Types of Mathematical Averages

When computing means, the type of mean you need to use depends on the type of data you are analyzing.

arithmetic mean

For example, the arithmetic mean of 2, 5, and 14 is (2+5+14)/3 = 7. 

The essential property of any mean is that it must fall between the highest value and the lowest value.

Geometric Mean

(x₁·x₂·...x_n)^1/n

For example, suppose a business's profits grow by 25% one year, and by 45.8% the next year. To find the average yearly percent growth rate, you must take the geometric mean of 1.25 and 1.458.

sqrt[(1.25)(1.458)]
= sqrt[1.8225]
= 1.35

Thus, the average growth rate over the two years was 35%. 

Compare this to the result you would get if you took the arithmetic mean of 25 and 45.8. Since (25+45.8)/2 = 35.4

Harmonic Mean

In science in business applications, the harmonic mean is used to average ratios. 

For two numbers x and y, the harmonic mean is 2xy/(x+y). 

For three numbers x,y, and z, the harmonic mean is 3xyz/(xy+xz+yz).

For n numbers, the harmonic mean is

n/(1/x₁ + 1/x₂ + ... + 1/x_n)

For example, suppose a man drives at a speed of 80 k/h for 100 kilometers (1.25 hours), and then drives at a speed of 40 k/h for the next 100 km (2.5 hours). The average speed of the car for the entire 200 km trip is total distance divided by total time. Since 200/(1.25+2.5) = 53.33, the average speed is 53.33 k/h. This is equivalent to the harmonic mean of 80 and 40. Observe:

2(80)(40)/(80+40)
= 6400/120
= 53.33

In business, investors use the harmonic mean to compute the average price/earning ratio of a stock portfolio. For example, suppose you have three stocks, and their P/E ratios are 8, 18, and 30. The average P/E ratio of the three stocks is 

3(8)(18)(30)/(144+240+540)
= 12960/924
= 14.026

Root Mean Square (Quadratic Mean)

The root mean square, aka quadratic mean, is used in many engineering and statistical applications, especially when there are data points that can be negative. The standard deviation of a set of numbers is an example of the root mean square. (It is the root mean square of the differences between each data point and the arithmetic mean.) If you have two numbers x and y, the quadratic mean is sqrt[(x² + y²)/2]. For n variables, it is

sqrt[(x₁² + x₂² + ... + x_n²)/n]

For example, suppose you have this set of numbers: -10, -5, -4, 1, 6, 7. The root mean square is

sqrt[(100+25+16+1+36+49)/6]
= sqrt(227/6)
= 6.15

which can be interpreted as the average positive value.

Contraharmonic Mean

The contraharmonic mean of x and y is (x² + y²)/(x + y). For n values, the contra- harmonic mean is

(x₁² + x₂² + ... + x_n²)/(x₁ + x₂ + ... + x_n)

For example, the contraharmonic mean of 1, 3, 5, and 7 is

(1+9+25+49)/(1+3+5+7) = 84/16 = 5.25

Other Means

[(x^p + y^p)/2]^1/p    (Power Mean)

[(x^p - y^p)/(p(x - y))]^1/(p-1)    (Stolarsky Mean)

        sqrt[(x² + xy + y²)/3] when p = 3

(x^p + y^p)/(x^p-1 + y^p-1)    (Lehmer Mean)

[(x^p + y^p)/(x^r + y^r)]^1/(p-r)

[(r(x^p - y^p))/(p(x^r - y^r))]^1/(p-r)

[(x^py^r + x^ry^p)/2]^1/(p+r)

(x - y)/(Ln(x) - Ln(y))    (Log Mean)

(xLn(x) + yLn(y))/(Ln(x) + Ln(y))

(x + sqrt(xy) + y)/3    (Heronian Mean)

(1/e)(x^x/y^y)^1/(x-y), e = 2.718281828....    (Identric Mean)

(e)(x^y/y^x)^1/(y-x)

(x^xy^y)^1/(x+y)

(x^yy^x)^1/(x+y)

Mean Inequalities

Some means are in a constant relationship to one another. If we denote the arithmetic mean of x and y by A, their geometric mean by G, their harmonic mean by H, their root mean square by R, and their contraharmonic mean by C, then the following chain of inequalities is always true

C ≥ R ≥ A ≥ G ≥ H

Cyclically adjusted price-to-earnings ratio

The cyclically adjusted price-to-earnings ratio, commonly known as CAPE or Shiller P/E, is a valuation measure usually applied to broad equity markets. It is defined as price divided by the average of ten years of earnings, adjusted for inflation.

Graham and Dodd noted one-year earnings were too volatile to offer a good idea of a firm's true earning power. Decades later, Yale economist Robert Shiller popularized the 10-year version of Graham and Dodd's P/E as a way to value the stock market.

Mebane Faber extended Shiller's work to include over thirty foreign markets around the globe in his paper Global Value: Building Trading Models with the 10 Year CAPE

Over seventy years ago Ben Jamin Graham and David Dodd proposed valuing securities with earnings smoothed across multiple years. Robert Shiller popularized this method with his version of this cyclically adjusted price-to-earnings ratio (CAPE) in the late 1990s, and issued a timely warning of poor stock returns to follow in the coming years.  We apply this valuation metric across over thirty foreign markets and find it both practical and useful, and indeed witness even greater examples of bubbles and busts abroad than in the United States.  We then create a trading system to build global stock portfolios based on valuation, and find significant outperformance by selecting markets based on relative and absolute valuation.

Summary chart from the paper:

- See more at: http://www.mebanefaber.com/2012/08/23/global-value-building-trading-models-with-the-10-year-cape/#sthash.Qk1lE2eG.dpuf

Efficient Markets

Efficient Markets

Robert Shiller, Eugene Fama, and Lars Peter Hansen shared the Nobel Prize for economics this past week “for their empirical analysis of asset prices that greatly improved our understanding of how financial markets work, when they seem to work well and when they seem to work otherwise.”

 As Ron Rimkus, CFA, and others have already pointed out, there’s some irony in the fact that Fama, the father of the efficient market hypothesis, does not recognize the existence of financial bubbles, while Shiller, author of Irrational Exuberance, has gained considerable fame for sounding alarm bells for overvalued markets.

Shiller, of course, is also well-known for his use of the cyclically adjusted P/E as a measure of stock market value. Investors who are cautious on US equities today are quick to point to the CAPE, or Shiller P/E, as clear evidence of an overvalued stock market. Market bulls prefer to look forward and the promise of strong earnings, and they've been richly rewarded for their optimism over the past two years. This divergence of opinion is what makes a market truly efficient.