have that \(\bar{w}=E (\tilde{w})\in \mathcal {C}\,\).

### Theorem 2

The out-of-sample performance of \(w^*=\Sigma ^{-1}\mu /\alpha \) amounts to

Further, we have that

so that

Hence,

The decomposition of \(\mathcal {L}_\alpha (\tilde{w})\) into \(\mathcal {B}_\alpha (\tilde{w})\) and \(\mathcal {V}_\alpha (\tilde{w})\) follows immediately from

Since \(\Sigma \) is positive definite we have that

and from

it follows that \(\mathcal {L}_\alpha (\tilde{w})\ge \mathcal {L}_\alpha (\bar{w})\,\). Moreover, since \(\tilde{w}\in \mathcal {C}\) and \(\mathcal {C}\) is a convex set, it holds that \(\bar{w}=E (\tilde{w})\in \mathcal {C}\). If \(\tilde{w}\) is a mixed strategy, Jensen’s inequality implies that \(\mathcal {V}_\alpha (\tilde{w})>0\,\), i.e., the inequality is strict. Vice versa, if the inequality is strict, it must hold that \(\mathcal {V}_\alpha (\tilde{w})>0\,\). This is only possible if \(\tilde{w}\) is mixed.

### Theorem 3

First of all, we have that

where

Moreover, it follows that

where

This means

### Theorem 4

Since \(\tilde{w}\) and its copies \(\tilde{w}_1,\tilde{w}_2,\ldots ,\tilde{w}_n\) are irrelevant, it follows that

and thus

i.e.,

Since \(u\big (W\,(1+r+x)\big )\) is strictly concave in \(x\), we have that

with probability one, but since \(\tilde{w}'_1R,\tilde{w}'_2R,\ldots ,\tilde{w}'_nR\) are not almost surely identical, it follows that

with positive probability. This means

Finally, since \(\mathcal {C}\) is convex and \(\tilde{w}_t\in \mathcal {C}\) for \(t=1,2,\ldots ,n\), we have that \(\hat{\bar{w}}=\frac{1}{n}\sum _{t=1}^{n}\tilde{w}_t\in \mathcal {C}\).

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### Cite this article

Frahm, G. A theoretical foundation of portfolio resampling.
*Theory Decis* **79, **107–132 (2015). https://doi.org/10.1007/s11238-014-9453-0

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### Keywords

- Asset allocation
- Mean–variance analysis
- Noise trader
- Out-of-sample performance
- Portfolio resampling
- Resampled efficiency
- Signal trader

### JEL Classification

- Primary G11
- Secondary D81