January 26, 2022

My private SmallTrades Portfolio is a Roth Account in which I no longer make annual contributions and cash withdrawals. Calendar year 2021 marked the 14th year of active portfolio management.  Figure 1 shows the latest portions of year-end market value as 81% index-ETF, 18% U.S. stocks, and 1% money market. 

Fig 1.

Annual returns

My investment goal is to earn an annual rate of return above that of the benchmark Standard & Poors 500 Stock Index.  Unfortunately, the Portfolio usually underperforms the Benchmark.  

Fig 2.

Figure 2 presents the history of annual returns for the Portfolio (blue bars), portfolio holdings (yellow & red bars) and Benchmark (black bars).  In year 14, the Portfolio earned a 27% rate of return compared to the Benchmark’s 29% rate of return, thus underperforming the Standard & Poors 500 Index by a margin of 2%. The history of annual returns reveals devastating losses of market value in the first 6 years followed by encouraging gains in the last 8 years.  Speculation and hurried trading, augmented by the 2008 Recession in year 1, created multi-year losses.  After year 6, reduced speculation and slower trading produced multi-year gains.

14-year growth

Devastating losses of Portfolio value in the first 6 years precluded any chance of matching the cumulative growth of the Benchmark over 14 years.

Fig. 3.

Figure 3 displays the cumulative growth of annual returns as a chain of “unit” market values.  The unit market values of the portfolio (blue dots) are ratios of year-end market value to the initial market value at time 0.  After 14 years, the Portfolio’s compound annual growth rate reached 4% compared to the Benchmark’s 11%.

8-year growth 

To evaluate the cumulative growth of annual returns in the last 8 years, I reset the unit value to $1.00 at the end of year 6 and recalculated the succeeding unit values now displayed in figure 4.  

Fig. 4.

Less speculation coupled with longer holding periods enabled Stocks (red dots) to outperform the Benchmark (black dots) through year 10 and generally outperform the Portfolio (blue dots).  The Portfolio (blue dots) matched the performance of its ETF (yellow dots).  


Measurements of annual return and cumulative growth show that the SmallTrades Portfolio continues to underperform its Benchmark by a wide margin of 7% cumulative growth over 14 years.  The last 8 years produced encouraging results based on improvements in portfolio management.  To continue improving, but not wishing to leverage my investments, my choices are to invest in a growth index ETF and/or re-allocate assets to a higher portion of growth stocks in the portfolio.    

Copyright © 2021 Douglas R. Knight


June 14, 2021

After investing in mutual funds for several years, I began trying to earn an exceptionally high return of 30% in future years by trading profitable stocks in short periods of time. Could I outperform the stock market with short term trading? 

Short-termism is the habit of selling securities in the stock market after brief periods of ownership less than one year.  My short-termism is summarized in figure 1 for the past thirteen years.  According to the gold and green bar graphs, more short term than long term sales were made during the first ten years.  The gold dashed line shows a steady decline of 3-year moving averages for short term sales after the eighth year.  By the thirteenth year, the average number of short term sales fell below the average number of long-term sales (green dashed line).  The portfolio’s average number of year-end securities (blue dashed line) varied between 17 and 20 after the fifth year.   

Figure 1 legend.  The annual numbers of short-term sales (gold bars), long-term sales (green bars), and year-end holdings (blue bars) are represented by the height of the bars.  Dashed lines represent those numbers as 3-year moving averages. Each moving average is an average of the previous 3 years.

I compared my portfolio to the stock market using the “percentage return” measurement shown below in figure 2.  Heights of the bar graphs for the portfolio (blue) and stock market (black) changed according to the percentage change in market prices between the start and finish of the year.  Goal lines for “30%” and “-30%” represent exceptional gains and losses. At the end of 2008 (“year 1”) the portfolio lost 60% of its starting value compared to the market’s 37% loss.  During year 2, my successful trading in the resurgent stock market lifted the portfolio’s percentage return by 54%.  But successful trading had nothing to do with the portfolio’s 123% return in year 3.  Instead, I made a one-time transfer of additional stocks into the portfolio from another investment account.  Thereafter, with the exception of year 11, the stock market outperformed the portfolio as indicated by higher percentage returns.  Short term sales (gold bars) accounted for the portfolio returns of years 1, 2, and 5, but otherwise failed to account for the portfolio returns [nor did long term sales (green bars); unsold securities usually accounted for most of the percentage returns]. 

