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LO 65.3: Assess the impact of biases on reported returns for illiquid assets.

LO 65.2: Examine the relationship between market imperfections and illiquidity.
Many economic theories assume that markets are perfect. This means that market participants are rational and pursue utility maximization, that there are no transaction costs, regulation or taxes, that assets are perfectly divisible, that there is perfect competition in markets, and that all market participants receive information simultaneously. The reality, though, is that markets are imperfect.
Imperfections that encourage illiquidity include: Market participation costs. There are costs associated with entering markets, including
the time, money, and energy required to understand a new market. In many illiquid markets, only certain types of investors have the expertise, capital, and experience to participate. This is called a clientele effect. There will be less liquidity in markets that are suited to a limited number of investors and/or where there are barriers to entry in terms of required experience, capital, or expertise.
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Transaction costs. Transaction costs include taxes and commissions. For many illiquid
assets, like private equity, there are additional costs, including costs associated with performing due diligence. Investors must pay attorneys, accountants, and investment bankers. These costs can impede investment.
When acknowledging the existence of transaction costs (i.e., acknowledging that markets are imperfect), some academic studies assume that as long as an investor can pay the transaction costs (and sometimes these costs are large), then any investor can transact (i.e., any asset can be liquid if one can pay the transaction cost). However, this is not always true. For example, there are: Difficulties finding a counterparty (i.e., search frictions). For example, it may be difficult to find someone to understand/purchase a complicated structured credit product. It may also be difficult to find buyers with sufficient capital to purchase an office tower or a skyscraper in a city like New York. No matter how high the transaction cost, it may take weeks, months, or years to transact in some situations.
Asymmetric information. Some investors have more information than others. If an
investor fears that the counterparty knows more than he does, he will be less willing to trade, which increasing illiquidity. When asymmetric information is extreme, people assume all products are lemons. Because no one wants to buy a lemon, markets break down. Often liquidity freezes are the result of asymmetric information. Because investors are looking for non-predatory counterparties who are not seeking to take advantage of asymmetric information, information itself can be a form of search friction.
Price impacts. Large trades can move markets, which, in turn, can result in liquidity
issues for the asset or asset class.
Funding constraints. Many illiquid assets are financed largely with debt. For example, even at the individual level, housing purchases are highly leveraged. As a result, if access to credit is compromised, investors cannot transact.
Il l iq u id As s e t Re t u r n Bia s e s

LO 65.1: Evaluate the characteristics of illiquid markets. * 1
There are several characteristics that describe illiquid asset markets, including: 1. Most asset classes are illiquid, at least to some degree.
2. Markets for illiquid assets are large.
3.
Illiquid assets comprise the bulk of most investors portfolios.
4. Liquidity dries up even in liquid asset markets.
M o s t As s e t C l a s s e s Ar e Il l iq u id
All markets, even large-cap equity markets, are somewhat illiquid. It is clear, however, that some assets (e.g., real estate) are less liquid than others (e.g., public equities). Illiquid assets trade infrequently, in small amounts, and generally exhibit low turnover. For example, there are mere seconds between transactions in public equity markets with an annualized turnover rate greater than 100%. In contrast, over-the-counter (OTC) equities typically trade within a day, but sometimes a week or more may pass between trades, with annualized turnover of 25% to 35%. Corporate bonds trade daily, and municipal bonds typically trade semiannually. At the far end of the liquidity spectrum is institutional infrastructure with an average investment commitment of 50 to 60 years (up to 99 years), and art, with 40 to 70 years between transactions. There is negligible turnover in infrastructure. Turnover in residential real estate is about 5% per year, while turnover in institutional real estate is approximately 7%. Time between real estate transactions can range from months to decades.
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M a r k e t s f o r Il l iq u id As s e t s Ar e La r g e
The size of the U.S. residential mortgage market was $16 trillion in 2012. The institutional real estate market was measured at $9 trillion. In contrast, the market capitalization of the NYSE and Nasdaq combined was approximately $17 trillion. The total wealth held in illiquid assets exceeds the total wealth in traditional, liquid stock, and bond markets.
