7.1.1 Types of Benchmarks

The return on a benchmark is usually calculated as an average of the returns from a number of assets. There are two general types of benchmark returns that might be used in the analysis of fund performance: peer and index.

Peer benchmarks are based on the returns of a comparison or peer group. The peer group is typically a group of funds with similar objectives, strategies, or portfolio holdings. The group may include virtually all possible comparison funds, known as a universe group, or a sampling. Instead of using a peer group of similar funds, a comparison group may be formed that contains some or all of the underlying securities that a fund might have in its portfolio. Unlike indices, comparison groups and peer groups tend to be customized for the specific needs of an investor analyzing one or more holdings. Thus, a particular financial institution, such as a pension fund or a pension consulting firm, might create comparison groups to benchmark managers against similar funds. Often the mean or median return of the group is subtracted from the return of the fund being analyzed to estimate abnormal returns. Also, the return of a fund being analyzed might be displayed in a graph or table alongside all the returns from a comparison group, rather than simply summarized using the mean or median return. The returns of a fund relative to its peer group are often expressed as a ranking or percentile in relation to the group.

Indices such as the MSCI World Index, a highly diversified equity index including stocks from 24 developed countries, and the Russell 2000 Index are commonly used as benchmarks. Indices typically reflect weighted averages of the returns of a set of securities or funds. Indices tend to be used for a more general audience and are often available for use by a variety of investors to gauge the performance of an investment, a market, or a sector.

7.1.2 A Numerical Example of Simple Benchmarking

Exhibit 7.1 lists the returns of 10 funds over 20.5 years of actual monthly data. The first eight rows of data summarize the returns of eight hedge funds chosen mostly at random. The next two rows contain the data for two well-known and diversified equity mutual funds (labeled Fund I and Fund J). The last row contains the returns of the MSCI World Index, a highly diversified equity index that includes stocks from 24 developed countries but excludes stocks from emerging markets, thus rendering the index less worldwide than is suggested by its name. Exhibit 7.1 compares the average annualized return of each fund with the MSCI World Index as the benchmark, demonstrating that 9 of the 10 funds generated higher average performance, as shown in the final column. The calculations in the chart are rounded.

Focusing on the returns of Fund A, and assuming that the MSCI World Index is an appropriate benchmark for the fund, Exhibit 7.1 indicates that Fund A outperformed the MSCI World Index by 0.66% per year and did so with a standard deviation of returns (volatility) of 14.73%, a little less than the 15.46% volatility experienced by the MSCI World Index. Thus, Fund A outperformed the proposed benchmark on both a risk basis and a return basis.

7.1.3 Three Considerations in Benchmarking

Three related questions arise regarding the previous analysis of the risk and return of Fund A relative to its assumed benchmark, the MSCI World Index: (1) Is the benchmark appropriate, meaning that the risk and return drivers of the fund are similar to the drivers of the benchmark? (2) Did the fund outperform the benchmark to an economically and statistically significant degree? (3) Why did the fund outperform its benchmark?

A performance analysis attempts to determine whether deviations of an investment's returns from its benchmark were the result of having different risk exposures than the benchmark or were attributable to non-risk-related factors, such as superior management or luck.

These issues are related to the concepts within financial economics and statistics that form the heart of alternative investment analysis. A solid understanding of benchmarking requires a solid foundation in finance and statistics. Before delving further into benchmarking in general and the example of Fund A in particular, the next section briefly discusses the variety of models that can be used to better understand financial markets.

7.2.1 Normative versus Positive Models

Evaluating the potential effectiveness of an investment strategy is a key aspect of alternative investing. Understanding whether a strategy is based on normative reasoning, positive reasoning, or both is essential.

In financial economics, a normative model attempts to describe how people and prices ought to behave. A positive model attempts to describe how people and prices actually behave. For example, when a hedge fund manager implements a trade in an attempt to benefit from a forecasted change in prices, did the manager base that forecast on how prices should behave or on his observation of how prices have behaved in the past? This essential issue is at the heart of analyzing many trading strategies and is perhaps the most fundamental aspect of a trading model that should be understood.

Normative economic models tend to be most useful in helping explain underlying forces that might drive rational financial decisions under idealized circumstances and, to a lesser extent, under more realistic conditions. Normative approaches can be used to identify the potential mispricing of securities by identifying how securities should be priced. Trading strategies based on normative reasoning anticipate that actual prices will converge toward normatively derived values if the models are well designed. Arbitrage-free pricing models, discussed in detail in Chapter 6, are normative models.

