2022 Annual Report

January 19, 2023

The SmallTrades investment portfolio {appended} is actively managed to seek large profits from the stock market.  The goal is to exceed the U.S. Stock Market’s returns by investing in an exchange-traded stock-index fund {ETF} and a select group of stocks.  The stock market’s returns are measured by a benchmark index named the Standard-and-Poors 500 Total Return index.  


In past years, the portfolio returns generally underperformed the benchmark returns for reasons explained in the Annual Report for 2021.  The magnitude of underperformance is graphically displayed in figure 3 of the 2021 Annual Report.  After 14 years, the portfolio’s compound annual growth rate reached 4% compared to the benchmark’s 11%.


The portfolio is now 15 years old. This is a report of the investment outcome of year 15.  Figure 1 below displays the outcome as a series of annual rate-of-returns for the benchmark index [black bars] compared to the portfolio [blue bars] and portfolio assets [red and yellow bars].  

Figure 1: Annual returns.

Figure 1 shows frequent gains above 0% during the 6th to 14th years with exception of the years 11 and 15.  In year 15, the benchmark’s annual rate of return fell -18% [black bar in figure 1].   The portfolio fell -24% [blue bar], ETF declined -38% [yellow bar], and the stocks collectively rose +40% [red bar].  The remainder of this report will describe the events and partial success of year 15.  

Remodeled portfolio

The strategy for year 15 was to reinvest in a growth-index ETF and increase the investment in stocks. 

Figure 2: Market values.

Figure 2 summarizes the remodeling process by displaying bar graphs of monthly portfolio market values.  The height of each stacked bar graph represents a monthly portfolio market value comprised of stocks [blue bar], ETF [green bar], and cash [black bar].  Changes in the bar heights for January, February, and March represent the remodeling process.  To begin the process, December 2021’s ETF [SCHX] was replaced in January by a smaller investment in the growth-index ETF [SCHG].  I also sold 3 stocks [trading symbols NKLA, NVEC, RKLB] to collect extra cash.  January’s cash was used to buy shares of stocks in February and March.  In February, I purchased 3 stocks [trading symbols SMED, META, MKSI] plus additional shares of existing stock holdings.  The effects of trading were essentially complete by the end of March.  Excluded from figure 2 are the cash dividends of various holdings and privatization of SMED during April through November.  Cash dividends were selectively invested in stock shares while SCHG’s dividends were automatically reinvested.  Over the last ten months of year 15, the percentage market values of portfolio assets fluctuated between 28-33% stocks, 70-66% SCHG, and 1% cash.  During that same time period, the allocation of principal held steady at 1% cash, 24% stocks, and 75% SCHG.  By December 31, 2022, the up-trending blue bars in figure 2 indicated that the portfolio’s remodeled group of stocks sustained its increased market value despite shrinkage of SCHG’s market value.

Analysis of Year 15

If monthly variations in market value are firmly linked to stock market fluctuations, there should be no difference between percentage market values of the portfolio assets and benchmark index.  Otherwise, the differences would diverge with the passage of time  

Figure 3: Relative returns.

Figure 3 is derived from figure 2’s monthly market values of the stocks [blue bar] and ETF [green bar] compared with historic changes of the benchmark index [gray bar].  The monthly market values in 2022 are re-plotted as percentages of the initial market values on December 31, 2021; “100%” represents the initial market values.  The remodeling process in January-March of 2022 created a remarkable divergence of percentage market values with the portfolio’s stocks rising above 100% while the ETF and benchmark dropped below 100%.  After the remodeling period, stock market dynamics further reduced the market values of the ETF and benchmark index below their initial 100% without reducing the 140% gain of the portfolio stocks.  In other words, the net effect of active stock trading in the first quarter endured year 15’s Bear market.        

Plan for 2023

A pessimistic economic forecast for 2023 might entice other market participants to retreat from the stock market.  I plan to hold fast by maintaining my current 75/25 allocation of investment principal and maintain a long position on portfolio holdings.

Copyright © 2023 Douglas R. Knight 


2022 bear market, personal impact

July 4, 2022

One of the biggest risks of investing is that of encountering the inevitable “bear market” during a market cycle. Today’s stock market descended into a ‘bear market’ on June 13th, 2022, when the market index fell by 21% below its previous ‘high’.  My SmallTrades portfolio fell by 26% over the same time period [Fig. 1].  

