Good companies attract investors. They do so by selling a desirable product that sustains the company’s growth of sales and earnings. The growth of sales is a good measure of market success. Durable companies convert their sales invoices into cash and use the cash wisely. Accounting items such as the *free cash flow,* *sustainable growth rate, quick ratio, *and* debt-to-equity ratio* are easy measures of the company’s health and durability. Growth stocks should be assessed by the quality of the company.

## What is a good company?

December 15, 2017## Pictures from financial statements

July 4, 2017## PE, PEG, and PEGY Valuation Ratios

June 19, 2017The 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

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

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.

### 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.

### Conclusions

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).

### Endnotes

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.

### REFERENCES

1. How to Find P/E and PEG Ratios, by Thomas Smith, Investopedia LLC. http://www.investopedia.com/articles/fundamental-analysis/09/price-to-earnings-and-growth-ratios.asp?lgl=v-table

2. How to use the PE Ratio and PEG to tell a stock’s future, by the Investopedia Staff, updated March 17, 2016. http://www.investopedia.com/articles/00/092200.asp

3. What is the PEG Ratio? https://www.fool.com/knowledge-center/peg-ratio.aspx

4. Aswath Damodaran, Intrinsic Valuation in a Relative Value World. http://people.stern.nyu.edu/adamodar/pdfiles/country/relvalFMA.pdf .

5. PEG Ratio; From Wikipedia, the free encyclopedia. pages 6-7. https://en.wikipedia.org/wiki/PEG_ratio

6. How useful is the PEG Ratio? Joseph Khattab, April 6, 2006. The Motley Fool. https://www.fool.com/investing/value/2006/04/06/how-useful-is-the-peg-ratio.aspx

7. Price/Earnings to Growth- PEG Ratio. Investopedia LLC. http://www.investopedia.com/terms/p/pegratio.asp

8. PEG Ratios Nail Down Value Stocks, by Ryan Barnes, 11/24/2015. Investopedia LLC. http://www.investopedia.com/articles/analyst/043002.asp?lgl=v-table

9. Double your dollars. Selena Maranjian, September 7, 2010. The Motley Fool. https://www.fool.com/investing/value/2010/09/07/double-your-dollars.aspx

10. Payback period = double your money. Course 304: PEG and Payback Periods. Morningstar, 2015. http://news.morningstar.com/classroom2/course.asp?docId=3066&page=2&CN=C

11. The longer the payback period, the greater the risk. Course 304: PEG and Payback Periods. Morningstar, 2015. http://news.morningstar.com/classroom2/course.asp?docId=3066&page=3&CN=C

**Copyright © 2017 Douglas R. Knight**

## The Physics of Wall Street, by James Owen Weatherall

April 25, 2017Physics needs math, so does Finance. Then no wonder some curious physicists began to create math models of financial markets in the 19th Century, only to find the task more difficult than imagined. Advances during the 20th Century ulitmately generated great wealth among a group of saavy traders known as the “Quants”. Their sophisticated trading models worked for a while until widespread use by firms in ‘Wall Street’ caused the global financial system to collapse in 2008. New models continue to evolve today. This book pays tribute to scientists who tackled the problem of modeling financial markets.

### The evolution of market models

- Bechelier made the first price-distribution model of stock markets. His normal distribution only worked in the Paris Bourse where there was little variation of prices.
- Osborne postulated that returns, not prices, are normally distributed. His model of the ‘random walk hypothesis’ expanded the understanding of price variation.
- Mandelbrot claimed that distributions of financial markets are more variable than previously thought.
- Thorp, Black, and Scholes converted price-distribution models to algorithms for daily trading. Their options pricing model was adapted from Osborne. Black later described the shortcomings in his paper “The holes in Black-Scholes”.
- The Prediction Company of Farmer, Packard, and McGill used black box models to improve the Black-Scholes model. They capitalized on short-term inefficiencies in the market.
- Didier Sornette predicted market catastrophes.

Today’s market models are still imperfect!

### Early Probability Theory, 16th-17th centuries

1526: Cardono wrote an unpublished book on the theory of probability based on the odds of dice games. For example, what is the chance of rolling 2 dice for a sum of 10?

- mathematical odds: 3 outcomes in 36 tries, or 1 in 12.
- betting odds: 33 to 3, or 11 to 1. Bet $1 and either lose it or win $11 plus the refunded $1.

1654: Pascal and Fermot established the modern theory of probability based on various gambling games. They realized that probability is a chance, not a certainty. In the 20th century, it was realized that the a probability becomes a certainty when taking an infinite number of chances [Law of Large Numbers].

### Random behavior of market prices, 19th-20th centuries

Bacheleier invented mathematical finance in the late 19th century. His graduate thesis applied probability theory to market speculation. In his ‘efficient market’ theory, Bechelier assumed that future prices take a ‘random walk’ within limits that describe the graph of a bell-shaped curve. In other words, stock prices have a normal distribution with a stable average. Bechelier’s ‘random walk’ model spawned 2 books in the 20th century.

