Book review: Make Your Kid a Money Genius (even if you’re not)

October 16, 2017

Beth Kobliner, Simon & Schuster, New York, 2017.

about the author

Beth Kobliner is an authority on personal finance for youth as shown by her successful publications of a NY Times best-seller book (Get a Fiancial Life), staff writer for Money Magazine, and contributing author to national newspapers. She served on President Obama’s Advisory Council on Financial Capability for Young Americans.

In this book she offers financial advice for 6 age groups: pre-school, elementary school, middle school, high school, college, and young adult. Much of her advice is based on academic studies. It’s not a textbook for children.

relevant topics for children

Most parents will discuss any topic except money; yet parents are the principal influence on their kids’ financial behavior. Many of a child’s money habits are set by age 7. This book discribes useful ways (called “teachable moments”) for talking about money with children as they grow from age 3 to young adult. Here are the author’s general comments about relevant topics for all ages:

TRUST: Parents need to build the trust of their pre-school children by following through on parental promises.
PATIENCE: Some children are impulsive, others are patient. Patient people tend to save more money! Pre-schoolers can be taught to wait for things.
CHARITY: Raise a generous child. Sharing time and money allows children to feel grateful for what they have. They are ready to show kindness by age 4. Elementary school children begin to understand the needs of others. Teen volunteers can engage in community service for their school and community. Most college students won’t have spare money, but they can donate their time. It allows them to explore the nonprofit world. Parents should honor their child’s charitable work with the same committment as other achievements in life; but, don’t overpraise their charitable efforts.
ALLOWANCE: It doesn’t matter if you give your elementary school child an allowance, but if you do, don’t make household chores a pre-requisite for receiving the allowance, use the allowance to set spending rules, and give them control of their spending decisions.
WORK: Children need to do unpaid chores and do well in school. Advanced chores such as raking leaves and doing laundry are essential to raising a self-reliant child. Elementary school children want to earn money, in which case the parent decides whether or not to pay the child for special chores. Middle school children are able to earn money. Limit the high school student’s work week to 15 hours; school is more important.
DEBT: Start teaching the basic concepts of debt to pre-school children. Middle school children need to understand the minimum monthly payments of credit card debt and protect themselves from identity theft. High schoolers are interested in car loans and credit cards; prepare them for wise use of credit cards. Parents, don’t buy your child a car or cosign for a car loan! After college, adult children are likely to have student-loan debt, car debt, or credit card debt. Parents should neither dip into their own retirement savings nor cosign for a loan as ways of helping young adults manage debt!
SHOPPING: Children want to buy stuff without limits, so parents need to start setting spending limits on pre-school children and teach the concept of ‘living within your means’. Elementary school children how to avoid being victimized by advertising and peer pressure. Tweens should spend their own money, not their parents’. If a teen prefers to spend for personal items rather than save money, they should learn from mistaken purchases. The ‘money culture’ among college students with different incomes can produce embarassment, resentment, and other strong feelings. Emphasize that college-related expenses are essential and everything else is extra.
SAVING: Parents should not raid their child’s savings. Middle school children should have a supersafe account (e.g., savings account, money market account, or CD). High school students should save for college, it will boost their motivation. Young adults should have an emergency fund and make maximal 401K deposits.
INSURANCE: Insurance is necessary for financial protection against devastating expenses. Most bankruptcies result from unpaid medical bills. Teens should pay for their car insurance and minimize their insurance rate with a good driving record. College students must have health insurance for the rest of their lives.
INVESTING: Don’t postpone the habit of investing in stocks; it’s a good way to protect against inflation. Children should learn the fundamentals and start saving small amounts at a young age. When they are old enough to understand numbers and show an interest in how money ‘grows,’ provide them with numeric examples of compound interest. High schoolers should open a Roth IRA to begin growing money.
COLLEGE: Attending college is the best pathway to earning higher wages compared to entering the workforce with a high school diploma. Middle school is the time to start talking about college and high school is the time to prepare for the college admissions process. High schoolers are advised to save for college; their chores should give way to college prep and testing. Avoid large student loans by chosing a good, inexpensive college and doing a better search for grants and scholarships. There are 3 ways to save for college:

