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.

Things Successful People Do On Weekends

Things Successful People Do On Weekends

 

• Weekends are an important time to unplug from the day-to-day and get a chance to think more deeply about my company and my industry

• “Even when I’m technically not working, I’m always processing in the background and thinking about the company. 

• Weekends are a great chance to reflect and be more introspective about bigger issues

• Always spends weekends with his family

• “Even if I’m on the road on a Friday and have to be back in that same city the following week, I always come home no matter what.”

• “A successful person is usually one who has achieved a measure of happiness and fulfillment in their work, family, and spiritual life

• “Most successful people need to feel a sense of accomplishment and are self-motivated to tackle the next challenge.”

• Roy Cohen, career coach and author of The Wall Street Professional’s Survival Guide, believes success is often defined in two ways: Achieving and exceeding financial milestones or achieving great satisfaction through one’s work.

• “From my perspective as a career coach, real meaningful success bridges the two–great prosperity combined with real joy and passion for your work.”

Ratios

Ratios

Profit Margin (Net Margin)

Net Income divided by Revenues

Also known as Net Profit Margin

Gross Margin

Sales margins are often called gross profit margins

Return On Assets - ROA (return on investment)

The ROA is often referred to as ROI

The assets of the company are comprised of both debt and equity.

Note: Some investors add interest expense back into net income when performing this calculation because they'd like to use operating returns before cost of borrowing.

Return On Capital Employed - ROCE

ROCE should always be higher than the rate at which the company borrows, otherwise any increase in borrowing will reduce shareholders' earnings.

A variation of this ratio is return on average capital employed (ROACE)

Return On Equity - ROE

Return on Equity = Net Income/Shareholder's Equity

return on common equity (ROCE) = net income - preferred dividends / common equity

dividing net income byaverage shareholders' equity

Shareholder's equity does not include preferred shares.

• If new shares are issued then use the weighted average of the number of shares throughout the year.
• Averaging ROE over the past 5 to 10 years can give you a better idea of the historical growth.

Also known as "return on net worth" (RONW)

Pareto Analysis

Pareto Analysis

A technique used for decision making based on the Pareto Principle, known as the 80/20 rule. It is a decision-making technique that statistically separates a limited number of input factors as having the greatest impact on an outcome, either desirable or undesirable. Pareto analysis is based on the idea that 80% of a project's benefit can be achieved by doing 20% of the work or conversely 80% of problems are traced to 20% of the causes.

In its simplest terms, Pareto analysis will typically show that a disproportionate improvement can be achieved by ranking various causes of a problem and by concentrating on those solutions or items with the largest impact. The basic premise is that not all inputs have the same or even proportional impact on a given output. This type of decision-making can be used in many fields of endeavor, from government policy to individual business decisions.

Pareto Analysis

Pareto Analysis

A technique used for decision making based on the Pareto Principle, known as the 80/20 rule. It is a decision-making technique that statistically separates a limited number of input factors as having the greatest impact on an outcome, either desirable or undesirable. Pareto analysis is based on the idea that 80% of a project's benefit can be achieved by doing 20% of the work or conversely 80% of problems are traced to 20% of the causes.

Pareto Analysis

Pareto Analysis

A technique used for decision making based on the Pareto Principle, known as the 80/20 rule. It is a decision-making technique that statistically separates a limited number of input factors as having the greatest impact on an outcome, either desirable or undesirable. Pareto analysis is based on the idea that 80% of a project's benefit can be achieved by doing 20% of the work or conversely 80% of problems are traced to 20% of the causes.

How to make financial projections

How to make financial projections

In fact you cannot realistically evaluate your business concept without examining its potential profitability in advance of starting. 

To determine what financial return you may achieve you will want to create a projection of the profit or loss of the business for its first year. And you will want to estimate the amount of cash that will come into the business and what amount will go out. This is known as a cash flow projection.

Let's look at each of these projections.

Making a Sales Projection

One of the hardest tasks you face before you open for business is to estimate what your sales will be by month for the first year in business. This is particularly difficult because to gain the greatest use from this projection you should do it 3-6 months before you launch. There is no magic formula for making a sales projection, but we have some suggestions.

The first step in projecting sales is to do enough research into your business to determine if there is a seasonal pattern to the sales. Articles about your type of business, interviews with other business owners and financial statistics collected by trade associations are helpful in determining what level of sales is produced during different times of the year. Department stores, for example do more than 40% of the total years sales in November and December. Use this information to determine what percentage of sales you are likely to do each month of your first year.