- 1947: Samuelson published “Foundations of Economic Analysis”.
- 1964: Paul Cootner published Bechelier’s thesis in “The Random Character of Stock Market Prices”.

1959: An astronomer named Maury Osborne wrote “Brownian motion in the stock market”. Osborne dismissed the idea that stock prices have a normal distribution. Instead, the rate of return was normally distributed. His plot of stock prices was not bell-shaped, but lump-shaped with a long tail to one side. Osborne was first to show the importance of the log-normal distribution of prices to markets.

1960s: Benoit Mandelbrot discovered fractal geometry and adapted the consequences to understanding markets. Mandelbrot’s method described extreme market events.

1965: The issue was whether to analyze stock prices with Osborne’s or Mandelbrot’s method of analysis. Today’s consensus is that rates of return are fat-tailed with an unstable average.

1973: Burton Malkiel adopted Osborne’s work in a book called “A Random Walk Down Wall Street”.

Bachelor, Osborne, and Mandelbrot neither traded nor earned profits; they were academics.

### Hedging

1961: Edward O. Thorp beat the game of Roulette with a successful strategy. He later showed that math models could earn profits from financial markets by operating a hedge fund. Thorp believed that the stock market is the world’s biggest casino. Buying stock is betting that the price will rise and Selling stock is betting that the price will fall. The true price of a stock is where the odds of winning and losing are equal. He devised the ‘delta hedging’ strategy of picking the right mix of warrants and stocks to consistently earn a 20% annual return. The idea was to simultaneously short-sell warrants and buy underlying stocks. The stocks would soften the impact of a bad bet and augment the impact of a good bet.

1967: Thorp co-authored the book, Beat the Market. Jay Regan, a stock broker, partnered with Thorp to create the Princeton-Newport Partners hedge fund.

1969: Fischer Black derived a relationship destined to become the Black-Scholes-Merten model for the pricing of options. Black made quantitative finance an essential part of investment banking.

### Forecasting

1991: Physicists James Dayne Farmer and Norman Packard studied nonlinear forecasting. Given a chaotic process such as the financial market, their goal was to predict the next movement of prices.

1991: Farmer, Packard, and McGill formed The Prediction Company with the goal of profiting from Wall Street. They developed black box models of algorithmic trading which often worked for unknown reasons but also suffered unpredictable failures. It is still a mystery how market patterns are corrected.

1997: Didier Sornette, a geophysicist, studied the patterns of complex systems to predict critical events in the physical and social sciences. He filed a patent notice in 9/17/1997 that predicted a market crash the following month. Then he bought far-out-of-the-money Put Options to earn a 400% profit from his prediction.

### Reform

2008: The economic collapse of 2008 presented an opportunity to change how economists think about the world.

2009: Smolin, Weinstein, and others convened a conference of intellectuals to develop new models of economics. They failed to agree on the problems and solutions.

### Author’s Conclusions

All of the physicists’ models had successes and failures, but their works represent steps in the evolution of understanding markets. Financial modeling is an evolutionary process in which excellent assumptions can be destroyed by a change in market conditions. The realistic goal is to develop a model that provides a good answer at the moment. Why? One reason is that markets are evolving in response to economic growth, regulations, and innovation. Models ultimately fail!

## Stop losing value from a declining price

March 4, 2017### background

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:

**Company performance**. ‘Good’ companies attract investors. Conversely, ‘distressed’ companies repel investors.**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*.**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:

- 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].
- 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.

### conclusion

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, 2017Selling 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.

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.

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.

### Conclusion

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

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.

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

## 2016

January 14, 2017My *SmallTrades* portfolio holds stocks and four classes of exchange-traded index funds (ETFs).

### Investment plan

The goal is to outperform a reputable benchmark, the *Standard & Poors 500 Total Return Index*, on a sustained basis. The ETFs are diversified and rebalanced in order to partially offset the losses of a declining market. A small group of stocks are used to boost the investment returns.

### Performance

In FY2016 the portfolio’s market value increased by 8.3% due to a 9.1% gain in stock value and 8.1% gain in ETF value. Charts 2 and 3 illustrate the nominal (solid lines) and real (dashed lines) growth in unit value for shares of the portfolio, ETF group, stock group, and benchmark. The number of shares for each entity was the initial market value divided by $1 of U.S. currency. Assume that the initial unit value of $1 was a real value unaffected by inflation.

Chart 2 shows the pattern of unit-value growth for the benchmark (black lines) and portfolio (blue lines) since December 31, 2007.

