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The Dimensions of Stock and Bond Returns

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Here is an introduction to the Three Factor Model, by Mark Hebner:

Researchers did not have a very good idea about what sources of investment risk actually produced higher returns until 1992. They previously only had Sharpe's One-Factor Model to explain how investment returns were derived as seen in the figure below. Sharpe's One-Factor Model explained only about 70% of the returns of the stock market, but when two additional factors were considered, over 90% of the variability of returns of diversified portfolios can be explained.

In 1992, Eugene F. Fama of the University of Chicago and Kenneth R. French of Yale University developed a three-factor model to characterize and describe the relationship between risk and return for stocks and two factors for bonds. Their model for stocks is essentially an extension of Sharpe's One-Factor Model. Sharpe said that the amount of a portfolio invested in stocks is the most important determinant of return. The Fama/French model added two other fundamental determinants. Fama and French sought to determine the factors that best describe why there are differences among the returns of stock asset classes over long periods of time. They first studied the period starting in 1964, the year that reliable computer data was available. It was later updated and confirmed with data dating back to 1926.

Fama and French concluded that exposure to three risk factors — market, size, and price (book-to-market) — collectively do the best job pinpointing the sources of investment risk that account for stock market returns. Risk factors are sources of risk that the stock market seems to reward over the long run. Based on the Fama/French findings, these three risk factors constitute the dimensions of stock returns. To review the average annual returns of the three factors since 1928, see the figure below.

Depending on which study you review, these three risk factors combined explain between 90% and 96% (as seen in the figure below) of the returns of the market in U.S. Studies have been completed in international markets with risk factors unique to those markets, and similar conclusions were obtained. These findings suggest that an investor's investment performance in comparison to the stock market or other investors depends almost entirely on the percentage of stocks (market factor) held in a portfolio, and more specifically, the amount of small stocks (size factor) and high book-to-market ratio stocks (value factor) held.

Market Risk Factor

The first risk factor in the Fama/French Three-Factor Model is the amount of exposure to the overall stock market or the market risk factor. Exposure to this factor is determined by the amount of a portfolio that's invested in or exposed to stocks. The greater this exposure, the higher the return in comparison to U.S. Treasury bills.

To help you remember the concept of Market Risk versus 30 day T-Bill Risk, we created this painting.


Size Risk Factor

The second risk factor in the Fama/French model is the amount of exposure to small company stocks or the size risk factor. Exposure to this factor is determined by the amount of a portfolio that is invested in small company stocks. The greater this exposure, the higher the return in comparison to large company stocks. 

Small company stocks have small market capitalization. The market cap is determined by multiplying the total number of shares times the price per share. These stocks are generally perceived as riskier than large company stocks because small companies have fewer financial resources and more uncertain earnings than large companies. Small companies are also less able to survive prolonged periods of economic downturns. Even when small companies have good track records, these track records aren't very long, adding more uncertainty and greater risk to their stocks. Because investing in small company stocks is riskier, investors demand a higher rate of return.

 It's important to understand that the average historical returns of small-cap company stocks have significantly outperformed large company stocks. The figure below plots the deciles (one-tenth buckets) of U.S. companies sorted by size over the period from 1928 to present. Note that a fairly clear line exists between the less risky large-cap stocks in decile 1 and the very risky microcap stocks in decile 10. However, in shorter time periods they don't always outperform large company stocks. In fact, the size risk factor fluctuates unpredictably. This is consistent with the Random Walk Theory of changes in stock prices.

To help you remember the concept of Size Risk, referring to the difference between large companies and small companies, we created this painting.


Value Risk Factor

The third risk factor in the Fama/French model is the amount of exposure to low priced stocks, which is measured by a book-to-market (BtM) value ratio. The book value of a company is just an accounting term for its net worth, its assets minus its liabilities. The market value of a company is its price per share times the number of shares outstanding. This risk factor is known by several different designations. It has been referred to as the value factor, BtM factor, style factor and price factor. Note that charts referring to it may have any of these designations. The most current designation is the price factor, referring to the low prices of these stocks compared to a company's book value or to other stocks.

Exposure to the price factor is determined by the amount of a portfolio exposure to high BtM stocks. In other words, when a stock's market price is less than its book value, the BtM ratio is greater than one. The greater the exposure to the price factor, the higher the historic and expected return in comparison to low BtM stocks. High BtM companies usually have low earnings and experience other signs of financial distress. Investors don't like these stocks for these reasons. As a result of their poor track records, the market drives down the prices of these stocks. This naturally makes them riskier to investors. 