  1. 529 Savings Plan. the earnings are tax-free for educational purposes and there is no income cap for donations. If your child rejects the plan’s participating schools, then rollover the savings to another education account, change the beneficiary to another child, or withdraw the savings with penalties.
  2. Coverdell Account. the earnings are tax-free for educational purposes, but the annual contribution is limited to $2,000 when a married couple earns less than $220,000 annually. The savings can be used for elementary school and high school educational purposes.
  3. Custodial accounts are available at banks and mutual funds. If the child’s earnings exceed $1,050, they are taxed at the child’s tax rate for the next $1,050, then at the parent’s tax rate for higher amounts until age 19 (age 24 for full time students). Colleges count the account balance as the child’s asset and expect 20% of the balance to go toward college expenses. That means less aid for the family.

The college student’s priorities are studies, a paying job [working students feel more invested in their education!], and employment after college. Does your college graduate want to start a career or go to grad school? Parents, don’t jeopardize your financial security to pay for their grad school.

the author’s curriculum and activities for pre-school children


EXPLAIN PATIENCE.  Patience, trust, and generosity are personal traits that facilitate financial success; pre-school training should help develop those traits.  Impulse buying reduces funds for tomorrow’s purchase! Always consider tomorrow’s purchases before buying on impulse.
Teach your children to perservere. Explain that you can’t always get what you want! Advise them to be patient (‘self control’) and wait for something.

EXPLAIN WORK.  Instill a work ethic; chores are a part of life.  Explain how you work to earn money  Convey the idea that a job is a source of pride and dignity

EXPLAIN SHOPPING.  Explain that you have to pay for things in cash (money, check) or card (debit, credit); the credit card is one way to pay (at the risk of incurring debt!).  Teach them to distrust advertising!   Explain how ads are produced with actors, scripts, colors, etc.  Explain the risks of materialism

DISCUSS CHARITY.  Be respectful of families with different levels of household income by explaining that “some people have plenty and others not enough”. Help bring your child closer to people of need by avoiding the terms “poor” and “rich” when discussing family wealth.
Worthy causes? Maybe your child needs guidance. What would they like to change in the world? Who would they like to help?

DESCRIBE THE HISTORY OF INSURANCE.  Ancient shipowners created a fund to pay for losses from shipwrecks. Their bookkeeper became the insurance company.

INTRODUCE INVESTING.  The basic concept of investing is to spend time and effort to produce something good (e.g., turning flour into bread).


PRACTICE PATIENCE.  Encourage them against skipping to the head of the line.  Discuss a communal effort (e.g., family savings pot) to save for a family reward (pizza, water park, etc.)

LEARN ABOUT WORK.  Perform household chores.  Discuss the jobs of people you know.

DISCUSS CHARITY.  Consider these charities: , , , , ,

general goals after pre-school

ELEMENTARY SCHOOL. Continue the development of personal traits by adding an adult perspective on respect for other people. School children need to start managing their money and protecting it. They are vulnerable to harmful attack from many directions, including identity theft from their online accounts. Give parental guidance.

MIDDLE SCHOOL. Tweens are vulnerable to marketing campaigns, overspending, and credit card debt. Give parental guidance.

HIGH SCHOOL. Reduce the time spent on household chores. The teen’s main job is graduating from high school with a good education. Teen’s also need to prepare for higher education or entering the workforce. Give parental guidance.


Kobliner’s book contains credible advice for the homeshooling of children in financial matters. There is much more information in her book than I’ve been able to summarize in this article. Be sure to visit her web site, “Money as You Grow”, for additional help and perspective (


Pictures from financial statements

July 4, 2017

These 20 graphs form a pictorial essay of  a company’s financial statements.  This large company survived the Recession of 2007-8!














Copyright © 2017 Douglas R. Knight

PE, PEG, and PEGY Valuation Ratios

June 19, 2017

The stock market lists several thousand stocks which have a variety of prices in relation to company profits. Company managers decide how they will use the profits to either pay dividends to shareholders or retain the earnings to build shareholders’ equity (ref. 1). The retained earnings may be used in ways that ultimately raise or lower the market price of the stock. Consequently, bankers and brokers pay attention to quarterly reports of the company profits measured as earnings per share (EPS).

Investors want to know whether the stock is priced too high (“overvalued”) or too low (“undervalued”) compared to the EPS. Professional analysts assist investors by preparing the PE, PEG, and PEGY valuation ratios.