In v e s t o r H o l d in g s o f Il l iq u id As s e t s
The home is often an individuals most valuable asset. As a result, illiquid assets represent approximately 90% of total wealth, not counting human capital, the largest and least liquid asset for many individual investors. High net worth individuals in the United States even typically allocate 10% of portfolios to fine art and jewelry, known as treasure assets. High net worth individuals in foreign countries hold an average of 20% in treasure assets. Institutional investors have also increased allocations to illiquid assets over the last 20 years. University endowments have increased allocations of illiquid assets to approximately 25%, up from 5% in the early 1990s. Pension funds have increased allocations to approximately 20%, up from 5% in 1995. In general, investors hold sizeable amounts of illiquid assets.
Liq u id it y Ca n D r y U p
In stressed economic periods, such as during the 20072009 financial crisis, liquidity can dry up. For example, money markets froze (i.e., repurchase agreement and commercial paper markets) during the crisis as investors were unwilling to trade at any price. Residential and commercial mortgage-backed securities markets, structured credit markets, and the auction rate securities market, a market for floating rate municipal bonds, also became illiquid during the crisis. The auction rate securities market is still frozen, more than six years later. Major liquidity crises have occurred at least once every 10 years across the globe, in conjunction with downturns and financial distress.
M a r k e t Im p e r f e c t io n s

LO 64.9: Describe potential explanations for the risk anomaly.
A comprehensive explanation for the risk anomaly is elusive. It has been speculated that the true explanation is some combination of data mining, investor leverage constraints, institutional manager constraints, and preference theory.
Some academics have wondered if the risk anomaly is the result of data mining. Ang et al. (2006) found that the risk anomaly appears during both recessions and expansions. Frazzini and Pedersen (2014)6 found that low beta portfolios have high Sharpe ratios in U.S. stocks, international stocks, Treasury bonds, and corporate bonds. Cao and Flan (2013)7 also found evidence of the risk anomaly in option and commodity markets. The argument of data mining is not well supported.
Another possible explanation is the prevalence of leverage constrained investors. This is sometimes an occurrence with institutional investors, but it is very much a constraint of retail investors. Since certain investors are leverage constrained, meaning that they cannot borrow funds for investing, they choose to invest in stocks with built-in leverage in the form of high betas. The additional demand for high-beta stocks will bid up their respective prices until the assets are overvalued and they deliver a decreased risk-adjusted return with regard to lower beta stocks. This same theory works to lower the prices of low beta stocks and, therefore, results in higher risk-adjusted returns due to lower entry prices.
Institutional managers also have constraints that could help to explain the risk anomaly. Consider a scenario with two competing portfolios. Portfolio A has positive alpha because the portfolio is undervalued, while Portfolio B has a negative alpha because it is overvalued. In a perfect world, an investor would buy (go long) Portfolio A and short sell Portfolio B to capture the perceived alphas. Many institutional investors will have constraints against short selling. Most also have tracking error constraints that only permit a specified deviation from their benchmark. Under either of these constraints, an institutional investor would not be able to capture the alpha that they think exists. One solution for the tracking error constraint is to change the benchmark or the tracking error tolerance bands, but this can be a difficult process requiring formal approval from the investment committee of the fund.
Sometimes investors simply have a preference for high-volatility and high-beta stocks. This could occur because their capital market expectations are very bullish, so they want to amplify their returns. The end result is that investors buy the higher-beta investments and bid up their prices to the point where future returns will be much lower. There will always be a group of investors that desire to shun safe and boring lower-volatility stocks. The good news is that this creates less emotionally driven entry points for long-term investors who desire lower volatility.
6. Andrea Frazzini and Lasse Heje Pederson, Betting Against Beta, The Journal o f Financial
Economics 111, no. 1 (2014): 1-25. Jie Cao and Bing Han, Cross Section of Option Returns and Idiosyncratic Stock Volatility, The Journal o f Financial Economics 108, no. 1 (2013): 231-A9.
7.
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Investors holding heterogeneous preferences (disagreeing on investment potential) and having investment constraints could explain a portion of the risk anomaly. Hong and Sraer (2012)8 found that when disagreement is low and investors are long-only constrained, then the CAPM holds the best. When disagreement is high, some investments become overpriced and future returns are decreased. Significant disagreement can lead to an inverse relationship between beta and returns.
Harrison Hong and David Sraer, Speculative Betas, NBER Working Paper 18548, November 2012.