Second, go back to your business description and marketing strategy to describe which one or two of your products will produce the majority of your sales. Determine what the "unit of transaction" will be for them. For example, if you are a consultant the unit of transaction is fees/hour or fees/day. A retail store might have a unit of transaction of so many boxes of a product/transaction. Using the promotional strategy you created earlier, estimate how many units you might sell in month one of your business; month two and so on.

The third step is converting you unit sale estimate into dollars. By now you should have developed your pricing--either a single fee per hour or a price list of various products or services and their prices. Multiply each month's unit sales estimate by your fee per unit or use an average of your different prices. This produces your dollar sales projection per month. 

Estimating Your Expenses

If you have completed your research on start-up and monthly operating expenses, you should have a reasonably good idea of your expense categories.

Expenses come in two varieties:

1. Fixed expenses. These are cost that you have whether you have any sales or not.
2. Variable expenses: Money you spend to get new customers and to make and deliver your product or service are your variable expenses.

Some expenses must be paid each month; others are due just once per quarter or possibly twice per year. In a vertical column next to Month I of your sales projections, write in "Operating Expenses" and fill in each fixed and variable expense you feel you will pay your first year.

Calculating Profit

There are three kinds of profit you deal with in business:

There are three kinds of profit you deal with in business:

1. Gross Profit. This is the dollars of sales left after paying for the materials or inventory used to produce the sale. In businesses selling their time, labor or knowledge their gross profit is usually equal to their sales, since there is no cost of product involved.
2. Operating Profit. This is what is left of gross profit after you pay your monthly operating expenses (including your personal compensation). Interest on loans and taxes are not deducted. 
3. Net Profit. This is your sales income left after all expenses, interest costs and taxes to your state and Uncle Sam. This is the profit left to put back into growing your business and to possibly pay yourself a bonus!

Sales often materialize more slowly than you estimated and your expenses often run higher than you estimated.

The Cash Flow Projection

The projection form used for cash flow looks very similar to that used for the profit and loss projection.

• Starting Cash. Unlike the Profit and Loss Projection, where we are only interested in sales and expenses each month, the Cash Flow Projection also takes into consideration the amount of cash you initially invest in your business. This amount is filled in the "Starting Cash" block on the projection form.
• Cash Sales.
• Credit Sales Collected. After your first month in business, you will hopefully receive payment for sales where you granted credit. When you receive the cash you record it in this row. 
• Monthly Cash Flow. To arrive at the final outcome of the cash that came into your business in a month's time and the money that was paid out, you take the total of the "Starting Cash" for the month, cash sales, credit sales collected, any other cash in, such as interest and subtract the total of cash paid out during the same month. The result is the "Monthly Cash Flow."
• Accumulated Cash Flow - After the first month, you add this months cash flow-positive or negative--to the previous months cash flow to see how you are progressing throughout the year.

Relationship between Research and Teaching

Understanding the Relationship between Research and Teaching

Research should and does influence teaching (and vice versa).

Teaching enhances the development of students, research advances the development of new knowledge, and service contributes to the growth of nonacademic, professional, or college and university communities.
Analysts who perceive that teaching and research enhance each other argue that active researchers are informed and engaging teachers and that teaching stimulates faculty creativity and enthusiasm for research.

Economic theory suggests that teaching and research are complementary. Because they use many of the same resources, facilities, and personnel, producing teaching and research together is more efficient than producing each separately.
Similarly, individual faculty may improve their efficiency and productivity if they sometimes engage in activities that accomplish both teaching and research goals at the same time. 

Those who define teaching and research in terms of classroom instruction and publications are less likely to perceive a positive relationship between the two faculty roles than those who define the roles more broadly.

TOP 10 FOREX BROKERS

TOP 10 FOREX BROKERS   COMPANY 1. Interactive Brokers 2. TradeStation Securities, Inc. 3. FXCM / Forex Capital Markets LLC 4. Global Futures Exchange & Trading Company, Inc. 5. A. Packard Trading 6. FCStone LLC Futures Direct 7. Interbank FX, Member of Monex Group 8. Robbins Trading Company 9. Lightspeed Trading / Lightspeed Financial Inc. 10. Beverly Hills Capital

Momentum tracking and Trend-following

Momentum tracking and Trend-following

Mel Widner, PhD.

