## The joy of stock returns

August 5, 2018

A good reason for investing in stocks is to earn more money than the interest paid by a bank account or savings bonds.  Some investors ignore their stocks until it’s time to cash in.  Most prefer to watch the growth of returns, in which case they need to know the total return and holding period.

### Total Return

Stock profits depend on the capital gains and dividends.  A capital gain is the amount earned when the current stock price exceeds its purchase price.  A capital loss is the amount lost when a current price is below the purchase price.  The capital gain (or loss) is “unrealized” if the investor doesn’t sell the stock or “realized” if the investor sold the stock.  Some companies make occasional cash payments, called dividends, to their stock holders.

Total Return is the total profit from your stock investment.  It represents the stock’s change in market value combined with all dividends you received from the company.  In equation 1, the change in market value is equal to “market value – total cost”.

total return = market value – total cost + dividends

• Market value is the combined value of all shares owned at the current market price (market value = current price * volume; volume is the number of shares).
• Total cost is the value of all shares purchased (cost = purchase price * volume) during the holding period
• Holding period is the period of stock ownership.

Trading fees are ignored in order to simplify this discussion. In actual transactions, trading fees reduce the market value and increase the cost by small amounts so as to reduce the total return by a small amount.  The impact of trading fees on profits is lower in larger transactions.  For example, a \$5.00 trading fee is 5% of a \$100 purchase compared to 0.5% of a \$1,000 purchase.

### Return on Investment (ROI)

ROI is the basic measurement of profitability (ref 1).  It is the ratio of total return to total cost (equation 2).

ROI = total return/total cost

Significance: the ROI shows how much profit you earned from every invested dollar.  If the ROI were 0.2/1, which is 20%, then you earned 20 cents per invested dollar.

“Price performance” (equation 3) may not measure the ROI.  Price performance = (price2 – price1)/price1, where price1 is the earlier number and price2 is the later number.  Price 1 may neither be the purchase price nor the only purchase price.  ROI includes all purchase prices.

### Rate of Return

The rate of return measures profitability with respect to time.  Think of if it as the ROI for the holding period (equation 4).

Rate of Return = ROI/holding period

Don’t forget that the ROI compares profit to cost when the time period is anchored to the date of the initial purchase.

ISSUE: The rate of return is most precise when there is just one purchase.  Serial purchases require a more complicated calculation of the “annual return”.

### Annual return

The annual return is a number that represents the average rate of growth per year of the holding period.  The annual return has several important properties:

1. It doesn’t change during the holding period.
2. It’s a geometric average, not an arithmetic average.  The graph of geometric growth is a curved line (“exponential”) rather than a straight line (“linear”).
3. The geometric average represents the phenomenon of compounded growth known as “compounded interest”.

Stock investors are interested in 2 types of compounded growth:

1. The compound annual growth rate (“CAGR”) of a single purchase.
2. The internal rate of return (“IRR”) for a series of purchases.

There are free calculators which are posted online to determine the CAGR (ref 2) and the IRR (ref 3).

ISSUE: The annual return is usually inaccurate during the first year of compounded growth and becomes more accurate over longer time periods.

### Rule of 72

The payback period is something to celebrate!  It’s the point when the investment doubles your money.  Payback is measured by the ratio of total cost to the rate of return.  Or, it can be estimated by the Rule of 72 (equation 5).

Rule of 72 = 72/assumed annual return

Significance: the Rule of 72 is used to forecast the holding period needed to double your money (ref. 4).  For example, assume that your total return will grow at a constant rate of 10% per year [approximately the same rate as the growth of the U.S. Stock Market].  The expected payback period is 7.2 years (7.2 = 72/10).

### Summary

The total profit of your stock investment is called the total return.  The simplest way to measure profitability is to calculate the ROI with equation 2.  The ROI is insensitive to time until you calculate the rate of return with equation 4, which allows you to compare the profitability of several stocks in your portfolio.  The annual return of compounded growth is a refined measurement of your calculated rate of return.  After a holding period beyond one year [to avoid the chance of considerable inaccuracy], the annual return may be determined with an online calculator for a single investment (CAGR) or serial investments (IRR).

### References

1. Return on Investment (ROI). https://www.investopedia.com/terms/r/returnoninvestment.asp

2. Compound annual growth rate (CAGR) calculator.  http://www.moneychimp.com/calculator/discount_rate_calculator.htm  .

3. XIRR calculator to calculate IRR of non-periodic cash flows. https://www.free-online-calculator-use.com/xirr-calculator.html   .

