Mutual Fund Risk Ronald N. Kahn

Mutual Fund Risk Ronald N. Kahn Ronald Kahn is Director of Research at BARRA. Since joining BARRA in 1987, he has published numerous articles on investment management and has written (with Richard Grinold) the book Active Portfolio Management: Quantitative Theory and Applications. Ron was a Senior Consultant in the Research Group from 1987 until 1989, when he became Manager of Special Projects. In 1991 he was named Director of Research. Ron received an A.B. in Physics, summa cum laude, from Princeton University in 1978. He received a Ph.D. in Physics from Harvard University in 1985 for his work on computer simulations of the very early universe. From 1985 to 1987, he was a postdoctoral fellow in the Physics Department at the University of California, Berkeley. Combining his work on technical subjects with a long-standing interest in writing, Ron has authored articles appearing in The New York Times, Newsweek, and other publications. BARRA 1997

Table of Contents Introduction.......................................................... 1 Quantitative Measures of Mutual Fund Risk: An Overview..................... 5 History and Perspective......................................... 5 The Uses of Risk Analysis........................................ 6 Defining Risk................................................. 7 Three Approaches to Forecasting Risk............................. 12 Other Issues................................................. 13 Recommendations............................................ 14 Figure 1-1................................................... 15 Forecasting Mutual Fund Risk: Current Holdings or Past Performance?.......... 17 Description of the Study....................................... 18 Analysis..................................................... 20 Conclusions and Recommendations.............................. 26 Data and Analytics Sources..................................... 27 Table 2-1.................................................... 28 Table 2-2.................................................... 29 Table 2-3.................................................... 30 Figure 2-1................................................... 31 Figure 2-2................................................... 32 Figure 2-3................................................... 33 Bibliography......................................................... 35

Introduction

INTRODUCTION In 1994 many mutual fund investors experienced staggering and unexpected losses. Amid the resulting public outcry, the U.S. Securities and Exchange Commission considered how to better inform the public of investment risk, and asked for public commentary. The response, from many thousands of individual investors, far surpassed the SEC s expectations. The public has considerable interest in mutual fund risk. As the leading provider of risk models and analytics to the institutional investment community, BARRA responded with two reports to the SEC. The first, Quantitative Measures of Mutual Fund Risk: An Overview, outlined the basic issues and choices as informed by our twenty years of modeling risk. The second report, Forecasting Mutual Fund Risk: Current Holdings or Past Performance? followed up with a study comparing methods for forecasting mutual fund risk. This BARRA Research Insights document collects these two reports. While the topic specifically focuses on mutual fund risk, we believe the issues and conclusions will interest our institutional clients. Quantitative Measures of Mutual Fund Risk: An Overview Institutional investors rely on risk analysis for three purposes: forecasting the risks of their current portfolio, measuring past investment risks to help interpret historical performance, and constructing new portfolios to optimally trade off expected return and risk. Individuals investing in mutual funds face these same challenges, and so they too can benefit from risk analysis. All definitions of risk arise from the distribution of returns. All single risk measures such as standard deviation, downside risk, shortfall probability, and value at risk attempt to summarize the information on returns and probabilities contained in the distribution of returns. Of these, the standard deviation is the most stable over time, and hence the most accurately forecastable. Conditional risk measures such as beta and duration measure returns conditional on a particular factor, and are inadequate single risk measures. Institutional investors have chosen the standard deviation as their risk definition. Its benefits include stability and the availability of associated powerful BARRA 1997 1

BARRA RESEARCH INSIGHTS econometric forecasting tools. By assuming a normal distribution, the standard deviation can be reported as a shortfall probability or a value at risk. Three approaches to forecasting risk include historical standard deviation, applying style analysis to historical returns to separately forecast style and selection risk, and analyzing portfolio holdings. The third approach, analyzing holdings, is the most accurate and the most costly. It is the standard choice of institutional investors. The SEC must decide on an appropriate risk definition and a forecasting methodology. We recommend using the standard deviation as the definition of risk and the analysis of portfolio holdings as the most accurate forecast of risk. Forecasting Mutual Fund Risk: Current Holdings or Past Performance? In July 1995, we recommended that the SEC define mutual fund risk as the standard deviation of the return, and that they require mutual funds to report risk based on the current fund holdings and not on the fund s past performance. We report here on a new study of 29 representative equity and bond mutual funds which clearly demonstrates that current holdings are more accurate than past performance for forecasting mutual fund risk, based on three separate tests. The first test showed that among the funds studied, negative returns would seldom have surprised investors if they had known the fund risks based on the fund holdings. In many cases, negative returns did surprise them based on past performance. These surprises occurred even though our second test found no statistically significant evidence of average over or under prediction of risk for either method. Finally, the third test identified a much stronger relationship between portfolio-based forecasts and subsequent risk than between historical forecasts and subsequent risk. Furthermore, this study implies that a large discrepancy between portfolio risk and historical risk constitutes a significant warning sign about future fund performance. 2 BARRA 1997

INTRODUCTION These results support our recommendation that the SEC require mutual funds to report risk based on current fund holdings. They also point out that the public would benefit from portfolio-based risk reports, even if the SEC does not require them. Hence we are making a third recommendation to the SEC. Mutual funds should report detailed holdings on a quarterly basis, and the SEC should make these data readily available to the public in machine readable form. BARRA 1997 3