Figure 2 legend.  “Percentage return” of the “portfolio” and “market” represents the change in total value for each entity at the beginning (value1) and end (value2) of one year [return = 100(value2 – value1)/value1].  “Percentage return” of sales represents the net profits of “short-term sales” and “long-term sales” earned during the year as percentages of total portfolio value at the end of the year.  The investment goal of 30% return is represented by the upper black line.   

Figure 3 (below) clearly shows that my portfolio failed to achieve the goals of earning a 30% CAGR and outperforming the stock market.  The portfolio’s thirteen-year CAGR of 7.9% was 1.9 percentage points below the stock market’s 9.8% CAGR and 22.1 percentage points below the desired 30% CAGR.  In terms of cash value, a $1,000 sample of baseline investment ultimately grew to $2,690 in the portfolio (blue line) compared to $3,370 in the stock market (black line).  At the stated “30% target” growth rate (dashed line), every $1,000 invested at baseline would theoretically grow to $4,000 in merely 5 years.  The portfolio’s final surge in the last two years came from reinvesting 80% of the total value into an index fund that tracks the stock market.

Figure 3 legend.  “Multiple” is the ratio of year-end value to baseline value.  Baselines of the “Portfolio” and “Market” occurred at the close of trading in 1979 (time 0).  Year-end value of the “Market” was reported by the Standard and Poors 500 Total Return Index. Multiples below 1.0 are losses and above 1.0 are gains.

Short-termism wasn’t the only reason I failed to earn a 30% annual return, but I believe it is the main explanation; here’s why:

  1. Fluctuating prices in the market prevent accurate timing of returns. Technical analysis of stock prices doesn’t guarantee accurate timing of trades.
  2. The fundamental analysis of stocks for short term trading is costly in terms of time spent on research and subscription fees for research reports.  
  3. Frequent stock purchases can be an overwhelming task without the aid of an effective screening program and computer-assisted analyses.  A problem with screening programs is the slow turnover of attractive stocks over a period of months rather than weeks.  I began to run out of new ideas after a few months.  
  4. A portfolio with too few or too many stocks is unlikely to beat the market.  Underperformance of one or more stocks among a few securities incurs the burden of recovering losses before earning high returns.  Too many stocks likely dilute the returns.
  5. I lost the opportunity to earn higher returns by selling good stocks too soon.  Behavior analysts offer the opinion that long term buy-and-hold strategies offer a better chance of earning annual returns than short term sales strategies.  
  6. Investing in distressed companies at low prices incurs an extended period of time needed for the company to recover its financial health and desired growth of earnings.
  7. Cash is necessary to purchase new securities, but too much cash dilutes the profits from short-term trading [please see “cash penalty” in the Appendix].

Conclusion.  Bad choices and hurried trading most likely explain my subpar performance.  The worst choice was purchasing shares of LEH (the listing of Lehman Brothers Holdings Inc. in the New York Stock Exchange) in year 1 when respected analysts warned against investing in such a deeply indebted company [more explanation in the Appendix].  And, hurried trading is a risky business due to unpredictable pricings and unexpected market declines.  It’s less risky to purchase undervalued stocks of good companies and wait for however long it takes to sell those stocks at overvalued prices.  A collection of healthy stocks protects from the inevitable decline of some companies. 

Appendix: My trading behavior

30% goal.  A compounded monthly return of 2.25% should yield a 30% annual return, which produces a compound annual growth rate (“CAGR”) of 30% when repeated for a period of years.  


  • net monthly returns of 2.25%
  • Every sale is promptly replaced by a security that continues earning a monthly return of 2.25%.
  • Negligible ‘cash penalty’ [cash penalty: every dollar of uninvested cash doubles the required monthly return from the same amount of invested cash].

My strategy was to sell stocks above cost by analyzing price charts.  Frequent trading produced mixed results during the first two years (figure 2).  Several trades earned sizable short term profits, notably stocks listed as GOOG (19%) and SOHU (14%).  Other trades earned disasterous short term losses, notably LEH (-27%), SOL (-26%), and NVDA (-35%).  The 2008 Recession occurred during this time period and undoubtedly contributed to my 60% loss in year 1.  Year 1 ended with a portfolio of 4 stocks and $15 (~0%) in cash.  During year 2, successful trading in a resurgent stock market lifted the portfolio’s annual return by 54%.  The year-end portfolio held 27% cash and 73% securities which were comprised of stocks and one real estate investment trust (REIT).