  • PE standardizes the market value of a stock for ease of comparison with other stocks.
  • PEG refines the valuation of stocks by adjusting PE to the growth rate of company earnings. PE is in equilibrium with the growth of earnings at PEG = 1.
  • PEGY adjusts PEG to stocks with high yields of dividends.


PE is the ratio of stock price (P) to company earnings (E). Formula 1 is used to calculate the PE (ref. 1,2):

PE = P / E,          formula 1

P is the price per share and E is the EPS accumulated over a 12-month period. For more information, please see notes at the end of this article.

example: if one share of stock is priced at $50 and the company’s annual EPS is $5, then 50/5 equals 10/1. The PE is 10.

The timing of company earnings determines whether the PE is labeled as “trailing” or “forward”.
Trailing PE is the current price per share divided by the EPS accumulated from the past 12 months or past 4 quarters. Trailing PE is based on known quantities.  Forward PE is the current price divided by the accumulated EPS expected for the next 12 months or next 4 quarters. Forward PE is an uncertain forecast of future market value based on the management’s or analyst’s expectation the EPS for the next 12 months.

PE represents the market value of all shareholders’ claims to $1 of annual EPS, past or future. The market value is judged to be high (“overvalued”) or low (“undervalued”) compared to an arbitrary estimation of fair value. There are several ways of determining a fair value.

  • Compare the stock’s PE to an average PE of the industry, market, or historical record (ref. 3).
  • Normalize the PE to the company’s rate of earnings growth, which generates the PEG ratio. By convention, the PE is fairly valued when PEG = 1 (ref 5,6).
  • The least practical method is a comparison to some theoretical PE that is not readily available to most investors (ref 1,4).


PEG is the ratio of PE to G (ref. 1,7,8; formula 2):

PEG = PE / G,          formula 2

G is the compound annual growth rate of EPS over a time period of 3-5 years, perhaps even longer in special cases. For more information, please see endnotes.

example: if one share of stock is priced at $50 with an annual EPS of $5 and a 10% compound annual growth rate of EPS, then 50/5/10 equals 1.00. The PEG is 1.00.

PEG measures the market value of a stock relative to the company’s rate of earnings growth (ref. 7,8). The theoretical equilibrium between market value and the rate of EPS growth occurs at PEG = 1.0. PEGs below 1.0 suggest undervaluation and those above 1.0 suggest overvaluation (ref. 9).

Trailing- and Forward PEGs represent the stock’s market value relative to past and future eras (ref. 7,8). Trailing PEG is a factual measurement of market value provided that the EPS was measured during the past year and the EPS growth rate occurred during the past several years. Forward PEG is an uncertain prediction of market value based on the company’s expected earnings for next year and an analyst’s forecast of earnings growth for the next several years.

examples: when the forward PEG is above 1.0, the market expectation of growth exceeds consensus estimates and the stock is overvalued (ref. 8). If PEG is below 1.0, the stock is undervalued (ref. 2,7).

Limitations of PEG (ref. 3,8):

  • The EPS growth forecast may be invalid.
  • Another variable besides PEG could add or subtract value to the investment. For example, PEG ignores the value of a cash-rich company.
  • An overvalued company, for example one with a PEG of 3.0, might still be a stable investment despite its low rate of earnings growth.
  • PEG is best suited for stocks that don’t pay dividends; otherwise, calculate the PEGY.


Some investors prefer high-yield ‘value’ stocks rather than low-yield ‘growth’ stocks. High yield stocks typically pay higher dividends at lower EPS growth rates (e.g., the stocks of utility companies). PEGY includes dividends in its valuation ratio for high-yield stocks (ref. 8; formula 3).

PEGY = PE / (G+Y),          formula 3

Y is the stock’s dividend yield. Dividend yield is the ratio of the annual dividend per share to the price per share.

example: if one share of stock is priced at $50, the annual EPS is $5, the compound annual growth rate of EPS is 8%, and the dividend yield is 5%, then what are PEG and PEGY?
PEG = 50/5/8 = 1.25. PE is overvalued if the high dividend is excluded.
PEGY = 50/5/(8+5) = 0.77. PE is undervalued if the high dividend is included.