BARRA RESEARCH INSIGHTS 4 BARRA 1997

Quantitative Measures of Mutual Fund Risk: An Overview

QUANTITATIVE MEASURE OF MUTUAL FUND RISK: AN OVERVIEW Risk analysis has been central to investing since at least the 1950 s, when Harry Markowitz showed mathematically exactly how diversification reduced risk. Since then, risk analysis has developed into a very powerful tool relied upon by institutional investors. And because the goals of individual investors buying mutual funds do not fundamentally differ from those of institutional investors, they too could benefit from investment risk analysis. This report will review how institutional investors have analyzed risk, and then discuss the advantages and disadvantages of three particular approaches to mutual fund risk for individual investors. BARRA has been a pioneer and leader in developing these applications and forecasting methods over the past 20 years, and this is where we can most significantly contribute to this debate. The report will also include a brief historical perspective, review the many definitions of risk, and discuss several issues of general concern for risk analysis. History and Perspective Harry Markowitz began the modern era of risk analysis in the 1950 s with his work on portfolio selection. He emphasized that the portfolio was the fundamental entity for investment analysis because it contained the sum total of all investments. What ultimately matters is not whether an individual investment goes up or down, but whether the portfolio goes up or down. Markowitz then showed mathematically how diversification could greatly benefit the portfolio over individual investments. A portfolio could have the same expected return as a set of individual investments, but much less risk, because the risks of individual assets tend to cancel. Initially, Markowitz detailed results were of more theoretical than practical interest because the technology did not yet exist to implement his mathematical ideas. Only with advances in computer technology and risk modeling have these ideas become feasible and available to institutional investors. This early work by Markowitz was expanded upon by others, especially by Sharpe, Lintner, Treynor, and Mossin in their work on the capital asset pricing model (CAPM). This model explicitly connected expected returns to risk. In equilibrium, investors are rewarded based on the market risk they assume. Markowitz demonstrated the importance of understanding risk in building BARRA 1997 5

BARRA RESEARCH INSIGHTS portfolios. The developers of the CAPM theorized that investors should only expect returns in proportion to the nondiversifiable risk they take on. The Uses of Risk Analysis Institutional investors today have developed three uses for risk analysis. These uses concern the present, the past, and the future. Risk analysis for the present looks at risk of the current portfolio. How volatile is the current portfolio? How closely will its return match the S&P500 return or a bond index return? What bets does it contain? How does it compare to other portfolios? Risk analysis applied to the past helps interpret past investment performance. Did past returns arise through skill or luck? What were the past returns when adjusted for risk? What investment policies were most and least successful? Risk analysis applied toward the future helps construct new portfolios, trading off the expected returns to individual assets against the risk of various combinations of those assets. Portfolio construction is a very intensive use of risk analysis, in that it must contrast risk and return for a large multitude of possible investment choices. In particular, it requires not only the risk of each asset, but also the correlation of each asset with every other asset. Along with these three uses for risk analysis, we should carefully understand two terms used in these contexts: forecast risk and realized risk. Risk analysis for the present (current portfolio risk) and risk analysis for the future (portfolio construction) rely on forecast risk: forecasts of what might happen in the future. Past risk analysis (performance analysis) relies on realized risk: the risk actually incurred as measured by actual movements in portfolio value. Past realized risk may or may not be a good forecast of future risk. Mutual fund investors would, presumably, use risk analysis to compare different funds and to help understand relative fund performance. This paper will focus on what is most relevant for these applications. 6 BARRA 1997

QUANTITATIVE MEASURE OF MUTUAL FUND RISK: AN OVERVIEW Defining Risk All definitions of risk arise fundamentally from the probability distribution of possible returns. This distribution describes the probability that the return will be between 1% and 1.01%, the probability of returns between 1.01% and 1.02%, etc. For example, Figure 1-1 displays the realized distribution of monthly returns for the Fidelity Magellan Fund, based on its performance from January 1973 to September 1994. According to this distribution, 26% of the Magellan Fund monthly returns fell between 2.5% and 7.5%. The distribution of returns describes probabilities of all possible outcomes. As such, it is complicated and full of detail. It can answer all questions about returns and probabilities. It can be a forecast, or a summary of realized returns. Conceptually, it applies to every fund type: equity, bond or other. Unfortunately the distribution of returns is too complicated and detailed in its entirety. Hence all the definitions of risk will attempt to capture in a single number the essentials of risk more fully described in the complete distribution. Each definition of risk will have at least some shortcomings, due to this simplification. Different definitions may also have shortcomings based on difficulties of accurate forecasting. Let s discuss these proposed risk definitions in turn. The standard deviation measures the spread of the distribution about its mean. Investors commonly refer to the standard deviation as the volatility. The variance is the square of the standard deviation. For the Magellan Fund, the realized monthly standard deviation is 6.3% and the mean is 1.6%. If these returns were normally distributed, then two-thirds of the monthly returns would have fallen within 6.3% of the mean, i.e. in the band between -4.7% and 7.9%. In fact, 73% of the Magellan Fund returns were in that band, reasonably close to the normal distribution result. The Magellan Fund s annual mean and standard deviation were 19.2% and 21.8% respectively. Roughly two thirds of the fund s annual returns were in a band from -2.6% to 41%. BARRA 1997 7

BARRA RESEARCH INSIGHTS As the standard deviation decreases, the band within which most returns will fall narrows. The standard deviation measures the uncertainty of the returns. Standard deviation was Harry Markowitz definition of risk, and it has been the standard in the institutional investment community ever since. It is a very well understood and unambiguous statistic. It is particularly applicable to existing tools for building portfolios. Standard deviations tend to be relatively stable over time (especially compared to mean returns and other moments of the distribution), and econometricians have developed very powerful tools for accurately forecasting standard deviations. Critics of the standard deviation point out that it measures the possibility of returns both above and below the mean. Most investors would define risk based on small or negative returns (though short sellers have the opposite view). This has generated an alternative risk measure: semivariance, or downside risk. Semivariance is defined in analogy to variance, based on deviations from the mean, but using only returns below the mean. If the returns are symmetric, i.e., the return is equally likely to be x percent above or x percent below the mean, then the semivariance is just exactly one-half the variance. Authors differ in defining downside risk. One approach defines downside risk as the square root of the semivariance, in analogy to the relation between standard deviation and variance. The Magellan Fund had a realized semivariance of 21.6, which was 55% of its variance of 39.5. According to Figure 1-1, the distribution extended slightly farther to the left (negative returns) than to the right (positive returns), and so the semivariance was slightly more than half the variance. Downside risk clearly answers the critics of standard deviation, by focusing entirely on the undesirable returns. However, there are several problems with downside risk. First, its definition is not as unambiguous as standard deviation or variance, nor are its statistical properties as well known, so it isn t an ideal choice for a universal risk definition. Second, it is computationally challenging for large portfolio construction problems. 8 BARRA 1997