The unit value of both entities declined in year 2008 and began to recover in year 2009. The benchmark (black lines) recovered in year 2011 while the portfolio (blue lines) is still struggling to recover [notes 1,2]. The effect of inflation was to devalue real growth (broken lines) compared to nominal growth (solid lines). The real unit value signifies the purchasing power of the investment. The investment has greater purchasing power than uninvested money when the real unit value exceeds $1.

Chart 3 shows the result of implementing the current investment goal [note 2] with a small group of stocks (red lines) and large group of ETFs (blue lines). In chart 3, the initial unit value was re-calculated on December 31, 2013.

Since 2013 the stock group clearly outperformed the benchmark (black lines) and ETF group. The success of the Stock group is attributed to investing in ‘good’ companies for the long term [note 3].

### Stock group

Chart 4 shows the market sector and market cap diversity of the stock group defined in chart 1.

Several stock trades were made during FY2016 to improve the chance for success.

Closings:

- Alibaba Group (BABA), for 10% capital gain, to exit the Chinese market.
- Geely Automobile (GELYF), for 14% capital gain, to exit the Chinese market.
- Corning Inc. (GLW) for no gain.
- iRobot Corp. (IRBT) for 10% capital gain.
- ITC Holdings (ITC) for 14% capital gain, due to the stock’s delisting.
- Stericycle (SRCL) for 34% capital loss, to stop further loss.

Purchases:

- Biogen (BIIB), an innovative biotechnology firm.
- Cal-Maine (CALM), a leading producer of shelled eggs.
- Express Scripts Holdings (ESRX), a large mail order pharmacy
- Royal Bank of Canada (RY), a well-capitalized bank.

### ETF group

Chart 5 shows the distribution of asset classes among the ETFs. All asset classes drifted from an allocation plan of 30% stocks, 30% REITs, 20% bonds, and 20% gold [note 4].

The *SmallTrades* portfolio’s primary strategy for risk management is holding a large group of diversified ETFs that are rebalanced to correct a significant allocation error. In theory, a significant drift of asset classes occurs when one asset class surpasses a 28% allocation error. At the end of FY2016, the existing allocation errors (blue bars) were within 24% error limits (red dashed lines) as illustrated in Chart 6.

Chart 6 reflects the portfolio’s response to an incline in equity markets compared to decline of the bond and gold markets. History has shown that a decline in equity markets tends to be offset by a rise in the bond and gold markets.

### Plan for FY2017

The *SmallTades* portfolio will continue to be actively managed for long term success. The ETFs will be rebalanced anytime there’s a 24% allocation error or a modification of the ETF holdings. I would like to own fewer large cap stocks in favor of small- and mid-cap stocks issued by good companies with potential growth of earnings.

### Notes

- On 12/31/2007, the portfolio held a group of actively managed mutual funds in a tax-deferred Roth account. Since then there have been no cash deposits or withdrawals and the portfolio still resides in the Roth account. During 2007-2010 the mutual funds were traded for stocks in an attempt to earn a 30% annual return by process of turning over short term ‘winners’. Several mistakes led to a big loss: A) after a couple of short term capital gains from Lehman Brothers Inc., I ignored the dangers of that company’s large debt and lost $45,000 during its decline to bankruptcy. B) substantial long term profits from good companies were lost by selling holdings for short term profits. I was trying to earn a quick 30% annual rate of return and immediately re-invest in the next set of winners. It was too difficult to identify the next winners. C) day trading also prevented a 30% return. It was a game of chance that I played without a strategy and I was fortunate to break even. D) a trial of investing in leveraged ETFs resulted in losses due to negative compounding. Leveraged ETFs were very high-risk investments that I made without a sound strategy.
- I abandoned the goal of a 30% annual rate of return in 2012 by adopting a more realistic, but still aggressive, goal of outperforming the benchmark. That same year, I changed my investment strategy to that of holding a mixed portfolio of 80% broad market ETFs and 20% stocks for the long term.
- ‘Good’ companies attract and retain investors for many years. I search for profitable companies with growth potential that are undervalued by the stock market. My search methods include reading reputable sources of business news, participating in investment club discussions, using stock screeners, and attending investor conferences. I include and exclude stocks by reading analyst reports, financial statments, SEC filings, and market analyses. Valuation critieria help me decide if the stock price is worth paying.
- Prior to March, 2016, five ETFs were allocated to four asset classes with each asset class holding 25% of the combined market value. Since I don’t depend on making withdrawals from the
*SmallTrades*Portfolio, I increased my exposure to global stocks and REITs by decreasing my exposures to investment-grade bonds and gold bullion. The new allocation rule was 30% stocks, 30% REITs, 20% bonds, and 20% gold. Any drift in allocation to a 24% error will be rebalanced.

Copyright © 2017 Douglas R. Knight

## R-squared, the linearity of investment returns.

December 24, 2016[updated 12/25/2016: R^{2} is a useful measure of indexing]

The R-squared (R^{2}) 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 R^{2} 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 R^{2} of 0.997.

