Financial health

May 17, 2020

Long-term investors depend on their stocks to remain viable during economic recessions.  In today’s Coronavirus Pandemic, businesses of all sizes are losing income from the forced reduction of consumer spending, which may destabilize companies to the brink of bankruptcy.  Investors can assess stability by reviewing the financial health of their companies.

Financial health is the ability to pay all obligations in a timely matter.  Credit ratings and analyst reports use propriety methods to measure financial health.  You can independently rate the financial health of a public company using a single numerical score from 0 to 10 based on liquidity and solvency; the higher the score, the healthier the company (eq. 1).

equation 1:    Health = Liquidity + Solvency

Liquidity

Liquidity refers to the ease of converting current assets into cash for payments of current liabilities.  Current assets are considered convertible to cash within one year.  Some assets are more liquid than others. Savings accounts, checking balances, money market funds, and receivables [i.e., customers’ IOUs] represent liquid assets. Inventory [i.e., unused supplies and unsold products] is considered an illiquid asset.  Current liabilities are the costs of paying business expenses such as wages, payables, and interest on short-term credit.  The following ratios provide useful measurements of liquidity:

  • Current ratio = Current assets / Current liabilities.
  • Quick ratio = (Current assets – Inventory) / Current liabilities
  • Interest coverage = EBIT / Interest  [EBIT is the company’s earnings before accounting for the charges of interest and tax; EBIT is a measure of recurring income]

 

liquidity

chart 1

Solvency

Solvency refers to the liquidation value of a company in case the company must pay all of its short-term and long-term liabilities. I use the shareholders’ equity [aka net worth or book value] as a common denominator for the measurement of solvency.  Solvency ratios and free cash flow provide useful measurements:

  • Debt-to-Equity = Long-term debt / Shareholders’ equity.
  • Financial Leverage = Total assets / Shareholders’ equity.
  • Free Cash Flow = Operating cash flow – Capital expenses

 

solvency

chart 2

Examples

Chart 3 displays health scores for a list of companies identifiable by stock tickers; they are the current holdings of my investment club.  The data were calculated with the formula in eq. 1 using open source data for liquidity and solvency.  Three stocks received low health scores of 2.

health

chart 3

From chart 3, I selected five strong competitors of VZ and CMCSA to determine if the low health score represents a larger group of 7 competitors listed in the trading sector of Communication Services.  The additional competitors are listed below in chart 4. Three of the additional competitors matched the scores of CMCSA and VZ, inferring that most companies in that select group operate with low liquidity and solvency.

competitors

chart 4

Another comparison was made using a sample of stocks with an open-source, proprietary grade of low financial health (chart 5).  One stock, JCP, recently filed for chapter 11 bankruptcy.

others

 chart 5

Risk management

The health scores in chart 3 are based on historical data at least 3 months old.  Stocks with the lowest scores are considered more unstable.  If, in your informed opinion, there’s a credible risk of bankruptcy and delisting, you can protect your investment by either selling the stock or placing a stop-loss order on it.

Conclusion

Open-source financial data can be used to assess the risk of potential bankruptcy and delisting among publicly traded stocks, especially during an economic recession.  Combined assessments of liquidity (chart 1) and solvency (chart 2) additively form a health score of 0 to 10, with lower scores implying poor financial health.  The scoring system is easy to implement, but unreliably predicts financial failure of public companies with low scores.  Additional fundamental analysis of the company is strongly recommended and meanwhile, if you wish to protect your investment from a substantial loss, place a temporary stop-loss order on the holding.

Copyright © 2020 Douglas R. Knight 

 

 

 

 

 

 


Stock Market Prices

April 1, 2020

SUMMARY:

The Stock Market is a place where professional traders arrange cash-for-stock transactions between buyers and sellers.  Other securities are sold in the Market, but stocks occupy the vast majority of listed securities—(securities are investment contracts worth money, of which stocks represent shares of ownership in companies).

Every transaction is called a Trade.  Regular trades involve the buyer’s payment of cash for securities offered by the seller.  Buyers and sellers propose trades to their brokers who then send the proposals (orders) to professional traders.  Market rules require traders to fill orders at the next available price, either the highest Bid of a buyer or lowest Ask of a seller, depending on the type of trade.  The general trend of prices among many trades is calculated as the Market Index.  Investors should prepare trading orders carefully with awareness of the potential consequences.

