## Abstract

Investors have traditionally relied on mean–variance analysis to determine a portfolio’s optimal asset mix, but they have struggled to incorporate private equity into this framework because they do not know how to estimate its risk. The observed volatility of private equity returns is unrealistically low because the recorded returns of private equity are based on appraised values, which are serially linked to each other. These linked appraisals, therefore, significantly dampen the observed volatility. As an alternative to observed volatility, some investors have argued that private equity volatility should be estimated as leveraged public equity volatility, because private equity companies are more highly levered than publicly traded companies. However, this approach yields unrealistically high values for private equity volatility, which invites the following question: Why isn’t the appropriately leveraged volatility of public companies a reasonable approximation of private equity volatility? This article offers an answer to this puzzle.

**TOPICS:** Private equity, volatility measures

**Key Findings**

▪ Why isn’t the appropriately leveraged volatility of public companies a reasonable approximation of private equity volatility? The authors look for clues in the public markets where they find no association between volatility and leverage, counter to what financial theory would suggest.

▪ The evidence suggests that the relationship between leverage and volatility is hopelessly obscured by a variety of confounding effects in both public and private markets.

▪ This article arrives at the counterintuitive conclusion that private equity volatility is similar to public equity volatility despite its higher leverage. The likely explanation is that privately held companies are inherently less risky and thus able to bear greater leverage.

Investors struggle to incorporate private equity into mean–variance analysis because they do not know how to measure its volatility. They recognize that the observed volatility of private equity returns is unrealistically low because returns are estimated from appraised values that are serially dependent.^{1} This serial dependence arises because appraisals rely, in whole or in part, on stale information from events that occurred prior to the valuation date. For example, appraisers may reflect the values of comparable privately held companies that were bought or sold in the past. Or, they may anchor company valuations to their own prior values. Because each period’s appraisal incorporates values from prior periods, the returns of an appraisal-based index are artificially smooth, akin to a moving average.

Some have suggested that private equity volatility should be estimated as leveraged public equity volatility, because private equity firms are more highly levered than public firms.^{2} But this approach produces unrealistically high values, which is puzzling. Why isn’t the relative volatility of private equity related to its relative leverage? Because the public market is considerably less opaque than the private market, we approach this question by looking for clues from the public equity market.

To begin, we show how leverage should affect volatility theoretically. We then document the extent to which private equity violates this theoretical relationship. We also simulate the effect of leverage on volatility, which underscores the disconnect between leverage and private equity volatility. We then turn to the public market in search of clues about the relationship between leverage and volatility. We first sort public firms by their explicit leverage and compare these differences to differences in volatility. We then regress volatility on leverage both across time and cross-sectionally, controlling for a wide range of confounding factors. These regressions fail to show a relationship between leverage and volatility. So, we examine several individual companies anecdotally to gain a better understanding of the apparent disassociation between leverage and volatility. By induction, we extend our findings to private equity and propose an approach for measuring private equity volatility, which, in our view, yields realistic values.

## LITERATURE REVIEW

It is common practice for investors to estimate the volatility of private equity buyouts by applying a leverage multiple to public equity volatility. This approach is also espoused widely in the literature:

▪ L’Her et al. (2016) estimate the net debt-to-enterprise value ratio for buyouts to be 55% on average versus 25% to 30% for public companies. They extrapolate this finding to recommend that public market returns be scaled up to adjust for this leverage differential to compare with buyout performance.

▪ Stafford (2015) notes that a typical private equity transaction increases debt-to-enterprise value from roughly 30% to 70% and that an investor in public markets would therefore need to invest about double the capital to generate returns similar to those of private buyouts.

▪ Chingono and Rasmussen (2015) equate private equity buyouts to investments in levered (public) small-cap value companies.

▪ L’Her, Karthik, and Desrosiers (2017) state that “1.2 should be viewed as a minimum beta estimate for buyout funds,” which they attribute to the observation that buyouts have significantly higher leverage multiples than public firms. Axelson, Sorenson, and Stromberg (2014) and Harris, Jenkinson, and Kaplan (2016) make similar assertions with leverage-adjusted beta estimates for buyouts ranging between 1.5 and 2.7.