Often, people do not behave in adherence to the rational prescriptions of economic theory. Positive economic models try to explain past behavior and then predict future behavior. Positive economic models are often used to try to identify mispricing of securities by recognizing patterns in actual price movement. Technical trading strategies are based on positive economic modeling.

Alternative investment analysis uses both normative and positive modeling. The effectiveness of models should not be judged solely on the reality of their assumptions or on their ability to explain the past. Primary attention should also be given to their ability to predict the future. Both normative and positive models can be useful in understanding and predicting future behavior.

7.2.2 Theoretical versus Empirical Models

An issue related to normative and positive modeling is theoretical and empirical modeling. As introduced in Chapter 6, theoretical models describe behavior using deduction and assumptions that reflect well-established underlying behavior. For example, the price of simple options can be deduced through a number of underlying assumptions, including that financial markets are perfect, that stock prices follow a particular process, and that arbitrage opportunities do not exist. Empirical models are primarily based on observed behavior. For example, the relationship between the observed prices of option trades and particular underlying variables might be analyzed through time and fitted with an approximation function. Whether the theoretical approach or the empirical approach is more effective in explaining and predicting behavior depends on the complexity of the relationships and the reliability of the data.

Theoretical models tend to explain behavior accurately in more simplified situations, in which the relationships among variables can be somewhat clearly understood through logic. Arbitrage-free models are developed by theory.

Empirical models tend to explain complex behavior relatively well when there are many data points available and when the relative behavior of the variables is fixed or is changing in predictable ways. For example, an empirical model might be better than a theoretical model in the case of a frequently traded but extremely complex security with many overlapping option features. In such a complex case, the most accurate models might simply fit curves to the relationships based on observations of past data. The numerous and complex attributes of the security may make theoretical modeling impractical.

Alternative investing tends to lend itself more to empirical models than to theoretical models. The reason is that alternative investments tend to be characterized by illiquidity, changing risks, dynamic strategies, or other complexities that can foil theoretical modeling. Empirical modeling in the midst of such complexities, however, may also be inadequate, especially when data are limited.

7.2.3 Applied versus Abstract Models

The distinction between applied and abstract modeling is perhaps the easiest distinction in understanding research methods. Applied models are designed to address immediate real-world challenges and opportunities. For example, Markowitz's model, which is an applied model, provides useful insights for accomplishing diversification efficiently. Many Markowitz-style models are used throughout traditional and alternative investing to manage portfolios. Most asset pricing models are applied models.

Abstract models, also called basic models, tend to have applicability only in solving real-world challenges of the future. Abstract models tend to be theoretical models that explain hypothetical behavior in less realistic scenarios. For example, a model might be constructed that describes how two people with specific utility functions might bargain with regard to prices in a world with only two people and two risk factors. Eventually, abstract models can lead to innovative applications.

Models in alternative investing, especially those described in this book, are applied models. They are intended and used for solving immediate real-world problems, such as managing risk and evaluating potentially profitable investment opportunities.

7.2.4 Cross-Sectional versus Time-Series Models

Both cross-sectional and time-series models are used throughout economic modeling in alternative investments. Cross-sectional models analyze behavior at a single point in time across various subjects, such as investors or investments. Time-series models analyze behavior of a single subject or a set of subjects through time. When a data set includes multiple subjects and multiple time periods, it is often called a panel data set, and it is analyzed with a panel model. Panel data sets combine the two approaches by tracking multiple subjects through time and can also be referred to as longitudinal data sets and cross-sectional time-series data sets.

For example, consider a researcher analyzing returns on REITs (real estate investment trusts) using a particular REIT index where the index is simply an arithmetic average of the returns of each of the REITs. At first, the researcher builds a model that explains the index returns of the REITs through time using such variables as changes in Treasury rates, mortgage rates, and equity prices. This time-series model might tell the researcher how the average REIT returns are explained in terms of various market factors. The researcher might then use a cross-sectional model to attempt to explain why the long-term average returns of various REITs differed. The researcher might regress the long-term returns of the individual REITs against such variables as geographic region, property type, and leverage. If the researcher put all of the short-term returns for each time period and for each REIT into a single data set and econometric model, it would be a panel study.

