Your portfolio is (probably) not efficient

I want to have a rational asset allocation so I did a bit of reading about modern portfolio theory.
There is this nice idea of plotting all the risky assets on two axis: risk vs. return
Then for each fixed level of return, you can find the unique combination of assets that minimizes the risk. That defines a point for each return levels, and gathering all these points you get an “efficient frontier”, where all the “good” portfolios are. You want your portfolio to be on this frontier if you’re a rational investor.

A further claim is that the Total Stock Market portfolio (like VT) lies on the efficient frontier. That means that if you only have VT in your portfolio, it’s a bit boring, but also maximally efficient. So you should give your own shoulder a deserving pat.

But this last point is actually very wrong.
Let me explain: a portfolio on the efficient frontier is defined by the weights that are associated to each asset, and the only constraint on these weights is that they have to sum up to 1.

Since these weights are somewhat random numbers only constrained by “sum = 1”, one would assume that some of these weights can be negative.
In Impossible Frontiers, two finance professors show that indeed as soon as you have more than > 100 assets to pick from it becomes extremely likely that some of the weights are negative numbers, and that this is the case for all the portfolios on the efficient frontier.

But, by definition the Total Stock Market portfolio has positive weights on every assets. So it does not lie on the efficient frontier. And as long as your portfolio in not shorting at least one stock, it also cannot be on the efficient frontier.

So none of our portfolios are efficient. I find this thought very liberating :slight_smile:

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Take what you read online about MPT with a grain of salt until you have played with it yourself.

It’s a cute theoretical tool and teaches a few important lessons about diversification, but is not particularly useful for practical portfolio optimization at least in the form that’s commonly taught.

It requires you to have the clairvoyance of knowing expected future returns and correlations between assets. And it’s quite sensitive to small perturbations in these inputs - small changes may produce very different output recommendations from the optimizer. So there’s no the efficient frontier, it’s only as good as the inputs you feed into the model - which you’d normally fit from historical data but correlations may change overnight especially during crisis.

By whose definition? And where did such huge assets classes as debt, commodities and real estate go?


Knowing that MPT is okay to explain diversification but doesn’t travel much further like @kilyn said, the more accurate claim would be to say that the “market porfolio” lies on the efficient frontier. This theoretical market portfolio contains the whole market, not just the shares on VT since other assets (and asset classes!) are not taken into account.

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Thanks @Ed_Waadt and @kilyn for correcting me. In the paper Impossible Frontier the authors indeed talk about the “market portfolio” (the portfolio of all assets in which each asset’s weight is proportional to its total market capitalization).

If you believe the conclusion of that paper, then the “market portfolio” cannot be on the efficient frontier as it’s not shorting any asset and all the portfolios on the efficient frontier must be shorting at least one asset.

As @kilyn said the real efficient frontier is essentially unknowable. Furthermore, even if an oracle were to tell us the correct expected returns and covariances, the frontier moves every day.
So holding a portfolio on it would never be cost-effective.

The point is not that VT was not efficient. Its not possible to have a fully efficient portfolio, but this is not due to the point observed in the misleading paper stated above but simply a matter of information and timing challenges. The point is two-fold:

  • assuming efficient markets, its impossible to systematically beat the market; BUT
  • the market doesnt consist of shares only but all liquid assets with a risk premium
    => An efficient Portfolio consists of an unknown blend of risky asset classes; a shares only investment introduces inefficiencies

When we go into details, we need to consider that the paper is flawed. Whilst an assets with marginal zero weight may be hardly imagined, a negative weight one could only be explained with an asset that is either about to go bust; or that will experience a material de-valuation. In any other scenario, a negative weight does not make sense. In such scenario (bust or de-valuation) where assets risk or return would go belly up; such assuming efficient markets implied an immediate reduction / voiding of the assets market cap to adjust to the change in return/risk. A negative weight therefore only makes sense for a i) marginally small amount of time required for Mr Market to absorb the information, ii) in case of a market inefficiency or iii) asymetric information.

This actually highlights the underlying issue which is that both return and risk may not be determined but only retrospectively observed and approximated. Hence, its impossible to establish an ideal portfolio as such would need to be based on unknown risk and return data. Assuming efficient markets however, the market will constantly translate retrospectively observed data and resulting adjustment to approximate risk/return into a change in the assets market cap; leading to a situation where a market cap represents the best possible proxy to an efficient portfolio given „known“ data. Indexing is therefore the best alternative to beeing the allknowing Mr. Market himself.

What puzzles me is that many investors in this forum go all-in on shares only. Going 100% in shares is simply a speculation on a certain market segment (shares) of the entire market (risk carying assets). The rationale behind 100% assets is maximized return but not a maximized risk weighted return. The same logic could be applied further by cutting shares into additional segments that historically warrant higher returns like small caps or Nasdaq. Whilst small cap premium might (depending on the research) be historically observed, they are not guaranteed as there is no undisputed fundamental reason for the premium to prevail. Same applies to Shares vs. e.g. Real Estate returns. What if Mr. market at any given point in time and after centuries decides that RE shall have the higher return?

Taking the Shares vs. Small Caps example to showcase the point. Whilst many investors probably agree that Small Caps might imply higher prospect returns than All Shares. Who would dare to claim that their risk weighted return was higher? Hard to do… right? Even if currently observed (which is not the case) its even harder to claim that such was just given and prevailed. The exact same thought could be applied on Shares vs. All Risky Assets. Would anyone dare to claim that shares had a higher risk weighted return than e.g. 25% Shares, 25% Real Estate, 25% Bonds and 25% Gold? We could still claim so but huh… just looking at recovery time from max drawdowns; that might be bold. And even if we could claim so… how could we justify that this overperformance prevailed?