Stocks with a low BtM ratio have low book values relative to their market prices and are termed growth stocks. Investors favor growth stocks because they're perceived to be great companies and therefore are less risky. They often represent successful companies with strong track records and healthy earnings. They can also represent companies that have had no earnings but offer a product or service that is deemed to have a high potential for future growth.

The Nobel Prize-winning contribution made by Merton Miller provides a framework for better understanding the connection between the price risk factor and stock returns. Miller set forth a simple but profound notion: the cost of capital to a company equals the expected return to an investor who holds its stock. A company's cost of capital is reflected by the price and book value of its shares or the amount that investors are willing to pay to receive a dollar of book value (a higher share price implies a lower cost of capital). The Figure below proves out Miller's Nobel-prize winning research. The figure plots the long-term risk and return characteristics for the entire U.S. stock market as divided by book-to-market ratios in five quintiles. As you can clearly see, the low-book-to-market companies (numbers 1 and 2) produced lower returns that came with a lower risk. The 20% of all U.S. companies with the highest book-to-market (number 5) were perceived to be in the greatest distress, and consequently paid a higher cost of capital (return) to their investors.

To help you remember the concept of Value Risk, referring to the difference between growth companies and value companies, we created this painting. The painting illustrates the difference in the allure of hot products like the early days of cell phones from Motorola to the bland appeal of a can of Spam from Hormel. Over the long haul, stocks that sell closer to their book values have had higher returns as shown in the Figure above.


To explain why we chose these two stocks, see the chart below.

For an explanation of Book Value versus Market Value, watch this video from the Khan Academy.

Suppose that a value company and a growth company each approach a bank for a loan. Which company will have to pay the higher cost of capital (the higher interest rate) on its loan? The value company will pay the higher cost because its future is less certain and the bank will need to charge extra interest for taking the extra risk that the company won't be able to pay back its loan. Thus, the riskier the company's stock, the higher the cost of capital paid by a company.

Because the market perceives a value stock to be riskier, it drives down their price so that the expected return is high enough to make it worthwhile for investors to hold, despite the extra risk they take when buying it. In this way, stock prices adjust, (the market sets the price at a discount, so its expected return is higher) to reflect the perceived riskiness of the stock. This ensures that the stock will be purchased, even though growth companies have better earnings prospects and generally seem safer.

The key to understanding the connection between the price risk factor and stock returns lies in focusing on the market price of a stock. A high BtM ratio suggests that the market values the stock less than the stock's accountants. This is usually because the stock has poor earnings as well as other indications of financial distress. This makes the stock riskier. As a result, investors demand a higher rate of return to compensate them for the risk that a high BtM stock will do worse than expected, go bankrupt, and end up as one of the "stocks in a box."

A 1987 study1 compared the investment performance of a portfolio of 29 growth stocks to one with 29 value stocks. The growth stocks represented companies that were stronger and healthier than value stocks by every economic measure, including return on total capital, return on equity, and return on sales. The value stocks represented companies that had little profitability, terrible management, and a bad image. Yet, the study found that the value stocks outperformed the growth stocks, 298% to 182%, over the five-year period of 1981 to 1985.

This means that investors earned higher returns by owning the stocks of companies that did poorly. That seems counterintuitive to most investors, since they tend to think that healthy stocks are better investments than distressed stocks. After all, if investors ask for a stock tip, they want to hear the name of the next Microsoft, not a stock with poor earnings. The fact is that investors should be interested in great investments (value stocks), not great companies (growth stocks).

The figure shown above illustrates the relationship between expected return and the three factors of market, size, and value. Since we are only discussing stocks, the market exposure is not shown on this plot. The higher the investment plots in the top right quadrant, the higher the expected return. The full page figure seen below actually puts values of average expected returns of various indexes over the market return. The blue circles on the plot represent various indexes. The total market index is plotted at the center of the cross hatch, which is a 0,0 on the scale of size and value. The return in the top right corner is the highest return on the plot. The dotted diagonal line represents indexes that would have the same return as the market return. Also note that large growth stocks have negative average returns relative to the total market return. Large and safe companies have lower risk associated with them and therefore have lower returns compared to smaller companies. Note that a portfolio similar to IFA Index Portfolio 100 plots with an expected average return of about 3% over the total market return (before advisory fees). Because Index Portfolio 100 has international indexes in the mix, this is not entirely accurate, but it does give us an idea of how this type of analysis works.