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Ke y Co n c e pt s
LO 64.1 The capital asset pricing model (CAPM) states that there should be a positive relationship between risk and return. Higher risk, as measured by beta, should have a higher return. The low-risk anomaly appears to suggest the exact opposite. This anomaly finds that firms with lower betas and lower volatility have higher returns over time.
LO 64.2 Alpha is the average performance of an investor in excess of their benchmark. Excess return is often called active return, and the standard deviation of the active return is known as tracking error.
The ratio of active return to tracking error is called the information ratio, which is one way to easily rank competing investment alternatives.
O L
a
If an investor is using the risk-free rate as their benchmark, then their alpha is any return earned in excess of the risk-free rate, and the best risk-adjusted return measurement is the Sharpe ratio.
Sharpe ratio =
LO 64.3 A benchmark is very important for investment comparisons. If the benchmark is riskier than the investment in question, then both the alpha and the information ratio will be too low. The best combination for a benchmark is for it to be well-defined, tradeable, replicable, and adjusted for the risk of the underlying pool of investments.
LO 64.4 Grinolds fundamental law of active management suggests a tradeoff between the number of investment bets placed (breadth) and the required degree of forecasting accuracy (information coefficient).
I R r s I C x Vb R
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An investor either needs to place a large number of bets and not be very concerned with forecasting accuracy, or he needs to be very good at forecasting if he places only a small number of bets.
LO 64.5 The traditional capital asset pricing model only accounts for co-movement with a market index. Multifactor models, like the Fama and French three-factor model, add other explanatory factors in an attempt to better predict the alpha for an asset. Multifactor models have been shown to enhance the informational value of regression output. The Fama-French three-factor model is expressed as:
Ri – RF – a + (3i>MKT x (Rm R f ) + 3i,SMB x (SMB) + 3i?HML x (HML)
This model adds a size premium (SMB) and a value premium (HML) to the CAPM single-factor model. A momentum effect (UMD) could also be added to help explain excess returns. This factor suggests that upward trending stocks will continue their upward movement while downward moving stocks will continue their downward trend.
LO 64.6 Style analysis is a form of factor benchmarking where the factor exposures evolve over time. The traditional Fama-French three-factor model can be improved by using indices that are tradeable, such as the SPDR S&P Value ETF (SPYV), and incorporating time-varying factors that change over time.
LO 64.7 Alpha is computed using regression, which operates in a linear framework. There are nonlinear strategies that can make it appear that alpha exists when it actually does not. This situation is encountered when payoffs are quadratic terms or option-like terms. This may be a significant problem for hedge funds because merger arbitrage, pairs trading, and convertible bond arbitrage strategies all have nonlinear payoffs.
LO 64.8 The volatility anomaly and the beta anomaly both agree that stocks with higher risk, as measured by either high standard deviation or high beta, produce lower risk-adjusted returns than stocks with lower risk.
LO 64.9 A comprehensive explanation for the risk anomaly is elusive. It has been speculated that the true explanation is some combination of data mining, investor leverage constraints, institutional manager constraints, and preference theory.
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Co n c e pt Ch e c k e r s
1.
2.
3.
Which of the following statements is correct concerning the relationship between the low-risk anomaly and the capital asset pricing model (CAPM)? A. The low-risk anomaly provides support for the CAPM. B. The notion that the low-risk anomaly violates the CAPM has not been proven
empirically.
C. The low-risk anomaly violates the CAPM and suggests that low-beta stocks will
D. Both CAPM and the low-risk anomaly point to a positive relationship between
outperform high-beta stocks.
risk and reward.
Which of the following statements is not a characteristic of an appropriate benchmark? An appropriate benchmark should be: A. B. C. well-defined. D. equally applied to all risky assets irrespective of their risk exposure.
tradeable. replicable.
Grinolds fundamental law of active management suggests that: A.
investors should focus on increasing only their predictive ability relative to stock price movements.
B. sector allocation is the most important factor in active management. C. a small number of investment bets decreases the chances of making a mistake
and, therefore, increases the expected investment performance.
D. to maximize the information ratio, active investors need to either have high- quality predictions or place a large number of investment bets in a given year.
4.
Why would an investor include multiple factors in a regression study?