QUANTITATIVE MEASURE OF MUTUAL FUND RISK: AN OVERVIEW Third, to the extent that investment returns are reasonably symmetric, most definitions of downside risk are simply proportional to standard deviation or variance and so contain no additional information. To the extent that investment returns may not be symmetric, there are problems forecasting downside risk. Return asymmetries are not stable over time, and so are very difficult to forecast. Realized downside risk may not be a good forecast of future downside risk. Moreover, we estimate downside risk with only half of the data, losing statistical accuracy. Shortfall probability is another risk definition, and perhaps one closely related to intuition for what risk is. The shortfall probability is the probability that the return will lie below some target amount. For example, the realized probability of a Magellan Fund monthly loss in excess of 10% was 3.4%. Shortfall probability has the advantage of closely corresponding to an intuitive definition of risk. However, it faces the same problems as downside risk: ambiguity, poor statistical understanding, difficulty of forecasting, and dependence on individual investor preferences. Forecasting is a particularly thorny problem, and it s accentuated the lower the shortfall target. At the extreme, probability forecasts for very large shortfalls are influenced by perhaps only one or two observations. Value at risk is similar to shortfall probability. Where shortfall probability takes a target return and calculates the probability of returns falling below that, value at risk takes a target probability, e.g., the 1% or 5% lowest returns, and converts that probability to an associated return. For the Magellan Fund, the worst 1% of all returns exceeded a 20.8% loss. For a $1,000 investment in the Magellan Fund, the value at risk was $208. Value at risk is closely related to shortfall probability, and shares the same advantages and disadvantages. Where does the normal distribution fit into this discussion of risk statistics? The normal distribution is a standard assumption in academic investment research and is a standard distribution throughout statistics. It is completely defined by its mean and standard deviation. Much research has shown that BARRA 1997 9

BARRA RESEARCH INSIGHTS investment returns do not exactly follow normal distributions, but instead have wider distributions, i.e., the probability of extreme events is larger for real investments than a normal distribution would imply. The above five risk definitions all attempt to capture the risk inherent in the true return distribution. An alternative approach could assume that returns are normally distributed. Then the mean and standard deviation immediately fix the other statistics: downside risk, semivariance, shortfall probability, and value at risk. Such an approach might robustly forecast quantities of most interest to individual investors, using the most accurate calculations and a few reasonable assumptions. More generally this points out that the choice of how to define risk is separate from the choice of how to report risk. For example, if standard deviation is the definition of risk, mutual funds could convert and report this as shortfall probabilities or even discrete letter grades. Other suggested risk measures are variants of the above definitions. A bar chart of realized annual returns can provide a feeling for fund risk, because it is related to the distribution of returns and contains approximately the same information. By itself the bar chart includes too many individual numbers to facilitate easy comparisons across funds. This is the same problem with the distribution of returns. A graph of daily risk over time can show how risk has changed historically, but it is no longer a single number defining risk. Beyond the above unconditional measures of risk, there exist two conditional measures of risk: beta and duration. Beta is commonly used for equity funds, and duration is commonly used for bond funds. Beta estimates return conditional on a market return. For example, the beta of the Magellan fund is 1.2. If the S&P500 drops 5%, the Magellan Fund should drop 6%, or 1.2 times 5%. This use of beta for estimating conditional returns is quite well established and beyond controversy. The CAPM model uses beta to connect expected returns to risk, and that connection is in dispute. 10 BARRA 1997

QUANTITATIVE MEASURE OF MUTUAL FUND RISK: AN OVERVIEW Duration measures expected bond return conditional on interest rate movements. For example, a bond fund with a duration of 5 will drop 5% if interest rates rise 1%. These two conditional measures of risk are less valuable than the unconditional measures we have already discussed. First, they traditionally apply only to bond or equity funds separately. Individual investors may want to compare the risks of an equity fund and a bond fund, and so a unified definition is highly desirable. Second, conditional risk measures focus attention on only one source of risk, the S&P500 or interest rates respectively for beta and duration. They ignore many other sources of risk in these investments. Gold stocks exhibit low betas, and yet they are certainly not low risk. Junk bonds can exhibit low durations, yet their default risk is quite high. As single measures of risk, these conditional quantities are insufficient. Other statistics which often arise in discussions of risk are not risk measures per se, but rather return to risk measures. These include Jensen s alpha, Sharpe s ratio of return to standard deviation, Treynor s ratio of return to beta, and the Information Ratio of return relative to the benchmark to standard deviation relative to the benchmark. These measures may be useful for understanding performance, but they are not relevant for portfolio risk analysis. Is a single number defining risk sufficient? As we have seen, each of these risk measures attempts to reduce the complete distribution of returns to a single manageable number. Simplicity dictates that one number has great advantages and certainly facilitates comparison of different funds. More generally, one risk number can suffice for understanding a fund s risks, comparing that to the risk of other funds, and helping to understand relative fund performance. Since these are the interests of individual investors, single numbers can adequately capture fund risk. Overall, the best candidates for a single number defining fund risk, identical for stocks or bonds, are standard deviation, shortfall probability, or value at BARRA 1997 11

