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.


February 3, 2015

The ‘challenging’ investment goal of the SmallTrades Portfolio (“portfolio”) is to outperform the Standard and Poors 500 Total Return Index (“benchmark”) on an annual basis.  My investment strategy is to rebalance a diversified set of index ETFs that generate 80% of the portfolio market value and to speculate in stocks that create 20% of the portfolio market value.  The year-end portfolio holdings are shown in the following table:

Table 2 2013

In 2013 the portfolio market value increased at the annual rate of 2.1% while the benchmark surged upward at the annual rate of 32.4%.   The following chart shows that since 2007, the portfolio has not recovered from the impact of the 2008 Recession and continues to underperform the benchmark.  The portfolio underperformed its benchmark in 2013 because every exchange traded fund (“ETF”) lost value.

Fig 6 2013

The data are year-end market values relative to the market value on 12/31/2007.

I am optimistic that the index ETFs will gain value over the long term.  The speculative trading of stocks might be improved by implementing better stock screens for undervalued securities.

Douglas R. Knight

Conditional orders for stock trades

September 6, 2014

[The basic concept of ‘price diligence’ was inserted on 3/20/2015.  The MockTrades planner was updated on 12/11/2014]


Individual investors typically use a market order to buy and sell stocks, ETFs, and REITs.   Market orders activate the trade immediately at the next available price; this controls the time rather than the price.  You can exert more control over trading conditions by placing a conditional (a.k.a. advanced, automated) order.  Conditional orders are considered a form of ‘price diligence’ because they specify the share’s price for activating the trade at any time during approximately 60-90 days, depending on your brokerage firm’s expiration date.  The conditional order controls the price rather than the time of the trade.  After reading this article, you may wish to download SmallTrade’s MockTrades4 for free; it’s uniquely designed to account for market volatility when planning a conditional order.

The trading arena; a simplified description

Stocks and other securities are traded for cash by an auction process in the stock market.  The participants are investors who make offers, brokers who generate orders, and traders who finalize the orders.  Individual investors submit offers to brokers through a computer terminal or by direct communication.  Use of the computer terminal gives the illusion of directly participating in the auction process, but there are several intermediary steps occurring at the speed of light along with a few brief delays.  The first delay is at the brokerage firm where all offers are filtered, recorded, and transmitted to traders in the stock market.  At any moment in the stock market, there are millions of orders to buy and sell securities.

The broker’s computer program, called a trading platform, provides investors with a market quote plus commands for placing a trading order.  The market quote reports a current purchasing price (“ASK”), sales price (“BID”), last-traded price, and latest number (a.k.a. volume, quantity) of traded shares (a.k.a. units).  Valid trading orders are announced to all market participants and filled at the next available price. The next available price is determined by an auction of the available shares for which a trader serves as the auctioneer.  The price can fluctuate between trades and is not protected from fluctuation until the order is filled.  Brokers and traders are always paid a fee for their services.  Custodians are hired by brokers to store traded securities in electronic accounts on behalf of the investors.

Trading orders

As an individual investor, you may place one of several types of trading orders in your broker’s trading platform.  The simplest, called a market order, merely specifies the number of shares to be traded.  The market order is filled immediately unless a time delay is imposed by an undersupply of available shares.

A limit order specifies the preferred price of the trade.  This order is filled when the next available price either matches the limit price or provides a better price {in other words, the limit price is YOUR minimum selling price or maximum purchasing price}.  There is only one opportunity for the limit order to be filled after it is placed in the market.  In rare instances, an unusual market event may shift the next available price out of your limit’s price range, in which case the order is cancelled without an exchange of cash for shares.  Your only recourse is to place a new limit order.  The limit order is used to protect from shifting prices.

Stop and stop-limit orders are useful for protecting against losses of investment capital.  The stop is a specific price at which your trading order will be submitted as a market order.  Your stop is stored in the exchange’s computer until it is submitted or expires at the end of a time period called the time-in-force (“TIF”).   The market order is then filled immediately at the next available price, which may shift to a better or worse price than the stop’s price.  The stop-limit is a specific price at which your trading order will be submitted to the stock exchange as a limit order.  It too has an expiration date.  The limit order protects your trading price until the order is filled, cancelled by the expiration date, or cancelled by an unusual market event. Your only recourse to a cancelled order is to place a new stop-limit order.

Trailing stop orders provide opportunities to seek a better trading price in the market.   The trailing stop is a specific point spread (1 point equals $1) by which the stop price will trail the movement of market prices toward a better price.  In general, the trailing stop can move toward a better price but won’t move toward a worse price.  The trailing price resides above or below the market’s price by the cash value of the point spread.  The trailing price adjusts upward with market prices when you offer to sell shares and downward when you offer to buy shares.  Reversal of the market price initially stops the adjustment of the trailing price and eventually triggers the submission of a market order.   The trailing stop order is stored in the brokerage firm’s computer until its expiration date or the submission of a market order.

Setting the stop

Trading platforms are designed to set stops above the ASK and below the BID of a market quote.  For example, when you offer to purchase shares with a stop or trailing stop order, a market order is submitted when the market’s ASK increases to the stop price.  Or when you offer to sell shares with stop-limit order, a limit order is submitted when the market’s BID decreases to the stop price.

The likelihood that daily fluctuations in market price will trigger any type of stop order depends on the width of the margin between the stop price and market price.  Exceptionally narrow margins may trigger an order the same day and exceptionally wide margins may cause the order to expire before being triggered.  The width of the margin depends on your trading strategy.  What is your acceptable price range for buying and selling a security?  Do you prefer using the same margin for all stops or customizing the margin based on price volatility?  The SmallTrades MockTrades4 is a free, useful worksheet (in Microsoft’s Excel™ format) for customizing the stop margin of your trading order.