4. Brian Beers. What is the Rule of 72? https://www.investopedia.com/ask/answers/what-is-the-rule-72/ , 1/2/2018.

## Ways to invest in stocks

July 19, 2018

There are thousands of  investors who want to own ‘good’ companies that avoid ‘trouble’.

• they invest in stock shares [stock shares are equal units of part ownership]
• a good company
• operates a profitable, growing business
• avoids financial distress and regulatory penalties

Investors purchase and sell shares in the stock market.  They hope to sell their stock at a desirable price and may also receive cash rewards from companies that pay dividends.  Investors earn a profit (called a capital gain) when the sales price is above their cost of investment or lose money (called a capital loss) when the sales price is below cost.

### Stock Analysis

Two ways of evaluating a stock are called technical analysis and fundamental analysis.  Technical analysis measures the performance of share prices and share volumes in the stock market.

• Shares are units of part ownership which are traded in the stock market.
• Price: the price of a share in the stock market.
• Volume: the total number of shares traded in the stock market

Fundamental analysis evaluates the business performance of a company by way of searching through its quarterly and annual filings.  The business description, financial statements, and CEO’s annual letter to shareholders are important sections of the filings.

• CEO: Chief Executive Officer; top manager of the company.
• Filings: periodic reports to shareholders that are required by the U.S. Securities and Exchange Commission (SEC).

Business performance is also assessed by the company’s market share and competitive advantage within its industry.  This information is available online.

### Investment Strategies

The most common investment strategies for stocks are swing trading, value investing, and growth investing.

Swing trading (cyclic trading) uses brief upward or downward trends in share prices to determine when to buy or sell stocks.  The typical holding period is from one day to several weeks.  The investor hopes to earn a capital gain (–if seeking a profit–) or capital loss (–if seeking to reduce the short term capital gains tax–).  The investor uses either a technical analysis or guesswork to judge the price trend.  The main risks of incurring a loss are due to price volatility and taxation of returns.

• Hold: to own.
• Short term: one year or less.
• Short term capital gains tax: the taxation of a capital gain at the regular income tax rate.
• Price volatility: the random fluctuation of prices based on the market forces of supply and demand.
• Return: the profit or loss from an investment.

Value investing seeks a capital gain by purchasing the stock at an unusually low price (e.g., 60% of intrinsic value) and then selling it at approximately double the purchase price.  The holding period depends on the length of time for the stock price to become profitable. During the holding period, an investor will receive any dividends paid by the company.  The informed investor uses a fundamental analysis to assess the quality of the company and the intrinsic value of its stock.  The causes of an unusually low price include a market downtrend (e.g., economic recession) and poor company performance.  The main risks of incurring a loss are due to an eventual delisting of the company and taxation of returns.

• Intrinsic value: the share price calculated by a professional analyst’s secret formula.  However, you can estimate the intrinsic value as the net worth of the company (book value) per share, based on the idea that a wealthy investor could acquire the company at its intrinsic price by puchasing all shares of stock at the book value per share.
• Dividend: a cash reward paid to share holders from the company’s profits or cash reserves.
• Delisting: removal of the stock from the stock market for various regulatory reasons, including bankruptcy of the company.

Growth investing is a long term strategy for using the upward momentum of share prices to earn a capital gain. The capital gain is earned by simply holding the stock and reinvesting all dividends.  The rule of 72 estimates the holding period needed to double the purchase price of the stock at an assumed rate of annual return.  The growth investor uses a fundamental analysis of the company and market valuation to judge the fairness of the stock price.  The main risks of incurring a loss are due to deterioration of the company, decline in market value, and taxation of the returns.

• Long term: after one year.
• Momentum: an upward trend of share prices.
• Rule of 72: [ Years to double the price = 72/percentage annual rate of return ] For example, a 15% annual rate of return will double the share price in 4.8 years.
• Annual rate of return: a constant percentage change in value every year that accelerates the growth of an investment; CAGR is an acronym for the annual rate of return.
• Valuation: the art of judging if the price is low (discounted, undervalued) or high (expensive, overvalued).

## 2017

January 1, 2018

chart 1. SmallTrades Portfolio in 2017.

Chart 2 shows the diversification of ETFs as measured by percentages of year-end market values among ETF classes.

chart 2. Diversification of ETFs in 2017.

Chart 3 shows the diversification of stocks among 8 market sectors as measured by percentages of year-end market value for each stock sector and the ETFs.

Chart 3. Distribution of stocks and ETFs by market sectors.

Chart 4 shows the distribution of stocks according to market capitalization.

Chart 4. Combined market capitalizations.