## THE PRIVATE EQUITY LEVERAGE PUZZLE

Private equity firms are highly levered, yet their observed volatility is lower than that of public equity. Even after adjusting for the smoothing effect introduced by appraisals, private equity volatility is significantly lower than the product of public equity volatility and the relative leverage of private equity to public equity.^{3}

In principle, equity volatility should scale directly with leverage absent other effects, as shown in Equation (1).

1where σ_{E} equals equity volatility, *L* equals leverage (assets/equity), and σ_{A} equals asset volatility.

Axelson et al. (2013) show that buyout firms have approximately two times more leverage than their public counterparts. Therefore, based on Equation (1) and assuming similar asset volatility between the two groups, we would expect buyout equity’s volatility to be roughly two times that of public equity’s volatility. This is not what we observe empirically.

Exhibit 1 compares public and private equity volatility over the period December 1996 through September 2018. We use the S&P 500 Index to represent public equity and the buyout segment of the State Street Private Equity Index to represent private equity.^{4} We measure the volatility of quarterly total returns for the S&P 500 and quarterly internal rates of return (IRRs) for private equity. To account for the smoothing effect appraisals have on private equity valuations, we also measure the volatility of overlapping annual and triennial returns.^{5} The first column reports public equity’s volatility, the second set of columns shows the expected volatility of private equity, assuming it is twice that of public equity (given that it has two times more leverage), and the third set of columns reports private equity’s actual volatility.

Contrary to the expectation that leverage should have a one-to-one relationship with volatility, Exhibit 1 shows that leverage has no apparent effect on private equity volatility. Actual private equity volatility does increase over longer measurement intervals, which is a function of its serial dependence. But it falls short of public equity volatility, where we would expect it to be twice as large. Therein lies the puzzle.

## SIMULATED EFFECT OF LEVERAGE ON VOLATILITY

To support the validity of our expectation from Equation 1 of a one-to-one relationship between leverage and volatility, we simulate the theoretical association of leverage and volatility using the following steps.

1. We form 500 hypothetical new “companies” that each consist of investment in one of the S&P 500 stocks as the sole “asset.”

2. Each year, we introduce random amounts of leverage (measured as the asset-to-equity ratio), modeled from a distribution with a mean of 1.5 and a lower bound of 1, to each hypothetical company and record the monthly returns of these levered companies over time.

^{6}Thus, we have added leverage mechanically to each position in a way that is uncorrelated to all other variables.3. We record the annual volatility of each new company.

4. We repeat steps 2–3 to generate results for a 28-year period.

5. We run panel time series and cross-sectional regressions (specified later).

The results of this experiment are shown in Exhibit 2.

The leverage coefficient of 1.01 in Exhibit 2, Panel A, is consistent with a one-to-one relationship between leverage and volatility through time. When a given company’s leverage goes up from one year to the next, its volatility increases proportionally. The leverage coefficient of 0.99 in Panel B reveals the analogous one-to-one relationship in the cross-section of companies. In any given year, the synthetic companies with higher leverage have higher volatility than those with lower leverage.

These simulations lend unambiguous credence to the notion that volatility should scale directly with the degree of leverage. We are therefore left with the puzzle that even though private equity has twice the leverage of public equity, its volatility adjusted for smoothing is no greater than public equity volatility. To resolve this puzzle, we turn to the public market to look for clues about the empirical relationship between leverage and volatility.

## THE EFFECT OF LEVERAGE ON VOLATILITY IN THE PUBLIC EQUITY MARKET

We base our analysis of leverage and volatility within the public equity market on the constituents of the S&P 500 from March 2001 through October 2018. Within groups of stocks with similar levels of fundamental risk, we form capitalization-weighted portfolios according to leverage. In this initial experiment, we use borrowing costs as a proxy for fundamental risk. Specifically, each month we form 16 portfolios by 1) sorting stocks into quartiles according to their borrowing cost^{7} (a proxy for the fundamental risk of their underlying businesses, irrespective of the degree of leverage) and then 2) within each fundamental risk quartile, sorting stocks into quartiles according to their leverage (asset-to-equity ratio). We record the monthly returns of these 16 portfolios over the full sample.