A large and growing body of time-series analysis focuses on the way an asset's unexplained price risk, measured as its volatility, might change through time. Examples of this approach include autoregressive conditional heteroskedasticity (ARCH) and generalized autoregressive conditional heteroskedasticity (GARCH) analyses, which focus on time-series behavior such as potential patterns through time in the variance of unexplained return.

7.2.5 Importance of Methodology

The primary purpose of this section has been to describe how to identify the nature of a model using four distinctions: normative versus positive, theoretical versus empirical, applied versus abstract, and cross-sectional versus time-series models.

It is important to be able to understand investment analysis from a methodological perspective in order to better organize and compare investment strategies. For example, one manager may have identified a profitable trading opportunity by specifying the proper equilibrium price of an asset and recommending trades when the actual price deviated from the ideal price. Another manager may have detected a statistical pattern to actual trading on the last day of each month and used that as a signal for trades. The first manager used a theoretical and normative model, whereas the other manager used an empirical and positive model. Both models were applied. By evaluating these managers in the context of their methods, an analyst may be able to better evaluate the prospects for investment success.

7.3.1 Single-Factor Market Model Performance Attribution

The purpose of this section is to demonstrate the use of a single-factor market model for performance attribution. Whether using a simple benchmark approach or a formal asset pricing model, virtually every investment professional who is evaluating performance must adjust for risk. Hence, every professional is explicitly or implicitly using an asset pricing model. This section is designed to (1) equip analysts with more advanced and robust approaches than simple benchmarking, and (2) facilitate an explicit recognition of the assumed relationship between risks and return that underlies a return attribution analysis.

The example uses a single-factor approach for simplicity, even though many alternative assets would likely benefit from a more robust multifactor method. The single-factor market model approach is similar in many ways to the capital asset pricing model (CAPM); however, an important distinction should be kept in mind. The CAPM describes efficiently priced assets wherein the expected returns of all assets are directly and linearly related to their market betas (i.e., all assets have equal Treynor ratios). In practice, performance attribution is used to evaluate assets presumed to have potential price inefficiencies (i.e., different levels of risk-adjusted performance). The ex post form of the CAPM was given in Equation 6.3 and is repeated here:

In the CAPM, the error term on the far right-hand side of the equation is presumed to have a zero mean. The risk-free rate can serve as the intercept or, as is represented in the equation, can be subtracted from the asset's return to form its excess return. A single-factor market model allows an intercept that is not equal to the riskless rate and that can indicate abnormally high or low returns due to mispricing. Thus, a difference between a CAPM model and the single-factor market model is whether consistently abnormal returns are allowed to be captured in the intercept term or are disallowed due to a presumption of informational market efficiency.

The first component of asset i's realized return is from the effect of systematic risk: the effect of the realized return of the market portfolio. The error term is the effect of idiosyncratic risk. The equation can be used to perform a single-factor return attribution by inserting the known returns and estimating the unknown terms: the beta and the error terms. Given an estimate of the security's beta, the equation may be used to estimate the idiosyncratic returns of the security.

The idiosyncratic return of asset i represents the portion of asset i's realized return that is not attributable to its market risk. In the previous example, the estimated idiosyncratic return of asset i in year t was –1%. Asset i's performance benefited from the higher-than-expected returns of the overall market and its high beta, but it suffered a small setback (–1%) from the combined effects of the idiosyncratic effects.

Attempts by investment managers to earn superior risk-adjusted returns may be viewed as attempts to construct portfolios for which the idiosyncratic return is positive. Asset i seemed at first to have performed well with its 16% realized return, but after risk adjustment, its realized return was found to be 1% lower than it should have been, given its level of risk.

This example illustrates the return attribution process for a single security. Return attribution can be similarly performed for the returns of portfolios of securities and for investment funds. Performance attribution can be used to indicate whether the manager generated superior risk-adjusted returns or whether the returns can be attributed to other factors. As with any exercise involving randomness and unobserved components such as beta, the results of this return attribution analysis are estimates. For example, because the beta of the stock would typically be an estimated value with some level of error, the estimate of the attribution of returns into the portions due to systematic and idiosyncratic risks would similarly be subject to error. Further, another analyst may have measured the security's beta differently or may have used different returns to represent the risk-free and market portfolio returns.

