Measuring volatility

Unbiased estimates of asset class risk and return are essential inputs into the investment selection and portfolio planning process. However, private equity presents a challenge to many sophisticated investors looking to incorporate even the most basic risk metrics into traditional mean-variance portfolio optimisation frameworks and other risk analytic tools.

This is because private company valuations are subject to infrequent appraisal pricing, while observed periodic returns appear smoothed even under prevailing fair market value accounting standards. Unadjusted periodic returns of private equity may underestimate volatility and correlation to public markets – so mean-variance portfolio optimisation models that use risk inputs based on unadjusted returns frequently suggest overallocation to illiquid private equity investments.

This article examines what might cause investors to underestimate the volatility of private equity and looks at the statistical properties of unadjusted historic private equity returns in comparison to public market returns. It then discusses potential methods to estimate private equity volatility that correct for the smoothness in unadjusted observed returns. These estimates may prove useful in mean-variance portfolio optimisation and other risk control models.


Private equity portfolio companies face the same conditions that affect public companies, such as purchase prices, financing conditions and macroeconomic growth. Observed private equity returns appear to correlate positively with public market returns, but with significantly less volatility. Simple linear regression models of observed private equity returns relative to public market returns subsequently imply a positive beta of less than one.

The apparent lower volatility of private equity may be counter-intuitive to investors. Champions of the asset class often cite high returns, but rarely discuss low volatility as a benefit. Sceptics might argue that private equity is just leveraged public equity. Should leverage not increase volatility? That is not what the raw data reveal.

The distributions of quarterly private equity and public market returns, from 1990 to 2013, are compared in Figure 1. As private equity invests globally across a variety of industries, the MSCI All Country World Index (ACWI) serves as a suitable public market benchmark (shown here with gross dividends reinvested). The statistical difference between both return streams is highlighted in Table 1. Consistent with academic research on the return premium for private equity investments, the annualised private equity return provides a significant premium above the annualised public equity return. While the correlation between the return streams is 71 percent, the observed annualised standard deviation of private equity is 64 percent of the public market volatility.

Figure 1.

Figure 2 demonstrates heteroskedasticity in the return streams of both private equity and public markets, meaning that the standard deviation in the samples is not constant over time. The rolling five year standard deviation of private equity clearly increases due to crisis, such as in 2008, and dissipates during recovery periods. Although the overall level of volatility in the market may change, the ratio of the volatility of private equity to public markets typically remains in a tight band between 60 percent and 80 percent. Reviewing the idiosyncrasies of private equity accounting is necessary in order to understand the drivers of the low volatility of this asset class.

Figure 2.


To practitioners familiar with the public markets, private equity may appear highly unusual. The general partner (GP) provides information on the value of limited partner (LP) holdings on a quarterly basis. Although a GP must mark its portfolio holdings to market, it can use some discretion in selecting a value within a reasonable range for unlisted assets. Funds are audited once per year and the auditing party verifies the valuation methodology. Unlike hedge fund managers, private equity GPs only receive carried interest payments on realised investments, so there is little direct monetary incentive to write up unrealised assets aggressively. Therefore, GPs tend to hold unlisted assets at a value close to that of the prior quarter, provided that this remains within a reasonable range of fair value. The result is a lower observed volatility in private equity that does not necessarily reflect lower risk.

We can quantify the extent of this phenomenon by asking: what happens to that portion of new information that is not incorporated into that quarter’s valuation? The answer is that the information must be incorporated in subsequent periods, certainly before an asset is sold. Each quarter’s return represents some new information from that quarter and some portion of information leftover from prior quarters. The result is that the reported return of a private asset in the current quarter tends to be in the same direction (that is, positively correlated) with that asset’s return in the prior quarter. This correlation of an asset’s return with its own prior return is known as serial autocorrelation. The magnitude of autocorrelation in private equity returns is 0.41 compared to 0.05 for the MSCI ACWI. This is strong evidence that the lower volatility of private equity is the result of appraisal pricing.


