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Do you speak riskese? Residents of China speak Chinese, citizens of Japan speak Japanese, lawyers speak legalese and top-notch investment advisors, casino statisticians, and insurance underwriters speak riskese. Riskese is the language that’s used to discuss topics of risk, return, time, and correlation. Risk, return and time are all intertwined. Higher exposure to the right risk factors leads to higher expected returns. The longer you hold a risky investment, the more likely you will obtain the long-term expected return. However, because of “random drift,” risk is very unpredictable in the short run, but it can be quantified far more accurately than gut feelings and intuition in the long run. For example, you can flip 10 heads in a row with a coin, but there is still a 50/50 chance that you will flip heads the next time and in the long run. Remember that if there is no risk, there is no reason that you can expect a higher return than Treasury bills, which have paid an annualized return of 3.8% per year for the last 70 years, just 0.5% over inflation. High risk exposure is like a scream inducing roller coaster, with soaring highs and stomach churning lows. On the roller coaster, the greater the ups and downs, the greater the returns... measured in thrills. The same thing applies to investing. However, not everyone has the “capacity” for such “exposure” to risk. In this step the concepts of risk, return and time will be explained.
Standard deviation, as used by investors, is a statistical measure of the historical volatility of an index, stock, mutual fund or portfolio, usually computed from a minimum of 36 monthly returns. More specifically, it is a measure of the extent to which numbers deviate from their average. It also quantifies the uncertainty in a random variable, such as historical stock market returns. To be precise, a standard deviation is the root-mean-square deviation of values from their average, or the square root of the variance. Figure 8-1 illustrates standard deviation. One standard deviation away from the average in both directions on the horizontal axis (the green area) accounts for approximately 68% of the annual returns in a time period. Two standard deviations away from the mean (the green and blue areas) account for approximately 95% of the annual returns. And three standard deviations (the green, blue, and red areas) account for approximately 99.7% of the outcomes, or for returns in a certain period for investments. For normal distributions, the two points of the curve which are one standard deviation from the mean are also the inflection points, which is a point on a curve where the curvature changes signs.
Figure 8-1a shows the approximate bell curve of the S&P 500 Index over the last 82 years, where the annualized return is about 10% and the Standard Deviation is 20%. Based on these characteristics, about 68% of the annual returns have been between -10% and +30%, while 95% of the annual returns fall between -30% and +50%.
Figure 8-1
In the field of finance, standard deviation represents the risk associated with a security (stocks or bonds), or the risk of a portfolio of securities (including actively managed mutual funds, index mutual funds, or ETFs). Risk is an important factor in determining how to efficiently manage investments because it determines the variation in returns on the asset and/or portfolio and gives investors a mathematical basis for investment decisions (the basis for mean-variance optimization). The overall concept of risk is that as it increases, the expected return on the asset should increase as a result of the risk premium earned – in other words, investors should not expect a higher returns on an investment without that investment having a higher degree of risk, or uncertainty of those returns. When evaluating investments, investors should always estimate both the expected return and the uncertainty of that return. Standard deviation provides a quantified estimate of the uncertainty of those future returns, the average of which is the expected return. For example, let's assume an investor had to choose between two stocks. 1) The Walt Disney Company (DIS) over the 20 years (1988-2007) had an annualized return of 10%, with an standard deviation of 26% (a return/risk ratio of 0.385) or 2) Hewlett Packard (HPQ), over the same period, had annualized return of 12%, but a higher standard deviation of 36% (a return/risk ratio of 0.333) (see the chart below). Figure 8-1a Enlarge On the basis of risk and return, an investor may decide that The Walt Disney Company is the better choice, because Hewlett Packard's additional 2% points of return is not worth the additional 10% standard deviation (greater risk or uncertainty of the expected return). Hewlett Packard has a lower risk/reward ratio (0.333 for HP versus 0.385 for Disney) and is more likely to fall short of the initial investment under the same circumstances, and is estimated to return only 2% more on average. In this example, The Walt Disney Company is expected to earn about 10%, plus or minus 26% (a range from 36% to -16%), about two-thirds of the future annual returns. When considering more extreme possible returns or outcomes in the future, an investor should expect results of up to 10% plus or minus 78% (3 x one standard deviation), or a range from 88% to -68%, which includes outcomes for three standard deviations from the average return (about 99.7% of probable returns). Actually, neither stock is a good choice because an S&P 500 index fund over the same period earned 11.63% annualized average return with a much lower standard deviation of 13.48% (a higher return/risk ratio of 0.863), because of the diversification benefit of 500 stocks versus one stock, making it always the statistically preferred investment over individual stocks. If you diversified into other global markets and small value indexes, in addition to the S&P 500, like Index Portfolio 100, the average annualized return in this same period was 13.39%, with a standard deviation of 13.49% (a very high return/risk ratio of 0.993), making it always the statistically preferred investment over the S&P 500 and anything else that we can find.