Payback period

Besides measuring market value, the PE and PEG also predict the stock’s payback period. A payback period is the amount of time needed for the accumulation of company earnings to match the original amount of investment. If all accumulated earnings were paid to investors, which is unlikely, the payout would provide a 100% return. Longer payback periods represent riskier investments, especially when the company is still establishing its market position or competing with innovative companies (ref. 9,10,11).

The PE ratio also represents a payback period measured in years.

example: if a stock is priced at $50 per share and the EPS is $5 per share every year, then $50/share divided by $5/share/year equals the payback period of 10 years. The same units of $/share cancel each other in the numerator and denominator.

The PE payback period is the time needed for an accumulated EPS to equal the original share price, assuming the EPS remains constant during the accumulation period. Most companies don’t repeat the same EPS every year.

The PEG payback period accounts for the desired phenomenon of EPS growth. The PEG payback period is the number of years that the growth of earnings accumulates enough value to match the original investment (ref. 9,10).

example: if a stock is priced at $50 per share, the EPS is $5 per share, and the EPS growth rate is 10%, it would take 7 years for the EPS to accumulate a value of the original stock price of $50. The PEG payback (7 years) is earlier than the PE payback (10 years) due to the 10% rate of earnings growth.


Company earnings are a strong determinant of stock value. PE, PEG, and PEGY ratios represent the stock market’s valuation of company earnings. Don’t rely solely on company earnings to judge the investment value of stocks. Also assess the business performance and company value (ref. 2,6).


Formula 1: PE = P / E

  • P is the current auction price for a share of common stock listed in the stock market. The auction price fluctuates often depending on when a trading order is filled at an agreeable price between buyer and seller. Analysts typically use the closing price of the latest trading day to calculate the PE.
  • E is one year’s accumulation of the company’s earnings per share of common stock [EPS]. EPS represents the company’s net income divided by its outstanding shares and fluctuates at quarterly intervals. Any guaranteed payments of dividends to shares of preferred stock automatically reduce the EPS before calculation of the PE. EPS depends on the analyst’s choice between GAAP- and non-GAAP earnings and choice between basic and diluted outstanding shares.

Formula 2: PEG = PE / G

  • PEG fluctuates with frequent changes of PE and infrequent changes of G.
  • G is the EPS growth rate, which is the compound annual growth rate of EPS for a time period of at least several years. Although G is measured as a percentage change of EPS per year, the common practice is to ignore the units of measurement when calculating PEG.
  • Trailing PEG is the trailing PE divided by the G for past earnings.
  • Forward PEG is the forward PE divided by the G for future earnings.


1. How to Find P/E and PEG Ratios, by Thomas Smith, Investopedia LLC.
2. How to use the PE Ratio and PEG to tell a stock’s future, by the Investopedia Staff, updated March 17, 2016.
3. What is the PEG Ratio?
4. Aswath Damodaran, Intrinsic Valuation in a Relative Value World. .
5. PEG Ratio; From Wikipedia, the free encyclopedia. pages 6-7.
6. How useful is the PEG Ratio? Joseph Khattab, April 6, 2006. The Motley Fool.
7. Price/Earnings to Growth- PEG Ratio. Investopedia LLC.
8. PEG Ratios Nail Down Value Stocks, by Ryan Barnes, 11/24/2015. Investopedia LLC.
9. Double your dollars. Selena Maranjian, September 7, 2010. The Motley Fool.
10. Payback period = double your money. Course 304: PEG and Payback Periods. Morningstar, 2015.
11. The longer the payback period, the greater the risk. Course 304: PEG and Payback Periods. Morningstar, 2015.

Copyright © 2017 Douglas R. Knight

Stop losing value from a declining price

March 4, 2017


The market value of your stock equals your principal (i.e., the amount you invested) plus any profit or loss from price fluctuation. The market price that moves below what you paid to purchase the stock will produce a loss of principal if you sell the investment. Here are several risk factors that may drive stock prices downward:

  1. Company performance. ‘Good’ companies attract investors. Conversely, ‘distressed’ companies repel investors.
  2. Industry performance. Business cycles can affect the sales of products from an entire industry. For example, sales of new automobiles declined during the Recession of 2008.
  3. Market cycles. Aside from business performance, the entire stock market is subject to periods of declining prices due to massive selloffs by investors.