BARRA RESEARCH INSIGHTS risk; and the standard deviation is easiest to forecast accurately. However by assuming a normal distribution, we can convert the standard deviation to a shortfall probability or a value at risk. Three Approaches to Forecasting Risk We have discussed the uses of investment risk analysis and described alternative risk definitions. Let s now focus on forecasting risk for mutual funds, and define risk as standard deviation. There are three potential approaches for forecasting mutual fund risk, including using historical realized risk, forecasting risk based on past returns, and forecasting risk based on the current portfolio. The first and simplest method forecasts risk using the historical standard deviation. This approach has several advantages. It is cheap and easy to calculate, and it is completely unambiguous. Everyone will agree on the standard deviation of the past 36 or 60 monthly returns. This approach has its problems however. It does assume constancy of risk over time. It cannot handle a new fund manager, new assets, changing leverage, or changing levels of market volatility. The second approach forecasts risk based on past returns, using style analysis developed by Sharpe. First, an analysis separates historical returns into two components: that due to several investment style indices, and that idiosyncratic to the fund. Then the forecast risk will depend on the risk of the style indices (forecast using advanced econometric methods), and the risk of the fund s idiosyncratic returns. This approach is not as cheap, easy, or unambiguous as the realized standard deviation. Its advantage is that it can account for changing volatilities. It still assumes some constancy of portfolio exposures over time. While this approach is somewhat harder and more expensive to implement, many vendors including BARRA provide this technology. Different vendors will provide somewhat different answers. 12 BARRA 1997

QUANTITATIVE MEASURE OF MUTUAL FUND RISK: AN OVERVIEW The third approach to forecasting risk uses actual portfolio holdings, as well as sophisticated econometric forecasts of the standard deviations and correlations of all the portfolio s assets. BARRA pioneered this approach 20 years ago, and today we also have competitors who provide such risk forecasts. This is the standard approach to forecasting risk for the most sophisticated institutional investors. The great advantage of this approach is that it provides an exact analysis of the current portfolio, accounting for changing market conditions, with no assumptions about constancy of portfolios over time. If a manager never owned a CMO before, but now has 50% of his portfolio in CMO s, portfolio based analysis will accurately forecast portfolio risk. The disadvantages of this approach are that it requires more data the portfolio holdings and it requires expensive and proprietary analytics. Institutional investors have decided that the advantages of this approach are worth the costs. Other Issues There are two other issues pricing and gaming which are worth covering briefly in this overview of risk analysis. We have seen many examples of funds which own very illiquid securities of uncertain value. The fund values such securities based on quotes potentially from only one dealer. Unfortunately this quote may not resemble a real offer when you need to sell. In general, funds marked-to-market based on modeled prices will tend to exhibit reduced risk relative to the true risk they are facing. It may also be possible for funds to work around any particular approach to forecasting risk, once the approach is known. They could change portfolio exposures compared to history, window-dress the portfolio for end of month analysis, or potentially even purchase special options to reduce volatility when measured based on month-end prices. More generally the choice of a risk definition can influence fund strategy. For example if downside risk becomes BARRA 1997 13

BARRA RESEARCH INSIGHTS the standard definition, then portfolio insurance may return to popularity: it is a strategy specifically designed to minimize downside risk. Recommendations Clearly the SEC must make two decisions concerning mutual fund risk. One, what is the appropriate definition of risk? Two, how should funds forecast risk for reporting? Deciding on a risk definition will involve trading off ease of estimation, forecastability, and intuition for the individual investor. It may include a separate choice of what to forecast and how to report it. We recommend the standard deviation as the best overall definition of risk. Choices for how to forecast risk vary based on cost, complexity, and accuracy. We recommend analysis of portfolio holdings as the most accurate forecast of risk. 14 BARRA 1997

QUANTITATIVE MEASURE OF MUTUAL FUND RISK: AN OVERVIEW 40% 35% 30% 25% 20% 15% 10% 5% 0% Magellan Fund January 1973 September 1994-25% -20% -15% -10% -5% 0% 5% 10% 15% 20% Return Figure 1-1 BARRA 1997 15

BARRA RESEARCH INSIGHTS 16 BARRA 1997

Forecasting Mutual Fund Risk: Current Holdings or Past Performance?

FORECASTING MUTUAL FUND RISK: CURRENT HOLDINGS OR PAST PERFORMANCE? Interest in mutual fund risk is very strong currently. The public s investment in mutual funds stands at an all time high. Yet, with the stock market reaching record levels and memories of staggering losses in bond funds in 1994, the public wants to know the risk of its investments. The SEC has received thousands of responses to its request for public comments on reporting requirements for mutual fund risk. In July 1995, we made two recommendations to the SEC concerning mutual fund risk reporting. First, the SEC should define risk as the standard deviation of the return. Second, mutual funds should report risk based on the current fund holdings and not on the fund s past performance. This is the best forecast of risk in the future, which is what the investing public cares about most. We have recently completed a study related to this second recommendation. We have tested which is more accurate: forecasting risk based on current holdings or past performance. For several reasons, we expect risk based on current holdings to be more accurate. Fund risk can change over time as markets change in volatility, or as the fund changes its leverage, asset holdings, or manager. Current portfolio-based risk forecasts avoid all these problems. In addition, historical performance can also be misleading due to insidious statistics. Imagine a world of thousands of equally high risk funds with random returns. One out of every 32 of those funds should exhibit superior performance in every one of the past 5 years, just based on pure luck. When investors choose funds based on past performance, they focus on a set of funds that includes both true outperformers and a large number of these statistical flukes. In this imaginary world, which may not differ much from the real world, historical performance may be a very poor predictor of future risk. This study looked at a particular set of equity and bond mutual funds. For each fund, at a historical analysis date, we used two methods to forecast risk. First, we used commercially available models to forecast risk based on the fund holdings on that date. Second, we used the prior three years of monthly returns to calculate a historical standard deviation. We then looked at each fund s performance over the subsequent year. Our key concerns included how BARRA 1997 17