We next evaluate the impact of leverage while controlling for differences in fundamental risk across firms. Specifically, within each risk quartile, we compare the volatility of the portfolio of firms with the highest leverage to the volatility of the portfolio of firms with the lowest leverage. These ratios are reported in Exhibit 3 (light gray bars). For comparison, we also show the ratio of leverage for the same portfolios (dark gray bars). For example, on average, within the universe of firms with the lowest fundamental risk (quartile 1), the most levered firms have two times more leverage than the least levered firms, and the volatility of their returns is 1.3 times greater.

If leverage functions as expected, the volatility ratios should be of similar magnitude to the leverage ratios. However, for every quartile of fundamental risk, they are about half of what we would expect. Moreover, the rightmost bars report results for portfolios that ignore fundamental risk (they are single-sort portfolios formed on leverage alone). If borrowing costs effectively proxy for fundamental risk, we would expect a stronger relationship between leverage and volatility for the double-sort portfolios than for the single-sort portfolios. Again, this is not the case.

We repeat the experiment using two alternative proxies for fundamental risk: level of cash flows and volatility of cash flows.^{8} All else equal, we presume that companies with higher and more stable cash flows relative to their book value are inherently less risky. Exhibit 4 summarizes the average volatility and leverage ratios of the double-sort portfolios. Again, we find that the volatility ratio of highly levered firms to firms with low leverage does not come close to matching the ratio of their respective amounts of leverage.

Next, we resort to regression analysis to determine if differences in volatility correspond to differences in leverage. Our data comprise the S&P 500 constituents from January 1990 through December 2017. We exclude firms with no leverage as well as annual equity volatilities that reside in the 5% right tail of the distribution.^{9} We run two panel regressions: one that captures the time series relationship between equity volatility and leverage and another that captures the cross-sectional relationship. We use the natural log of each variable in order to transform equity volatility into a linear combination of leverage and asset volatility.

For the time series panel regression, we include a constant for each stock (firm fixed effects) and use market volatility to represent asset volatility, as shown in Equation (2).

2where σ_{i,t} equals volatility of firm *i*’s equity over calendar year *t*, σ_{M,t} equals volatility of the market (S&P 500) over calendar year *t*, and *L*_{i,t−1} equals leverage (assets/equity) of firm *i* at end of year *t* – 1.

For the cross-sectional panel regression, shown in Equation (3), we include a constant for each year (year fixed effects).

3where σ_{i,t} equals volatility of firm *i*’s equity over calendar year *t* and *L*_{i,t−1} equals leverage (assets/equity) of firm *i* at end of year *t* – 1. In both regressions, we adjust the standard errors to account for possible correlations across firms.

Exhibit 5 shows results from the time series panel regression. In addition to the baseline regression described earlier, we also report leverage coefficients for three variants. First, we control for changes in the average amount of leverage in the market from year to year. Second, we exclude financial and utility companies, recognizing that their capital structure may differ from other sectors. Third, we exclude the Global Financial Crisis period of 2008–2009, recognizing that extraordinary events during this period may distort our results. Yet, in each case, we are unable to reject the null hypothesis that the leverage coefficient equals zero.

Exhibit 6 shows the cross-sectional results. In this case, we introduce four variants, each designed to control for the underlying volatility of a company’s assets. First, we again exclude financial and utility companies from the analysis, as we did with the time series regression. We also control for the three measures of fundamental risk introduced in Exhibits 3 and 4.

Again, we see no evidence that volatility scales with leverage. The coefficients are not only insignificant, they are in the wrong direction as well. Given this evidence of the lack of association between leverage and volatility in the public equity market, it is unsurprising that private equity volatility shows no correspondence to leverage. In the next section, we examine two companies in search of anecdotal evidence to explain the lack of correspondence between leverage and volatility.