The risk of an extreme loss can be prevented by setting a stop-loss price (“stop”) to sell part or all of your shares.

ways of setting the stop

The systematic way is quite simple. If the market value is below your invested principal, then select an absolute loss or a fraction of the principal. Examples:

  1. Absolute loss. Suppose you invest $5,000 in 100 shares of stock (i.e., $50/share) and you can tolerate a loss of $1,000 should the price start to fall. Regardless of future prices, you choose to stop the decline at $1,000 below the original $5,000 value. In this example, the stop would be $40/share [stop = (value – loss)/shares = ($5,000 – $1,000)/100].
  2. Fraction of value. Suppose you can tolerate a 10% loss from an investment originally valued at $5,000 for 100 shares.  Ten percent is one-tenth of 100, which is equivalent to a decimal number of 0.10. The stop would be $45/share [stop = (1.00 – decimal)*value/shares = (1.00 – 0.10)*$5,000/100].

The technical way is based on the stock’s historical prices. If you want to minimize the chance of a sale, set the stop at the lowest price from the past 5-10 years. Beware that setting the stop at a historical low may incur a steep loss. Other ways involve the more complicated analyses of trendlines, moving-average lines, or price statistics.

Another way is to adjust the price gap (gap = market price – stop) to the growth of capital gains. As the market price increases over time, choose a narrow gap to protect the capital gain or a wide gap to reduce the chance of a trade. Generally speaking, widening the price gap will reduce the chance of a trade at the risk of incurring a bigger loss.

add a limit price (“limit”) for extra protection

A brokerage firm will enforce your stop order for 30-90 days depending on the firm’s trading platform. The firm’s computer activates the order when the latest market price reaches the stop. The order is then filled at the next available price. In a chaotic market, the price could plunge below your stop to an exceptionally low value at the next available trade, resulting in a bigger loss than you planned. You might be able to prevent this result by setting a limit slightly below the stop. The trading order would be filled somewhere within the stop-limit price zone unless the transaction is cancelled, unfilled, when the next available price dips below the limit. The limit helps protect the extent of your loss.

who should worry about an extreme loss?

Nobody’s immune, but long-time investors have the least concern. Investment strategies such as dollar-cost-averaging and automatic-dividend-reinvestment plans will help protect against damages from periodic bear markets. Short- and intermediate-time investors are at greater risk for incurring an extreme loss from market down-cycles. For example, families who are saving to pay college fees or to buy a home risk big losses from a bear market.


Stop orders are used to set the price for buying or selling exchange-traded products such as stocks, ETFs, and REITs. This article discussed the use of a stop-limit order to sell a stock in a declining market. Brokerage firms may restrict the duration of stop-limit orders to 30-90 days after which the order is cancelled without a transaction until you renew the order. Periodic renewals allow you to reconsider your strategy in light of the prevailing price trend. In a downtrend, simply renew the order. In an uptrend, you may wish to protect a growing profit by resetting the stop-limit order to higher prices. Click on this link to skimming a profit for another perspective on protecting a growing profit.

Copyright © 2017 Douglas R. Knight

Skim the profit?

February 7, 2017

Selling all or part of a profitable investment is a tough choice to make.  On one hand, holding the investment allows time to accumulate a high return, but at the risk of losing profit in the market’s next big decline. On the other hand, selling portions of the investment to ensure a profit today will diminish the future return.

Both choices are easy to illustrate by imagining a stock investment that pays no dividends.  Assume there is a consistent growth of stock price and that no additional shares are purchased after the original purchase. The profit is skimmed by selling part of the investment when its market value grows to twice the original purchase.  Repeat the process every time the market value doubles until the investment is closed.  Chart 1 illustrates the skimming of a $1,000 investment.

chart 1, Market values.  $1,000 was invested in a good growth stock that paid no dividends. A generous 15% annual return doubled the market value every 5 years. The HOLD strategy (black squares) was to avoid selling for 20 years. The SELL strategy (green circles) was to sell half the shares every time the market value doubled. There were no trading fees.

After 5 years, the investor could claim a profit of $1,000 on the original $1,000 investment. Then the choices would be to close the investment at $2,000, withdraw only the $1,000 profit and wait for more (green circles), or withdraw nothing and wait for a bigger profit (black squares). The largest profit is made by waiting 20 years.