If you want to calculate the standard deviation of a certain stock or index, start by calculating the average return (or arithmetic mean) of the security over a given number of periods, like 20 years or more. For each year, subtract the average return from the actual return for that year, which will give you the "variances". Square the variance in each period. The larger the variance in a period, the greater risk the security carries. Calculate the average of the squared variances and then calculate the square root of this average variance and you will get the standard deviation of the investment in question. The arithmetic mean is used to calculate the standard deviation, but when comparing returns of stocks or indexes, it is customary to use the annualized average return, not the arithmetic mean return. For more information on Standard Deviation see the Wikipedia page.
Volatility or standard deviation relative to the market, also known as beta, is one way to look at risk. There are three risk factors that have explained 95% of the variability of stock market returns, dating back to 1929. (The 23 page academic study that supports this can be found here.) According to a landmark study by Eugene Fama and Kenneth French in 1992, the three factors that explain 95% of the variability of stock market returns are Beta (Market Risk); Market Capitalization, (Size Risk); and Value (High Book to Market Ratio Risk). Small value stocks have the highest expected return. In Step 9, you will see the historical difference in the performance of various asset classes. Evidence of the accuracy of the Three Factor Model is represented below. Since 1927, the increased risk of owning small size and high value companies has delivered higher returns. Non-US and emerging markets provide opportunities for the diversification of more small and even more value-tilted stocks in addition to additional country risks which all results in higher cost of capital for those firms and higher expected returns for investors. An investment like Index Portfolio 90 is highly volatile. Figure 8-2 shows the annual returns over 35 years in the top graph and a histogram based on the average return and standard deviation on the bottom graph. Figure 8-3 shows a low standard deviation investment, Index Portfolio 5. Note the lack of volatility over the 35 years on the top graph and the narrow distribution of the bell curve on the bottom graph.
Figure 8-4
A probability distribution is a mathematical function that describes the probabilities of possible outcomes. For example, if two dice are rolled, the range of possible outcomes or the possible results of the dice toss are two through 12. The corresponding probability distribution for the dice toss is reflected in Figure 8-4. The mean of a probability distribution is its average or expected value. Figure 8-5 shows a distribution of 600 monthly returns of Index Portfolio 90, which is a histogram of simulated past results over the last 50 years. The average monthly return was 1.14% and the monthly standard deviation was 4.01%. Based on the average return and standard deviation of long-term historic data, a probability distribution of future outcomes can be estimated. Figure 8-6 provides a comparison of a more narrow histogram of monthly returns. The lower risk level of Index Portfolio 30 is illustrated in the narrower range of past monthly returns.
Mean reversion is a tendency for certain random variables to remain at or return over time to a long-run average level. For example, interest rates and broadly diversified indexes tend to be mean reverting. Individual stock prices, mutual fund manager performances, and short-term market performance tend not to be mean reverting. Large diversified portfolios of stocks, such as the CRSP 1-10 Total Market Index or the S&P 500 Index rates of return tend to be mean reverting. They may be high or low from one year to the next. However, over 80 years, the rate of return of the S&P 500 has tended to average in the 10% range plus or minus 20% two-thirds of the time. It is impossible to tell for certain if a variable is mean reverting by looking at its performance over any short period of time, such as a three or five-year track record of a stock, time, manager or style. This is because a tendency toward mean reversion may only reveal itself over very long horizons of 20 years or more. Figure 8-7 illustrates the difference between mean reverting and non-mean reverting outcomes.
Expected return refers to the expected or average rate of return of an investment. The term refers to a theoretical future performance, and it is definitely not a guarantee. The expected return of an index can only be derived from its very long-term historical past performance. Because returns are full of uncertainty, higher variables and the actual return for short periods of time is unpredictable, but the expected return remains the same. Only the uncertainty or standard deviation of expected returns changes with time.
An investment’s expected return is simply the middle value of the probability distribution or bell-shaped curve, which is shown as 5% in Figure 8-8. Investors are constantly surprised by short-term results, which will look nothing like the distribution below. In practice, sophisticated investors often base their expected return and volatility assumptions on historical returns of 20 years or more of an index or asset class. Watch this explanation of the expected value by the Khan Academy.