I. To attempt to improve the adjusted R2 measure. II. To reduce the r-stat value on the respective regression coefficients. A. I only. B. II only. C. Both I and II. D. Neither I nor II.
3.
Which of the following characteristics is a potential explanation for the risk anomaly? A. Investor preferences. B. The presence of highly leveraged retail investors. C. Lack of short selling constraints for institutional investors. D. Lack of tracking error constraints for institutional investors.
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Co n c e pt Ch e c k e r An s w e r s
1. C The low-risk anomaly violates the CAPM and suggests that low beta stocks will outperform
high-beta stocks. This has been empirically proven with several studies. The CAPM points to a positive relationship between risk and reward, but the low-risk anomaly suggests an inverse relationship.
2. D An appropriate benchmark should be well-defined, replicable, tradeable, and risk-adjusted. If the benchmark is not on the same risk scale as the assets under review, then there is an unfair comparison.
3. D Grinolds fundamental law of active management focuses on the tradeoff of high quality predictions relative to placing a large number of investment bets. Investors can focus on either action to maximize their information ratio, which is a measure of risk-adjusted performance. While sector allocation is a very important component of the asset allocation decision, Grinold focused only on the quality of predictions and the number of investment bets made.
4. A An investor should consider adding multiple factors to the regression analysis to potentially improve the adjusted R2 measurement, potentially increase the tests of statistical significance, and to search for a benchmark that is more representative of a portfolio s investment style.
5. A Potential explanations for the risk anomaly include: the preferences of investors, leverage
constraints on retail investors that drive them to buy pre-leveraged investments in the form of high-beta stocks, and institutional investor constraints like prohibitions against short selling and tracking error tolerance bands.
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The following is a review of the Risk Management and Investment Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in:
Il l i q u i d A s s e t s
Topic 65
Ex a m Fo c u s
This topic examines illiquid asset market characteristics and the relationship between illiquidity and market imperfections. Reported return biases are discussed as well as the illiquidity risk premium within and across asset classes. For the exam, understand that all markets, even highly liquid markets such as commercial paper, can be illiquid at some points in time. Also, know the three biases that impact reported returns of illiquid asset classes (survivorship bias, sample selection bias, and infrequent sampling). Finally, understand the factors that influence the decision to include illiquid asset classes in a portfolio.
Il l iq u id As s e t M a r k e t s

LO 64.8: Compare the volatility anomaly and beta anomaly, and analyze evidence of each anomaly.
Using data from 19261971, Haugen and Heins (1975)3 found that over the long run, stock portfolios with lesser variance in monthly returns have experienced greater average returns than riskier counterparts. Ang, Hodrick, Xing, and Zhang (2006)4 tested whether increased volatility, as measured by standard deviation, has a positive relationship with returns and Sharpe ratios. They organized their data, which comprised monthly return data from September 1963December 2011, into quintiles and controlled for numerous variables including leverage, volume, bid-ask spreads, dispersion in analysts forecasts, and momentum. They observed a volatility anomaly which shows that as standard deviation increased, both the average returns and the Sharpe ratios decreased. For the lowest three quintiles, the average return was above 10%, but declined to 6.8% for quintile 4 and to 0.1% for the quintile with the highest volatility. Likewise, Sharpe ratios declined from 0.8 for the lowest volatility quintile to 0.0 for the highest volatility quintile. It was found that the most volatile stocks produce the lower returns, while the least volatile stocks performed the best.
When the capital asset pricing model (CAPM) was first tested in the 1970s, a positive relationship was found between beta and expected returns. Numerous academics have since retested this relationship with interesting results. Ang et al. (2006) found that stocks with high betas tend to have lower-risk-adjusted returns. Organizing monthly return data from September 1963December 2011 into quintiles, they found that the Sharpe ratio fell from 0.9 for stocks with the lowest betas to 0.4 for stocks with the highest betas. This beta anomaly does not suggest that stocks with higher betas have low return because they do not. It means they have lower Sharpe ratios (risk-adjusted performance) because higher betas are paired with higher volatility as measured by standard deviation, which is the denominator in the Sharpe ratio.
