Eugne Fama Jr.

The Error Term

Eugne Fama Jr.

Investment planning is about structuring exposure to risk factors. In any period, random variation can make it hard to focus on the factors that drive returns. It is therefore important to expect statistical "noise" along the way and not let it unduly influence policy. The Fama/French multifactor model helps distinguish the systematic factors in returns from the random noise (Exhibit 1).

Exhibit 1
Three-Factor Model
 

 

Book-to-market ratio (BtM) is the ratio of a firm's book value of equity to its market value of equity. Book value of equity is determined by the firm's accountants using historic cost information. Market value of equity is determined by buyers and sellers of the stock using current information.
 
Expected return ("E(R)") is the mean value of the probability distribution of possible returns.
 
Variance2) measures the dispersion of a return distribution. It is the sum of the squares of a return's deviation from the mean, divided by n. The value will always be >=0, with larger values corresponding to data that is more spread out.

The three priced risk factors in equity returns are market, size, and book-to-market (BtM). All are "priced" because markets compensate investors with expected returns for taking them. The error term in the model (e(t)) captures all the residual, non-priced risk. In a well-diversified portfolio this can include variance from country or industry weights, differences in holdings, and even volatility from individual stocks. Such risks have no expected return.

Factor exposure determines expected return, but there are infinite ways to achieve any given factor exposure. For example, suppose you want mid-cap exposure. You can get it by holding mid-cap stocks. You can also get it by holding no mid-cap stocks at all but instead holding a combination of tiny stocks and huge stocks. Both portfolios might have identical factor exposures and identical expected returns, but inter-period returns are likely to differ dramatically.

Such differences are the result of residual error. They do not increase or decrease returns because the variance has no "direction"-differences from random holdings tend to average to zero through time. But since residual error is risk nonetheless, it is worth minimizing by building portfolios with similar composition to the target universe.

Even then, portfolios with identical factor exposures will behave differently, as long as there are differences in their underlying securities.

The beta coefficient (β) measures an investment's relative volatility or impact of a per-unit change in the independent variable (market) on the dependable variable (portfolio) holding all else constant.

Exhibit 2 shows three portfolios with the same factor exposures. For simplicity's sake the target is the Total Stock Market published by CRSP. The example portfolios have virtually identical, market-like factor exposures: around 1.00 on beta, 0.00 on size, and 0.00 on BtM.

Exhibit 2
Portfolio Combinations
January 1990-October 2001
 
 
S&P
500
Index

Russell
1000
Index

CRSP
1-10
Index

CRSP
9-10
Index

Russell
2000
Index

Russell
3000
Index
Russell
3000
Growth
Index
Market     100%        
Portfolio 1           100%  
Portfolio 2   90%     10%    
Portfolio 3 75%     10%     15%
               

 

  Size
Co-efficient (s)
BtM
Co-efficient (h)
Market 0.00 0.00
Portfolio 1 -0.04 0.04
Portfolio 2 -0.02 0.03
Portfolio 3 0.00 0.04
     

 

  Market Portfolio 1 Portfolio 2 Portfolio 3
Monthly        
Average Return 1.04% 1.03% 1.03% 1.04%
Standard Deviation 4.24% 4.23% 4.24% 4.30%
Tracking Error to Market 0.00% 0.31% 0.36% 0.54%
Maximum Over 0.00% 1.55% 1.68% 2.99%
Maximum Under 0.00% -1.28% -1.57% -1.46%
         
Annualized        
Average Return 12.54% 12.41% 12.41% 12.53%
Standard Deviation 14.69% 14.67% 14.70% 14.90%
Tracking Error to Market 0.00% 1.07% 1.23% 1.88%
         
Rolling 6-Month Cumulative Return Difference
(Rolling 6-Month Cumulative Return Tracking Error)
   
Maximum Over 0.00% 2.62% 2.67% 3.06%
Maximum Under 0.00% -1.84% -2.09% -3.72%
 
S&P data courtesy of © Stocks, Bonds, Bills and Inflation YearbookT, Ibbotson Associates, Chicago (annually updated works by Roger C. Ibbotson and Rex A. Sinquefield).
Russell data courtesy of Russell Analytic Services.
CRSP data courtesy of the Center for Research in Security Prices, University of Chicago.

Portfolio 1 is the Russell 3000. This index is so much like the market that most of the institutional world uses it for a market benchmark. Portfolio 2 is a blend of large cap stocks (Russell 1000) and small cap stocks (Russell 2000). Portfolio 3 is a blend of S&P 500 and micro-cap stocks (CRSP 9-10), adding a large cap growth index (Russell 3000 Growth).

Standard deviation (σ) is the statistical measure of the degree to which an individual value in a probability distribution tends to vary from the mean of the distribution.