BARRA RESEARCH INSIGHTS those subsequent returns compared to the two different forecasts of risk, and how accurate were these two risk predictions. In the remainder of this report, we will describe the details of the study, analyze the results, and provide conclusions and recommendations. Description of the Study The study began by choosing a particular set of mutual funds. We focused on funds that mainly invested in U.S. stocks or U.S. bonds. We excluded balanced funds, municipal bond funds, junk bond funds, and international funds. We also excluded funds not in existence prior to 1989. Beyond this, we chose a small set of funds relative to the universe of available mutual funds. It is difficult and costly to acquire accurate fund holdings historically, and we must process each portfolio by hand. We do not currently have automated procedures that input the portfolio holdings data to risk analysis systems. For equity mutual funds, we used the portfolio holdings data provided by Morningstar. However, Morningstar did not provide bond portfolio holdings in sufficient detail to allow risk analysis. Hence, we acquired bond portfolio holdings from CDA Spectrum BondWatch. This firm specializes in providing such data to brokers and traders, to facilitate trading by identifying who owns which securities. They have not designed their databases for historical analysis; however, they do maintain some records of historical bond portfolio holdings. Morningstar and CDA Spectrum BondWatch obtain mutual fund holdings information directly from the mutual funds, on a voluntary basis. While most funds cooperate, and some even provide very frequent updates, for others the holdings data in these commercial databases are far out of date. The SEC requires mutual funds to report their holdings semiannually, within 60 days of their fiscal year end and 6-month anniversary. These filings are not synchronized from fund to fund. The commercial services do not obtain holdings data directly from the SEC, possibly because they can obtain more timely information from the majority of funds. 18 BARRA 1997

FORECASTING MUTUAL FUND RISK: CURRENT HOLDINGS OR PAST PERFORMANCE? Considering these obstacles to analyzing mutual fund holdings, we used four criteria to actually choose the funds in this study. We first obtained a list of funds that are the most popular constituents of 401(k) plans. Stephen J. Butler of Pension Dynamics Corporation, a firm that specializes in establishing and servicing independent 401(k) plans, provided this list. We then augmented this list with other large funds, based on assets under management according to the Morningstar database. These two first criteria guaranteed that we would cover the funds of most interest to the general public. Since this is a study of forecasting fund risk, we then included funds with recent poor performance. In particular, we identified funds whose performance in 1994 was much worse than their performance in previous years. These three criteria led us to an initial list of 42 funds. Then, based on the fourth criteria data availability this list shrank to the 29 funds finally included in our study. We believe these criteria result in a reasonable and interesting, if not exhaustive, set of test funds. Table 1-1 lists the funds including their type, holding date, and reason for inclusion in the study. Since we did not test each of the thousands of existing mutual funds, we do not claim that this is the definitive scientific study of mutual fund risk. However, given the obstacles to completing such a comprehensive and definitive study, this study has focused on a representative and interesting set of funds. For each fund, we then calculated several statistics, starting with the Morningstar database of fund returns and the data we collected on fund holdings. First, we converted the Morningstar total returns into excess returns each month. Here, we define excess as the return the above the risk free Treasury Bill rate. This is consistent with standard risk analysis practice where the risky component of the return is that above and beyond the return of a risk free instrument, in this case Treasury Bills. We define risk as the standard deviation of these excess returns, which by convention we annualize in this study. This means that when we calculate standard deviations of monthly returns, we multiply them by the square root of twelve to convert them to analogous standard deviations of annual returns. BARRA 1997 19

BARRA RESEARCH INSIGHTS With these definitions in mind, we then calculated four quantities that are at the core of this study. We first calculated the realized excess fund return over the one year period following the analysis date. Second, we calculated the fund risk based on the holdings on the analysis date. We calculated this using commercially available software provided by BARRA. Third, we calculated the historical risk, the annualized standard deviation of the monthly returns over the three year period up to the analysis date. Finally, we calculated the realized risk as the standard deviation of monthly returns over the one year period following the analysis date. For each fund, we used a somewhat different analysis date. In particular, we chose a date for which we had portfolio holdings near the end of 1993 and the beginning of 1994. The one exception was the Steadman American Industry Fund, where our most recent fund holdings corresponded to July 31, 1991. We chose the 1993/1994 period to allow at least one year of data beyond the analysis date to observe subsequent performance. We also know that 1994 was a particularly bad year for bond fund performance, and so an appropriate period to study risk prediction. Because our analysis date corresponds exactly to the date of the portfolio holdings, the performance of the portfolio-based risk analysis may be slightly better than an investor could have achieved historically. This is because investors do not receive portfolio holdings until after the holding date. We ignore that effect here because we are trying to show the potential of this approach. Analysis We ran three tests of the accuracy of these risk forecasts. The first two tests compared the realized excess returns after the analysis to the two different forecasts of risk: one based on the prior three year history, the other based on the current portfolio holding. What does the standard deviation, our definition of risk, tell us about subsequent returns? This question has both a technical answer and an intuitive 20 BARRA 1997

FORECASTING MUTUAL FUND RISK: CURRENT HOLDINGS OR PAST PERFORMANCE? answer. The technical answer is that, assuming normal distributions, twothirds of the returns should lie within 1 standard deviation of the mean, and 95% of the returns should lie within 2 standard deviations of the mean. A larger standard deviation means that the possibility of large negative returns is greater. In more intuitive terms, investors will be surprised when their returns are more than 2 standard deviations from the mean. With accurate risk forecasts, such surprises should occur only 5% of the time. What mean excess return do we use for this analysis? Several choices are justifiable. For this study we will consider the mean to be zero. Since we mainly care about negative surprises, this is a conservative choice. An excess return of minus 2 standard deviations is clearly a negative surprise. Since investors assume positive mean returns, even excess returns above that will be negative surprises. We can also justify using a mean of zero since mean excess returns tend to be much smaller than standard deviations (roughly one third the size) over a one year horizon. To quantify this notion of surprise, we define the standardized outcome ( outcome for short) as the ratio of the realized excess return to the forecast risk. Each outcome of -2 or less is a negative surprise. With accurate risk forecasts, outcomes greater than 2 in magnitude should occur only 5% of the time. Table 2-2 displays for each fund the realized excess return, the two risk forecasts, and the outcomes calculated based on the two separate risk forecasts. We have sorted Table 2-2 by the historical outcomes, and highlighted the surprises in bold. Given that the study includes 29 funds, we would expect to see only one or two outcomes greater than 2, and only 9 or 10 outcomes greater than 1. Assuming normal distributions, we would not expect to see even one observation greater than 3 in magnitude. Using the portfolio holdings to predict risk, we see a pattern roughly corresponding to this expectation. We see no outcomes greater than 3, only one outcome greater than 2 in magnitude, and 76% of the observations less than 1 in magnitude. BARRA 1997 21