Repeated smoothing of data gives a spectrum of trends that, when plotted in color, have the appearance of a rainbow. Observe them all, pick your time frame, and act accordingly. 

One approach to finding gold is momentum tracking and trend-following. Trend-following is a common, if not the most common, timing method used by traders and investors.

The underlying premise is that prices have momentum and inertia and will continue in the same direction until something occurs fundamentally to change that.

Top 2013 Law School Rankings

Law School Ranking

Entrepreneurs Get Better with Age

by Whitney Johnson 

When my mother turned forty, we threw her a tongue-in-cheek funeral-themed surprise party, festooning the living room with paper tombstones engraved with Rest in Peace. That party theme is now a laughable conceit — forty then was older than forty now. Almost. In today's world, there is still a bias against older people — employers in particular often think (in their mind) what Shark Tank's Kevin O'Leary is fond of saying to entrepreneurs he doesn't like, "You are dead to me." If we're being honest, we probably agree with O'Leary. Who of us hasn't said, "I'm looking for someone young and hungry." The implication is clear: If you aren't young, you have nothing to contribute.

According to famed developmental psychologist Erik Erikson, as we grow older, hunger for meaning animates us, making us more alive. His theory explains that each healthy human passes througheight stages of development from infancy to adulthood. The seventh stage of development typically takes places between the ages 40-64 and centers around generativity, a period not of stagnation, but of productivity and creativity, including a strong commitment to mentoring and shoring up the next generation. Individuals in this developmental stage are supremely motivated to generate value, not just for themselves, but for others, asking the question: What can I do to make my life really count?

There are loads of both anecdotal and empirical data to support this idea of accelerating creativity in our middle years. Take Cheryl Kellond, for example, one of forty women we recently profiled in 40 Women Over 40 to Watch. Kellond, 43, founded Bia Sport, a GPS sports watch that records time, current heart rate, sending the data straight to an online profile. It also comes with a panic button that gives women who work out alone peace of mind. With traditional sources of financing unavailable, Kellond raised her first round of capital ($408,000) on Kickstarter — the ultimate in creative financing. Then there's Linda Avey, who at age 46 started breakout company 23 and Me, a direct-to-consumer company that gives people access to their genetic data. At age 53, Avey is on her second start-up, Curious, a tool that gives people tools to ask questions about their health through data aggregation and sharing.

Research suggests Kellond and Avey aren't one-offs. "The average age of a successful entrepreneur in high-growth industries such as computers, health care, and aerospace is 40. Twice as many successful entrepreneurs are over 50 as under 25. The vast majority — 75 percent — have more than six years of industry experience and half have more than 10 years when they create their startup," says Duke University scholar Vivek Wadhwa, who studied 549 successful technology ventures. Meanwhile, data from the Kauffman Foundation indicates the highest rate of entrepreneurship in America has shifted to the 55-64 age group, with people over 55 almost twice as likely to found successful companies than those between 20 and 34.

The over-40 crowd is also more likely to do work that matters not just for themselves, but also future generations. For example, Jacki Zehner, 47, the youngest woman to become a partner at Goldman Sachs, is pouring her post-forty life into philanthropy on behalf of women and girls as CEO of Women Moving Millions. Carol Fox, 69, has devoted her golden years to the China-U.S. Philanthropy project, teaching Chinese billionaires how to extend their circle of caring beyond family. While photojournalist Paola Gianturco, 73, igniting an activist grandmother movement, inspiring grandmothers across the world to become involved in education, health and human rights. In learning about these inspiring individuals, it's easy to see why research indicates that a 55-year-old and even a 65-year-old have more innovation potential than a 25-year-old: innovators really do get better with age.

Just as larger businesses provide economic stability to society in the form of higher pay, better medical care, and retirement, experienced workers provide intellectual and emotional ballast in the workplace including innovation expertise. Think about it — disruptive innovation is about playing where no one wants to play (low-end), or has thought of playing (new market). As individuals move into Erikson's seventh developmental stage, creating something new isn't just a "nice thing to do" — it is a psychological imperative. The urge to create, to generate a life that counts impels people to innovate, even when it's lonely and scary. Data notwithstanding, some of the companies among us will continue allow these individuals to fall into the arms of independent work, if we don't give them the boot first. The smart companies — and my money is on you — will harness this hunger of the underserved, ready-to-serve corp of talent, and upend the competition.