September 2020

The Mathematics of Active Equity Management

Active equity managers believe they can ‘beat the market’ or, if they invest conservatively, earn market like returns but with less risk.

The performance of these managers has been under the spotlight over the last few years, and more and more retail and institutional investors are switching to passive equity funds.

In this blog, we use mathematics to demonstrate why it’s been so difficult for active equity managers to beat the market, and why pension trustees hoping that their actively managed equity funds will do so in the future are likely to be disappointed.

A Simple World and Stock Market

Imagine a Simple World where the stock-market has only three constituents, Stocks A, B and C. All three stocks have market capitalisation of £1,000, which gives the market in this Simple World a value of £3,000.

Further imagine this Simple World has a Large Passive Investor who holds half the market. The Large Passive Investor holds each stock in the same manner as that stock is represented in the market. Our Simple World also has a number of small Boutique Managers, who invest actively and hope to outperform the market.

After three years Stock A quadruples in value to £4,000, while Stocks B and C remain unchanged at £1,000 each. The market itself has doubled in value to £6,000. As the Large Passive Investor invests passively in each stock, he participates in the markets’ surge and his investments also double in value.

Between them, the Boutique Managers’ hold the other half of the market, which means that the average actively managed pound also doubles in value. However, as the Boutique Managers’ charge higher fees, the return net of fees on the average actively managed pound is lower than the market return.

It’s worth pointing out that this will be the case over any time-period and in all market conditions. Comments and opinions in the investment press that ‘…active management is better over the long run…’ or ‘…active management is better in a bear market…’ are wrong.  As long as both invest in the same universe of stocks, the laws of arithmetic mean the return on the average actively managed pound will always be the market return less active management fees.

We know that the investments of our Boutique Managers above, taken as a whole, have doubled in value.  But what could the returns on each individual Boutique Manager look like?

By definition, an active investor does not hold the index and will only invest in a proportion of all the stocks in an index. Let’s imagine that the Boutique Managers’ in our Simple World hold one stock each. That is, they hold one of Stocks A, B or C.

A few very skilled (or lucky) Boutique Managers’ will have had the foresight to hold Stock A. These managers will have earned a gross return of 300%, with their investments quadrupling in value. However, the majority will hold Stocks B or C.  As they don’t have any exposure to Stock A they miss out on the markets’ surge, earning a return of 0% and charging their clients higher fees while doing so.  

The Impact of Positive Skewness

The chart below sets out the returns on the Dow Jones Industrial Index of stocks in the three years to 30 June 2020:

The Dow Jones shares an important feature to the market in our Simple World above; – positive skewness. That is, the existence of a select few ‘Big Winners’ with exceptionally good performance (Stock A in our Simple World. Apple and Microsoft in the Dow Jones).

Positive skewness occurs because, while the most a stock can lose is 100%, potential positive returns are well above 100%. In a positively skewed distribution, the average is always higher than the median. (The average return on the Dow Jones over this three year period was 41% while the median return was 31%).

Positive skewness means that the odds are stacked against active managers. As they only hold a proportion of the index, an active manager risks missing out on exposure to the exceptional performance from a small proportion of stocks.

A select few active equity managers will have been skilled (or lucky enough) to pick, in advance, the tech stocks (Facebook, Apple, Google, et al) that have driven markets over the last decade. These managers will have enjoyed incredible success. However, the majority will necessarily be under-weight these Big Winners with performance very poor in comparison.

Positive skewness is a long-standing feature of equity markets, re-occurring over different time periods and across different jurisdictions. For as long as this feature continues, and active managers continue to charge fees, the rules of mathematics mean that the vast majority of active equity manager will under-perform the market, and this under-performance increases with skewness.  

Modelling the Experience of a Pension Trustee

As demonstrated above, an active equity manager hoping to beat the market will need exceptional skill to pick the industries that will out-perform in the future, and the best companies in those industries. Positive skewness means only a select few can be successful.

A pension trustee hoping to invest in active management faces another hurdle. They need to identify, in advance, the select few managers that are skilled enough to beat the market.

The temptation will be to look at past performance, but investment managers rarely beat the market for long and past performance is a poor guide. At the very least the trustee will need to:

  • Close their eyes to all the slick marketing material and dubious statistics produced by the investment industry
  • Find some way to distinguish between manager luck and skill
  • Understand the manager’s style, biases, incentives and expenses
  • Monitor their selected manager and consider how changes in the fund size and staff could impact future performance

Even after all this, positive skewness and the laws of mathematics mean that the majority of trustees must be unsuccessful and their actively managed equity funds will under-perform the market.  

The difficulty of picking Big Winners, and identifying the few managers who can, means that the experience of a typical trustee will likely be the same as that of a portfolio of randomly selected stocks.

Using Monte Carlo simulations, we model the frequency that a random portfolio of stocks would have beaten the Dow Jones Index over the last three years. Our analysis assumes that fees for active management average 0.5% of assets each year.

The results show trustees are likely to underperform the index. This happens because of the impact of fees, but also because a small portfolio of randomly selected stocks is likely to be underweight Big Winners.

Lies, damned lies and investment manager performance statistics

Trustees are often provided flattering statistics on the performance of active managers. How does this reconcile with our mathematics above which shows most should underperform?

Let’s revisit our Simple World again to demonstrate some of the errors the industry makes when producing statistics on investment performance.

We’ve already seen that the market in our Simple World doubled in value, which means a return of 100% for the market and the Large Passive Investor.

Now let’s imagine that there are four Boutique Managers. As before, each Boutique Manager invests in one of Stocks A, B and C. Let’s assume two invest in Stock A (and earn a return of 300%) and the other two invest in Stocks B and C respectively (and earn a return of 0%). The average return on of these four Boutique Managers is 150% - which is larger than the market return of 100%!  

This happens, of course, because our calculation doesn’t allow for the weighting that each Boutique Manager has in the market.

Why did I assume two Boutique Managers hold Stock A but only one each in Stocks B and C? Because this is what is typically done in practice! Our Simple World may well have started with an equal number of Boutique Managers in each stock, but the industry typically excludes closed funds and bust managers (which will have had poorer performance) from these calculations.


Much of a pension trustees time and expenses are associated with active investment management.

Actively managed funds have higher fees to begin with. But they also need more input from Investment Consultants to research and advise on manager selection, monitor performance and developments at the selected manager(s) and advise on the replacement of under-performing manager(s). Additional advisory expenses are incurred if the trustees make tactical asset allocation decisions.

The truth is, active management is very difficult and, once costs are accounted for, the contribution from active management may be far less than trustees believe.

Please contact us if you would like to discuss this further.