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Finance 3303

Mean & Standard Deviation

Arithmetic Mean, Geometric Mean &

Standard Deviation.

These calculations are particularly important to investments as they explain relationships of investment returns.

Very simply, this is an arithmetic (simple) average of numbers. The arithmetic mean is one of the first statistical relationships that we are exposed to. When an exam is returned to the class, the "average grade" is usually announced or requested. Why? We are accustomed to react to that statistic. When a "low" average grade is announced as the average, it is usually followed by disappointment from the class. The average has nothing to do with a persons individual score, but people will react that way regardless. Arithmetic averages are used in investments to give an idea as to how an investment has performed.

Statistically, the arithmetic mean becomes the expected value. We are schooled to think that way. When a market professional is asked "how they think the market will perform next year?" they will probably have an idea of how to answer that question by reflecting on past performance "on average" and then add a bit of what they believe is different in the year to come that may change that average.

Geometric Mean has a more involved calculation. It calculates the average annual compounded return of the set of numbers. It is the more appropriate measure of changes in wealth. It is used to calculate the changes in wealth.

Use this example: A person deposits $100 into an account earning 6% interest, compounded annually. At the end of the first year, $6 is deposited into the account by the bank; payment for their use of the money for one year. After the second year, the account holder is owed another $6 PLUS interest on the first $6; this is nothing new, this is compound interest.

Let's say that a mutual fund earns 14% in year one; 9% in year two; and 18% in year three. The example assumes and implies that the difference in the account in year one was measured at the beginning of the year and again at the end of year, the difference was 14%. Say $1000 was deposited at the beginning of the year and a balance of $1140 was observed at year end, that would constitute a 14% return [$1000( 1 + .14)]. Could the account balance have fluctuated during the year? Of course it did, a mutual fund balance fluctuates every day.

The $1140 balance is now the beginning balance of year two. At the end of the second year, a balance of $1242.60 would indicate a 9% return. It would give 9% return on the original $1000, but also 9% on the $140 of interest from the first year. [1140( 1 + .09)].

We would use the same approach to calculate the third year: $1242.60( 1 + .18) = $1466.27.

At the end of the third year, your fund balance would be $1466.27. If you had not made annual calculations you would know one thing: That you invested $1000 at the beginning of year 1; three years later, you have $1466.27. Is this good? People ask themselves that question constantly in investing. We would use the Geometric Mean to find the Average Annual Compounded Return for that period.

To calculate the average annual compounded return, first add one to each of the annual returns, then multiply these numbers together, using the future value of a single sum formula: (1+ .14)(1+ .09)(1+ .18). = 1.46627. Secondly, take the root of this number, the cube root in this case because we have three observations. [Adding one to the annual returns is called the Return Relative.]

1.46627.33333 = 1.136068. Lastly, since we "added a one" to each value in the previous paragraph, we have to take the one away to finish the problem. Thus, the average annual compounded return or Geometric Mean of this problem is 0.136068, 13.6068%.

It says that the investor earned an average annual compounded return of 13.6068% to grow $1000 into $ $1466.27 in three years. We know that none of the three years in question returned 13.6068%; the actual returns were 14%, 9% and 18% respectively. The calculations that we made created an Average, a Geometric Average, an average annual compounded return.

Let's say that a person begins investing with $1000. They say that their portfolio returned 100% the first year and lost 50% the second year. They announce proudly that "their average rate of return was 25%." [(100-50)/2].

As we mentioned earlier, the arithmetic mean calculation that was performed in the last paragraph, is NOT the tool to use to calculate the change in wealth, we should use the geometric mean. Here's why.

Look at the example. If $1000 earned a 100% return in the first year, the investor would have $2000 at year-end. The next year, they lose 50% (or half). They are back to the original $1000 investment. They had a ZERO rate of return for the two years. NOT a 25% average return as they reported.

If we use a geometric mean, the true change in wealth will be calculated.

First, add one to each year's return and multiply them together. (1+ 1)(1+ -.5)= (2)(.5)=1. Then, take the square root of the result (the square root, because we have two observations). The square root of one is one. Lastly, subtract one from that result. One minus one is ZERO. That is the change in wealth of this investment as we said before!

Standard Deviation is a statistical technique used against a set of number in an attempt to glean a relationship out of those numbers. It can be used to forecast the outside temperature on a given day to measuring anticipated investment returns from one period to another.

It uses tools that we learned earlier on this page in this calculation.