Given private equity’s high ratio of return to volatility, an unconstrained, classic meanvariance model will greatly favour allocation to private equity over public markets. However, this is not a viable option for many institutions due to the illiquid nature of private equity investments. One way to avoid model outputs that allocate significant capital to private equity is to set a simple allocation constraint based on an investor’s ability to handle illiquidity. This is a prudent method to determine potential private equity exposure, but severely constraining model outputs largely defeats the purpose of using mean-variance optimisation for allocation decisions. Other options involve estimating the adjusted, or ‘de-smoothed’, volatility of private equity for use as an input in optimisation models.

Some practitioners may use brute force methods to arrive at a higher volatility input for private equity. In fact, the author has seen many portfolio consultants simply ignore observed return data for private equity altogether, and instead use volatility inputs for the asset class that approach 35 percent just to force a model to suggest less exposure! This type of adjustment to model inputs is a blunt and unsophisticated
method to prevent over-allocation.

Fortunately, there are more robust analytical methods that can be used to arrive at adjusted private equity volatility.

One option is to use the volatility of annual returns rather than quarterly returns as a model input, as GPs have shown evidence of a valuation ‘true up’ occurring once a year. The ratio of private equity to public equity volatility is roughly 50 percent during the first three quarters of each year, and increases to 93 percent during the fourth quarter. This supports the notion that larger movements in valuation occur during end-of-year fund audits. As such, the volatility of annual returns provides a higher standard deviation estimate than annualising the volatility of quarterly returns. However, using annual returns to estimate volatility greatly decreases available data points for analysis, thereby increasing the effect of noise in a data set. Stronger methods can be found by turning to research on the effects of stale pricing in other illiquid asset classes.

In real estate, the infrequent appraisal of properties leads to significant serial autocorrelation, and similar phenomena have been revealed in the return streams of hedge funds. In these asset classes, a common approach has been to adjust, or ‘de-smooth’, the observed returns to create a new set of returns which are more volatile. The adjusted returns may more accurately capture the volatility risk of the underlying asset class than the observed returns.

A mathematical model for one quarter frequency autocorrelation correction is shown below.


Using this approach, the observed value at time t can be expressed as a weighted average of the ‘true’ value at time t and the observed value at time t-1. The long-term autocorrelation of the asset class is used as the weighting mechanism. As presented previously, the autocorrelation of private equity in our sample set is 0.41 compared to a public equity autocorrelation of 0.05. The expression above yields an adjusted return series for private equity with close to zero first order autocorrelation and significantly higher measured volatility.

Figure 3 highlights the distribution of ‘de-smoothed’ private equity returns versus reported private equity returns and the MSCI ACWI public market benchmark. The statistics related to the adjusted private equity returns are shown in Table 2. The process of adjusting returns has created a quarterly return distribution that is closely aligned with the public market return distribution. This adjusted private equity return stream has a 14 percent expected return and a 73 percent correlation with public markets, which is relatively unchanged from the unadjusted return stream. The primary effect of autocorrelation correction is to increase the standard deviation estimate, which is now approximately equivalent to the standard deviation of the MSCI ACWI.

Figure 3.


The danger of underestimating the volatility of private equity is real. Hamilton Lane analysis reveals significant autocorrelation in the historic series of returns. However, making flawed assumptions or taking modelling shortcuts to prevent private equity allocation can be a costly mistake given private equity’s long-term outperformance of
public markets (by a significant margin).

As demonstrated here, single period autocorrelation correction can be an effective tool for investors looking to estimate the ‘true’ volatility of private equity – which is roughly equivalent to the volatility of public market benchmarks. This powerful conclusion may empower portfolio managers to endorse private equity as a significant
part of their total exposure.

Griffith Norville is a vice president with Hamilton Lane’s Research team, where he is responsible for analysing private equity investment strategies and market trends for a global firm with over $29 billion in discretionary assets under management.

This is an excerpt of a chapter from PEI’s new book Private Equity Mathematics. More information about the guide can be found here.