Fig.1 bear market

Figure 1 shows a greater loss of market value in the portfolio, compared to its index, due to losses incurred by the portfolio’s holdings.  Here’s what happened:

In January, 2022, last year’s broad market ETF [ticker SCHX] and two stocks with disappointing share prices [tickers NVEC and NKLA] were replaced by a growth sector ETF [ticker SCHG] and three established stocks [tickers FB, MKSI, and SMED].  Residual cash was added to several stocks as needed to equalize the distribution of stock investments.  Consequently, the distribution of invested principal shifted from 80% ETF-20% stocks to 70% ETF-30% stocks in three months.  Figure 2 illustrates the sequential transfer of cash from the ETF to the stocks  [Fig. 2].  

Fig. 2 reallocation

Meanwhile, a market-wide decline in prices diminished the market value of my portfolio over the entire six-month period [Fig 3].  

Fig. 3 impact

All relative values started at 1.00 on December 31st and subsequently changed due to the combined effects of portfolio revision combined with the 6-month onset of the ‘bear market’.  Parallel declines of unit values in the stock market (black line), portfolio (red line), and ETF (green line) are consistent with a general downward trend of share prices driven by pessimistic trading in the stock market.  A transient surge of my stock values (blue line) reflects infusions of cash into several stock investments during the first 2 months of portfolio revision.

Delistings might add to losses during the ‘bear market’.  Suppose two of my stocks with the worst market performance were delisted on June 1st.  Estimated losses of -5.1% stock value and -1.6% portfolio value would deepen the June 13th losses described in figure 1.  The diversification of my stock investments protects from a steeper loss caused by any delistings. Hopefully my portfolio will remain intact in the near future and recover its growth trend after the “2022 bear market“.

Copyright © 2022 Douglas R. Knight 


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

Alpha is a point on a straight line, plus more.

December 22, 2016

{update on 12/23/2016: the significance of technical and operational alpha}

Alpha (⍺) is the cherished -but overrated- measurement of superior investment. Here are several interpretations:

  • A measurement of how well an investment outperforms its market index (ref 1).
  • The observed characteristic of a mutual fund that predicts higher fund performance (ref 2).
  • A portfolio’s return that’s independent of market returns (ref 3).
  • The excess (or deficit) return compared to the market’s return. Used this way, ⍺ is called Jensen’s Alpha.

Alpha represents a unique risk of outperforming the market’s returns. It is classically calculated as the “Y intercept” of a straight line attributed to the CAPM model (see appendix). In the last century, famous investors outperformed the market either by choosing exceptional investments or by investing in exceptional market sectors. The investment could be a single security (e.g., a stock) or a portfolio of capital assets (e.g., a mutual fund) (footnote 1, refs 1, 2). Now in this century, those alledged ‘alpha’ strategies are increasingly difficult to achieve. There’s an emerging sentiment among investors to avoid wasting time and money on attempting to outperform the market, the so called “loser’s game”. The current “winner’s game” is to seek ‘beta’ (refs 1, 2, 4, 5).

‘Beta’ is the portfolio’s return generated by market returns. Therefore, beta represents the risk of earning the market’s returns. The beta statistic, β, is currently calculated and reported by financial research firms as a coefficient for the incline of a straight line attributed to the CAPM model (see appendix).

Straight line of imaginary returns

(refs 5-8)

A straight line of imaginary returns presumably offers the best possible comparison of investment returns to a market index (footnote 2). ‘Returns’ and ‘performance’ are interchangeable terms that indicate the direction and movement of prices over time. An investment’s rate of return is calculated as the percentage change in price at regular intervals of time [likewise, the market’s rate of return is a percentage change in value of the market’s index at regular intervals of time]. Any rate of return is easily converted to a risk premium by subtracting the guaranteed interest rate for a Treasury bill (“T bill”). The risk premium is an investor’s potential reward for purchasing a security other than the T bill.

The straight line is drawn on a graph that shows actual measurements of investment returns plotted against market returns. The returns may either be measured as the rate of return or the risk premium depending on the goal of analysis. In the following chart, black dots represent a series of investment returns plotted against corresponding market returns.


The blue line of imaginary returns is the best possible comparison of investment returns to market returns. The position of the line on the graph is governed by its incline (β) and intersection (⍺+ε) with the vertical axis.