Competitive prices

The Stock Market is designed to set prices for securities at an agreeable price among competitive Bids and Asks –(the buyer’s price is called a Bid and the seller’s price is an Ask).  The agreed price varies according to the prevailing action of trading orders in which Buy orders are filled at the lowest available Ask and Sell orders are filled at the highest available Bid.  In a ‘seller’s market’, the buyers’ surplus demand for securities raises prices for the sellers.  Examples include the rising prices in rallies and bull markets.  In the ‘buyer’s market’, the sellers’ surplus supply of securities lowers prices for the buyers.  Examples include the falling prices in corrections and bear markets.  

Market Index

The Market Index is a singular value which represents the prices of many securities traded in stock exchanges [see index methodology in the appendix].  Graphs of the market index display daily fluctuations (volatility), trends, and market cycles.  The trend of a market index is useful in several ways:

  • Analysis of supply-and-demand: An increased demand for shares drives prices upward and conversely, an increased supply of shares drives prices downward.  The index follows the price trends.
  • Benchmark: Investors like to know if the prices of their holdings are performing better or worse than the market index.
  • Passive management: Index funds (e.g. ETFs)  are investment portfolios designed to match the performance of an index.
market indices, 5y

Chart 1.  The ‘Dow’ represents stock prices of 30 large companies traded in the New York Stock Exchange and Nasdaq market.  The ‘S&P 500’ represents market capitalizations of 500 large companies traded in U.S. exchanges (market capitalization is the sum of prices for all shares of a given stock).  The ‘Nasdaq’ is calculated from market capitalizations of all companies listed in the Nasdaq market.

Chart 1 displays the parallel behavior of 3 popular indices; they are broad market indices by virtue of describing the price volatility and trends of many stocks listed in the Market.  Small fluctuations represent daily values reported at the close of the trading day.  Large fluctuations display short and long cycles of market activity.  A long market cycle consists of one “bull” and “bear” market in succession.  Long bull markets create a general upward trend of market prices that endures several market cycles.

Trading Orders

In contrast to the market index, which represents many stocks, the quote represents one stock.  Quotes are widely published in the media and brokerage firms.  A typical broker’s quote shows the last traded price, traded volume, and opposing prices (bid & ask).  

Investors place a trading order by consulting their broker or employing the broker’s online trading platform.  In a trading platform, the investor completes an order form with the following information:

  1. Ticker.  The trading symbol of the desired security
  2. Action.  Buy or Sell
  3. Volume.  Quantity of units (shares) to be traded
  4. Type.  Method for filling the order (Market versus conditional)
  5. Price.  conditional or Market.

The basic types of orders are Market and conditional.  Market orders are filled immediately at the next available price, but the investor is unable to specify the price.  Conditional orders enable the investor to specify the price of a future trade within a period called the “time-in-force” (typically 60 days).  Conditional trades are activated at the specified price and filled at the next available price. 

Limit and Stop are two types of conditional orders available to most stock investors.  A Limit is the preferred price for a Buy or Sell order.  The Limit order is activated when a future market price matches the Limit price.  The activated trade is then filled at the same price or a more favorable price; but, if the next available price becomes unfavorable due to price fluctuation, the Limit order is cancelled unfilled.  A Stop is the specified price of a Sell order.  The Stop order is activated at the specified price and then filled by a Market order.  The seller has no control of the price after a Stop order is activated.

A Trading Story

Two fictional investors named ‘Green’ and ‘Red’ decided to place opposite trading orders for the same security on March 5th when the quoted price was $49.  ‘Green’ thought the price would eventually drop and wanted to buy 100 shares for a bargain at the Limit of $45.  The intended bargain was a $400 reduction of investment cost.  ‘Red’, who owned 100 shares, thought the future price would drop for a loss.  “Red’ wanted to prevent a deep loss by selling 100 share at the $45 Stop.  ‘Red’ would be happy if the Stop order were never activated, but just in case prices declined, the loss of $400 could be tolerated.  The outcomes are illustrated in Chart 2.

trading orders

Chart 2: The Fate of 2 Conditional Orders.  Red and Green symbols represent respective Stop and Limit orders made on March 5th.  The dashed line indicates that both orders remained-in-force until activated and filled on March 9th.  An overnight crash of prices halted trading at the start of the March 9th trading day.

