Short-termism

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.  

Assumptions:

  • 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 


Rates of return

March 20, 2015

Preview

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.

Return

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%.

CAGR

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.

IRR

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.

IRRanalysis

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.

IRRinterpretation

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

TimeDistortionOfCAGR,IRR

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.

Conclusions

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

References

  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.

The internal rate of return (IRR) of a portfolio

December 18, 2014

Summary

Since investment portfolios have multiple cash flows, their performance is typically measured by the internal rate of return (“IRR”)refs 1-3.  IRRs are widely used to plan and analyze financial projects.  The planning process called capital budgeting won’t be discussed in this article.  The purpose of this article is to describe the analytical use of IRRs in evaluating profitability.  Generally speaking, the positive IRR reflects a profit and the negative IRR reflects a loss.  Higher IRRs infer more profitable investments, but the analyst is cautioned to examine the investment’s return as well as its IRR refs 1-2!!  There are three steps to computing an IRR.

  1. calculate the present value of every cash flow
  2. find the net present value
  3. find the IRR (the IRR is a specific discount rate that sets the net present value to 0).

[Click on this link –calculate_IRR– to download an IRR calculator.]

Present value

Analysts evaluate the history of multiple cash flows by finding the time value for each cash flow.  Time value is measured by converting the future value of each cash flow to its present value.  In hindsight, the present value is the initial cash payment and all future values are subsequent cash flows.  Equation 1 shows how one present value is estimated from one future value over the time span labeled N.  The present value depends on its discount rate, R.

present value = future value/(1+R)N                    Eq. 1

Discount rate

The process of discounting an item means to reduce its price or market value.  In equation 1, the discount rate (R) is the rate at which the known future value reverts to its theoretical present value.  The practical significance of the discount rate depends on its intended use.  In financial planning it reflects the risk of an investment as influenced by interest rates, inflation, and the uncertainty of time ref 2.  In the hindsight analysis of a portfolio, the discount rate represents the rate of return for a given cash flow.

Net present value (NPV)

The sum of all present values in a portfolio is the theoretical cash balance called net present value (NPV)refs 2-3.  The positive NPV reveals a profit and the negative NPV reveals a loss.

Internal rate of return (IRR)

The IRR is a specific discount rate that sets the net present value to 0.  As such, it represents the time value of all cash flows in a portfolio.  It also reflects the rate of return of the portfolio.  The IRR is calculated by a trial-and-error process of computing net present values for different discount rates.  In the appropriate set of trial discount rates, net present values will vary from negative to zero to positive or positive to zero to negative depending on the cash flows.

Example

Suppose $1,000 was invested every 6 months and the stockbroker charged a $7 trading fee each time.  After 21 months, the total market value grew to $4,436.46.  Was the IRR 10%?

Time span in years (N) Item  Cash flow
0 Investment + trading fee -1,007
0.5 Investment + trading fee -1,007
1 Investment + trading fee -1,007
1.5 Investment + trading fee -1,007
1.75 Market value 4,436.46

 

NPV = sum of present values

= (PV at N=0) + (PV at N=0.5) + (PV at N=1) + (PV at N=1.5) + (PV at N=1.75)

= (-1,007/(1+.10)0) – (1,007/(1+.10)0.5) – (1,007/(1+.10)1) –( 1,007/(1+.10)1.5) + (4,436.46/(1+.10)1.75)

= -1,007 -960.79 -915.75 -873.51 +3,757.05

= 0

Yes, the IRR was 10%.  The NPV was $35.13 at 9% IRR, $0 at 9.985% IRR, and -$35.18 at 11% IRR.

Applications

Periodic reports.  The IRR increases when cash inflow increases, cash outflow decreases, and time is compressed ref 1.   The passage of time will decrease the IRR when all cash flows are static.  Consequently, any increase in 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.

Comparisons.   Be cautious about using the IRR to compare different investments ref 1.   For one reason, higher IRRs don’t always identify higher returns.  Two projects with different cash flows may have the same IRR, yet one project yields a higher return at the time of comparison.  For another reason, the compression of time tends to raise the IRR and promote a false sense of security.  Project A’s exceptionally high IRR for a brief time period may not be sustainable in the long run.  Project B’s lower IRR over a longer time period may be sustainable.  Be sure to examine the payouts as well as the IRRs when comparing investments ref 1!!

U.S. Tax Code

The calculation of IRR is indifferent to tax rules for reporting an investment’s cost basis.  The LIFO and FIFO rules have no effect on calculations of IRR.

Miscellaneous

IRR vs CAGR.  Both the IRR and CAGR measure an investment’s rate of return.  The CAGR measures an initial and final cash flow over one time period.  The IRR is a more flexible measure due to its capability of analyzing multiple cash flows over time ref 4.

ERR.  The IRR is sometimes called the economic rate of return (ERR)ref 3.

IRR computation.  The trial-and-error determination of IRR is applicable in all situations, but it can be simplified to a single step when all cash flows are constant ref 2.

Two IRRs.  For mathematical reasons, an investment project with delayed cash outflows may have two IRR’s of widely different values ref 1.   The practical significance of the higher IRR is uncertain.

References

  1. Baker, Samuel L. Perils of the internal rate of return.  Economics interactive tutorials, University of South Carolina.  12/5/2009.  ©2000.
  2. A.A. Groppelli and Ehsan Nikbakht. Barron’s FinanceFifth Edition.  2006, Barron’s Educational Series, New York.
  3. Grayson, Linda.  Internal Rate Of Return: An Inside Look.  © 2014, Investopedia, LLC.
  4. Fuhrmann, Ryan C.   What are the main differences between  compound annual growth rate (CAGR) and internal rate of return (IRR)?  © 2014, Investopedia, LLC.

Performance measured by GAGR

November 6, 2011

Summary

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.

Return

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.

Performance

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

References

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


Geometric mean, Arithmetic mean

November 6, 2011

Geometric mean

Geometric mean1 is a representative statistic for the set of products expressed in Eq. 3.

Geometric mean = (R1*R2*RS*RN)(1/N)

Equation 3

Equation 3 states that geometric mean is the Nth root of a product.   The Nth root is (1/N), with N being the number of terms in the product.  The product’s terms are labeled   R1 for the 1st term, R2 for the 2nd term, RS for subsequent terms, and RN for the final term of the series.

The calculation of compound annual growth rate (CAGR) is a special case of the geometric mean in which Eq. 3 takes the special form (V1/V0* V2/V1 * VN/V(N-1))(1/N).  The latter formula factors out to the (VN/V0)(1/N) term used in Eq. 1.

Arithmetic mean

Arithmetic mean1 is a statistic for the set of sums expressed in Eq. 4.

Arithmetic mean = (V1+V2+VN)/N

Equation 4

According to Eq. 4, arithmetic mean is the quotient of a sum.  The numerator is a sum of values and the denominator is the number of values in the sum.

Copyright © 2011 Douglas R Knight

References

1.  Geometric mean.  Copyright © 2011 Investopedia ULC.  http://www.investopedia.com/terms/g/geometricmean.asp#axzz1ceLXsjvF.


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