Since all of these portfolios have similar factors, they have similar expected returns. From January 1990 to October 2001, they also happen to have virtually identical realized returns (this will not always happen). Standard deviations of all four portfolios are similar, as are tracking differences (volatility of the premium) versus the market. All three portfolios are valid ways to capture diversified market-like exposure.

Yet residual variance still affects the returns. The maximum over- and under-performance versus the market in both monthly and cumulative six-month periods is significant for all three portfolios. Even the Russell 3000 Index, a portfolio we would expect to track the market, has a month where it under-performed by 128 basis points and an entire six-month span where it under-performed by 184 basis points. In spite of their varied structures, the other portfolios have similar highs and lows. The dispersion of securities among priced factors is not what causes these periodic differences-they result from residual error unrelated to systematic factors. This error is random: sometimes it's positive and sometimes negative. The periods of over-performance and under-performance tend to cancel each other out through time.

Exhibit 3 shows how wide the six-month cumulative difference due to residual error seems in plotted form. Tracking error like this can be distracting, but since the expected differences average to zero, managing this error should not take priority over managing the paying factors in returns. For many investors, foremost among these factors is taxes.

Exhibit 3
Tracking Error of Rolling Six-Month Cumulative Return
(Portfolio n - Market)
January 1990-October 2001
 
 
S&P data courtesy of © Stocks, Bonds, Bills and Inflation YearbookT, Ibbotson Associates, Chicago (annually updated works by Roger C. Ibbotson and Rex A. Sinquefield).
Russell data courtesy of Russell Analytic Services.
CRSP data courtesy of the Center for Research in Security Prices, University of Chicago.

Most individual investors should consider taxes. After all, the expected return that comes from factor exposure has a wide variance around it. You will not get the average annual return every year. You will, however, be "asked" to pay taxes every year. The expected impact of taxes might be the most reliable explanatory factor in returns-the "known quantity."

The latest advance in multifactor engineering takes taxes into account. Dimensional has developed an algorithm that builds portfolios with targeted exposure to systematic factors like size and BtM, while "optimizing" the underlying set of securities in an attempt to harvest capital losses and minimize dividends. The tax-managed versions of strategies can have specific factor exposures that are identical to non-tax-managed alternatives or any other investor preference. But, as in the examples discussed above, the returns of the tax-managed portfolios and non-tax-managed alternatives will differ through time simply because the underlying securities differ. As in the earlier examples, these differences are random.

The only reason to make an example of tax-managed strategies is that tax management is such a worthwhile reason to accept random error. Differences between holdings in a tax-managed portfolio and its target universe exist because they lessen the tax burden. The long-term disadvantage of the error this causes is uncertain, but the strong advantages of tax-conscious investing are as certain as taxes themselves. In other words, investors should accept "noise" around the returns of some benchmark (which in the end is another arbitrary portfolio) in exchange for seeking stronger returns after taxes.

Exhibit 4 shows two portfolios that, like those in the earlier example, target the market. As before, ten-year returns are shown, but this time simulating the effect of tax optimization. Case 1 applies moderate dividend management and Case 2 applies stronger dividend management.

Exhibit 4
Dividend Management
Model / Backtested
 
  CRSP 
Total Market 
Index 
Dividend
Management
Case 1
Dividend
Management
Case 2
Annualized Average Monthly Return 13.94%   14.57% 15.11%
Annualized Average Dividend Yield 2.14%   1.62% 0.86%
Annualized Standard Deviation of the
Monthly Returns
14.54%   14.94% 16.31%
Annualized Standard Deviation Tracking
Error to Market
  1.56% 3.41%
Correlation with Market 100%   99.47% 98.21%
Maximum Monthly Over-performance   2.16% 2.66%
Maximum Monthly Under-performance   -0.94% -2.93%
Pre-Tax Growth of $1 3.09    3.35    3.57   
After-Tax Growth of $1 2.75    2.98    3.26   
Pre-Tax Annualized Compound Return 13.7%   14.3%   14.7%  
After-Tax Annualized Compound Return 12.8%* 13.4%   14.1%  

* No capital gains.

 

  CRSP
Total Market Index
  Dividend
Management Case 1
       
  Return Dividend
Yield
  Return Dividend
Yield
Return
Difference
Yield
Difference
7/90-12/90 -7.74% 1.68%   -8.58% 1.27% -0.85% -0.41%
1991 33.59% 3.64%   35.85% 2.78% 2.26% -0.87%
1992 9.04% 2.79%   10.0% 2.13% 0.96% -0.66%
1993 11.50% 2.70%   10.23% 2.06% -1.28% -0.64%
1994 -0.60% 2.56%   0.89% 1.86% 1.50% -0.70%
1995 35.71% 2.83%   35.53% 2.12% -0.18% -0.71%
1996 21.27% 2.23%   22.74% 1.73% 1.47% -0.50%
1997 30.42% 1.96%   31.43% 1.53% 1.01% -0.43%
1998 22.55% 1.59%   24.85% 1.24% 2.30% -0.35%
1999 25.12% 1.41%   25.01% 1.02% -0.10% -0.39%
2000 -11.04% 1.08%   -10.49% 0.79% 0.54% -0.29%
1/01-6/01 -6.16% 0.48%   -6.13% 0.35% 0.03% -0.13%
Maximum Annual Over-performance 2.30%    
Maximum Annual Under-performance -1.28%    

