These 20 graphs form a pictorial essay of a company’s financial statements. This large company survived the Recession of 2007-8!
Copyright © 2017 Douglas R. Knight
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 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.
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):
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.
Besides measuring market value, the PE and PEG also predict the stock’s payback period. A payback period is the amount of time needed for the accumulation of company earnings to match the original amount of investment. If all accumulated earnings were paid to investors, which is unlikely, the payout would provide a 100% return. Longer payback periods represent riskier investments, especially when the company is still establishing its market position or competing with innovative companies (ref. 9,10,11).
The PE ratio also represents a payback period measured in years.
example: if a stock is priced at $50 per share and the EPS is $5 per share every year, then $50/share divided by $5/share/year equals the payback period of 10 years. The same units of $/share cancel each other in the numerator and denominator.
The PE payback period is the time needed for an accumulated EPS to equal the original share price, assuming the EPS remains constant during the accumulation period. Most companies don’t repeat the same EPS every year.
The PEG payback period accounts for the desired phenomenon of EPS growth. The PEG payback period is the number of years that the growth of earnings accumulates enough value to match the original investment (ref. 9,10).
example: if a stock is priced at $50 per share, the EPS is $5 per share, and the EPS growth rate is 10%, it would take 7 years for the EPS to accumulate a value of the original stock price of $50. The PEG payback (7 years) is earlier than the PE payback (10 years) due to the 10% rate of earnings growth.
Company earnings are a strong determinant of stock value. PE, PEG, and PEGY ratios represent the stock market’s valuation of company earnings. Don’t rely solely on company earnings to judge the investment value of stocks. Also assess the business performance and company value (ref. 2,6).
Formula 1: PE = P / E
Formula 2: PEG = PE / G
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
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.
Today’s market models are still imperfect!
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?
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].
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.
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.
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.
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.
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.
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!
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:
The risk of an extreme loss can be prevented by setting a stop-loss price (“stop”) to sell part or all of your shares.
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:
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.
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.
Nobody’s immune, but long-time investors have the least concern. Investment strategies such as dollar-cost-averaging and automatic-dividend-reinvestment plans will help protect against damages from periodic bear markets. Short- and intermediate-time investors are at greater risk for incurring an extreme loss from market down-cycles. For example, families who are saving to pay college fees or to buy a home risk big losses from a bear market.
Stop orders are used to set the price for buying or selling exchange-traded products such as stocks, ETFs, and REITs. This article discussed the use of a stop-limit order to sell a stock in a declining market. Brokerage firms may restrict the duration of stop-limit orders to 30-90 days after which the order is cancelled without a transaction until you renew the order. Periodic renewals allow you to reconsider your strategy in light of the prevailing price trend. In a downtrend, simply renew the order. In an uptrend, you may wish to protect a growing profit by resetting the stop-limit order to higher prices. Click on this link to skimming a profit for another perspective on protecting a growing profit.
Copyright © 2017 Douglas R. Knight
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.
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.
The accumulated cash balance will vary according to the annual rate of return (appended chart 3), the amount skimmed (appended chart 4), and the payment of dividends (appended chart 5). In every condition, the total profit of the HOLD strategy exceeds the total profit of the SELL strategy.
On the question of whether or not to skim profits, skim if you need cash in the next 5-10 years. Otherwise, don’t sell without reassessing the investment or using a risk management scheme. The question of selling for a loss was excluded from this discussion; that’s a different topic.
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
My SmallTrades portfolio holds stocks and four classes of exchange-traded index funds (ETFs).
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.
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].
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.
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.
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.
Copyright © 2017 Douglas R. Knight
[updated 12/25/2016: R2 is a useful measure of indexing]
The R-squared (R2) statistic describes a pattern of plotted data with respect to a straight line. R-squared is called the coefficient of determination (ref 1,2).
The black dots in figure 1 represent investment returns that are poorly related to market returns. There is a random distribution of investment returns with respect to market returns. The blue line is an inadequate representation of the relationship simply because there is no relationship. The R2 score for this distribution is 0.03. Conversely, the black dots in figure 2 show the ‘herding’ of data around a straight line.
Figure 2’s investment returns are highly related to market returns with an R2 of 0.997.
The R2 score represents the degree of alignment of data to a best-fit line as determined by regression analysis. The lowest possible score of 0 indicates a random pattern of data with absolutely no alignment. The highest possible score of 1 represents complete alignment.
The product of R2 X 100 represents the percent of variation in investment returns that are related to market returns (ref 1,2). In other words, R2 measures the relavance of the best-fit line to a set of data. Relavance increases as the R2 score varies from 0 to 1.
The lowest score of 0 defies any financial analyst to draw a meaningful line for investment returns as they relate to market returns. In figure 1, the incline (β) and Y-intercept (⍺) of the blue line are unreliable measurements of investment performance.
The highest R2 score of 1.00 identifies a straight line of near-perfect predictions of returns. Any R2 above 0.75 identifies a straight line for making predictions of returns. Lower scores represent increasingly random events. In figure 2, the incline (β) and Y-intercept (⍺) are reliable measurements of investment performance.
R-squared is an excellent measure of index fund performance. Websites for index mutual funds and ETFs publish R2 as a measure of alignment between fund returns and the market index. Funds that have an R2 score of nearly 1.00 track the index very closely.
1. Lain Pardoe, Laura Simon, and Derek Young. STAT 501, Regression Methods. 1.5- The coefficient of determination, r-squared. Pennsylvania State University, Eberly College of Science, Online courses. https://onlinecourses.science.psu.edu/stat501
2. R-squared. 2016, Investopedia http://www.investopedia.com/terms/r/r-squared.asp?lgl=no-infinite