Chart 2 illustrates the accumulated cash balances of the HOLD and SELL strategies.

chart 2. The cash balances of the strategies in chart 1 are illustrated in this chart using the same symbols for data points. After 20 years, all remaining shares are sold for cash. The end point of each graph is the final cash balance.

chart 2, Cash balances. The cash balances of both strategies in chart 1 are illustrated in this chart using the same symbols for data points. The proceeds from every sale were held in a cash account and allowed to accumulate for 20 years.  After 20 years, the remaining shares were sold for cash. The end point of each strategy is the final cash balance.

After closing the investment in 20 years, the accumulated cash balance would be $16,367 from the HOLD strategy and $5,045 from the SELL strategy.

Alternate conditions

The accumulated cash balance will vary according to the annual rate of return (appended chart 3), the amount skimmed (appended chart 4), and the payment of dividends (appended chart 5).  In every condition, the total profit of the HOLD strategy exceeds the total profit of the SELL strategy.


On the question of whether or not to skim profits, skim if you need cash in the next 5-10 years. Otherwise, don’t sell without reassessing the investment or using a risk management scheme.  The question of selling for a loss was excluded from this discussion; that’s a different topic.

Appendix: Tables of cash balances

Charts 3-5 are tables of cash balances that represent profits from an imaginary investment of $1,000. The choices for taking a profit were to HOLD the investment for 20 years before liquidating the account or to SELL profitable portions of the investment.  Assume there were no trading fees.

Chart 3 shows that a 15% annual rate of return earned a bigger profit than a 7% annual rate of return.  Furthermore, the HOLD strategy earned a larger profit than the SELL strategy at both rates of return.

chart 3.

chart 3, Rate of return.  $1,000 was invested in a good growth stock that paid no dividends. No shares were purchased after the original investment. The 20-year cash balance (cells) was only affected by the annual rate of return (rows) and liquidation strategy (columns).  The HOLD strategy did not sell shares for 20 years.  The SELL strategy sold half the shares whenever the market value doubled in size during the 20 year period.  The 7% rate permitted 1 selling period and the 15% rate permitted 4 selling periods. 

Chart 4 illustrates the effect of skimming 50%, 100%, or 150% increments of market value.

chart 4.

chart 4, Increments of market value. $1,000 was invested in a good growth stock that paid no dividends. The investment’s annual rate of return was 15% and no shares were purchased after the original purchase. The cash balances (cells) accumulated every time period (rows) among 3 different increments of market value (columns). The HOLD strategy did not sell shares for 20 years. The ‘rule’ for the SELL strategy was to sell a portion of shares when the market value grew by approximately 50% every 3 years ($521), 100% every 5 years ($1,011), or 150% every 7 years ($1,660).

The HOLD strategy outperformed the SELL strategy. With the SELL strategy, waiting longer to skim bigger profits accumulated a larger cash balance after 20 years. Why? The bigger profits were less frequent, which had the effect of preserving the investment’s principal for longer time periods.

Chart 5 reveals a surprising effect for skimming profits from reinvested dividends.

chart 5.

chart 5, Dividends. $1,000 was invested in a good growth stock that paid a 2% dividend on every share. No shares were purchased after the original investment unless the dividends were automatically reinvested. The cash balances (cells) accumulated with the passage of  time (rows) among 3 types of investments (columns). The HOLD strategy did not sell shares for 20 years. The SELL strategy removed half the remaining shares every 5 years.

There were no surprises in the HOLD strategy. Reinvested dividends accumulated the largest cash balance over 20 years. However, reinvested dividends accumulated the lowest cash balances in the SELL strategy. Why? Slightly more shares were sold every 5 years from ‘reinvested dividends’ compared to ‘no dividends’. Yet the same number of shares were sold from ‘cash dividends’ compared to ‘no dividends’. The cash dividends directly augmented the cash balances.

Copyright © 2017 Douglas R. Knight

R-squared, the linearity of investment returns.

December 24, 2016

[updated 12/25/2016: R2 is a useful measure of indexing]

The R-squared (R2) statistic describes a pattern of plotted data with respect to a straight line. R-squared is called the coefficient of determination (ref 1,2).