On March 9th, a market crash activated both orders at the moment trading was halted by a circuit breaker.  When trading resumed, ‘Green’ bought 100 shares at the very favorable price of $41, even $4 cheaper than the intended $45 Limit.  ‘Green’s’ bargain was $800 instead of $400.   ‘Red’ sold 100 shares at the very unfavorable price of $41, $4 below the intended $45 Stop.  ‘Red’s’ original market value of $4,900 dropped by $800 instead of $400 after the activated Market order filled at the next available price of $41.

–Lesson: Limit orders protect a preferred range of transaction prices.  Market and Stop orders don’t protect the transaction price.

References

  1. Trading orders, Invsetopedia.com
  2. Supply and demand, Investopedia.com
  3. Stock quote, Investopedia.com
  4. DJIA index methodology, Investopedia.com. 
  5. S&P 500 index methodology, Investopedia.com.
  6. Do ‘Circuit Breakers’ Calm Markets or Panic Them?: QuickTake. Nick Baker & Sam Mamudi.  The Washington Post 3/19/20, WashingtonPost.com 

Appendix: Index methodology

The stock Index is a special sum of weighted prices for many stocks, (w * Price) of many, listed in the stock market. The sum is divided by a special divisor, D.

Index(w * Price) of many ÷ D

The stocks, their weighting factor (w), and the divisor (D) are proprietary definitions of the Index provider.  Repeated calculations of the Index create a string of values that reveal the general volatility and trend of stock prices.

Copyright © 2020 Douglas R. Knight


Language of the Stock Market

February 29, 2020

Summary: New investors might find it helpful to understand the basic language of the Stock Market.  In this article I discuss the basic vocabulary as it relates to practical ideas for personal investing.  Links are provided for further reading about a particular topic.

Investment returns

An investment is the payment of capital to earn a return.  The return is a gain (or loss) of value in the investment.  Taxes on returns are regulated by the Internal Revenue Service (I.R.S.) and local governments.   

  • Principal: the amount of money invested.
  • Capital: the cash or goods used to generate income.
  • Capital gain (or loss): the increase (or decrease) in cash value of an asset.
  • Dividend: a company’s cash payment to its stockowners.   
  • Interest: the borrower’s cash payment to the lender that is added to the principal of the  loan.

Investment portfolio

Financial assets are potential sources of income for investors.  Asset classes are groupings of assets that earn income in uniquely different ways.  The most popular asset classes are Equities and Fixed Income Securities.  Equities earn income by the sale of a security (e.g., shares of a Stock).  Fixed Income Securities earn guaranteed interest (e.g., bonds) or guaranteed dividends (e.g., preferred stocks).  Securities and investors are regulated by the Securities & Exchange Commission (S.E.C.). 

  • Securities: contracts that require an investment of money to generate profits from the efforts of other people. 
  • Stock: a security that represents part ownership of a company.  
  • Common stock: a security that entitles its owner to vote on important issues, collect dividends, and earn capital gains from the stock market.   
  • Preferred stock: a security that entitles its owner to receive dividends before dividends are paid to owners of the company’s common stock.  Preferred stockowners have no voting rights.    
  • Bond: the debt that requires a company to return an investor’s principal, plus interest, by the date of maturity.    

A Portfolio is the investor’s collection of financial assets called holdings.  By comparison, an Investment Fund is a portfolio of financial securities which are professionally managed on behalf of the fund’s investors.  Famous examples are mutual funds and exchange-traded funds (ETFs).  An actively managed portfolio generally seeks to earn higher returns than one which is passively managed.  The passively managed portfolio seeks to duplicate the returns of a market index.  

Glossary:

  • Market index: a hypothetical portfolio designed to measure the value of a market or market segment. 
  • ETF: an Investment Fund that sells shares of the fund in the stock market.  Index ETFs are passively managed. 

Stock market

A new stock is issued in its primary market.  The primary market is a private assembly of the company’s founders, venture capitalists, and third parties such as banks and advisors.  The stock may later be sold by public auction in the secondary market.  The secondary market is the familiar stock market where millions of investors, —like us!—,  trade cash for stocks and other exchange-traded securities (e.g., ETFs). 

