Some of You Will Sell or the Achilles Heel of the Arithmetic of Active Management

I’ve made no secret of my fascination with the active/passive debate. I don’t have a dog in this fight: my perspective is that of an active individual DIY investor. There is strategy diversification in my own portfolio: 40% of my total portfolio is in what I call a “passive strategy” where the equity allocation consists entirely of low-cost index funds; nearly all of the rest is in an “active strategy” where the equity allocation consists of individual stocks. However, since this secular change from active to passive has the potential to affect security pricing and the persistence of trends everyone should care deeply. Last time, we took a look at possible float manipulation, today the attention is on the corner stone of the argument for passive indexing. I don’t think I came close to disproving it, only weakened it by considering real people who make suboptimal decisions. That seemed appropriate in a year when the father of behavioral economics won the Nobel memorial prize.

Risk Tolerance or Why People Sell at Bottoms

A typical questionnaire for an investor’s risk tolerance includes questions such as: “If the stock market drops 30%, will you A) Sell everything, B) Sell half, C) Do nothing, D) Buy more stocks”. In my opinion, these questions are worth less than the paper they’re printed on. People don’t sell because the market has dropped 30%; they sell because after the 30% drop the market looks to be dropping another 30% ‐ a virtual certainty according to the perma-bears whose advice they wish they had heeded. In addition, serious market declines often don’t not appear out of thin air, rather they tend to be accompanied by real economic malaise: in 2000-2002 it’s the mal-investments of the dot com era and closure of countless “new economy companies” and those that service them; in 2008-2009 it was the collapse of the housing bubble and the veritable brink-of-collapse of the global financial system. Imagine the following scenarios and ask how many would sell:

  • A nuclear attack on US soil.
  • US losing its superpower status which has underwritten its prosperity since WWII.
  • Total collapse in Japan/EU. Sovereign debt crisis. US 10Y treasure yields above 5%.
  • Banking crisis, bank bail-in
  • Wide spread job loss including one’s own

This is not at all a prediction that any of the above may actually happen, but rather that a questionnaire or even thoughtful introspection during times of peace and prosperity can never capture the emotional and psychological stress investors will be under in a real economic crisis. Time and again, many who know they should, and think they can hold through a bear market ended up selling at the worst time. In fact wide-spread capitulation is a prerequisite for bear-market bottoms. Nobody sells? Fine, the bear market continues until a lot of people do — this is the way it works.

Time Horizon for Most Investors is Shorter Than Expected

The long term stock market average is often plotted as an uninterrupted bottom-left to upper-right squiggle. “In the long run, stocks always go up.” is the prevailing mantra. The rub is the definition of the “long run”. As I discussed in The Arithmetic of Prudence, for a typical worker saving for retirement, what really matters is the the return during two decades — one on each side of the retirement date. During the accumulation phase, the money weighted return necessarily depends more on when the portfolio value is greatest, i.e. near retirement. The closer the investor is to the distribution phase the less he can rely on the “long run”, the less he can afford to suffer through a severe draw down. Success of the withdrawal phase largely depends on the period immediately after retirement, i.e. the dreaded sequence of return problem. No matter how one slices it, two decades is not long term given the natural gyrations of the stock market. The standard recommendation of “invest for the long run” should be understood as an expediency — most people are even more terrible at being a short term trader — than a truth.

Note that on average there is 1 bear market per decade (source).

The Arithmetic of Active Management and Its Achilles Heel

Nobel laureate William Sharpe made the observation that in the aggregate active investors and passive indexers must both own the market portfolio, hence the passive indexer must outperform, in the aggregate, due to lower fees. That is the gist of the arithmetic of active management. It’s tautologically true and has proven to be the bane of active fund managers everywhere. I have only seen this argument applied to stocks and that’s where our discussion will remain.

Index funds are a great invention and Jack Bogle deserves all the accolades going his way. I’m all for the little guy getting the most he can and sticking it to the banksters in the process. It can only be a good thing if LBYM, invest early and often, passive indexing become part of the ethos &mdas; Americans consume too much and save too little IMO. However, thinking individuals must also realize there is no such thing as an unmitigated good: by-passing financial advisors and active management also removes gate keepers and return dispersion during bear markets. The issue with the popularity of index funds is the synchronicity of selling and the echo of negative emotions during market duress. Even die-hard Bogleheads readily admit it is not easy to hold through a bear market (thread 1, thread 2). As the democratization of investing via passive indexing gathers more adherents we can expect a fair number to lack the fortitude to hold through a down turn. Such is the Achilles heel of the arithmetic of active management if there ever is one: the passive indexer must hold the market portfolio to earn the market returns; if he sells, all bets are off.

Let me recast this argument which I’ll call the Corollary of the Venditor (Latin: seller).

  • In general and among the passive indexers there are two types: type A who holds through market ups and downs; and type B who makes poor timing decisions: hesitates to buy when the market is trending up, and/or sells at market bottoms.
  • Type A indexers receive market returns; type B indexers must receive below-market returns on account of their poor timing.
  • Therefore, passive indexers as a whole receives below-market returns.
  • Active investors who make up the rest of the market, must receive above-market returns equal to that surrendered by the type B indexers.

It’s not necessary to estimate the proportion of type A and B indexers, only that B is non-zero. There are type B active investors as well, but their losses accrue to other active investors not type A indexers. Nothing above dictates how the return among active investors is distributed. This is not a defense of actively managed funds with high fees that eat into investor returns. The obvious recommendation for passive indexers is not to be distracted by emotions but that’s like saying everybody should eat healthy and exercise. Lastly, the net alpha of active investors comes from poor timing decisions of some passive investors, in other words timing counter to type B investors is a path to above-market returns. This counter move can be achieved by allocation adjustment during bull/bear markets, market timing in the traditional sense with active managed funds, or even market timing with index funds (I know you were wondering where these guys are).

In summary, I believe the Corollary of the Venditor points to an a priori opportunity for the aggregate active investors to achieve above-market returns. The advantage is probably not large enough to overcome the fees of active fund managers per SPIVA records but the active individual investor can claim his due by: cost-reduction (fees, trading and taxes), differentiated positioning (concentration, factor tilts, active share), and above all adaptation to macro cycles that occur on the order of years or a decade. These are precisely the things I practice and write about in this blog.

This post is not meant to be a detraction of passive indexing which is still the best recommendation for most people. Type B investors exist irrespective of the investment vehicle so they may as well pay less fees while invested. Studies have shown small out-performance by active mutual funds before-fees and now we have an explanation for it being a systemic effect. With that we can reject the more fundamentalist passive indexers who claim there is only one way to invest, who also tend to be over-allocated to equities. Though I remain grateful — a fair number of them and their converts will end up being type B investors without whom people like me will have diminished hope of out-performance.


  1. Sharpe, W. “The Arithmetic of Active Management.” Financial Analysts’ Journal, Vol. 47, No. 1 (1991), pp. 7-9. Link
  2. Pedersen, L., “Sharpening the Arithmetic of Active Management” (2016, SSRN 2849071) Link Talks about the effect of trading.
  3. Vertes, D., “Active vs. Passive Investing and the ‘Suckers at the Poker Table’ Fallacy” Link I believe is saying the same thing, but boy is it a long read.