Does estimated superior performance indicate skill on the part of the manager? First, there is the issue that the performance analysis may contain estimation errors due to flaws in the return attribution process, such as model misspecification. If systematic risks were ignored or misidentified, the returns will not be accurately attributed to risks. But even if this estimate could be considered accurate and reliable, an important issue remains: Was the estimated superior return generated due to skill or to luck? That important issue should be addressed statistically and is discussed in detail in Chapters 8 and 9.

7.3.2 Examining Time-Series Returns with a Single-Factor Market-Based Regression Model

The ex post form of the single-factor market model can be used in a time-series model to better understand and estimate the effects of systematic risk and idiosyncratic risks through time. This section focuses on using a single-factor model. Multifactor models could be similarly used and should typically be used for alternative investments, but we are focusing on a single-factor model because of its relative simplicity. An estimated single-factor time-series model is typically written like this:

(7.1)

R_it is the return for the asset in period t, and R_mt is the return for the market. The equation's parameters (a and B) are usually estimated using a regression method, which is performed over a set of time periods for a particular asset (i).

The Greek letters α, β, and ϵ tend to be used to represent the true and unobservable variables, and the Latin letters a, B, and e are used to represent the estimates of those variables from a statistical procedure (such as a regression, as discussed in Chapter 9). Thus, ϵ_it in the theoretical model represents the true portion of asset i's return attributable to the effects of idiosyncratic risks in time period t, whereas e_it in equation 7.1 represents the researcher's estimate of ϵ_it using a statistical analysis, in this case a regression, and a particular model.

If the CAPM describes returns perfectly, then empirical tests of Equation 7.1 should indicate the following: (1) the intercept of the regression equation, a, should be statistically equal to zero; (2) the slope of the regression equation, B_i, should be statistically equal to the true beta of the asset; and (3) the residuals of the regressions, e_t, should reflect the effects of idiosyncratic, asset-specific risks. But many alternative assets may trade at prices that depart from perfectly efficient prices. Hence, analysts often use this time-series approach and interpret statistically nonzero intercepts as a signal of asset mispricing. Statistical testing using linear regression is an important and multifaceted subject. The statistics of linear regression are summarized in Chapter 9 and explained in more detail in books about quantitative and statistical techniques for finance and investments.

7.3.3 Application of Single-Factor Benchmarking

The application of single-factor performance attribution in section 7.3.2 used the market portfolio as the factor. This section illustrates the use of the investment's benchmark as the single factor.

We return to the example in the beginning of the chapter summarized in Exhibit 7.1. In the exhibit, the performance of Fund A was analyzed by directly comparing its return (as well as the return of the other funds) to the return of the MSCI World Index, assuming that the MSCI index served as a reasonable benchmark. This section examines the performance of the funds using the same benchmark but with a single-factor model framework, which allows each fund to have a different sensitivity or beta with respect to the benchmark/factor. In other words, the previous simplistic benchmarking example implicitly assumed that the beta of each fund with respect to the benchmark was 1. This section allows that beta to depart from 1 depending on the observed sensitivity of each fund's return to the return of the benchmark.

Equation 7.2 illustrates the concept of benchmarking with a single-factor linear regression model, in which the benchmark takes the place of the market factor:

(7.2)

where R_t is the return of a fund in period t, R_f is the riskless rate, a is the intercept of the regression (usually viewed as an estimate of the average overperformance or underperformance of the fund through time), B is the sensitivity of the fund's return to the benchmark's return (which is typically expected to be near 1 for a benchmark with equivalent risk to the fund), R_benchmark,t is the return of the fund's benchmark, and e_t is the fund's estimated idiosyncratic return above or below its risk-adjusted return.

Exhibit 7.2 shows results from this time-series analysis of the returns of Fund A as the variable on the left-hand side of Equation 7.2) R_t) and the MSCI World Index as the market and benchmark return (R_benchmark,t) on the right-hand side. Fund A's return has an estimated beta of 0.68. This means that rather than containing the same level of systematic risk as the index, Fund A's return tended to move only 68% as far as the market each time the market moved. The analysis appears to magnify the favorable implications of Exhibit 7.1. The estimated annual performance of Fund A in Exhibit 7.2 was 1.3% higher than would be expected in a perfectly efficient market, as indicated by the intercept. Thus, a single-factor risk-adjusted analysis indicates that Fund A outperformed its benchmark by 1.3% per year while taking only 68% of the systematic risk of that benchmark.