Why not simply going even more „passive“ than just shares only? Clearly… its impossible to establish a market cap global allocation over all risky assets. Besides, such would be screwed given market unequalities and incentives, e.g. tax relieve on real estate. So the second best to an all passive global allocation is probably to use observed, estimated risk/return as well as correlation figures from investable asset classes and to build a corresponding portfolio that gets close to the desired target return yet optimizes risk and return.

And what if you not just want to get close to but actually the maximum return? Well… its still more efficient to use such optimal risk/return portfolio that is somewhat close to the desired target return… and then simply leverage or de-leverage accordingly. Leverage doesnt need to imply debt, hefty interest or margin calls (dont do that) but can be self-funded. Latest beyond the first 10 years of work, most people establish both an investment portfolio as well as a fairly moderate safety cushion that within reason could be used for „self funded leverage“.

If you still insist on max return and don‘t have such safety cushion for self leverage… in my experience you then either are at the beginnig of your investment journey where future investments compared to current investments are major… aka your future income constitutes kind of a low risk asset that you use to re-balance against (so you follow a ballanced approach without noticing as your alternative otherwise would have been to on day 1 take a loan equivalent to your next 10 years savings and immediately invested all). Or if you already had certain wealth where your future investments p.a. did not exceed e.g. 10% of your current investments and you still invested 100% on shares… well then good luck with your investment strategy and I wouldnt want to comment on what I thought of such approach. But just remember that max returns of shares over all risky assets are observed but not guaranteed and you highly likely experience a worse risk weighted return than possible.

Or in simple words… do never go 100% all in on shares; unless you anual new investments materially exceed 10% of your current investments.


The Tea Lover

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In the paper I mentioned, having a negative weight for one asset simply means taking a short position on that asset. The authors give an example on page 8 after Proposition 2: with only 3 assets you can already build cases where portfolios on the efficient frontiers have negative weights. (This other paper has a similar example).

In their analysis the authors of the first paper show that as the number of assets increases, all the portfolios on the efficient frontier are going to have at least one negatively weighted asset. Hence the market portfolio cannot lie on the efficient frontier (because as you said that portfolio cannot have negative weights).

The paper aims at disproving that the market portfolio is efficient. It’s a critic of that conclusion of a strong version of the efficient market hypothesis.

Does this amount to the same as doing factor investing, e.g., small cap (VBR)? When you leverage your portfolio you increase both the return and the volatility If, as you mentioned, you manage to get a risk-free loan with no interest, both return and volatility are simply multiplied by your leverage. If you invest in small cap, you also in effect increase both return and volatility. Is there a difference (aside from comparing the sharpe ratios)?

First thank you very much for this very interesting post.

And i would add something i read on bogleheads: indexing (with low cost etf) is a strategy offering 100% safety not to underperform the market, while active investing is 100% sure to not always overperform.

The 2. Säule (Pensionskasse) is a not negligible part of total portfolio and in most cases by far not 100% shares (max 40% i would guess). So in real life in CH you will never be all in. Or did you mean something else?

Guess my criticismn on the paper was not fully clear. The reason why you would want to go short on an asset, when building an efficient portfolio, was either:

  1. the asset‘s value was beyond the fair value established given „known“ return/risk data. This is exactly the point above where we talk about either an interim adjustment anomaly that dissolves once Mr. Market prices in the „known“ risk and return, market inefficiency we should for the simplicity assume was not present or asymetric information. Assuming efficient markets, these effects are neglible and should be ignored.
  2. an attempt to indirectly build up leverage. From an efficiency point of view, such however doesnt make much sense. Assuming efficient markets, its more efficient to directly lever but not to shortsell risky assets.

When we consider the above, and the fact that return/risk may not be measured but only observed and estimated… we may conclude that the idea of marginal zero weight assets is questionable and that negative weight assets in an efficient portfolio does not make much sense. Hence, please disregard the paper. This puts market cap weighted indexing back to its position as the best proxy to an efficient portfolio… if only we knew the entire asset universe, asset caps and distortions introduced e.g. by tax treatments :slight_smile:.

With regard to maximizing targeted return vs. risk weighted return; the example would be that instead of investing 50% of your wealth only but all of such in shares where we expect a max. return of 6%… its more efficient to invest 60% of your wealth but these 60% in a multi-asset portfolio that you set up with a target return of 5% only. Meaning that you use your safety cushion to lever yourself up a bit. Whilst your resulting target return in both cases was 6%, the risk/volatility of the „Not-Only-Shares“ Portfolio was lower.

With regard to Pension Fund as a „Non-Shares-Asset“. Remember that the objective of investing in multiple asset classes is two-fold. On one hand, we would want to tab on to different sources of risk premium; which might potentially be more efficient than the risk/return profile of shares. But even if the risk/return was worse than the one of shares; combining two asset classes can improve efficiency given diversification resulting from low correlation. These efficiency gains are further accentuated in case we can apply smart re-balancing. Unless we talk about an 1E solution or a vested benefits account, a pension fund will not provide you with the ability to indirectly re-balance your portfolio. We therefore lose a material source of efficiency if our multi-asset solution consists of Non-Shares in the form if a Pension Fund only. Your asset allocation should encompass all your assets that you may both invest and controll. Therefore, I propose to consider 3a as part of an overall asset allocation but at the same time to disregard the pension fund. Do not count your pension fund as money market / bonds as you cant truly re-balance. Therefore, make sure that your remaining assets (as per the above) are not 100% in shares unless you are at the early stages of investing.

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