Interestingly, CAPM does not predict that lagged betas (measured over previous periods) should produce higher returns. It does predict that investors should find a contemporaneous relationship between beta and expected returns. This means that stocks with higher betas should also have higher returns during the same time period when the beta was measured. This is a confirming, not a predictive, metric. Following this logic, if investors could reliably predict future betas, then they could more accurately predict future expected returns. The trouble is that historical betas are not good predictors of future betas. Buss and Vilkov (2012)5 estimated future betas using implied volatility measures in option pricing models and found some improvement over using historical betas. The beta anomaly is less a mystery as it is a challenge to find a reliable way of predicting future betas to improve the risk perspective of beta.
3. Robert A. Haugen and A. Janies Heins, Risk and the Rate of Return on Financial Assets: Some
Old Wine in New Bottles, Journal of Financial and Quantitative Analysis 10, no. 5 (1975): 775-84.
4. Andrew Ang, Robert J. Hodrick, Yuhang Xing, and Xiaoyan Zhang, High Idiosyncratic
Volatility and Low Returns: International and Further U.S. Evidence, Journal o f Financial Economics 91 (2009): 1-23.
5. Adrian Buss and Grigory Vilkov, Measuring Equity Risk With Option-Implied Correlations,
The Review of Financial Studies 25, no. 10 (2012): 3113-40.
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Potential Explanations for the Risk Anomaly

LO 64.7: Describe issues that arise when measuring alphas for nonlinear strategies.
Alpha is computed using regression, which operates in a linear framework. There are nonlinear strategies, such as uncovered long put options, that can make it appear that alpha exists when it actually does not. An uncovered long put option has a payoff profile that is L-shaped (nonlinear), but applying traditional regression tools will yield a positive alpha, which does not exist in reality. This situation is encountered when payoffs are quadratic terms, like R^ or are option-like terms, such as max(Rt, 0). This can be a significant problem for hedge funds, because merger arbitrage, pairs trading, and convertible bond arbitrage strategies all have nonlinear payoffs.
One reason that nonlinear strategies yield a false positive alpha is because the distribution of returns is not a normal distribution. Certain nonlinear strategies will also exhibit negative skewness in their distribution. This will increase loss potential in the left-hand tail and make the middle of the distribution appear thicker. Skewness is not factored into the calculation of alpha, which is an issue for nonlinear payoff strategies.
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Vo l a t il it y a n d Be t a An o m a l ie s

LO 64.6: Explain how to measure time-varying factor exposures and their use in style analysis.
Style analysis is a form of factor benchmarking where the factor exposures evolve over time. To illustrate time-varying factors, consider four investments: (1) LSV Value Equity (LSVEX), (2) Fidelity Magellan (FMAGX), (3) Goldman Sachs Capital Growth (GSCGX), and (4) Berkshire Hathaway (BRK). Figure 4 shows the regression data from monthly returns on all four funds using the Fama-French three-factor model plus the UMD factor. The key difference between this information and data already presented is that the time period has been adjusted to January 2001 through December 2011.
Figure 4: Regression of Excess Returns for Multiple Funds
Alpha (a)
t-stat
Market beta ((3j MKT)
t-stat
SMB beta (3i SMB)
t-stat
HML beta (3i HML)
t-stat
UMD beta ((^u m d )
t-stat
LSVEX 0.00% 0.01 0.94 36.9 0.01 0.21 0.51 14.6 0.2 1.07
FMAGX -0.27% -2.23 1.12 38.6 -0.07 -1.44 -0.05 -1.36 0.02 1.00
GSCGX -0.14% -1.33 1.04 42.2 -0.12 -3.05 -0.17 -4.95 0.00 -0.17
BRK 0.22% 0.57 0.36 3.77 -0.15 -0.97 0.34 2.57 -0.06 -0.77
This data presents a different story about these funds than earlier. The only calculated alpha that is statistically significant is for Fidelity Magellan, but it is a 3.24% (= 0.27% x 12) in annualized terms. This was not good news for Fidelity investors, although it is time
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constrained to a period that ended in 2011. Berkshires alpha is nicely positive, but for this time period, it is not significant. According to the HML beta factors, LSV Value Equity is indeed a value-focused investment. The data also shows that FMAGX is a leveraged play on the market with a 1.12 market beta. The UMD beta confirms that none of these four funds are momentum plays.