Implications of the Fama and French Five-Factor Model

Most investors are really only guessing which managers or asset classes will outperform the market. They encounter vast inefficiencies in trying to pick winners from among thousands of money managers and mutual funds. However, with the introduction of the Fama and French model, these costly efforts can be entirely eliminated by investing in Fama and French designed index funds.

Some mutual fund companies such as Dimensional Fund Advisors (DFA) have taken advantage of the Fama and French research by offering a full assortment of index funds, including low price and small company index funds.

Investing in these funds is the most efficient and effective way to maximize exposure to the three risk factors that generate 95% of the market returns. For example, DFA offers investors value index funds that are structured to (1) maximize exposure to the size and price risk factors and (2) diversify that exposure as much as possible. This building block approach to building portfolios is a cleaner and more consistent way of managing money.

How a portfolio is structured for optimal exposure to the three risk factors determines how well the portfolio performs relative to other portfolios. Portfolio structure refers to the indexes the portfolio holds and in what proportions. The Fama and French findings offer guidelines to investors for effectively allocating indexes within a portfolio. The allocation decision is crucial, since the degree of exposure to the three risk factors for equities and two additional factors (term and default) for fixed income accounts for nearly all the returns earned by diversified portfolios of stocks and bonds. That's why investors should focus on properly structuring their portfolios rather than trying to pick winning stocks or managers.

Changing the Definition of "Alpha"

The Fama and French Three Factor (Five Factors with bonds) Model changes the definition of alpha, as seen in the figure below. According to the one-factor CAPM, alpha is the amount by which an active money manager outperforms a broad market index. The Fama and French Three Factor Model defines alpha for equities more precisely as the return an active manager achieves above the sum of the portfolio's expected return due to all three equity risk factors. Alpha measures a manager's skill in earning a return that couldn't have been achieved by indexing the same exact risk exposure as the portfolio run by the manager. In short, did the money manager earn anything above the indexed return?

A portfolio can be plotted anywhere on the cross hair map, and it's easy to calculate its expected return. For example, a small-cap manager may overweight value stocks relative to a benchmark, such as the Russell 2000 Small-Cap Index. As a result, the manager outperforms it. But if the extra return was simply compensation for taking additional non-diversifiable market risk, why should the manager get credit? The job of an active manager is to consistently outsmart the millions of other traders who get the same news at the same second, and through this process provide additional returns that can't be achieved through indexing. This is exactly what the alpha is in the Three-Factor Model. Investors should insist that a manager outperforms a three factor risk adjusted benchmark before crediting him with an alpha return. After all, active manager fees are supposed to pay for predicting the future of stock prices, not for taking additional market risk from low cost index funds.

So, what "positive alpha" managers have been doing with the one-factor CAPM measurement model is just systematically subjecting their clients to two additional risk factors - size (small company) and high BtM value (distress). Thus, what's showing up as alpha (skill) is nothing more than a measurement error. If the performances of active managers are compared between CAPM and the Fama and French model, there are radical changes in the outcomes. Any evidence of manager skill just vanishes under the Fama and French model. The formula for this type of analysis is summarized in the figure above.

Even though active managers focus on alpha, the amount of return due to alpha from stock picking or market timing is random, and on average is expected to be negative. It turns out that alpha is nothing more than a myth perpetuated by the improper measurement of a manager's performance.

Higher Expected Returns of Value and Small Company Stocks

Long-term investment data makes it clear that value stocks outperform growth stocks and small company stocks outperform large company stocks, as seen in the U.S and Non-U.S. returns data in the figure below.

But there has been some debate as to what causes these stocks to outperform large company stocks. Why are there differences in the expected returns of these indexes?

In one corner of the ring are those who say that value and small company stocks outperform because investors mistakenly price the value of the future earnings of distressed companies too low. This is the "market inefficiency" view. That is, investors see the poor earnings and high risks of value and small company stocks and decide that they are worse investments than they really are. As a result, the market sets erroneously low prices for these stocks. In effect, the combination of all market participants' opinions is wrong, and they agreed on a price that undervalues these stocks. When value and small company stocks then go up, the market is surprised. If investors guessed wrong in the past, presumably they should learn from their mistakes and guess right in the future. But, according to the market inefficiency point of view, investors will continue to repeat these mistakes in the future, thereby allowing other investors such as certain professional stock pickers to outperform them and the market. The market inefficiency view holds that the value and size risk factors turned up by Fama and French aren't really fundamental sources of risk, just opportunities for stock picking. 