The black dots in figure 1 represent investment returns that are poorly related to market returns. There is a random distribution of investment returns with respect to market returns. The blue line is an inadequate representation of the relationship simply because there is no relationship. The R2 score for this distribution is 0.03. Conversely, the black dots in figure 2 show the ‘herding’ of data around a straight line.


Figure 2’s investment returns are highly related to market returns with an R2 of 0.997.


The R2 score represents the degree of alignment of data to a best-fit line as determined by regression analysis. The lowest possible score of 0 indicates a random pattern of data with absolutely no alignment. The highest possible score of 1 represents complete alignment.

The product of R2 X 100 represents the percent of variation in investment returns that are related to market returns (ref 1,2). In other words, R2 measures the relavance of the best-fit line to a set of data. Relavance increases as the R2 score varies from 0 to 1.

The lowest score of 0 defies any financial analyst to draw a meaningful line for investment returns as they relate to market returns. In figure 1, the incline (β) and Y-intercept (⍺) of the blue line are unreliable measurements of investment performance.

The highest R2 score of 1.00 identifies a straight line of near-perfect predictions of returns. Any R2 above 0.75 identifies a straight line for making predictions of returns. Lower scores represent increasingly random events. In figure 2, the incline (β) and Y-intercept (⍺) are reliable measurements of investment performance.

R-squared is an excellent measure of index fund performance.  Websites for index mutual funds and ETFs publish R2 as a measure of alignment between fund returns and the market index.   Funds that have an R2 score of nearly 1.00 track the index very closely.


1.  Lain Pardoe, Laura Simon, and Derek Young. STAT 501, Regression Methods. 1.5- The coefficient of determination, r-squared. Pennsylvania State University, Eberly College of Science, Online courses.
2.  R-squared. 2016, Investopedia

Alpha is a point on a straight line, plus more.

December 22, 2016

{update on 12/23/2016: the significance of technical and operational alpha}

Alpha (⍺) is the cherished -but overrated- measurement of superior investment. Here are several interpretations:

  • A measurement of how well an investment outperforms its market index (ref 1).
  • The observed characteristic of a mutual fund that predicts higher fund performance (ref 2).
  • A portfolio’s return that’s independent of market returns (ref 3).
  • The excess (or deficit) return compared to the market’s return. Used this way, ⍺ is called Jensen’s Alpha.

Alpha represents a unique risk of outperforming the market’s returns. It is classically calculated as the “Y intercept” of a straight line attributed to the CAPM model (see appendix). In the last century, famous investors outperformed the market either by choosing exceptional investments or by investing in exceptional market sectors. The investment could be a single security (e.g., a stock) or a portfolio of capital assets (e.g., a mutual fund) (footnote 1, refs 1, 2). Now in this century, those alledged ‘alpha’ strategies are increasingly difficult to achieve. There’s an emerging sentiment among investors to avoid wasting time and money on attempting to outperform the market, the so called “loser’s game”. The current “winner’s game” is to seek ‘beta’ (refs 1, 2, 4, 5).

‘Beta’ is the portfolio’s return generated by market returns. Therefore, beta represents the risk of earning the market’s returns. The beta statistic, β, is currently calculated and reported by financial research firms as a coefficient for the incline of a straight line attributed to the CAPM model (see appendix).

Straight line of imaginary returns

(refs 5-8)

A straight line of imaginary returns presumably offers the best possible comparison of investment returns to a market index (footnote 2). ‘Returns’ and ‘performance’ are interchangeable terms that indicate the direction and movement of prices over time. An investment’s rate of return is calculated as the percentage change in price at regular intervals of time [likewise, the market’s rate of return is a percentage change in value of the market’s index at regular intervals of time]. Any rate of return is easily converted to a risk premium by subtracting the guaranteed interest rate for a Treasury bill (“T bill”). The risk premium is an investor’s potential reward for purchasing a security other than the T bill.

The straight line is drawn on a graph that shows actual measurements of investment returns plotted against market returns. The returns may either be measured as the rate of return or the risk premium depending on the goal of analysis. In the following chart, black dots represent a series of investment returns plotted against corresponding market returns.


The blue line of imaginary returns is the best possible comparison of investment returns to market returns. The position of the line on the graph is governed by its incline (β) and intersection (⍺+ε) with the vertical axis.