BARRA RESEARCH INSIGHTS If we use the historical risk to predict future risk, the results do not look nearly as encouraging. We see one outcome greater than 3 in magnitude, five outcomes greater than 2 in magnitude, and then 72% of the outcomes less than 1 in magnitude. An outcome of 3 is a big surprise. Using portfolio-based risk forecasting, just 3% of these realized returns were surprising, whereas using historical risk over 17% of these realized returns were surprising. In this first test, portfolio-based risk forecasts significantly outperform historical risk forecasts. Let s look at some examples of where large negative returns surprised investors, considering the historical record. One example is the Heartland U.S. Government Securities Fund, a fixed income fund. Prior to December 31, 1993, its three year historical standard deviation was 5.88%. That record would not prepare the typical investor for the -13.3% excess return it experienced over 1994. That turns out to be a -2.26 outcome. However, looking at the Heartland Portfolio at the end of 1993, the portfolio-based risk forecast was 11.15%, almost double the historical risk forecast. Investors would still be unhappy with their large negative return, but in this case it would not have surprised them. Using this risk forecast, the return would have only corresponded to an outcome of -1.19. Another example is the Excel Value Fund, which has become the Centurion Tactical Asset Allocation Fund. This is an equity fund. At the end of 1993, its historical risk was 10.00% and its portfolio-based risk was 18.25% once again, almost twice the historical risk. The realized excess return over 1994 was -28.19%. Using the portfolio-based risk analysis, this was an outcome of -1.54. Using historical risk analysis, this was an outcome of -2.82 a very big surprise. Yet a third example is the Piper Jaffray Institutional Government Fund, a fixed income fund subject to considerable media coverage in 1994 due to its precipitous losses. From April 1, 1994, through March 31, 1995, the fund suffered an excess return loss of 20.72%. This was a surprise whether you used portfolio-based risk or historical risk; however, it was a much bigger surprise using the historical risk. The portfolio-based risk was 42% above the historical forecast and led to an outcome of -2.23, instead of an outcome of -3.17. 22 BARRA 1997

FORECASTING MUTUAL FUND RISK: CURRENT HOLDINGS OR PAST PERFORMANCE? Of course in this analysis we can avoid surprises by consistently overpredicting risk. Accurate risk forecasts should lead to few surprises, not zero surprises. Fortunately, the outcomes also allow us to test whether we have overpredicted risk. This was our second test. Each time series of outcomes should have a standard deviation of 1. This standard deviation is effectively the ratio of realized to forecast risk. If this ratio exceeds 1, we have underpredicted risk, and if it is less than 1, we have overpredicted risk. Either way, we must determine whether the ratio is statistically distinguishable from 1. Unfortunately, we do not have time series of outcomes for each fund, but rather just cross sectional data: one outcome for each fund. However, if these outcomes are uncorrelated across funds, the standard deviation of these cross sectional outcomes should also equal 1. In general we expect some correlations across mutual fund returns especially for returns in the same month but this is the only test we can perform with our cross sectional data. Using this test, we found no statistically significant evidence that either method over or underpredicted risk on average. The two approaches perform somewhat similarly under this second test. The third comparison of forecast and realized risk directly compared the two forecasts to the standard deviation of the excess returns over the following year: the realized risk. Figure 2-1 is a scatterplot of the realized risk against the portfolio-based risk forecast. Figure 2-2 is a scatterplot of the realized risk against the historical risk forecast. In each figure, we have distinguished equity and bond funds. To test which risk forecast is more accurate, we fit a line through each comparison of realized and forecast risk. A perfect forecast will have an intercept of 0 and a slope of 1. We do not expect perfection, but would like to see an intercept statistically indistinguishable from 0, a positive slope, and a high correlation between the forecasts and realizations. Table 2-3 displays these statistics for both the historical and portfolio-based risk forecasts. BARRA 1997 23

BARRA RESEARCH INSIGHTS Table 2-3, as well as Figures 2-1 and 2-2, clearly show that portfolio-based risk outperforms historical risk in this third test. The R 2 statistics (square of the correlation between forecasts and realizations) are higher for the portfoliobased risk, which also exhibits an insignificant intercept (t-statistic less than 2) and a very significant slope. The historical forecasts show a significant intercept and a slope less significant that the slope for the portfolio-based forecasts. This third test demonstrates a much stronger relationship between portfoliobased forecasts and subsequent risk than between historical risk and subsequent risk. Given the superior performance of portfolio risk over historical risk, it is interesting to see just how different are these two numbers. Figure 2-3 is a scatterplot of historical risk versus portfolio risk. On this plot, we explicitly mark the portfolios which experienced surprises according to the historical forecasts. They are all funds for which the portfolio risk exceeded the historical risk. Figure 2-3 and this study imply that a large discrepancy between portfolio risk and historical risk is a significant warning about future performance. At the same time, not all funds with differing historical and portfolio risk experienced subsequent surprises, so it is worth understanding how these discrepancies arise. In general, historical and portfolio risk can differ if market risk has changed, if the current portfolio differs in style from historical portfolios (perhaps due to a new manager), or if the historical performance did not accurately reflect the true risk (i.e,. the fund was lucky). We can examine some of the funds in this study, to understand why historical and portfolio risk differed. In particular, let s look at three surprising funds discussed above Heartland U.S. Government Security, Excel Value, and Piper Jaffray Institutional Government plus the Vanguard Index 500 fund, which did not exhibit a surprise. The three surprise funds had the same manager and kept to the same investment strategy over the historical period and through at least the one year period following the analysis date. The Heartland U.S. Government Security fund 24 BARRA 1997

