William Bernstein

"If your broker [or investment advisor] is not familiar with the concept of standard deviation of returns, get a new one."

— William Bernstein, The Intelligent Asset Allocator, 2001

Charles Ellis

"The average long-term experience in investing is never surprising, but the short term experience is always surprising. We now know to focus not on rate of return, but on the informed management of risk."

— Charles Ellis, Investment Policy, 1985


"The probable is what usually happens."

— Aristotle, Lady Luck, The Theory of Probability, 1963

Gary Belsky

"Odds are you don't know what the odds are."

— Gary Belsky and Thomas Gilovich, Why Smart People Make Big Money Mistakes, 2000

Marquis de Laplace

"The most important questions of life are, for the most part, really only problems of probability."

— Marquis de Laplace, Lady Luck, The Theory of Probability, by Warren Weaver, 1963


Do you speak riskese? Citizens of Japan speak Japanese, lawyers speak legalese, and leading investment advisors, casino statisticians and insurance actuaries speak riskese. Riskese is the essential language of investing and is used to discuss topics of risk, return and time. Returns and risk go hand in hand. You cannot expect high returns without taking risk. People are perfectly comfortable talking about the returns portion of the investment process, but how squeamish do they get when they realize they may lose money on their investments? After the crash of 2008, people are far more aware of this possibility, but they are still looking for that perfect investment with small risk and big returns. People are also still looking for a weight loss pill that will allow them to continue eating country fried steak, massive cinnamon buns and ice cream on a regular basis. Neither exists.


Lack of Understanding

One of the primary deterrents to investors earning market rates of returns is their lack of understanding of the relationship between risk and return. The natural tendency of investors is to want returns without the risk. Because risk is the source of returns, investors would be better served to be more concerned with the risk level of their investments.

The overall concept of the risk/return relationship is that when risk increases, the expected return on the asset should also increase as a result of the risk premium earned. Investors should not expect high returns on an investment if they aren't taking a high degree of risk. If you bet on last year's defending champions to win the World Series, your neighborhood bookie will give you much lower odds than if you bet on Major League Baseball's cellar dweller from last season. You are obviously taking a lot more risk by putting your money on the last place team.


Risk Defined

Blaise Pascal

Modern finance began with the realization that risk needed to be measured and managed. In 1654, French mathematicians Blaise Pascal and Pierre de Fermat tried to predict the future outcome of a game of chance. Their questions led to Pascal's Theory of Probability, which quantifies the numerical likelihood of future events. Pascal's Triangle was the foundation for learning how to manage the uncertainty of future outcomes, such as investment returns.

Every investment carries an expected return. The risk of an investment is quantified by the degree to which the returns of the investment deviate from the average return during specific periods of time. Higher risk investments carry a wider range of short-term outcomes but also carry higher expected returns, compensating investors for withstanding short-term volatility. In contrast, investments that have had a narrow range of outcomes over long periods of time are expected to provide more consistent returns with the trade-off of lower returns. For example, an all-bond index portfolio has provided a small but consistent return, while an all-equity index portfolio has provided a larger but erratic return. Higher expected returns are the reward for an investor’s willingness to accept this volatility. In other words, risk is the source of returns and therefore should be embraced in appropriate doses.


Standard Deviation of Returns

An effective method to measure the deviation of investment returns from the average is the standard deviation of returns. Standard deviation provides a statistical measure of historical volatility and sets forth a distribution of the ranges of probable outcomes. In investing, measuring standard deviation of returns shows the extent to which returns (daily, monthly or annual) are distributed around the average return, estimating a range of probable outcomes and establishing a likely framework of risk and return trade-offs.

The normal distribution in the form of a bell-shaped curve shown in Figure 8-1 illustrates the concept of standard deviation. The curve represents the set of outcomes. In this case, let's say the outcomes are the monthly returns of an investment. The yellow area covered in one standard deviation away from the average in both directions accounts for approximately 68% of the outcomes in a period. The area within two standard deviations from the average, the yellow and green shaded areas, accounts for 95.6% of outcomes, and the area up to three standard deviations away from the mean, illustrated by the yellow, green and orange shaded areas, accounts for 99.7% of all outcomes. The higher an investment's standard deviation, the greater the chance that future returns will lie farther away from the average return.

Figure 8-1


Francis Galton
Probability Machine
IFA's Probability Machine

Francis Galton, an English mathematician who was an expert in many scientific fields, created his "Quincunx" machine to demonstrate how a normal distribution is formed through the occurrence of multiple random events. He expressed his fascination with this phenomenon by stating, "I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the ‘Law of Frequency of Error.' It reigns with serenity… amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along."1

I recently commissioned the creation of the Probability Machine to better educate investors about the probability of outcomes that result from a series of random events.