### Performance

My investment goal is to outperform the “Benchmark” Standard & Poors 500 Total Return Index, yet my portfolio has never outperformed the Benchmark (chart 5).

Chart 5. Portfolio performance.

Chart 5 shows growth trends for the benchmark (blue dashed line) and portfolio (solid blue line) since 2007 [the benchmark represents a passively managed, buy-and-hold investment; my portfolio is an actively managed investment].  On the Y axis, a unit value of \$1.00 was assigned to both the total market value of the Portfolio and the Benchmark on December 31, 2007. Ratios of subsequent market- and benchmark values to the 2007 baseline are displayed line plots on the chart.

In 2014, my investment policy was modified to buy stocks of good companies and hold them for the long term. Chart 6 shows the result of my stock investments (red line) compared to the Benchmark Index (blue line) and ETF investments (red dashed line). The unit value of \$1.00 was calculated on December 31, 2013. Since then, the stock group clearly outperformed the Benchmark and ETFs.

Chart 6. Stock and ETF performances.

### Risk Management of ETFs

Broad-market index ETFs are primarily protected against stock losses by the passive management of investment portfolios which mimic the composition and performace of reputable market indices.

ETFs are secondarily protected by rebalancing significant allocation errors as described in the SmallTrades Portfolio’s strategies for risk management. In theory, a significant drift of asset classes occurs when one asset class surpasses a 24-28% allocation error. My preferred allocation of ETF market values is 30% stocks, 30% REITs, 20% bonds, and 20% gold bullion.

A perfect allocation of ETFs would result in 0% allocation error.  Furthermore, allocation errors would reflect disproportional gains or losses of market value.  Chart 7 shows the year-end allocation errors (blue bars) and error limits (red dashed lines) of my ETFs. There was growth of the Global Stocks ETF and decline of the remaining ETFs. Any allocation error that exceeds an error limit (red dashed line) should trigger trades that rebalance the ETFs to the preferred allocation.  My ETFs were not rebalanced in 2017.

Chart 7. ETF allocation errors in 2017.

### Risk management of Stocks

My stocks are primarily protected against risks of steep loss by diversification of the market sectors, as illustrated in the preceding chart 3. The second line of defense is stop-loss orders.  In keeping with the investment goal of holding good stocks for the long run, I set ‘stops’ at a wide margin to prevent recent market fluctuations from triggering an unwanted sale.

### Plan

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. In 2017, I failed to sell large cap stocks in order to buy good small cap and mid cap stocks. Consequently, 60% of the total market capitalization of my stock portfolio was in the Large Cap category.  In 2018, I would like to reduce the Large Cap category to 40% total market capitalization and boost the market capitalization of small- and mid cap stocks issued by good companies with potential growth of earnings.

### Portfolio history

1. 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 a Roth account.
2. During 2007-2010 the actively managed mutual funds were traded for stocks in an attempt to earn a 30% annual return by process of turning over short term ‘winners’.  Four mistakes led to a big loss:
3. mistake #1: after a couple of short term capital gains from Lehman Brothers Inc., I ignored the dangers of the company’s large debt and lost \$45,000 during Lehman’s decline to bankruptcy.
4. mistake #2: substantial long term profits from good companies were lost by selling holdings for short term profits. My strategy was to earn a quick 30% in the first year and re-invest in the next winners. It was too difficult to identify the next winners.
5. mistake #3: day-trading was a game of chance that I played and managed to break even; meanwhile, good stocks grew in value.
6. mistake #4: a trial of investing in leveraged ETFs resulted in losses due to negative compounding.
7. I abandoned the goal of a 30% annual 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 index ETFs and 20% stocks for the long term. ‘Good’ companies attract and retain investors for many years. I will search for profitable companies with growth potential that are undervalued by the stock market. My search methods include reading reputable sources of business news, partiicipating in an investment club, using stock screeners, and attending investor conferences. Then 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.
8. Prior to March, 2016, five ETFs were allocated to four asset classes with each asset class holding 25% of the combined market value. Since my retirement income didn’t depend on making withdrawals from the SmallTrades Portfolio, I increased my ETF exposures 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.

## What is a good company?

December 15, 2017

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.

## 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!

## 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

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 .
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

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

April 25, 2017

Physics 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:

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.

### 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.

## 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, 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.

### 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 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, 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, 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.

## 2016

January 14, 2017

My SmallTrades portfolio holds stocks and four classes of exchange-traded index funds (ETFs).

chart 1

### 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.

chart 2

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.

chart 3

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.

chart 4

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].

chart 5

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

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

1. 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.
2. 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.
3. ‘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.
4. 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.