Style analysis tries to solve some of the problems with standard multifactor regression. Unlike Fama and Frenchs untradeable SMB and HML indices, style analysis uses tradeable assets. For example, consider three funds: (1) SPDR S&P 500 ETF (SPY), (2) SPDR S&P 500 Value ETF (SPYV), and (3) SPDR S&P 500 Growth ETF (SPYG). These three exchange-traded funds (ETFs) are hosted by State Street Global Advisors and they all belong to the SPDR (pronounced spider) family of ETFs. Style analysis also adjusts for the fact that factor loadings (betas) change over time. A possible multifactor regression could be estimated for next periods expected asset return (Rt+1) as follows:
R t+ i = <*t + (3SPY,tSPYt+ i + PsPYV.tSPYYt+i + 3sPYG,tSPYGt+i + t+i This formula has an imposed restriction that all factor loadings (i.e., factor weights) must sum to one: - 3sPY,t + 3sPYV,t + 3sPYG,t 1 - 3sPY,t + 3sPYV,t + 3sPYG,t The time-varying portion of this equation comes into play with the respective factor loadings. This process uses estimates that incorporate information up to time t. Every new month (t + 1) requires a new regression to adjust the factor loadings. This means that the beta factors will change over time to reflect changes in the real world. Is s u e s w it h Al p h a M e a s u r e m e n t f o r N o n l in e a r St r a t e g ie s

LO 64.5: Apply a factor regression to construct a benchmark with multiple factors, measure a portfolio s sensitivity to those factors, and measure alpha against that benchmark.
Consider the CAPM formula, where E(R) is the expected return for asset i for a given level of beta exposure, and E(RM) is the expected return on the market:
E(Ri) = RF+(3[E(RM) – R F
If an investment has a beta of 1.3, then the following formulas demonstrate the algebraic evolution of this expression:
E(Ri) Rp +1.3[E(Rm) Rp] E(Rj) = Rp +1.3E(RM) 1.3(Rp) E(Ri) = 0.3Rp + 1.3E(RM)
In this example, the expected return on a $ 1 investment in asset i should be equal to a portfolio with a short position in the risk-free rate of $0.30 and a long position in the market of $1.30. Any return earned in excess of this unique blend will exceed our expectations and is, therefore, considered to be alpha. Using regression, the alpha is approximated as:
Ri,t Rf o l + (3(RM Rf ) + ijt
This exact process was conducted on Berkshire Hathaway stock over the period of January 1990 to May 2012 relative to S&P 500 Index. The results are shown in Figure 1. The monthly alpha coefficient is statistically significant at a 95% confidence level due to the absolute value of the /-statistic being greater than 2.0. Most regressions do not produce a statistically significant alpha.
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Figure 1: Regression of Excess Returns
Alpha Beta Adjusted R2
Coefficient 0.72% 0.51 0.14
t-Statistic
2.02 6.51
This regression implies the following CAPM equation:
Rg 0.49Rp +0.51Rm
According to these regression results, a customized benchmark of 49% in the risk-free asset and 51% in the market would produce an expected alpha of 0.72% per month for Berkshire Hathaway. That is 8.6% (0.72% x 12) of annualized expected alpha! Since alpha is the excess return above the actual return, Ri5 you can think of alpha using the following formula:
a = R; – [0.49R f + 0.51E(RM)] Professors Note: For Berkshire, it is important to note that their market capitalization has grown from less than $10 billion in the early 1990s to over $220 billion in 2012. In his Annual Letter to Shareholders for 2010, Warren Buffet told shareholders that the bountiful years, we want to emphasize, will never return. The huge sums of
capital we currently manage eliminate any chance o f exceptional performance.2 Thus, Berkshire Hathaway has acknowledged the law o f declining marginal returns due to scale.