The field of behavioral finance would add that these mispriced stocks are over or under reactions of investors to market news. A study by Eugene Fama titled "Market Efficiency, Long-Term Returns, and Behavioral Finance," indicates that this may be the case, but such reactions are random and therefore not a viable investment strategy. That paper and many other academic papers can be found on the Internet at

Eugene Fama and other proponents of efficient markets say that the higher expected returns of small and value stocks are compensation for bearing the greater risk.

According to this "market efficiency" view, greater risk and cost of capital of these firms creates higher expected return for investors, reflected in the lower prices relative to book value for value stocks, and the lower market capitalizations of small company stocks. Quite simply, there are differences in expected returns because there are differences in risk. If value and size truly are risk factors, their expected return premiums shouldn't disappear, even when more investors are informed about the favorable risk/return relationships. As a result, there shouldn't be a predictable decline over long periods of time in the probability distributions of future returns generated by these stocks, compared to the safer overall market returns. Remember that the expected returns for these risk factors have standard deviations of about 13%, so an expected return of about 4%, plus or minus 13% two-thirds of the time, is a very wide probability distribution.

Regardless of whether an investor thinks that the higher returns of value and small company stocks are a result of habitual mispricing (market inefficiency) or rational risk compensation (market efficiency), the conclusion is the same. It would behoove investors to include value and small company stock indexes in their diversified portfolios.

The Trade-Offs Between Risk and Return

Risk and return are inseparable. This means that investors must often face bedeviling trade-offs between risk and return. There's no way around these decisions, since they're required in order to build portfolios. For example, sometimes investors look at short-term CD rates. They like the certainty and stability of CD returns, but they feel they need to obtain higher returns. So, these investors turn to stocks. But, when they focus on the years of negative returns, they become uncomfortable because of their aversion to losses.

The result of all this is the "eat well/sleep well dilemma." That is, if investors want to eat well and earn higher returns with stocks, they need to be prepared to take more risk and go through the volatile roller coaster ride of fluctuations in the value of their portfolio. But if they want to sleep well, they must take less risk; that is invest in fixed-income investments such as bonds, and accept that they'll earn lower returns. Thus, the price of obtaining greater long-term accumulation of wealth with stocks is frightening fluctuations in the value of a portfolio. There really is no free lunch in investing. It's the same old story of risk and return trade-offs identified by Markowitz.

The Dimensions of Bond Returns: Term and Default Risk

Bonds are a component of investment portfolios because they dampen the volatility of stocks due to their low correlations to movements of stock prices. Bonds also provide short-term liquidity to investors with cash needs over a two to five-year period. There are two primary risk factors that explain bond returns. The first is the term factor, which is the difference between the returns of long-term government bonds and short-term Treasury bills. The annual average return for the term risk factor has been 2.48% for the 85 years from 1928 to 2012.

Term Risk Factor

While the term provides higher expected returns, the excess returns diminish significantly beyond a term of five years as can be seen in the figure below, so bonds with terms of more than five years should be avoided.

If investors keep terms or maturities short and default risk relatively low, they have more opportunity to capture the much higher expected returns from the size and value risk factors of stocks.

The figure below shows six different allocations of Long-Term Government Bonds and 30-Day T-Bills and the differences in risk and return of those various allocations. Since the period from 1928, there has been a clear relationship between the risk and return among these allocations.

To help you remember the concept of Term Risk, referring to the difference between short duration 30-day US T-Bills and long duration US government bonds, we created this painting.


Default Risk Factor

The second risk factor for Fixed Income is the default risk factor, which is associated with the credit quality of bonds. Instruments of lower credit quality are riskier than those of higher credit quality, thus yielding higher expected returns. Despite the August 2011 downgrade of U.S. Government debt by Standard & Poors, the market still assigns a higher default risk to corporations over the U.S. Government. The default risk factor refers to the additional expected return of corporate bonds over government bonds. The figure below shows the strong relationship between risk and return as the probability of default increases.

To help you remember the concept of Default Risk, referring to the difference between Long Term Corporate Bonds and Long Term Government Bonds, we created this painting.


The Five Dimensions of Risk Exposure

Now that we have the three risk factors of stocks and the two risk factors of bonds, we can look at the explanation of returns for balanced portfolios that include stocks and bonds. The charts below summarize all 5 Risk Factors.


Once we adjust the returns of equity mutual fund managers for their average exposure to the three risk factors, we discover that the factors explain almost all of the returns, leaving little, if any, alpha (skill).