7.3.4 Multifactor Benchmarking

Exhibits 7.1 and 7.2 summarize return attribution of Fund A using a single benchmark. The first exhibit implicitly assumes that the beta of Fund A with respect to its benchmark is 1, whereas the second exhibit allows the returns to be attributed to a different level of systematic risk than the benchmark. Now let's return to the example using a multifactor model: the Fama-French-Carhart model detailed in Chapter 6. Using an ex post form of that model, the performance of Fund A can be explained by three systematic risk factors, as indicated in Exhibit 7.3.

The estimated betas in Exhibit 7.3 indicate that Fund A's return included exposures to size, value, and momentum factors in addition to its exposure to the market index. Note that the annual idiosyncratic performance (the intercept) is now estimated as being 2.91% lower than would be obtained in a perfectly efficient market. What was previously estimated as a 1.3% positive alpha using a single-factor model is now estimated as a –2.9% alpha using multiple factors. Apparently, the intercept of Fund A using a single-factor model was erroneously identified as an indication of superior return rather than as compensation for the omitted risk exposures that the fund was incurring by investing in small-capitalization value stocks with a high degree of momentum. This indication that performance was inferior is in marked contrast to the estimated superior performance shown in Exhibits 7.1 and 7.2, using a simple benchmark and single-factor approach, respectively.

In an up market (i.e., a market in which major indices outperformed the riskless rate), the omission of systematic risk factors will result in an analysis that overestimates the risk-adjusted performance of assets positively exposed to the omitted risk factors and underestimates the performance of assets negatively exposed to the omitted risk factors. In a down market, the anticipated effect would be the opposite. Most long-term studies are more likely to be up markets, since risky assets on average outperform the riskless asset.

The analysis underscores the importance of methodology. Benchmarking is only as accurate as the model that implicitly or explicitly serves as its foundation. Thus, the study of methods and the careful selection of an appropriate method serve as key processes in the attribution of return performance.

The benchmarking examples of Exhibits 7.1, 7.2, and 7.3 illustrate the great difference between applying a single-factor model and applying a multifactor model. There are solid reasons to believe that alternative investing is especially exposed to systematic risk factors other than the market risk factor. Multifactor models provide a more robust basis for understanding and estimating the sources of the realized and expected returns of alternative investments. Multifactor models may allow analysts to better separate systematic risks from idiosyncratic risks and perhaps to separate superior performance based on skill from superior performance based on luck.

7.4.1 Why Not Apply the CAPM to Alternative Assets?

In an ideal world without market imperfections, with normally distributed asset returns, and with a stationary distribution for the returns of the market portfolio, assets should tend to be priced well using the CAPM. The CAPM collapses all of the potential complexities of investments into one simple assertion: All investors fully diversify into the same portfolio of risky assets, the market portfolio, which defines the one and only systematic risk factor. The CAPM separates systematic risk from idiosyncratic risk with a single factor, the return of the market portfolio, and specifies how expected and actual returns are determined. In a CAPM world, the only way that an investment manager can consistently earn higher returns is by taking more market risk, and any investor bearing idiosyncratic risk is acting irrationally. In a CAPM world, there is no distinction between traditional and alternative investment methods.

Are there solid theoretical reasons to believe that the CAPM does not hold for alternative assets? Can assets require higher returns for any risk other than the beta of an asset with the market portfolio? Does there need to be more than one risk factor in a performance attribution analysis? This section explores three primary reasons why the CAPM approach to investing may not work for alternative investing.

7.4.2 Reason 1: Multiperiod Issues

The CAPM is a single-period model, in which it is assumed that all investors can make an optimal decision based only on analysis of the outcomes at the end of the one period. Investors do so with assurance that by repeating the process of optimizing each single period's decision, the investor's lifetime decisions will be optimized. All investors are assumed to share the same one-period time horizon for their decision-making.

For the CAPM to hold in a world of multiple periods, it is usually assumed that, among other things, the market's return process behaves in similar patterns through time. If return distributions of securities or distributions of corporate earnings randomly change through time, Merton as well as Connor and Korajczyk demonstrate that additional systematic risk factors will emerge, and a single-factor approach will no longer hold.¹

For example, assume that the expected return of the market portfolio varies through time in relation to the average credit spread risk in the marketplace. In that case, it is possible that credit spread risk can become an additional factor, and the single-factor CAPM approach must be expanded into a multifactor model. Similar arguments for additional risk factors have been made if the variance or dispersion of the market changes through time in relation to an economic variable.