Almost everyone would agree that a value that fluctuates often, and wildly would have more variation than one that stayed mostly the same. For example, if the temperature outside would begin the morning at 60 degrees, jump to 85 degrees before lunch, plunge to 45 degrees in the afternoon and back up to 90 degrees before the end of day, would be difficult to predict its next move and would be said to be volatile. This compared to a temperature that began the day at 82 degrees, would rise to 90 degrees by mid-day, then end the day back at 82; this is less volatile.

In investments, a volatile investment is one that acts like the first temperature example. A volatile investment is usually described as a risky one. Risky, because it is hard to predict, hard to follow and gives the investor really rough ride.

Standard Deviation measures the variation or volatility of a set of numbers. In our example a set of investment returns.

Let's say that the market as defined by the Dow Jones Industrial Average (DJIA, or "the DOW") had the following annual returns: [These are actual returns.]

1999 +25.2%

1998 +16.1%

1997 +22.7%

1996 +26.9%

1995 +35.5%

1994 +3.6%

1993 +15.1%

1992 +5.7%

1998 +16.1%

1997 +22.7%

1996 +26.9%

1995 +35.5%

1994 +3.6%

1993 +15.1%

1992 +5.7%

To calculate the standard deviation we follow these steps:

1. Calculating the [arithmetic] average return. Add the returns in column 2, divide by the number of observations (years) in column 1.

2. Find the "dispersion around the mean." In column 3, take each individual return (X) and subtract the mean (x) from it. The column should add to zero, if it does not, there was an error made in the calculation.

3. Column 4. Square the differences around the mean. Simply square each answer in column three. This will rid the column of negative numbers. Add the column.

1 2 3 4

Year Return (%) (X-x) (X-x)^2

1999 25.2 25.2-18.85 = 6.35 40.3225

1998 16.1 16.1-18.85 = -2.75 7.5625

1997 22.7 22.7-18.85 = 3.85 14.8225

1996 26.9 26.9-18.85 = 8.05 64.8025

1995 35.5 35.5-18.85 = 16.65 277.2225

1994 3.6 3.6-18.85 = -15.25 232.5625

1993 15.1 15.1-18.85 = -3.75 14.0625

1992 5.7 5.7-18.85 = -13.15 172.9225

Year Return (%) (X-x) (X-x)^2

1999 25.2 25.2-18.85 = 6.35 40.3225

1998 16.1 16.1-18.85 = -2.75 7.5625

1997 22.7 22.7-18.85 = 3.85 14.8225

1996 26.9 26.9-18.85 = 8.05 64.8025

1995 35.5 35.5-18.85 = 16.65 277.2225

1994 3.6 3.6-18.85 = -15.25 232.5625

1993 15.1 15.1-18.85 = -3.75 14.0625

1992 5.7 5.7-18.85 = -13.15 172.9225

n=8 Mean = 18.85% Column should sum to ZERO Σ = 824.28

The bulk of calculation is complete. The process is tedious. There is obviously plenty of chance to make a keystroke error on your calculator, summing column 3 to zero makes a handy check on your progress.

4. The 824.28 is divided by the number of observations. 824.28/8 = 103.035. Statistically, this is called the VARIANCE.

5. Take the square root of the variance to "un-do" the process in column 4 when we squared each of the entries in column three. The act of taking the square root of the variance, is the result that we have labored hard to produce: the STANDARD DEVIATION.

______________

Standard Deviation = √ Σ (X-x)2 / n = 10.15.

Standard Deviation = √ Σ (X-x)2 / n = 10.15.

The Standard Deviation measures the total variability of the numbers in question. S.D. is used in many disciplines; it is a statistical process. When testing services grades an exam using the scantron forms, the standard deviation is printed at the bottom of the page. In finance or more specifically, in asset returns or stock prices, it is the most comprehensive measure of risk. As stock returns are studied, risk is a key ingredient, without understanding the risk associated with an investment, it would be impossible to set goals or for an investment professional to manage a portfolio.

INTERPRETATION: Using this data, what can we conclude? We or an investment professional, market analyst or economist could say: "The stock market is expected to return 18.85% in the year 2000." Expected because the arithmetic mean is 18.85% (remember that the arithmetic mean is the expected value). "Further more, there is a 66% chance that the market will have a return within one standard deviation (plus or minus) from the expected value." Therefore, the is a 66% chance (this is a statistical value that is given in the standard deviation calculation) that the market return for the next year will be between 8.699% and 29%. [Take the expected value of 18.85% and add one standard deviation to it to form 29%; then subtract one standard deviation from it, to get 8.699%.

5/27/16