⍺, the intersection

(refs 1-3, 5-8)

Alpha resides at the intersection of the theoretical line with the vertical axis for investment returns (chart). Since the vertical axis crosses the horizontal axis at 0% market returns, ⍺ is the theoretical investment return at 0% market returns. A positive value for ⍺ implies that the investment tends to outperform its market index. Likewise, ⍺ = 0 implies no inherent advantage of the investment and a negative value for ⍺ implies that the investment tends to move less than the market index.

There’s a degree of error (ε) involved in drawing the line of imaginary returns, which means that its intersection is defined by the term ⍺+ε. The ε declines when a series of returns lie close to the line. The chart shows plots for 2 different series of returns; one series of black dots and another series of white circles. Both series have an equally small ε as illustrated by the close alignment of data to each straight line. Alpha of the blue line is 0% return and ⍺ of the orange line is 5% return, both occuring when the market return is 0. The series of open-circle returns outperformed the series of black-dot returns by 5%.


(refs 1, 2, 4, 5)

Alpha measures how well an investment outperforms the market. Yesterday’s ‘technical’ ⍺, shown in the preceding chart, applied to measuring superior stock-picking skills.  Today, the technical ⍺ of stocks is not reported by the most popular financial websites.

Today’s ‘operational’ alpha is really a beta loading factor of multi-factor models (see appendix).  Operational alpha is more relevant to measuring the performance of actively managed mutual funds and investment portfolios. The investment goal of an actively managed mutual fund is to outperform its market index. Active management may be the “loser’s game” of paying excessive fees in contrast to passive management, which may be the “winner’s game” of paying minimal fees.


1. Capital assets are securities and other forms of property that potentially earn a long term capital gain(loss) for the owner.

2. The straight line has other names that precede my use of the term ‘imaginary returns’. The straight line is also called a regression line or security characteristic line (ref 6).


1. Larry E. Swedroe and Andrew L. Berkin. Is outperforming the market alpha or beta? AAII Journal, July 2015. pages 11-15.

2. Yakov Amihud and Rusian Goyenko. How to the measure the skills of your fund manager. AAII Journal, April 2015. pages 27-31.

3. Daniel McNulty. Bettering your portfolio with alpha and beta. Investopedia.

4. John C. Bogle. The little book of common sense investing. John Wiley & Sons Inc., Hoboken, 2007.

5. Investing Answers. Alpha Definition & Example. 2016.

6. Professor Lasse H. Pederson. The capital asset pricing model (CAPM). New York University Stern School of Finance. undated.

7. MoneyChimp. Regression, Alpha, R-Squared. 2016.

8. Invest Excel. Calculate Jensen’s Alpha with Excel. undated.

APPENDIX: models for pricing assets and managing portfolios

(refs 1-3, 5-8)

The original one-factor model was called the Capital Assets Pricing Model (CAPM). The single factor is market returns (M).  The investment returns (I) are predicted by a best-fit line with incline (βm) and intersection with the vertical axis (⍺ + ε) (equation 1).

I = ⍺ + ε+ βmM,     equation 1, CAPM

Subsequent series of three-factor and four-factor models were sequential upgrades of CAPM. Equation 2 is an example of a four-factor model for the risk premium of an investment fund (F) comprised of separate portfolios for the broad market (M), asset size (S), asset value (V), and asset momentum (U).

F = ⍺ + ε + βmM + βsS + βvV + βuU,     equation 2, four-factor model

⍺ is the excess risk premium attributable to skillful management of the Fund.
ε is the model’s error
βm, βs, βv, and βu are portfolio loading factors assigned by the Fund’s manager

The four-factor model offers a spectrum of possibilities.

  • During 1927-2014, the average annual returns of indices for the the four-factor model were 8.4% for the broad stock market, 3.4% for stock size, 5% for stock value, and 9.5% for stock momentum.  The sum of average annual returns, 26.3%, represented the alpha-threshold for superior fund returns (ref 1).
  • Passive management could be predicted by setting βm to 1.00, measuring the market index return, and setting the remaining loading factors to 0.  A market index fund would  be expected to generate a risk premium that matches the market index risk premium with an ⍺ of 0 and slight ε for tracking error.
  • Active management involves designing loading factors and portfolio assets to outperform the fund’s predicted returns.

Copyright © 2014 Douglas R. Knight

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