In 1993, Eugene Fama and Kenneth French extended the traditional CAPM-based regression to include additional factors. They controlled for the size effect (small companies tend to outperform large companies) and for the value/growth effect (value stocks tend to perform better than growth stocks). They formally labeled the size premium as SMB, which stands for small minus big (the return on small stocks minus the return on big stocks), and they represented the value premium with HML, which stands for high minus low (high book-to-market stocks minus low book-to-market stocks). The factors for SMB and HML are long-short factors. The small minus big factor can be visualized as:
SMB = $ 1 in small caps (long position) – $ 1 in large caps (short position)
In a similar manner, we can visualize high minus low as:
HML = $ 1 in value stocks (long position) $ 1 in growth stocks (short position)
2. Berkshire Hathaway Annual Letter to Shareholders, 2010.
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The Fama and French three-factor model is constructed as follows:
Ri – RF – a + (3i)MKX x (Rm Rp) + Pi,SMB x (SMB) + Pi5HML x (HML)
The SMB beta will be positive if there is co-movement with small stocks, and it will be negative if there is co-movement with large stocks. If a given asset does not co-move with either small or large companies (i.e., a medium company focus), then its beta coefficient will be zero. Likewise, the HML beta will be positive if the assets have a value focus, and it will be negative if the assets have a growth focus. Applying the Fama-French model to Berkshire Hathaway over the time period of January 1990-May 2012 yields the results displayed in Figure 2.
Figure 2: Fama-French Three-Factor Model Results
Alpha (a) Market beta ((3i MKT) SMB beta ((3i SMB) HML beta (P^h m i) Adjusted R2
Coefficient 0.65% 0.67 -0.50 0.38 0.27
t-Statistic
1.96 8.94 -4.92 3.52
The results in Figure 2 show several interesting aspects. First, the alpha declined slightly but is still very high. Second, the market beta rose from 0.51 to 0.67. Third, the SMB beta is negative, which suggests a large company bias. Fourth, the HML beta is positive, which suggests a value focus for the fund. The adjusted R2 also rose from 0.14 to 0.27, which suggests that SMB and HML do add value. Based on the results, the custom benchmark implied by the Fama-French three-factor model for Berkshire Hathaway is shown as follows:
Rg = 0.33(T-bills) + 0.67 x (market portfolio)
0.5(small caps) + 0.5(large caps)
+ 0.38(value stocks) 0.38(growth stocks)
All of the factor weights in this formula sum to 1.0, but adding the SMB and HML factors add explanatory ability to the regression equation. A test could also be added to account for the momentum effect, which is the theory that upward trending stocks will continue their upward movement while downward moving stocks will continue their downward trend. Thus, a fourth factor can be added to the Fama-French model. This fourth factor could be labeled as UMD, which stands for up minus down (upward trending stocks minus downward trending stocks). A positive UMD beta would suggest a focus on upward trending stocks, while a negative UMD beta would suggest a focus on downward trending stocks. As with the SMB and HML betas, a beta of zero suggests no relationship. Figure 3 shows the UMD factor added to the previous results. Using this data, it can be discerned that Berkshire Hathaway does not have exposure to momentum investing.
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Figure 3: Fama-French Three-Factor Model Results With UMD Factor
Alpha (a) Market beta (Pi MKT) SMB beta (0i SMB) HML beta (Pi HML) UMD beta (Pi UMD) Adjusted R2
Coefficient 0.68% 0.66 -0.50 0.36 -0.04 0.27
t-Statistic
2.05 8.26 -4.86 3.33
0.66
One core challenge with using the Fama-French model is replication of indices. Fama and French have created an SMB index and an HML index to increase explanatory power, but there is no way to directly trade an SMB or HML portfolio. These indices are conceptual and not directly tradeable. It is important to include only tradeable factors because the factors chosen will greatly influence the calculated alpha.
M e a s u r e m e n t o f Tim e -Va r y in g Fa c t o r s

LO 64.4: Describe Grinolds fundamental law of active management, including its assumptions and limitations, and calculate the information ratio using this law.
Portfolio managers create value, and potentially create alpha, by making bets that deviate from their benchmark. Richard Grinold formalized this intuitive relationship in the fundamental law of active management.1 This fundamental law does not provide a tool for searching for high IR plays, but it does present a good mechanism for systematically evaluating investment strategies. The law states that:
IR ~ IC x VBR
The formula for Grinolds fundamental law shows that the information ratio (IR) is approximately equal to the product of the information coefficient (IC) and the square root of the breadth (BR) of an investors strategy. The information coefficient is essentially the correlation between an investments predicted and actual value. This is an explicit evaluation of an investors forecasting skill. A higher IC score means that the predictions had a higher correlation (high-quality predictions). Breadth is simply the number of investments deployed.