[case history #1, LEH: My worst investment occurred in 2008 when I ignored the signs of a troubled company named Lehman Brothers Holdings Inc. to purchase its stock at several levels of declining share prices.  Instead of a profitable rebound, the declining stock was delisted from the market when the company filed for bankruptcy.  I felt demoralized by the end of 2008]

[case history #2, GOOG & NVDA: Had I kept these stocks for thirteen years, my final returns-on-investment would be 283% (GOOG) and 3,415% (NVDA).

In year 3, the one-time transfer of stocks from another account boosted my portfolio’s return by 123%.  My intent was to acquire additional cash from short term sales in order to buy interesting securities such as exchange traded funds (ETFs).  The ETFs offered a layer of protection against corporate bankruptcies and stock delistings.  Short term trading of securities continued during years 3-5 with the inclusion of sector ETFs and leveraged ETFs. The portfolio held 19% cash, 34% stocks, and 47% ETFs at the end of year 5.

My revised goal in year 6 was to consistently outperform the market by combining market-matching returns using ETFs with above-market returns using stocks.  I would do so with an 80% investment in ETFs and 20% investment in stocks.  Promising stocks from foreign countries were added to the portfolio and leveraged ETFs were abandoned. The remaining ETFs were distributed among four different asset classes: 30% stocks, 30% REITs, 20% investment grade bonds, and 20% gold bullion.  The occasionally rebalanced ETFs offered good protection at modest returns that underperformed the stock market.     

In year 12, I sold the diversified group of ETFs to reinvest in a single broad-market ETF that mimicked the U.S. stock market.  I no longer needed to rebalance the portfolio, with the added advantage of automatically reinvesting the fund’s dividends.  The new ETF enabled the portfolio to match the stock market’s resurgence in years 12-13 (figures 2,3).  The portfolio held 3% cash, 16% stocks, and 81% single-ETF at the end of year 13.

Copyright © 2021 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.  


  • 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”.


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.       


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 

Model Portfolios, updated

January 23, 2019

Portfolio Visualizer is a highly rated online tool for designing investments (ref. 1). I used it to backtest the model portfolios listed in the following chart:


Legend: The top row shows the trading symbols of six index funds selected to build the model portfolios in rows 2-5.  The portfolios were backtested from December 2018 to January 2010.  $1.00 was initially invested in each portfolio and allowed to grow in value to the final balances shown in the righthand column.  The performance benchmark is Standard&Poors 500 TR Index in row 6.

Four-sector models in rows 2-4 represented diversified investments in stocks (VT, VTI), real estate investment trusts (VNQ), investment grade U.S. bonds (AGG), and gold bullion (GLD).  Several observations:

  • Four-sector models outperformed the bond market as determined by comparing their balances to the $1.32 that would result from investing only in AGG.
  • Portfolio performance was affected by the percentages of the index funds. The final balance of  four-sector models increased with the total percentage of stocks (VT, VTI) and real estate (VNQ) investments. 
  • Four-sector models underperformed the benchmark.

The one-sector model in row 5 held diversified investments in U.S. stocks. SCHX is a proxy for U.S. large-cap stocks and VTI is a proxy for all U.S. stocks. Among models, only the final balance of this model surpassed that of the benchmark in row 6.


Four-sector models are ideal portfolios for making short term investments of 1-5 year time periods. The goal of four-sector models is to improve safety by reducing the downside risk of investing in one sector.

The one-sector model of diversified U.S. stocks is ideal for making long term investments of 10 or more years.


Last year’s SmallTrades Portfolio, in 2018, was a four-sector portfolio that underperformed the benchmark.  In 2019, the new SmallTrades Portfolio will hold a group of actively managed stocks plus the passively managed Schwab U.S. Large-Cap ETF (SCHX). The initial allocation will be 20% stocks and 80% SCHX.

Thesis: SCHX is designed and tested to match the performance of the benchmark. Successful management of the stocks will raise the portfolio’s total performance above that of the benchmark.