Stop and stop limit orders are typically used to protect against capital losses when market prices are declining.  Trailing stop orders can help maximize a sales price or minimize a purchase price.   Examples of these applications can be found online in the educational materials of brokerage firms.  This user-friendly version, “Secure your position”, was a free download provided by the Commonwealth Securities Limited (“CommSec”), Sydney, web site.

Copyright © 2014 Douglas R. Knight


Book review: The Richest Man in Babylon, by George S. Clason

January 5, 2013

George S. Clason. The Richest Man in Babylon. Penguin Books, New York  © 1955, .., 1926.

In 1926, author George S. Clason published a famous series of pamphlets on thrift and financial success that were distributed by financial institutions to millions of readers.  His book is a collection of inspiring parables about personal finance.   It’s a book of “cures for lean purses”.  Here’s a revised summary.

Laws of building wealth

Arkad was a wealthy, generous merchant, but his friends in youth either failed to learn the laws of building wealth or didn’t observe those laws.  Arkad cautioned Bansir, the craftsman, that people who inherit wealth tend to spend it away or live the miserable life of a miser.  For those without an inheritance, acquiring wealth requires time and study.

  1. Save 10% of earnings to create an estate
  2. Put the savings to work by investing in good opportunities
  3. Seek good advice for investing
  4. Losses and gains depend on the skill and experience of the investor
  5. Avoid schemes to earn wealth quickly

People who wait are less fortunate.  Begin investing at a young age.  When Basir asked Arkad’s advice on starting late in life, Arkad repeated the same rules, emphasizing to invest savings with greatest caution not to lose any returns.

Seven cures

Ancient Babylon sat on arid desert next to the Euphrates River.  All of Babylon’s wealth was man-made in this harsh environment.  Work was not reserved for slaves, it was also the friend of freemen.  There was a time when very few rich citizens acquired most of the gold while the rest of the population lived in poverty.  The King desired that all men should know how to acquire wealth so he consulted Arkad, the richest man in Babylon.  Arkad agreed to train teachersabout the 7 cures for an empty purse:

  1. Save 10% of your income
  2. Avoid needless expenses for luxuries.  Budget for necessary expenses.
  3. Multiply your savings with compounded returns
  4. Invest wisely, where the principal is safe.  Don’t make risky investments.
  5. Own your house
  6. Insure a treasure for retirement and protect your family.
  7. Cultivate skill and wisdom.  Pay your debts and prepare a will.

Making loans

Only lend money to people with good credit.  They will use the loan wisely and repay the loan.  The safest loans are to those who can fully repay by selling property or who have an assured income.  Borrowers who have no property or income need a friend to guarantee repayment.  Hopeless debt is a pit of sorrow.  Is the borrower credit worthy?  How will he use the loan?  Does he understand the business?  Does he have a plan?

Repaying loans

Young Takard was destitute, hungry, and in debt.  One of his lenders, a wise camel trader named Dabasir, confronted Takard with an allegory:  Dabasir was once indebted to many lenders until he determined to repay with income earned by hard, honest work.  Moral: repaying a loan requires determination.

Dabasir’s plan to repay debt was to allocate earnings in this way:

  • 10% into savings account
  • 70% to support the household
  • 20% to repay lenders

Protection of wealth

Ancient Babylonians used a small army to defend against the mighty army of Assyria.  The strong walls of Babylon protected the citizens and their treasures.  Today, we can’t afford to be exposed by inadequate protection.  The impregnable walls that protect our treasures from unexpected loss are:

  • Insurance
  • Savings accounts
  • Dependable investments

Buy ‘low’ and sell ‘high’, and other AXIOMS

April 16, 2012

[updated on 11/21/2013 with additional text; on 7/30/2014 with a counterargument; updated on 10/18/2016 with additional text and revised title]

AXIOM #1: Markets are a human necessity.  corollary:  There will always be a volatile stock market.

AXIOM #2: The most basic stock-investment strategy is to buy shares at a discount price and sell them at a premium price.  In other words, buy ‘low’ and sell ‘high’.  corollary:  Buy shares that are in ‘low demand’ and sell those in ‘high demand’.  The share price will increase with demand.  This is the fundamental principle of “Contrarian” investing.

AXIOM #3:  Good companies attract investors.

AXIOM #4:  Good companies reward investors.

Capitalism 101:  Companies sell common shares of stock to raise money for their businesses.  Investors buy the stocks in order to share the business profits.  The first-generation investors are venture capitalists in the primary market who aim to resell their shares to second-generation investors in the secondary market.  Second-generation investors hope to earn dividends from the company’s profits and capital gains from stock sales in the marketplace.  Capital gains are earned by selling stocks at a higher value than the total cost of purchase.  The stock market price is governed by forces of supply-and-demand.

After the companies are financed, why should they care about the shareholders? Because common shareholders are beneficial owners who have the right to 1) vote on issues of governance, 2) seek rewards, and 3) seek damages.  Shareholders also provide a measure of crowd-protection from hostile takeover of the company.

Good companies reward investors by paying dividends and/or raising the value of tradable shares (e.g., issuing stock splits, repurchasing shares, growing the business).  These rewards are endangered by stock price fluctuations, declining business profits, corporate greed, and additional uncertainties.


The buy ‘low’ and sell ‘high’ strategy is difficult to practice in times of financial panic when investors lose interest and tend to pull out of plunging markets.  At the very moment when their prospect for future returns is highest, investors are out of the market 1,2.


  1. Buttonwood: No easy answers | The conundrum of asset allocation. Jul 19th 2014. The Economist
  2. William Bernstein, Rational Expectations: Asset Allocation for Investing Adults.   2014.

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