# Step 8: Riskese

Understand how risk, return and time are related
8.2

## Definitions

8.2.1

### Standard Deviation

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 yellow area) accounts for approximately 68% of the annual returns in a time period. Two standard deviations away from the mean (the yellow and green areas) account for approximately 95% of the annual returns. And three standard deviations (the yellow, green, and orange 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. The higher an investment's standard deviation, the greater the chance that future returns will lie farther away from the average return.

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

Here are two youtube videos that explain standard devaition:

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. Figure 8-1a below illustrates this point.

For example, let's assume an investor had to choose between two stocks and an S&P 500 Index Fund.

1) General Electric (GE)
2) Proctor & Gamble (PG)
3) S&P 500 Index

Neither stock is a good choice because an S&P 500 index over the period shown in Figure 8-1a earned a similar annualized return with a lower volatiliy (uncertainty or standard deviation), because of the diversification benefit of 500 stocks versus one stock, making the S&P 500 Index always the statistically preferred investment over individual stocks. You can see from the chart that even stocks that had higher returns than the S&P 500 required higher levels of risk. If you drew a diagonal line along the IFA Index Portfolios, you can see that all stocks but WalMart fell below the line.

If you diversified into other global markets and small value indexes, in addition to the S&P 500, like Index Portfolio 100, the annualized return in this same period was higher, with a similar standard deviation, making Index Portfolio 100 always the statistically preferred investment over the S&P 500 and anything else that we can find. Please note that investors must first determine their risk capacity before making investment choices. Then they can determine which investments have the highest expected return for the standard deviation that is right for them. According to IFA, the maximum risk exposure for any investor should be Index Portfolio 100.

Figure 8-1a

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 100 is highly volatile. Figure 8-2A shows the annual returns over 50 years in the top graph and a histogram based on the average return and standard deviation on the bottom graph. Figure 8-2C shows a low standard deviation investment, Index Portfolio 10. Note the lack of volatility over the 50 years on the top graph and the narrow distribution of the bell curve on the bottom graph. Figure 8-2B shows a more moderate volatility portfolio.

Figure 8-2A

Figure 8-2B

Figure 8-2C

Another important statistical concept is the standard error of the mean, which indicates the degree of uncertainty in calculating an estimate from a sample, like a series of returns data. A standard error can be calculated from the standard deviation by dividing the standard deviation by a square root of n (with n representing the number of years measured). So with only 3 years of returns data on the S&P 500, the error in the average return is 2.6 times larger than having 20 years of data. See the Khan Academy video below for further description of the Standard Error of the mean.

8.2.2

### Probability Distributions and Histograms

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 100, which is a histogram of simulated past results over a 50-year period. This all-equity index portfolio carries a wide range of outcomes or a high standard deviation, maintaining an approximate normal distribution with a high level of short-term volatility. Figure 8-6 provides a comparison of a more narrow histogram of monthly returns. The lower risk level of Index Portfolio 10 is illustrated in the narrower range of past monthly returns. Index Portfolio 100 experienced greater price swings but had higher returns than Portfolio 10. This historical data support the presumption that investors who have the capacity to hold higher risks are expected to earn substantially higher returns.

Figure 8-5

Figure 8-6

8.2.3

### Mean Reversion

 Figure 8-7

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.

8.2.4

### Expected Return

 Figure 8-8

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.

8.2.5

### Simulated Passive Investor Experiences

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 for 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 hundreds of 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 more than 400 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