Trading orders

The stock market participants include Investors who make offers, Brokers who generate orders, and Traders who finalize orders.  The broker’s trading platform is a computer program that assists investors with placing trading orders.  The platform provides a market “quote” comprised of the current purchasing price (the “bid”), sales price (the “ask”), last-traded price, and latest number of traded shares (the “volume”).   On any day there may be millions of orders to buy and sell in the stock market.  Orders are filled at the market price determined by an auction of shares conducted by the broker’s trader.  Brokers and traders often charge a fee for their services.  Custodians are hired by brokers to store traded securities in electronic accounts on behalf of investors.   

The simplest trading order, a MARKET ORDER, specifies the number of shares to be traded.  Market orders are filled immediately provided the shares are available; otherwise, the order remains open until shares are available.  Conditional limit- and stop orders are stored in computers until activated or expired at the end of a period called the time-in-force.  The LIMIT ORDER requires an investor to specify a preferred price for the trade.  Limit orders are activated when the market price reaches the preferred price and then filled at the preferred price or a better price. Please be aware that a sudden market event could displace the market price outside the limit range of an activated order, in which case the limit order is cancelled unfilled. The STOP ORDER is activated at a specified price after which it is converted to a market order to be filled immediately regardless of the next available price. 

Stock market index

Analysts like to follow the price trend of stocks by graphing a representative number called the stock market index.  The index rises and falls at any moment according to fluctuations in share prices during stock market transactions.  An influential sales surge moves prices downward and a buying surge generally sends prices upward.

Daily index values are strung together to form an observable trend called the market cycle.  The long market cycle is comprised of a “bull” market followed by a “bear” market.  The short market cycle is either a rally or a correction. Spikes and crashes are brief events caused by a sudden, large change of the index (chart 1).  

  • Bull market: a 20% rise of the market index over 2 months or more.
  • Bear market: a 20% fall of the market index over 2 months or more.  
  • Rally: a rise of the market index due to a burst of buying that subsides after the money is spent.
  • Correction: a 10% decline of the market index over 2-10 days.
  • Spike: a sudden large upward or downward price movement.
  • Crash: a sudden correction that lasts 1-2 days.
  • Circuit breakers: programmed halts of trading designed to offset a downward plunge of stock prices.  

Chart 1. Long and short cycles of the Dow Jones Industrial Average (“DOW”).

market cycles, DJIA

In chart 1, the vertical scale shows values of the DOW Index during a 20 year time period shown by the horizontal scale.  The jagged line represents daily fluctuations of market prices. Green, red, and black symbols illustrate the timing of various market cycles and events.  The horizontal line of green and red segments portrays 4 long cycles of the DOW Index.  After the partial 1st cycle (Jan 2000-Jul. 2001), the complete 2nd (Oct. 2001-Aug. 2002) and 3rd cycles (Sep. 2002-Mar. 2009) show orderly sequences of bull and bear markets.  The nearly complete 4th cycle began with a very long bull market of eleven years (Mar. 2009- Jan. 2020) that recently reverted to a bear market at the time of this writing.  Chart 1 also shows short cycles of rallies (green triangles) and corrections (red triangles).  A few market crashes (black triangles) in Nov. 2008 and Mar. 2020 represent 1-2 day periods of a 10% drop in the Index.  Rapid declines of the Index by 7% in one day triggered temporary halts of trading (black circles) known as “circuit breakers”.

Diversification

Stocks are high risk investments with respect to potential capital gains (upside risk) and losses (downside risk).  Capital loss occurs when the company declares bankruptcy or its share prices decline.  Stock diversification, dollar cost averaging, and dividend reinvestment plans (DRIPs) are effective strategies for managing the common risks of stocks.  Monthly purchases of a Stock-index Fund accomplish these strategies.  Chart 2 illustrates the potential capital gains from investing in a Stock-index Fund that duplicates a broad market index such as the S&P 500.

Chart 2.  Historical prices of the S&P 500 Index.

dividend reinvestment

Assuming that the Fund matches the performance of the S&P 500 Indexthe difference between holding the original investment in the Fund without further action (red graph) and augmenting the holding with reinvested shares (blue graph) illustrates the potential benefit of a dividend reinvestment plan.  In this example, the benefit became ‘significant’ after 6 years.       

Postscript

Stock investing is a time-consuming process that might not interest many people who wish to put their money in the market.  They can save time (and money) by investing in a Stock-index Fund that provides an instant portfolio of diversified stocks for long term investment.