Multiperiod issues could affect both traditional and alternative investment pricing. However, the relatively dynamic nature of alternative investments and their unusual return distributions (e.g., structured products) tend to make this issue more important for the analysis of alternative investments.

A highly related issue involves investment uncertainty in generating cash to fund multiperiod liabilities and uncertain liabilities. Many financial institutions manage their portfolios with the goal of funding a stream of future liabilities rather than simply trying to control risk and return one period into the future. For instance, a major portion of a college's endowment portfolio might be managed to fund a major construction project. Given real-world costs of hedging risks, it would not be optimal for the college to hold the same diversified portfolio that an insurance company or a pension fund is holding. If the endowment's goal is to reduce the risk that the future construction costs will not be met, then the portfolio should account for the unique sources of risk affecting the matching of asset values with future liabilities. Further, the liabilities themselves can be driven by risk exposures other than a single market factor.

7.4.3 Reason 2: Non-Normality

The normal distribution can be specified with two parameters: mean and variance (or standard deviation). Traditional portfolio theory demonstrates that portfolios can be managed using these two parameters if returns are normally distributed or if investors have preferences that require analysis of only those two parameters. If returns are non-normal, then investors may be concerned about additional parameters, such as skewness and kurtosis.

Alternative investment returns often tend to skew to one side or the other or to have excess kurtosis, with fatter tails on both sides. Non-normality of returns tends to be greater for larger time intervals, and alternative investments by their nature tend to be illiquid and are less likely to be managed with short-term portfolio adjustments. Another reason for the non-normality of many alternative investment returns is the structuring of their cash flows into relatively risky and asymmetric patterns.

CAPM-style frameworks have been extended to include additional parameters that capture the non-normality of returns. Rubinstein, Kraus and Litzenberger, and Harvey and Siddique develop models that incorporate skewness.² Homaifar and Graddy develop a model that incorporates kurtosis.³ It should be noted that measures such as skewness and kurtosis can be difficult to estimate accurately and can change rapidly. However, difficulty in forecasting quantitative measures of high moments does not mean that higher moments are irrelevant.

7.4.4 Reason 3: Illiquidity of Returns and Other Barriers to Diversification

Illiquidity in this context refers to the risk of not being able to adjust portfolio holdings substantially and quickly at low costs. The idea within the CAPM that every investor should seek perfect diversification through holding the market portfolio is predicated on perfect liquidity. But in real life, there are substantial barriers to perfect diversification. First, transaction costs, taxes on the realized gains, and differential taxation on individual investments inhibit transactions and may offset the benefits of diversifying fully. Since many investors are unable to diversify well without substantial costs, they are exposed to risk factors other than simply the market factor, and therefore the CAPM may not adequately capture all the sources of risk that are priced.

Further, the illiquidity of a particular investment may be priced. In traditional investments, the notion of illiquidity often translates to the relatively minor illiquidity that a small stock might have in terms of inability to transact large quantities quickly without affecting share prices. But many alternative investments, such as private equity, private real estate, and some hedge funds, can have far more severe illiquidity that prevents positions from being liquidated for months or even years. In private equity, an investment might not only be difficult to liquidate but also might obligate the owner to contribute additional cash flows when demanded by the investment's general partner.

Illiquidity of many alternative assets restricts an investor's ability to adjust a portfolio continuously, including the manager's ability to control risks and manage cash. Further, the absence of regular competitively determined pricing of highly illiquid investments hampers valuation, risk measurement, risk management, and decision-making. Surely such illiquidity is undesirable to many investors. It is reasonable to expect that most investors prefer liquidity and would demand a risk premium for bearing the risk of illiquidity, which would make illiquidity another factor, in addition to the single factor of the CAPM.

The ultimate question of whether illiquidity is priced in is an empirical question. Pastor and Stambaugh provide empirical evidence that liquidity risk is related to higher expected stock returns using merely the differences in liquidity among publicly traded equities.⁴ Given the relatively small differences in liquidity between the stocks analyzed by Pastor and Stambaugh, the inference is that classes of alternative investments with high levels of illiquidity may need to be priced to offer substantially higher long-term returns (see also Khandani and Lo's analysis of illiquidity premia).⁵