FORECASTING MUTUAL FUND RISK: CURRENT HOLDINGS OR PAST PERFORMANCE? consistently bet on falling interest rates over this entire time period, investing heavily in long maturity securities which will experience large positive returns as rates drop. This was a risky bet which paid off during the historical period, but went sour in the year following the analysis date. The historical record did not reflect the riskiness of the consistent strategy. The Excel Value fund also followed a consistent strategy from at least when manager Jack K. Heilbron s tenure began in late 1990. He ran it as a value-tilted equity fund, with large bets on a declining equity market. At the end of 1991 his fund included put options (which protect against market drops) and a large position in gold stocks. As of the analysis date (December 31, 1993) his fund included many of the same value stocks, and the same large position in gold stocks. The fund consistently included certain large bets which the historical record did not reflect. The Piper Jaffray Institutional Government fund consistently invested in exotic mortgage derivative securities to enhance yield. As early as November 1992, Morningstar analyst Catherine Sanders wrote that historical risk has been mild. Investors should remember, though, that the past is no guarantee of the future. The fund uses some unusual and unpredictable securities, and it s tough to gauge all the potential risks of its offbeat strategy. According to this study, it is possible to gauge these risks by examining the portfolio. Once again, the fund s historical performance did not reflect the risks of the consistent strategy. In contrast to the above three funds, the Vanguard Index 500 fund did not experience a surprise. However, this is an index fund (it owns all the stocks in the S&P 500 index), and so, like the above three funds, it followed a consistent strategy. So why do the historical and portfolio risks differ? The answer here is that overall equity market risk reached historic lows during the time period of this study. The volatility of the U.S. equity market has averaged about 20% from the 1920 s to the present, and about 17% from 1973 to the present. However from 1991 through 1993 it was under 11%. Once again, the recent historical record did not accurately reflect the fund s risk. Here too, the portfolio risk provided a realistic assessment of the fund s risk, even though in this case the fund did not experience a surprise. BARRA 1997 25

BARRA RESEARCH INSIGHTS This last example implies that perhaps the historical record would better reflect true risk if the historical period extended beyond just three years. However, that is not true. The Vanguard Index 500 fund is almost unique in having an extremely long track record (extending even before inception for this passive fund). Active (non-index) mutual funds the vast majority of all funds have no such lengthy track record; not to mention no similar history of consistency of strategy. However portfolio risk can take into account the history of the assets which make up the portfolio, even if that portfolio has not always included them. Overall, for the funds in this study, historical risk and portfolio risk have differed mainly because the historical record has not accurately reflected the true investment risks. When portfolio risk significantly exceeds historical risk, investors should beware. Conclusions and Recommendations We have compared the accuracy of mutual fund risk forecasts based on historical performance and current fund holdings. The current fund holdings provide more accurate risk forecasts than the historical performance, according to three separate tests. Among the funds studied, negative returns would seldom have surprised investors if they had known the fund risks based on the fund holdings. In many cases, negative returns did surprise them based on past performance. These surprises occurred even though we found no statistically significant evidence of average over or under prediction of risk. Finally, there was a much stronger relationship between portfolio-based forecasts and subsequent risk than between historical forecasts and subsequent risk. These results support our previous recommendation to the SEC that it require mutual funds to report risk based on current fund holdings. They also interestingly mirror results we have reported elsewhere on the persistence of mutual fund performance. Evidently for both risk and returns, the record of historical returns does not suffice to predict the future. 26 BARRA 1997

FORECASTING MUTUAL FUND RISK: CURRENT HOLDINGS OR PAST PERFORMANCE? Furthermore, this study points out that the public would benefit from portfolio-based risk reports, even if the SEC does not require mutual funds to report these numbers. In this case, commercial vendors could provide these numbers. However, for commercial vendors to provide risk forecasts based on current holdings requires that mutual funds provide these holdings in a convenient form and in a timely manner. Fund managers would quite reasonably object to reporting which could reveal proprietary short term strategies. However, quarterly reports, filed 60 days after the end of the quarter, shouldn t adversely impact performance. Hence, we are adding a third recommendation to the SEC. They should strictly require that mutual funds report detailed holdings on a quarterly basis, and the SEC should make these data readily available to the public in machine readable form. Data and Analytics Sources We obtained data on fund returns and equity fund holdings from Morningstar, 225 West Wacker Drive, Chicago, IL 60606. We received data on bond fund holdings from CDA Investment Technologies, Inc., 1355 Piccard Drive, Rockville, MD 20850. The list of popular 401(k) plan funds came from Pension Dynamics Corporation, 985 Moraga Road, Suite 210, Lafayette, CA 94549. To analyze equity fund risk we used BARRA s U.S. Equity Risk Model (E2), and to analyze bond fund risk we used BARRA s U.S. Fixed Income Risk Model (B2). These models are available from BARRA, Inc., 1995 University Avenue, Berkeley, CA 94704. BARRA 1997 27