  1. Vikram Chandrasekhar, 2016.  What is the best tool to backtest a portfolio online?


The total return of a portfolio is estimated by the following formula:

RT = aRA + bRB+ cRC + dRD

For example, what is the estimated total return for the following portfolio?;  

25% VT + 25% VNQ + 25% AGG + 25% GLD

  • a, b, c, and d = 0.25.
  • RA = 7.19%, RB = 10.21%, RC = 3.13%, and RD = 1.37%.
  • RT = 0.25*7.19% + 0.25*10.21%+ 0.25*3.13% + 0.25*1.37% = 1.80% + 2.55% + 0.78% + 0.34% = 5.47%

By comparison, the Portfolio Visualizer  reported RT = 5.93% with a final balance of $1.68.  

Copyright © 2019 Douglas R. Knight 


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).


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.


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

Beta is the incline of a straight line

December 10, 2016

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


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

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

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


Beta (β)

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

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

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


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

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

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

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


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

Summary and advice

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

Copyright © 2016 Douglas R. Knight

Rates of return

March 20, 2015


The simple rate-of-return ( R ) is a measure of your investment’s profitability for any chosen time interval.  By comparison, the CAGR and IRR are rates of return that measure your investment’s profitability as if it were an orderly process with respect to time.  CAGR is the acronym for “compound annual growth rate”.  It is the constant rate at which an investment’s market value grows every year in a cumulative fashion.  IRR is the acronym for “internal rate of return”, which describes the performance of all cash flows in a financial project such as the individual investor’s program of dollar-cost-averaging or an investment club’s program of portfolio management.  IRR is an annualized rate-of-return when all time intervals are measured in years.


Any profit from your investment is called a return.  There are 2 types of return: realized and unrealized.  Realized returns are cash payments from dividends, interest, and sales.  Unrealized returns are the market values of reinvested dividends and unsold holdings.

return = market value – cost  [equation 1]

Example 1: Suppose you invested $100 and held the stock for 5 years until its market value grew to $201.  From equation 1, you determine that your return is $101.  If you sell it, it’s a realized return; otherwise, it’s an unrealized return.

Simple rate-of-return

The simple rate-of-return ( R ) is a measure of your investment’s profitability for any chosen time interval, but time is not an essential factor in the calculation (equation 2).

 R = return/cost [equation 2]

R is reported as a decimal number or a percentage.

Example 2: The cost of an investment was $100 and 5 years later the return was $101.  From equation 2, R = $101/$100 = 1.01.  Multiply the answer by 100 to find the percentage.  R = 100×1.01 = 101%.


The CAGR is a rate-of-return that measures your investment’s profitability as a growth rate.  Time is a factor in the calculation of CAGR (equation 3).

rate = (final/initial)(1/N) -1  [equation 3]

N is the number of events or time periods between

the initial and final values.

Example 3, simple R versus CAGR: The cost of an investment was $100 and 5 years later its final value was $201.  We know from example 2 that the simple R is 101%.  Using the growth rate formula from equation 3, we find that the CAGR is 15%.

Significance: CAGR is the acronym for “compound annual growth rate”.  It is the constant rate at which an investment’s market value grows every year in a cumulative fashion.

MATH: CAGR is a growth rate that describes the ‘future’ (or final) value of a single cash payment.  In contrast, the discount rate devalues a cash flow.  Both rates represent a common ratio that generates a geometric series of points aligned to a smooth curveref 1. Chart 1 illustrates the geometric series of an inflated and devalued investment.

 Chart 1.  Geometric series.

geometric series

In chart 1, the black circle represents a single investment.  The blue curve is a series of theoretical values related to the investment by a common ratio called the discount rate or the growth rate depending on the particular application.  The discount rate devalues the investment to lower values as a function of the time-period N.  The growth rate inflates the investment to higher values.  Both rates are calculated by the formula in equation 3.


Equation 3 is also used to calculate the IRR, an acronym for the “internal rate of return”.  The IRR is used to measure the profitability of investments with multiple cash flows.  It is a discount rate that balances all devalued cash flows in a financial project.

MATH: In the field of Finance, a devalued cash flow is called the present value.  The present value is found by revising equation 3 to calculate the initial value for time period N at a given discount rate.  The net present value of the project is the sum of all present values.  The IRR is the discount rate that sets the net present value to zeroref 2.  It is the best-fit discount rate found by an iterative process of trial and error.  The significance of the IRR will be discussed after working through example 4.