In the case of the market, the random events are news stories about a company or about capitalism in general. The random falling of the beads ultimately forms a normal distribution in the shape of a bell curve. The distribution of the beads bears a striking resemblance to the distribution of monthly returns shown in red behind the beads, also shown in Figure 8-2 which reflects 600 monthly returns (50 years) for an all-equity index portfolio. Like the Probability Machine's normal distribution, the all-equity index portfolio carries a wide range of outcomes or a high standard deviation. It maintains an approximate normal distribution that accumulates to a 1.13% average monthly return over the 600 months, but with a standard deviation of  4.68%, which is a high level of short-term volatility.

Figure 8-2

Note the comparison to Figure 8-3, which shows a lower risk index portfolio comprised of 90% fixed-income funds and 10% stocks with a narrow range of outcomes. The 100% stock index portfolio in Figure 8-2 experienced greater price swings but had higher returns. Over the simulated 50-year period, a dollar invested in the low-risk index portfolio would have grown to $21.02, while a dollar invested in the higher risk index portfolio would have grown to $440.38. This historical data supports the presumption that investors who have the capacity to hold higher risk are expected to earn substantially higher returns.

Figure 8-3


Not All Risks Are Rewarded

While higher expected returns can only result from accepting increased risk, not all risks are rewarded at the same rate. Financial economists have long sought to identify the factors that explain stock market returns. With the help of CRSP, ground-breaking progress has been made. As previously mentioned in Step 2, Nobel Laureate William Sharpe presented the Capital Asset Pricing Model (CAPM), a financial model that explains approximately 70% of all stock portfolio returns. CAPM enabled investors to quantify expected returns based on how investments fluctuate relative to the market as a whole. It concluded that investments which fluctuate more than the market as a whole carry more risk than the market and therefore should also carry higher expected returns. Sharpe asserted however, that some investments carry increased risk without providing the trade-off of higher expected returns. To clarify, he divided risk into two categories: systematic and unsystematic.

The entire market is exposed to unavoidable systematic risk, such as war, recession, inflation, and government intervention. In contrast, unsystematic risk refers to threats specific to individual companies, such as lawsuits, fraud and competition. A summary of these different risks is presented in the top portion of Figure 8-4. Systematic risk—the risk of investing in capitalism itself—has rewarded investors with an approximate 9.5% annualized return over the last 85 years. However, an investment in unsystematic risk, such as buying individual stocks, does not increase expected returns. Unsystematic risk should be avoided through diversification, thereby maximizing portfolio efficiency and returns at each level of risk.

Figure 8-4

Figure 8-5 illustrates the lack of increased expected returns when investors accept the additional concentrated and unsystematic risk of individual stocks. The additional risk of buying individual stocks is not expected to carry additional returns.

Figure 8-5


The Trade-Off Between Risk and Return

Even when all non-compensated risk has been eliminated from a portfolio, an investor cannot escape the systematic risks inherent in the market itself. As previously mentioned, history shows an investment in the market as a whole has delivered about 9.5% a year on average for the last 85 years, but not without substantial uncertainty. Risk and return go hand in hand. To obtain greater stock returns, the trade-off is suffering significant short-term volatility, such as investors experienced in 2008 and early 2009.


The Dimensions of Investment Returns

Sharpe's CAPM was widely held as the explanation of equity returns until 1992 when Eugene Fama and Kenneth French introduced their Fama/French Three-Factor Model, identifying Market, Size and Value as the three factors that explain as much as 96% of the returns of diversified portfolios. Fama and French analyzed the CRSP database back to 1962 to determine that equity returns can be explained by a portfolio's exposure to the market as a whole, as well as the exposure to small and value companies. Their data show that small and value companies have carried higher risk and that risk has been rewarded. These small and value excess returns have been shown to carry long-term persistence, but are not consistent in short periods of time. More than 85 years of risk and return data have confirmed their results.

Fama and French later expanded their model to include Fixed Income, identifying Term and Default as two additional risk factors that explain returns for fixed income. Thus, a Five-Factor Model was created, as shown in Figure 8-6.

Figure 8-6

The relationship that exists between these five risk factors and a portfolio's expected return serves as a framework for designing investment portfolios. Following are the five risk factors.