BARRA RESEARCH INSIGHTS Fund Type Holding Date Reason for Inclusion 20th Century Heritage Equity 12/31/93 Popular 401(k) fund 20th Century Ultra Equity 12/31/93 Popular 401(k) fund Alliance Bond Corporate Bond Fund Fixed Income 12/31/93 Bad 1994 performance American Mutual Fund Equity 12/31/93 Popular 401(k) fund Berger 100 Equity 9/30/93 Bad 1994 performance CGM Capital Development Equity 9/30/93 Bad 1994 performance Dean Witter US Govt. Securities Fixed Income 6/30/94 Large fund Dodge and Cox Stock Equity 12/31/93 Popular 401(k) fund Excel Value Equity 12/31/93 Bad 1994 performance Fidelity Contrafund Equity 12/31/93 Large fund Fidelity Growth and Income Equity 7/31/93 Large fund Fidelity Magellan Equity 3/31/93 Popular 401(k) fund Fidelity Retirement Growth Equity 5/31/93 Popular 401(k) fund Fidelity Select Industrial Material Equity 8/31/93 Great 1994 performance Heartland US Govt. Security Fund Fixed Income 12/31/93 Bad 1994 performance Invesco Industrial Income Equity 9/30/93 Popular 401(k) fund Investment Company of America Equity 12/31/93 Popular 401(k) fund Janus Fund Equity 6/30/93 Popular 401(k) fund Lord Abbett US Govt. Securities Fixed Income 5/31/94 Large fund Nationwide Bond Fund Fixed Income 6/30/94 Bad 1994 performance Neuberger and Berman Guardian Equity 9/30/93 Popular 401(k) fund Neuberger and Berman Manhattan Equity 7/30/93 Popular 401(k) fund Piper Jaffray Institutional Govt. Fund Fixed Income 3/31/94 Bad 1994 performance Security Income Fund Corp. Bond Fixed Income 12/31/93 Bad 1994 performance Steadman American Industry Equity 7/31/91 Bad 1994 performance T. Rowe Price Equity Income Equity 6/30/93 Popular 401(k) fund Vanguard Index 500 Equity 11/30/93 Popular 401(k) fund Vanguard Windsor Equity 10/31/93 Large fund Vanguard Windsor II Equity 10/31/93 Popular 401(k) fund Table 2-1 28 BARRA 1997

FORECASTING MUTUAL FUND RISK: CURRENT HOLDINGS OR PAST PERFORMANCE? Realized Analysis Excess Portfolio Historical Portfolio Historical Fund Date Return Risk Risk Outcome Outcome Piper Jaffray Institutional Govt. Fund 3/31/94-20.72 9.28 6.54-2.23-3.17 Excel Value 12/31/93-28.19 18.25 10.00-1.54-2.82 Security Income Fund Corp. Bond 12/31/93-11.97 8.37 4.37-1.43-2.74 Alliance Bond Corporate Bond Fund 12/31/93-16.29 8.67 6.49-1.88-2.51 Heartland US Govt. Security Fund 12/31/93-13.30 11.15 5.88-1.19-2.26 Steadman American Industry 7/31/91-15.70 27.20 18.55-0.58-0.85 20th Century Heritage 12/31/93-10.10 16.36 14.36-0.62-0.70 American Mutual Fund 12/31/93-3.70 10.45 8.05-0.35-0.46 Investment Company of America 12/31/93-3.87 14.49 9.69-0.27-0.40 Fidelity Contrafund 12/31/93-5.10 14.71 13.08-0.35-0.39 Berger 100 9/30/93-6.93 19.21 19.73-0.36-0.35 20th Century Ultra 12/31/93-7.50 22.89 23.71-0.33-0.32 Vanguard Index 500 11/30/93-2.92 14.32 10.64-0.20-0.27 Invesco Industrial Income 9/30/93-2.61 11.23 10.68-0.23-0.24 CGM Capital Development 9/30/93-4.86 17.24 21.03-0.28-0.23 Vanguard Windsor II 10/31/93-1.52 13.12 10.32-0.12-0.15 Janus Fund 6/30/93-1.68 12.49 13.46-0.13-0.12 Neuberger and Berman Manhattan 7/30/93-0.93 18.35 16.10-0.05-0.06 Dodge and Cox Stock 12/31/93 0.96 14.52 11.19 0.07 0.09 Neuberger and Berman Guardian 9/30/93 1.61 15.31 13.13 0.11 0.12 T. Rowe Price Equity Income 6/30/93 2.18 10.81 11.11 0.20 0.20 Vanguard Windsor 10/31/93 2.47 13.21 11.65 0.19 0.21 Fidelity Growth and Income 7/31/93 3.61 12.27 13.81 0.29 0.26 Fidelity Retirement Growth 5/31/93 9.52 14.29 16.71 0.67 0.57 Fidelity Magellan 3/31/93 9.60 14.21 15.57 0.68 0.62 Lord Abbett US Govt. Securities 5/31/94 3.69 7.63 5.6 0.48 0.66 Nationwide Bond Fund 6/30/94 8.05 9.32 6.11 0.86 1.32 Fidelity Select Industrial Material 8/31/93 22.04 15.57 13.82 1.42 1.59 Dean Witter US Govt. Securities 6/30/94 5.99 5.93 3.39 1.01 1.77 Table 2-2 BARRA 1997 29

BARRA RESEARCH INSIGHTS Realized versus Forecast Risk σ realized = a + b.σ forecast + ε σ forecast a b t-stat(a) t-stat(b) R 2 Portfolio 1.90 0.63 1.21 5.81 0.54 Historical 4.42 0.52 3.19 4.85 0.45 Table 2-3 30 BARRA 1997

FORECASTING MUTUAL FUND RISK: CURRENT HOLDINGS OR PAST PERFORMANCE? Realized Risk versus Portfolio Risk Realized Risk 25 20 15 10 5 0 0 5 10 15 20 25 30 Portfolio Risk Equity Funds Bond Funds Source: BARRA Figure 2-1 BARRA 1997 31

BARRA RESEARCH INSIGHTS Realized Risk versus Historical Risk Realized Risk 25 20 15 10 5 0 0 5 10 15 20 25 Historical Risk Equity Funds Bond Funds Source: BARRA Figure 2-2 32 BARRA 1997

FORECASTING MUTUAL FUND RISK: CURRENT HOLDINGS OR PAST PERFORMANCE? Portfolio versus Historical Risk Historical Risk 25 20 15 10 5 0 0 5 10 15 20 25 30 Portfolio Risk Non-surprises Surprises Portfolio = Historical Source: BARRA Figure 2-3 BARRA 1997 33

BARRA RESEARCH INSIGHTS 34 BARRA 1997

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