Consider an example of an investor who requires an IR of 0.50. If this investor wants to time the market using an index and plans to only make four investments during the year, then he would need an IC of 0.25 as shown:
0.5 = 0.25 x T i
What would happen if this same investor instead decided to deploy a stock selection strategy based on either value or momentum plays? These two strategies both involve taking a high number of bets every year. If they placed 200 bets in a given year, then they would only need an IC of 0.035 instead of 0.25. A lower IC means lower-quality predictions.
0.5 = 0.035×7200
Grinolds fundamental law teaches us about a central tradeoff in active management. Investors need to either play smart (a high IC shows high-quality predictions) or play often (a high BR shows a lot of trade activity). Essentially, investors can be very good at making forecasts and place a small number of bets, or they will need to simply place a lot of bets. 1. Richard C. Grinold, The Fundamental Law of Active Management, Journal o f Portfolio
Management 15, no. 3 (1989): 30-37.
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Grinolds framework ignores downside risk and makes a critical assumption that all forecasts are independent of one another. The Norwegian sovereign wealth fund has used Grinolds fundamental law in practice. Their philosophy is to take a high number of bets using a large list of entirely independent asset managers. This helps to keep forecasts independent and allows them to have reduced reliance on forecasting prowess while still endeavoring to achieve their benchmark IR goals.
In practice, it has also been noted that as assets under management go up, the IC tends to decline. This affects mutual funds, hedge funds, private equity firms, pension funds, and sovereign wealth funds alike. This is one reason why some mutual funds close to new investors and turn away new assets once they reach an internally set size.
Fa c t o r Re g r e s s io n a n d Po r t f o l io Se n s it iv it y

LO 64.3: Explain the impact of benchmark choice on alpha, and describe characteristics of an effective benchmark to measure alpha.
The choice of benchmark has a significant impact on the calculated alpha for an investment. Strictly benchmarking to an identifiable index, like the S&P 500 Index, assumes that an asset has a beta of 1.0. What if the true beta is some value other than 1.0? Consider an investment that has a beta of 0.73 and tracking error of 6.16%. The alpha for this investment could be estimated by regressing the excess return of the fund (Rt Rp) against the excess return of the benchm ark(R ^00 R p ) . In the following regression equation, you see a calculated alpha of 3.44% and a placeholder for error term (e ) because we never know, in advance, how an individual observation will deviate from our model at any point in time.
Rt RF = 0.0344 + 0.73(RtP500 – RF) + et
We can rearrange this formula to isolate only the expected return on our investment. Doing so, we find that our customized benchmark should actually be invested 27% in the risk-free rate and 73% in the S&P 500 Index. Using a benchmark that recognizes the investments beta as 0.73, we calculate an alpha of 3.44%, which translates into an IR of 0.5584 (= 0.0344/ 0.0616).
Rt = 0.0344 + 0.27(RF) + 0.73(Rt P50) + et
If this same investor were to wrongly regress their investment against only the S&P 500 Index, then they would calculate an alpha of 1.50%, which is incorrect because it assumes a beta of 1.0 when the actual beta is 0.73.
Rt = 0.015+ R tP500 + et
Using the wrong benchmark would produce an IR of 0.2435 (= 0.0150 / 0.0616). This suggests that using an incorrect benchmark will understate both the expected alpha and the IR. Inaccurate information may cause an investor to pass on an investment that they otherwise should have accepted.
This illustration leads an investor to wonder: what is the best way to choose a benchmark? An appropriate benchmark can be selected by applying a few different complementary standards. First, the benchmark should be well-defined. It should be hosted by an independent index provider, which makes it both verifiable and free of ambiguity. The S&P 500 Index and the Russell 1000 Index are both examples of well-defined large-cap indices. Second, an index should be tradeable. If the benchmark is not a basket of tradeable securities that could be directly invested in as an alternative, then the benchmark is not a very good comparison. Third, a benchmark must be replicable. This is closely related to the tradability standard. There are some benchmarks, like absolute return benchmarks, that are
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not feasible for an investor to replicate. If it cannot be replicated, then the tracking error will be very high. Fourth, the benchmark must be adjusted for risk. In the previous example, you can see that the alpha and the IR will be calculated too low if the risk level of the benchmark is too high for the investment in question.
Fu n d a m e n t a l La w o f Ac t iv e M a n a g e m e n t