### Significance

The R^{2} 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 R^{2} X 100 represents the percent of variation in investment returns that are related to market returns (ref 1,2). In other words, R^{2} measures the relavance of the best-fit line to a set of data. Relavance increases as the R^{2} 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 R^{2} score of 1.00 identifies a straight line of near-perfect predictions of returns. Any R^{2} 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 R^{2} as a measure of alignment between fund returns and the market index. Funds that have an R^{2} score of nearly 1.00 track the index very closely.

### References

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. https://onlinecourses.science.psu.edu/stat501

2. R-squared. 2016, Investopedia http://www.investopedia.com/terms/r/r-squared.asp?lgl=no-infinite

## 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%.

### Significance

#### (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.

### Footnotes

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).

### References

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. http://www.investopedia.com/articles/07/alphabeta.asp#ixzz4SYJ0rN9q

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

5. Investing Answers. Alpha Definition & Example. 2016. http://www.investinganswers.com/financial-dictionary/stock-valuation/alpha-43

6. Professor Lasse H. Pederson. The capital asset pricing model (CAPM). New York University Stern School of Finance. undated. http://www.stern.nyu.edu/~lpederse/courses/c150002/11CAPM.pdf

7. MoneyChimp. Regression, Alpha, R-Squared. 2016. http://www.moneychimp.com/articles/risk/regression.htm

8. Invest Excel. Calculate Jensen’s Alpha with Excel. undated. http://investexcel.net/jensens-alpha-excel/

### 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

## Beta is the incline of a straight line

December 10, 2016Beta (which is symbolized as β) is the incline of a straight line. Mathematicians would say the same thing another way, that beta is the slope of a regression line. Either way, β describes the tendency of investment returns to move with market returns. The investment is a security (e.g., stock, bond, mutual fund) that has a unit price. The market is a trading place for a large group of securities. The combined value of all securities is measured by a market index.

### Returns

Trading causes security prices to change during the passage of time, a process called price movement. Calculations of β require price movements to be measured as percentage returns. In table 1, the daily closing prices of a security and its market index are listed under the column heading “close”. Percentage daily changes in closing price are listed under the column heading “Return %”. Equation 1 is the formula used to calculate a return:

Return % = 100 x (current price – past price) / past price (equation 1)

Notice in table 1 that all prices are a positive number and that the market’s close is bigger than the investment’s close. However, the calculated returns are positive and negative numbers of similar size. The positive and negative returns represent up and down movements of prices. Table 1 has 3 pairs of investment and market returns with corresponding dates.

### Beta (β)

β may be calculated directly from a table of returns, but it’s more meaningful to analyze a scatter plot of returns. The scatter plot in figure 1 has a solid blue line derived from 5 years of daily returns represented by more than a thousand black dots. Each dot has a pair of corresponding returns on each axis.

The blue line offers the single-best comparison of investment returns to market returns. The incline of the blue line is β, which is calculated as a ratio of the lengths AC and BC of the dashed lines. Since AC and BC have equal point spreads of 5%, β is 1.00, which means that the investment and its market TENDED to move together at the same rate of return.

Notice that the black dots are closely aligned to the blue line, therefore excluding the random movement of returns. Consequently, the blue line is highly predictive of this particular investment’s past performance.

### Significance

β is a measurement that literally means for every percent of market return, the percent investment return TENDED to change by the factor of β. This is illustrated in figure 2.

The colored performance lines in figure 2 represent different investments. Each line offers the single-best comparison of investment returns to market returns. For the sake of graphic clarity, a large cluster of paired returns was not plotted as data points.

At β = 1.00 (black dashed line) the investment and market TENDED to move together at the same rate. At β >1.00 (yellow line), the investment performance was amplified by trading activity in the market. The yellow line’s β infers that the investment’s return was 1.72 times the market’s return. At β <1.00 (green line), the investment performance was diminished by market activity. The green line infers that the investment’s return was 0.86 times the market’s return. At β <0 (red line), the investment performance was reversed by market activity. The red line infers that the investment’s return was -3.86 times the market’s return.

Thus, β is a ‘pretend’ multiplier of market performance. Higher β ‘amplified’ the market performance, lower β ‘diminished’ the market performance, and negative β ‘reversed’ the market performance.

### Risk

Risk is the chance for a capital gain and capital loss. Betas greater than 1.00 tend to be riskier investments and those lower than 1.00 tend to be safer investments compared to performance of the market. Negative β infers a reversal of investment outcomes compared to market outcomes.

### Summary and advice

β is a statistic for past performance that describes the tendency of investment returns to move with market returns. When comparing the β of different investments, be sure to verify the time periods and market index used by the analyst. β is typically measured with weekly or monthly returns for the past 3-5 years.

Copyright © 2016 Douglas R. Knight