## ANECDOTAL EVIDENCE

The following analysis highlights why it is difficult to capture the expected relationship between leverage and volatility.

Exhibits 7 and 8 depict the relationship between leverage and volatility for two companies: Motorola and Clorox. They illustrate that volatility can be highly time-varying even during periods where leverage is relatively stable. These examples reveal why a time series regression cannot detect a relationship. Even surrounding dramatic changes in the leverage of both firms, we do not observe commensurate changes in volatility. In fact, these two companies both increased their leverage following steep *declines* in volatility. The degree of leverage is an endogenous, firm-specific decision; it stands to reason that low-risk companies incur more leverage, all else equal, precisely *because* they are more able to do so.

Exhibit 7 plots the relationship between the rolling one-year volatility of Motorola shares against its rolling asset-to-equity ratio. The volatility of Motorola stock varied widely between 1991 and 2013, reaching a high of 70% during the Global Financial Crisis and a low of approximately 20% in recent years. Yet, its leverage hardly changed at all during this period. Furthermore, when the company tripled its leverage between 2015 and 2018, there was no apparent impact on its volatility. There does not appear to be any discernible relationship between the leverage and volatility of Clorox, either, as shown in Exhibit 8. For example, the firm has gradually reduced its leverage since 2011 only to see its share price volatility rise.

Next, we explore some of the fundamental factors that hide the effect of leverage on volatility. We find that financial leverage is a poor indicator of effective leverage for firms with substantial non-debt obligations. Moreover, we find that fundamental risk is unobservable and difficult to proxy. We illustrate these issues in Exhibit 9 by analyzing six companies: Lincoln National, AIG, GM, McDonalds, GE, and Cummins. Our observations are as follows.

▪ Compared with the other companies, Lincoln National and AIG have relatively low leverage but high volatility. This is because they carry a lot of non-debt liabilities in the form of insurance reserves.

▪ Within the consumer discretionary sector, GM has lower leverage than McDonalds but higher volatility. This may reflect the substantial size of its pension fund.

▪ Within the industrial sector, GE has higher leverage than Cummins but lower volatility. It could be the case that GE is fundamentally safer. It has significantly lower borrowing costs; however, its profitability measures are similar to Cummins’ profitability measures.

Exhibit 9 highlights but a few of the countless factors that can affect a firm’s inherent risk and thereby complicate the relationship between its leverage and volatility. We submit that this complexity renders it difficult, if not impossible, to control systematically for these factors in public markets let alone in private markets where the required data are scarce.

## CONCLUSION

The observed volatility of private equity is unrealistically low compared with the volatility of public equity. Investors have recognized that this apparent low volatility arises from the fact that valuations are based on appraisals that are anchored to prior period valuations. This practice has the effect of smoothing returns. Investors can offset this effect by estimating volatility from rolling longer-interval returns, such as annual or triennial returns. With this adjustment private equity volatility is similar to public equity volatility. Of course, it will differ by sub-categories within private equity.

Be that as it may, many investors believe that private equity volatility should be much higher than public equity volatility because private equity is much more highly levered than public equity. We showed both theoretically and by simulation that volatility should scale directly with leverage. However, it turns out that private equity volatility, adjusted for smoothing, is approximately equal to public equity volatility despite its much greater leverage, which presents a puzzle. To address this puzzle, we resorted to analyzing the relationship between leverage and volatility in the public market, where data are easier to come by. Using sorts as well as time series and cross-sectional regressions, we were unable to detect any relationship between leverage and volatility that conforms even remotely to the theoretical relationship of leverage and volatility. To put it bluntly, our results were robustly insignificant.

Next, we sought to understand why we could not uncover a statistical relationship between leverage and private equity. First, we reviewed the time series properties of leverage and volatility for two companies. We discovered that leverage is often stable for long periods whereas volatility is highly time varying. This is the proximate cause of why we cannot detect a time series relationship.