Stocks are one of several investable asset classes.  People with short term goals should consider diversifying their portfolio with different asset classes.

Copyright © 2020 Douglas R. Knight 


2018

January 19, 2019
Once again, the SmallTrades Portfolio failed to outperform 
the Standards & Poor 500 TR Index ('benchmark'). In 2019, I
will replace five exchange-traded funds (ETFs) with a single ETF.

The SmallTrades Portfolio is actively managed within a tax-protected Roth IRA.  No cash has been added or removed from the account since the time of inception in 2007.  Figure 1 describes the portfolio and its investment strategy:

portfolio 2018 v3

Fig. 1. The holdings as of 12/31/2018.

The following strategies are used to earn capital gains:

  • The passive strategy is to collect dividends and capital gains from exchange-traded index funds (ETFs).  Each ETF is ‘passively’ managed to match the performance of a market index rather than ‘actively’ managed to outperform or underperform a market index.
  • The swing strategy is to buy the stock at a low price (‘bargain’) and sell it at a high price, however long the price-swing happens to occur.
  • The growth strategy is to purchase a reasonably priced stock and hold it until the company stops growing over several-to-many years.  The stock price should increase with the company’s profit.
  • The drip strategy is to buy a reasonably priced stock to collect dividends and reinvest them in additional shares of stock.  The beneficial effect of ‘drip’ increases as the stock survives several market cycles.

2018 Performance

Figure 2 shows the changes in value for every $1 invested in the Portfolio (solid blue line) and Benchmark (dashed blue line) after 12/31/2007.  The market value of the benchmark was consistently higher than that of the portfolio.

invested $ portfolio

Fig. 2.

 

In 2013, I replaced the Portfolio‘s mutual funds with ETFs that match the performance of 4 market sectors based on a model portfolio of global stocks, U.S. real estate investment trusts (REITs), U.S. bonds, and gold bullion.  I rebalanced the ETFs as needed and continued to actively manage a group of stocks.  Figure 3 shows annual fluctuations of the stock values (solid red line) and ETF values (dashed red line) as if $1 were invested in each group on 12/31/2013.

invested $ stocks

Fig. 3.

The benchmark (solid blue line) underperformed the stocks and outperformed the ETFs until 2018, when the benchmark surpassed both groups of investments (Fig. 3).

Why?

Several events in 2018 worked against the portfolio.

  • The U.S. stock market lost its collective annual earnings in the last quarter of 2018.  Most stocks declined in value.
  • Stop-loss trading orders triggered steep losses from 5 stocks in the portfolio.  Four were high-risk investments in small companies that failed to generate returns.  One investment was a large company with steadily declining earnings.
  • The 4-sector model portfolio predicted that the portfolio’s ETFs would collectively grow by nearly 9% every year, but instead they grew at half that rate, 4.4% annually.  The databases for the model portfolio were outdated (limited to the time period of 1997-2011) and have not been updated.

Plan

The new SmallTrades Portfolio will hold one index fund, the Schwab U.S. Large-Cap ETF (i.e., SCHX), and a group of stocks.  The SCHX is designed and tested to match the performance of the benchmark (more information in Model Portfolios, updated). The stocks will initially comprise 20% of the portfolio’s market value and they will be actively managed to outperform the SCHX.  Consequently, the portfolio’s growth should outperform the benchmark’s growth.

Copyright © 2019 Douglas R. Knight


What is a stock and how much is it worth?

August 21, 2018

A stock is an offering of part ownership in a company.  Each part, — called a share—, is worth the price that buyers are willing to pay.  

A new stock is sold for the first time in the primary market.  The primary market is a private one comprised of the company’s founders, venture capitalists, and third parties such as banks and advisors.  Venture capitalists take a big risk that the company might fail.  In return, they have considerable influence on how the company is governed and operated.  They hope to earn a generous profit from selling their shares.

The stock may be sold again in the secondary market by public auction.  The secondary market is the familiar stock market where thousands of investors, —like us—,  trade cash for stocks and other exchange-traded securities.  We also hope to earn a generous profit from selling shares. Some companies may occasionally choose to pay us a cash bonus called a dividend.

Wise buyers seek the best price for a good company.  The best price is determined by ‘valuation’ and the quality of the company is assessed by ‘fundamental analysis’.

Copyright © 2018 Douglas R. Knight


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.

Copyright © 2018 Douglas R. Knight


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