 

  CRSP
Total Market Index
  Dividend
Management Case 2
       
  Return Dividend
Yield
  Return Dividend
Yield
Return
Difference
Yield
Difference
7/90-12/90 -7.74% 1.68%   -12.65% 0.72% -4.91% -0.96%
1991 33.59% 3.64%   39.70% 1.64% 6.11% -2.00%
1992 9.04% 2.79%   11.68% 1.15% 2.64% -1.64%
1993 11.50% 2.70%   11.34% 1.12% -0.16% -1.58%
1994 -0.60% 2.56%   0.79% 0.98% 1.40% -1.59%
1995 35.71% 2.83%   33.90% 1.04% -1.81% -1.80%
1996 21.27% 2.23%   23.51% 0.94% 2.24% -1.30%
1997 30.42% 1.96%   30.52% 0.80% 0.10% -1.16%
1998 22.55% 1.59%   24.52% 0.65% 1.96% -0.94%
1999 25.12% 1.41%   30.04% 0.52% 4.92% -0.89%
2000 -11.04% 1.08%   -9.01% 0.39% 2.03% -0.69%
1/01-6/01 -6.16% 0.48%   -7.28% 0.17% -1.12% -0.30%
Maximum Annual Over-performance 6.11%    
Maximum Annual Under-performance -4.91%    
 
Data courtesy of the Center for Research in Security Prices, University of Chicago.

All performance information is based on a model/backtested simulation; the performance was achieved with the retroactive application of a model designed with the benefit of hindsight; it does not represent actual investment performance. The model's investment objective is to achieve long-term capital growth while attempting to minimize federal income taxes on returns. [The model's investment strategy is to purchase stocks on a market capitalization weighted basis and maximize the after-tax value of a shareholder's investment.] The model's performance reflects the reinvestment of dividends and other earnings, and is net of fees. There are limitations inherent in model performance. In particular, model performance may not reflect the impact the economic and market factors have had on the adviser's decision making if the adviser were actually managing client money. Past performance is no guarantee of future results, and there is always the risk that an investor may lose money.

Managing dividends, especially in an aggressive fashion, causes differences in underlying composition that affect tracking versus target portfolios. In this example, the moderate Case 1 had a maximum annual over-performance of 230 basis points and a maximum annual under-performance of 128 basis points versus the market. The aggressive Case 2 had a maximum annual over-performance of 611 basis points and a maximum annual under-performance of 491 basis points versus the market. Over- and under-performance in all cases is within the bounds of what you'd expect randomly.

Dividend yield is the contribution to annual total return that an investor earns by the receiving dividends. It is determined by dividing the dividend per share by the current stock price.

Both cases have the same factor exposures and the same expected returns as the market. Cases 1 and 2, however, have significantly higher after-tax expected returns, especially in periods where dividend yields are high. In 1991, for instance, simulated Case 1 would have saved 0.87% in taxable dividends and simulated Case 2 would have saved 2.00% in taxable dividends. The contribution from tax management is expected to be positive regardless of the direction or magnitude of the investment return. The resulting increase in after-tax compound return is simulated in Exhibit 4.

When investing in tax-efficient strategies, investors make an implicit trade-off between tracking benchmarks and managing dividends. Structuring a diversified portfolio according to expectations and preference requires us to acknowledge and try to understand these trade-offs. This is easier when we recognize that benchmarks and published indexes are fundamentally arbitrary. They experience random noise in their returns just like managed portfolios do. Many investors will want to tolerate this noise in exchange for portfolio engineering and tax structure.

As always, the multifactor model helps us frame the problem. It helps us target factor exposures rather than benchmarks and helps us distinguish systematic expected returns from random noise. Most of all, it gives us the tools to build focused portfolios and the perspective to stay disciplined during times when performance differs from the long-term expectation.


This article owes a huge debt to discussions with Dave Butler and especially Eduardo Repetto, who provided the data and lots of guidance.

 

This article contains the opinions of the author and those interviewed by the author but not necessarily Dimensional Fund Advisors Inc. or DFA Securities Inc., and does not represent a recommendation of any particular security, strategy or investment product. The author's opinions are subject to change without notice. Information contained herein has been obtained from sources believed to be reliable, but is not guaranteed. This article is distributed for educational purposes and should not be considered investment advice or an offer of any security for sale. Past performance is not indicative of future results and no representation is made that the stated results will be replicated.