What de Moivre Knew About Mega-Funds

Private Equity (PE) is truly a tale of two firms.  There’s a stark dichotomy between colossal multi-billion-dollar funds and the typical sub-billion-dollar fund.  While the process of buying, building, and selling companies to generate returns for limited partners (LPs) is mostly the same, the range of opportunities and solutions, not to mention the sheer difference in scale, is much greater for the “mega-funds”, generally defined as a fund over $5 billion.  These funds are capable of buy-and-builds, carveouts, and take-privates far beyond that of their much smaller counterparts, resulting in deals such as the KKR Upfield-Unilever transaction, Blackstone’s Refinitiv (Thomson Reuters) carveout, etc.  Others argue that mega-fund managers struggle to generate the same high returns as smaller funds due to lack of sizeable enough deals, too many opportunities to manage, etc.  

Over the past few years, we’ve seen a much larger proportion of capital invested in these mega-funds, which not only increases their incidence, but the individual fund size as well.  Mega-funds raised 45% of North American and 42% of European private capital between 2016 and 2018.  Blackstone Private Equity has raised a $22 billion fund in 2019 (still open), and a number of other firms have closed funds (or have hard caps set) at well over $10 billion, demonstrating the relative permanence of this industry shift.  For many institutional and high net worth investors, this begs the question “which is better?” and “where should I seek returns?”

Pitchbook, a Morningstar company, is a database platform that specializes in the amalgamation and analysis of private equity, venture capital, and M&A information and news.  It released a report in late September seeking to answer this question, examining PE fund returns over the 2000 to 2013 vintages and comparing mega-funds to small funds across quartiles.  Its headline “PE mega-funds have higher floors and lower ceilings than smaller vehicles” is essentially short-hand for “PE mega-funds have less variability”.  Investors with enough capital to ‘agonize’ over the decision to invest in a mega-fund vs several smaller funds probably take some consolation in reputable analyses from financial advisers, analysts, and scholars. However, those of us with a merely theoretical interest in the topic could have answered their questions with a brief examination of the data and knowledge of 1730s era statistics.

Abraham de Moivre (1667-1754) is a French mathematician renowned for his work in statistics, probability, and trigonometry.  Much of his work concerns mathematics higher than that of which we’ll use today, but ours is no less important.  Popularized by Howard Wainer as the “Most Dangerous Equation”, we will examine de Moivre’s equation on the relation between the standard deviation of the sampling distribution of the mean and population mean, or:

where X-bar is the standard deviation of the average of samples drawn from a population, sigma is the standard deviation of the population, and n is the number in our (equal) samples.  This means that if we drew samples of 100 from a population, we could expect the average variability of our samples to be one-tenth that of our population.

Now, back to the PE funds of the early aughts through Financial Crisis era.  What PitchBook’s analysts did was split all funds from this time period into two groups:  those with $5 billion or more in capital and those with less. Here, it’s worth noting that fund capital is different from firm AUM.  A PE firm could have 5+ sub-billion-dollar funds, but this wouldn’t qualify as a mega-fund. Then, they grouped the IRRs of these funds, looked at quartiles, and produced their averages.  This is where we saw mega-funds performing in a narrower band of returns, while smaller funds generate both much higher and much lower returns.

I acknowledged the value of this approach earlier in the article, but a quicker way to yield the same insights would be de Moivre’s equation.  We know that small PE funds have much higher variability (sigma) than large funds, despite similar mean returns.  This means that small funds will, by nature, be more likely to populate the extreme tails of a total PE fund return distribution than a mega-fund, regardless of how we sample them.  For more visual learners, the below distributions demonstrate the difference between low variance and high variance populations.

Put differently, because smaller PE funds have higher variability, it’s no surprise that the top performing small funds outperformed top performing mega-funds, and the worst mega-funds still outperformed the worst small funds.  

This means that our answer to the proposed questions earlier is not “smaller is better” or “bigger is better”, because PE firm choice will be idiosyncratic to the investor, based on capital needs, fund characteristics, risk aversion, legal concerns, and a gamut of other criteria.  What we can say, thanks to de Moivre, is that a recommendation based solely on the size of the fund would be entirely misleading.  

Citations

Wainer, Howard. (2007). The Most Dangerous Equation. American Scientist - AMER SCI. 95. 10.1511/2007.65.249.