We next reviewed the financial conditions of several companies. This analysis revealed that companies have several sources of implicit leverage, which obscures the effect of explicit leverage. We also observed that asset stability differs significantly across companies for a variety of business reasons, which confounds the effect of explicit leverage on volatility. This anecdotal evidence suggests to us that the relationship of leverage and volatility is hopelessly obscured by a variety of confounding effects.

We assert that the volatility we estimate from longer-interval private equity returns (to offset the effect of valuation smoothing) is the correct approximation of private equity volatility because it approximates the actual distribution of outcomes realized by private equity investors over longer horizons. When applied to the data, this approach yields the stubborn conclusion that private equity volatility is similar to public equity volatility despite its higher leverage. Why is this the case? It could be that buyout fund managers prefer to invest in companies whose underlying business activities are inherently less risky and can therefore bear higher leverage, which increases profits. Whatever the reason, our findings debunk the widespread misconception that private equity has higher volatility than public equity volatility as a result of its higher leverage.

## ENDNOTES

↵

^{1}For example, Anson (2013, 2016, 2017) finds that the serial correlation in private equity returns extends back three to four quarters. The observed volatility of private equity is also understated because it is based on returns net of performance fees, which attenuate upside deviations but not downside deviations. Because standard deviation does not distinguish between upside and downside deviations, performance fees reduce standard deviation but not downside risk, which is what matters. In our analysis, we ignore this effect because it is not that large; we have found in previous research that it misstates standard deviation by about 1%.↵

^{2}Not all privately equity investments are levered. In this article, we focus on buyout funds because they represent the largest segment of the private equity market overall and are of the greatest interest to large, institutional investors. Buyout investors employ more leverage, on average, than public companies.↵

^{3}Other studies, such as Anson (2016), find that de-smoothed private equity returns have higher volatility than public equities. Estimates vary based on the source of private equity data and the specific de-smoothing methodology. We therefore base our conclusions on the volatility of longer interval private equity returns, which we discuss later in the article.↵

^{4}The State Street Private Equity Index captures the pooled internal rate of return each quarter for 3,097 private equity funds in which State Street clients are limited partners. The IRR calculation incorporates quarterly net asset values (NAVs) for each fund as well as daily net cash flows that occurred throughout the quarter. The index captures a total commitment size of $3.1 trillion as of September 30, 2019.↵

^{5}An alternate approach is to estimate the serial correlation statistics of the return series with a regression and reverse engineer the estimated smoothing effects. The approach is often called de-smoothing. Measuring volatility from longer interval returns has the same effect because multi-period returns contain all of the effects of serial correlation. Longer interval returns have the advantage that they represent an actual investment outcome, and they do not require a model to estimate.↵

^{6}We draw from a log-normal distribution with a mean of 0.50 (representing the debt-to-equity ratio) and a lower bound of zero. Then we add 1 to the simulated values to put them in units of assets-to-equity.↵

^{7}We proxy borrowing costs with Worldscope item 08356, “Interest Rate - Estimated Average,” which is computed as interest expense divided by the sum of short-term debt, current long-term debt, and long-term debt. We exclude observations with zero borrowing costs, negative borrowing costs, and those that reside in the 5% right tail of the distribution. All data are from the Worldscope database obtained via Datastream.↵

^{8}We compute the level of cash flows as cash flow from operations (Worldscope item 18310A) divided by the book value of assets (Worldscope item 03501A). We compute the volatility of cash flows as the rolling five-year volatility of the level. For both variables, we exclude observations that reside in the 5% right tail of their respective distribution. All data are from the Worldscope database obtained via Datastream.↵

^{9}We remove the tails to prevent firms that are experiencing extreme volatility driven by crisis or extraordinary circumstances from having undue influence on our results.**DISCLAIMER**This material is for informational purposes only. The views expressed in this material are the views of the authors, are provided “as-is” at the time of first publication, are not intended for distribution to any person or entity in any jurisdiction where such distribution or use would be contrary to applicable law, and are not an offer or solicitation to buy or sell securities or any product. The views expressed do not necessarily represent the views of Windham Capital Management, State Street Global Markets®, or State Street Corporation® and its affiliates.

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