Example 4, IRR:  An investor paid $100 each year for 4 years to purchase and accumulate shares of a particular stock.  After 5 years the market value of all shares was $735.  Since the purchases were multi year cash flows, the IRR is a good choice for analyzing this investment.  In this example, the trial discount rate is 13.1%.  Table 1 (below) illustrates the analysis:

Table 1.  IRR analysis of the investment in example 4.


Row 1, N displays the time period in years for factor N of equation 3.  Row 2, ACTUAL is the series of investments that began with a $100 payment at time 0.  Additional $100 payments were made at the end of years 1 through 4 for a total cash outflow of $500.   The total market value of the investment was $735 at the end of the 5th year.  To determine the IRR, the present value ref 2 of every cash flow was calculated with the trial discount rate of 13.1% after rearranging equation 3 to solve for the initial value.  Row 3, DISCOUNTED is the series of present values for each cash flow in row 2.  Notice that the total present value of all cash outflows equals the discounted cash inflow of $396.65.  Therefore, the net present value is $0 and the 13.1% discount rate is the investment’s IRRRow 4, PROJECTED is the final value for each present value in row 3.  The final value is predicted by rearranging the terms of equation 3 and using the IRR’s 13.1% as a growth rate for the remaining time.  It’s no accident that the sum of final values in row 4 equals the $735 cash inflow in row 2.  Chart 2 (below) illustrates the growth curves for projected values.

Chart 2. Projected values for every cash outflow in example 4.


In chart 2, N is the time period in years.  Each black square depicts an investment of $100.  Each blue curve shows the predicted growth of the investment.  Every point on a curve is a future value and the endpoint at year 5 is the final value.  The final values are listed in row 4 of table 1.  They decreased as the years progressed because there was less time remaining for growth.

Significance:  The IRR is a rate-of-return that describes the performance of all cash flows in an investment.  The IRR is an annualized rate-of-return when all time intervals are measured in years.

Time distortion

A positive CAGR or IRR always shows a profit.  Conversely, a negative CAGR or IRR always shows a loss.  Higher CAGRs and IRRs imply more profitable investments, but beware that those with short holding periods may grossly misrepresent the long term performance of an investment.

Example 5, time distortion:  Suppose that four different $100 investments grew to $200 apiece.  From equation 1, we know that the return was $100 for every investment.  If the holding periods were 10, 5, 1, and 1/5th years, what were the annualized rates of return?

Table 2.  Annualized- and Simple Rates of return

for different holding periods


Legend.  Equation 3 is used to calculate the annualized rate-of-return when the unit of time is in years.  For this equation, the “Holding period” is the value of N and “Final/Initial” is the quotient of $200 divided by $100.  The 4th column is the annualized rate-of-return calculated by equation 3.  The 5th column is the simple rate of return calculated by equation 2.

High annualized rates are desirable, but don’t feel exuberant about an exceptionally high annualized rate-of-return.  As shown in table 2, the annualized rate-of-return might temporarily be inflated by a brief holding period.  It’s unlikely that a short term investment could sustain the 3,100%, or even 100%, annualized rate-of-return in the long run.

Significance:  The passage of time decreases an annualized rate-of-return when cash flows are static.  ANY increase in the CAGR or IRR over time indicates a profitable increase of cash inflow relative to cash outflow.  It’s always wise to verify this impression by checking the payout of the project.


In the investment world, rates of return are measurements of profitability.  Positive rates indicate profits and negative rates indicate losses.  All rates of return are sensitive to the volatility of market prices; they rise and fall with the market.  The annualized rates of CAGR and IRR are exquisitely sensitive to short time periods; don’t get exuberant about high annualized rates before checking the time period and potential payout.  In the long term, annualized rates tend to decline unless supported by dividend payments and capital gains.  An IRR that is holding steady during the passage of time is revealing an underlying growth in market value.

Copyright © 2015 Douglas R. Knight


  1. Donna Roberts, Geometric sequences and series. Copyright 1998-2012.  http://www.regentsprep.org/regents/math/algtrig/atp2/geoseq.htm
  2. A. Groppelli and Ehsan Nikbakht. Barron’s FinanceFifth Edition.  2006, Barron’s Educational Series, New York.