The Market as a Risk Factor

Market Factor
Market Factor

The first risk factor in the Fama/French Five-Factor Model is the market risk factor or the amount of an investor's exposure to the overall stock market compared to risk-free investments, such as the 30-day T-Bill. Investors take on market risk through all their different equity investments. Figure 8-7 plots the risk and return associated with the market risk factor for five allocations of the total U.S. stock market and U.S. Treasury bills. The highest market exposure, labeled 5, carries 100% exposure to the total U.S. market. The button labeled 0 is invested in 100% T-Bills. The chart reflects the differences in growth of $1 in the various market exposures over the 85-year period from January 1, 1928 through December 31, 2012.

Figure 8-7


Size as a Risk Factor

Size Factor

The second risk factor in the Fama/French model is the size risk factor, referring to the level of a portfolio's exposure to small company stocks. Small companies are more volatile and riskier than larger companies because they have less business diversification, fewer financial resources and greater uncertainty of earnings than their large counterparts.

The image for the size factor contrasts General Electric with the small cap company Acme Packet. It is obvious that Acme is a riskier investment than GE, therefore the cost of capital for Acme and the expected return for its investors should be higher.

Figure 8-8 plots the risk and return of the S&P and U.S. companies according to size over an 85-year period.

Figure 8-8


Value as a Risk Factor

Value Factor

The third risk factor in the Fama/French model is the value risk factor, which refers to the amount of a portfolio's exposure to value or low-priced stocks relative to their book value. Value is measured by the book-to-market (BtM) ratio. The book value of a company is 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. Stocks with higher BtM ratios are considered value stocks while stocks with lower BtM ratios are considered growth stocks. Figure 8-9 plots the risk and return characteristics of the value risk factor for the five quintiles of the U.S. stock market from 1928 through 2013.

Figure 8-9


Term as a Risk Factor

Term Factor

Fixed income is also an important component to an investment portfolio. Since stocks and bonds frequently move in opposite directions, holding low-volatility bonds provides good diversification and will therefore level out a portfolio's performance by dampening stock volatility and providing short-term liquidity.

The term (maturity) risk factor refers to the difference in returns between long-term government bonds and short-term treasury bills. Longer-term bonds are riskier than shorter-term instruments and have yielded higher returns over the 85 years ending in 2012. Figure 8-10 shows six different fixed income allocations and their differences in risk and return.

Figure 8-10


Default as a Risk Factor

Default Factor

The last of the five risk factors 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. Figure 8-11 shows the strong relationship between risk and return as the probability of default increases.

Figure 8-11


Implications of the Five-Factor Model

A summary of the average annual returns for all five risk factors is presented in Figure 8-12. Both the risk and average annual returns for all five risk factors are shown in Figure 8-13.

Figure 8-12

Figure 8-13

The research of Eugene Fama and Kenneth French serves as a guiding protocol for both individual and institutional investing. Their Five-Factor Model has revolutionized how portfolios are constructed and analyzed.

As you have learned, three of the five factors apply to equities. The Three-Factor Model is an invaluable tool for asset allocation and portfolio analysis. In his own words, Eugene Fama candidly explains the small-value story, among other subjects, in a November 2007 interview with The Region, the publication for the Federal Reserve Bank of Minneapolis. "So, small-cap stocks have higher average returns than large-cap stocks, and stocks with higher ratios of book value to market value have higher returns than low book-to-market stocks." He continues, "Low book-to-market stocks tend to be growth stocks. High book-to-market stocks tend to be relatively more distressed; they're what people call value stocks. That's given rise to what the finance profession—academic as well as applied—calls the size premium and the value premium. The value premium tends to be bigger," he added. "So, our model has three factors. Every asset pricing model says you need the market in there. Then they differ on how many other things you need. The CAPM says you only need the market. We basically say a minimum of two other factors seem to be necessary. And these two do a pretty good job."2

  • 1J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956. pg. 1482.
  • 2Douglas Clement, "Interview with Eugene Fama," The Region (a publication of The Federal Reserve Bank of Minneapolis), December 2007.
step 8riskeseblaise pascalpierre de fermatprobabilitypascal's triangletheory of probabilitystandard deviation. normal distributionprobability machinefrancis galtonquincunxwilliam sharpecapital asset pricing modelcapmsystematic riskcapitalismunsystematic riskmarketriskreturneugene famakenneth frenchthree factor modelcrspfive factor modelmarket factorfive-factor modelmarket risktreasury billssize factorexposuresmall companiesvalue factorportfolio's exposurebook-to-market ratiobtm ratiovalue stocksgrowth stocksterm factorfixed incomebondsdefault factorcreditgovernment bondscorporate bondsthree-factor model

Related Material