One problem for investors is the high level of error when drawing conclusions from a small sample of stock market data, like the last 3, 5 or 10 years. One way to obtain an improvement in the number of observed periods is to create rolling periods with monthly or annual data that overlap from one period to the next. The overlapping period returns do have statistical problems because each period contains a substantial amount of the same data. However, since monthly returns are random and have no correlation to each other, a rolling period of 144 consecutive months out of 600 months should be similar to a random sample of 144 months out of 600 months. This random sampling with replacement is another methodology of looking at a larger sample of returns called bootstrapping. A third method to increase the number of samples is Monte Carlo simulations, which simulate future outcomes of portfolios based on a long term historical average and standard deviation. Figure 8-9 illustrates how to simulate passive investor experiences through monthly rolling periods that are obtained using monthly returns data over 50 years. As you can see in the illustrated examples that Period #1 is the 15 years from January 1961 to December 31, 1975. Period #2 starts one month later on February 1, 1961 to January 31, 1976. Imagine this occurring 423 or more times. This analysis helps capture the simulated passive investor experiences of investors with 15-year holding periods who start their investments in July 1961, January 1963 or any month within the first 35 years of the 50+ year period. You may notice that those investors who started in March 1994 experienced the single lowest 15-year annualized return of 5.23%. Figure 8-9 also contains a rolling period analysis of Index Portfolio 100 over periods from 3 months to 50 years. If you look at the red highlighted row, you will see that it covers 15-year monthly rolling periods (180 months each) and there are 423 or more 15-year rolling periods. As you read across the red highlighted row, you will see the median (50th percentile) return, the range of returns (highest return minus lowest return), the median growth of $1.00, and the lowest/highest rolling period dates, returns and growth of $1.00, etc. This information provides investors with an idea of what has happened over the last 50 years and over varying holding periods, based on a large sample of similated investors. The 15-year period represents the recommended holding period for investors who score a 100 on the IFA Risk Capacity Survey.
Figure 8-9
In the following definition from Webster’s Dictionary risk is defined in terms of loss: “Exposure to the chance of injury or loss; a hazard or dangerous chance.” But, a more appropriate definition of risk for investors is “uncertainty of expected returns.” Most investors think of risk as some sort of loss. To the surprise of many investors, the potential for loss is also the reason they earn a return. “Loss aversion” refers to the concept that the pain of losing a sum of money is greater than the pleasure of gaining the same amount of money. This is incorporated into the optimization process that uses risk and return trade-offs of different asset classes to build portfolios. Research shows that investors are about twice as sensitive to investment losses as to gains. Risk is most commonly measured in terms of standard deviation or the volatility around a given average. Prior to the groundbreaking Fama/French research, stock market risk was measured as volatility around the average return of the total stock market. However, Fama and French added two more dimensions to the measurement of investment risk — size and price. Investors envision risk in several different ways. One way would be the worst case probability of a loss, such as the chance of not achieving an expected rate of return, not being able to readily obtain an expected amount of money at a specific time or the need to withdraw funds from investments when they are in a cumulative negative return position. Risk is one of the most avoided, least quantified and misunderstood subjects by those working in the financial services industry. This is unfortunate because the primary purpose of investment professionals is the intelligent management of financial risks and the alignment of an investor’s risk capacity with the appropriate exposure to financial risk or uncertainty. One dimension of Risk Capacity™ is an investor’s knowledge about risk, the more they understand it, the more capacity they have for risk. We face risk because nobody can consistently predict the future. After all, if we could see the future, there would be no risk. Wouldn’t it be nice to get next year’s Wall Street Journal today! Risk, return, and time are all interconnected. Higher exposure to the right risk factors leads to higher expected returns. In accordance with the law of numbers, the longer an investor holds a broadly diversified risky investment, the more likely a long-term expected return will be obtained. However, because of random drift, risk is very unpredictable in the short run, yet more accurately quantifiable than gut feelings and intuition in the long run. Random drift can be illustrated by flipping a coin and obtaining 10 heads in a row. There is still a 50/50 chance of heads the next time and every time in the future.
Figure 8-10
Individual stocks and bonds contain both systematic and nonsystematic risk. If investors hold the market portfolio of stocks like the Wilshire 5000, they have eliminated nonsystematic risk and they have not concentrated their portfolio on fewer stocks than the market. Concentration risk occurs when investors try to pick stocks and bonds that they think will outperform the market. Concentration of investments is akin to speculation and add risk, but provide no additional expected return. Members of the Dimensional team talk about the advantages of diversification.
Concentration risk comes from all active management strategies such as stocks, timers, managers or style picking. The opposite of concentration is diversification and therefore diversification is often referred to as the antidote to uncertainty. Figure 8-10 summarizes these concepts of riskese in a flow diagram. Figures 8-11, 8-11a and 8-11b illustrate the reasons to avoid concentration risk.
Figure 8-11
Figure 8-11b