Performance measured by GAGR

November 6, 2011


Financial success is typically discussed in terms of return, rate of return, and performance.  The compound annual growth rate (CAGR) is a good measure of an investment’s rate of return.  An investment “outperforms” or “underperforms” a market index according to the difference in CAGRs.


Investors hope to earn a profit called the return.  The two main types of return are cash distributions and capital gains.  Cash distributions include dividends and interest.  A realized capital gain(loss) is the actual return earned from an increase(decrease) in market value between times of purchase and sale.   An unrealized capital gain(loss) is an imaginary return calculated by the increase(decrease) in market value of an unsold investment.  The total return from an investment is the sum of its cash distributions, realized capital gains, and unrealized capital gains 1Significance: It’s important to know whether a market index measures the total return of the market (e.g., S&P 500 Total Return) or the price return of the market (e.g., S&P 500) 2.

Rate of return

The rate of return is a change in value with respect to time.  Annualized return is a return, or rate of return, from any time period that’s converted to an annual value 3.  Annualization may or may not account for the effects of compounding.  Example #1:  To annualize any monthly or quarterly rate of return without concern for compounding the returns, multiply the unannualized rate of return by 12 months/year or 4 quarters/year as appropriate for the time period.  This method provides only an estimation of the annual rate of return.  Example #2:  The annual rate of return is simply the ratio of yearly return to initial investment expressed as a percentage.  Example#3:  To annualize a compounded return over several preceding years, compute the compound annual growth rate (CAGR4,5.  VIDEO: CAGR

Figure 4 illustrates the use of CAGR to describe the 5-year growth of a market Index called the S&P United States 500 Total Return 1988 (SPTR).

Fig. 4 Rate of return measured by CAGR

Each datum (blue dot) is a spot value of the Index at the end of the last trading day of the year.  Lines connecting the data form peaks and valleys on the graph to illustrate the dynamic nature of the stock market.  The dashed line represents a smoothed continuum of Index values as if the Index grew at the appropriate CAGR of 2.29%.  Significance: Net growth of value occurs when CAGR is a positive value and net loss occurs when CAGR is a negative value.


Investment performance is best determined by comparing the investment return to an impartial standard value.  The standard value is either a numerical goal or the value of a market index.  The comparison is only meaningful when the investment return and standard return are based on the same,

  • class of financial assets
  • type of return (e.g., total return, price appreciation) 2
  • units of return (e.g., percentage)
  • time interval (e.g., annual)

Outperform” means that the investment return exceeds the standard return and “underperform” means that the investment return lags the standard return.  Investment performance is often measured by comparing the CAGR of an investment portfolio to the CAGR of an appropriate market index1.  For example, Fig. 5 shows that a Fund’s portfolio underperformed its market index.

Fig. 5 Performance measured by CAGR

Figure 5 compares the SPDR S&P 500 ETF Trust’s 5-year portfolio return (NAV) to the benchmark return of the S&P United States 500 Total Return 1988 (SPTR).  The 5-year returns were measured as CAGRs and performance was measured by the difference in CAGRs.

Concluding trivia

  • CAGR is a statistic that’s calculated as the geometric mean for a series of annual percentage returns.
  • The graph of CAGR is an exponential curve defined by the formula Y = X(1+CAGR)N . Y is the final value, X is the initial value, CAGR is a decimal number, and N is the number of years.
  • The generic rate of return (R ) applies to unannualized growth rates and financial applications such as the calculation of future values.

Copyright © 2011, Douglas R. Knight


1.  Kennon, Joshua. Evaluating Investment Performance. Calculating Total Return and Compound Annual Growth Rate (CAGR) http://beginnersinvest.about.com/od/investing101/a/aa081504.htm

2.  S&P 500: Total and Inflation-Adjusted Historical Returns.  Copyright © 2009 Simple Stock Investing. http://www.simplestockinvesting.com/SP500-historical-real-total-returns.htm

3.  Annualize.  Copyright © 2011 Investopedia ULC. http://www.investopedia.com/terms/a/annualize.asp#axzz1d326sqhX

4.  Annual return.  Copyright ©2011 Investopedia ULC.  http://www.investopedia.com/terms/a/annual-return.asp#axzz1Zkpsxjsb

5.  Compound Annual Growth Rate- CAGR.  Copyright ©2011 Investopedia ULC.    http://www.investopedia.com/terms/c/cagr.asp#axzz1Zkpsxjsb

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