⍺, the intersection

(refs 1-3, 5-8)

Alpha resides at the intersection of the theoretical line with the vertical axis for investment returns (chart). Since the vertical axis crosses the horizontal axis at 0% market returns, ⍺ is the theoretical investment return at 0% market returns. A positive value for ⍺ implies that the investment tends to outperform its market index. Likewise, ⍺ = 0 implies no inherent advantage of the investment and a negative value for ⍺ implies that the investment tends to move less than the market index.

There’s a degree of error (ε) involved in drawing the line of imaginary returns, which means that its intersection is defined by the term ⍺+ε. The ε declines when a series of returns lie close to the line. The chart shows plots for 2 different series of returns; one series of black dots and another series of white circles. Both series have an equally small ε as illustrated by the close alignment of data to each straight line. Alpha of the blue line is 0% return and ⍺ of the orange line is 5% return, both occuring when the market return is 0. The series of open-circle returns outperformed the series of black-dot returns by 5%.


(refs 1, 2, 4, 5)

Alpha measures how well an investment outperforms the market. Yesterday’s ‘technical’ ⍺, shown in the preceding chart, applied to measuring superior stock-picking skills.  Today, the technical ⍺ of stocks is not reported by the most popular financial websites.

Today’s ‘operational’ alpha is really a beta loading factor of multi-factor models (see appendix).  Operational alpha is more relevant to measuring the performance of actively managed mutual funds and investment portfolios. The investment goal of an actively managed mutual fund is to outperform its market index. Active management may be the “loser’s game” of paying excessive fees in contrast to passive management, which may be the “winner’s game” of paying minimal fees.


1. Capital assets are securities and other forms of property that potentially earn a long term capital gain(loss) for the owner.

2. The straight line has other names that precede my use of the term ‘imaginary returns’. The straight line is also called a regression line or security characteristic line (ref 6).


1. Larry E. Swedroe and Andrew L. Berkin. Is outperforming the market alpha or beta? AAII Journal, July 2015. pages 11-15.

2. Yakov Amihud and Rusian Goyenko. How to the measure the skills of your fund manager. AAII Journal, April 2015. pages 27-31.

3. Daniel McNulty. Bettering your portfolio with alpha and beta. Investopedia. http://www.investopedia.com/articles/07/alphabeta.asp#ixzz4SYJ0rN9q

4. John C. Bogle. The little book of common sense investing. John Wiley & Sons Inc., Hoboken, 2007.

5. Investing Answers. Alpha Definition & Example. 2016. http://www.investinganswers.com/financial-dictionary/stock-valuation/alpha-43

6. Professor Lasse H. Pederson. The capital asset pricing model (CAPM). New York University Stern School of Finance. undated. http://www.stern.nyu.edu/~lpederse/courses/c150002/11CAPM.pdf

7. MoneyChimp. Regression, Alpha, R-Squared. 2016. http://www.moneychimp.com/articles/risk/regression.htm

8. Invest Excel. Calculate Jensen’s Alpha with Excel. undated. http://investexcel.net/jensens-alpha-excel/

APPENDIX: models for pricing assets and managing portfolios

(refs 1-3, 5-8)

The original one-factor model was called the Capital Assets Pricing Model (CAPM). The single factor is market returns (M).  The investment returns (I) are predicted by a best-fit line with incline (βm) and intersection with the vertical axis (⍺ + ε) (equation 1).

I = ⍺ + ε+ βmM,     equation 1, CAPM

Subsequent series of three-factor and four-factor models were sequential upgrades of CAPM. Equation 2 is an example of a four-factor model for the risk premium of an investment fund (F) comprised of separate portfolios for the broad market (M), asset size (S), asset value (V), and asset momentum (U).

F = ⍺ + ε + βmM + βsS + βvV + βuU,     equation 2, four-factor model

⍺ is the excess risk premium attributable to skillful management of the Fund.
ε is the model’s error
βm, βs, βv, and βu are portfolio loading factors assigned by the Fund’s manager

The four-factor model offers a spectrum of possibilities.

  • During 1927-2014, the average annual returns of indices for the the four-factor model were 8.4% for the broad stock market, 3.4% for stock size, 5% for stock value, and 9.5% for stock momentum.  The sum of average annual returns, 26.3%, represented the alpha-threshold for superior fund returns (ref 1).
  • Passive management could be predicted by setting βm to 1.00, measuring the market index return, and setting the remaining loading factors to 0.  A market index fund would  be expected to generate a risk premium that matches the market index risk premium with an ⍺ of 0 and slight ε for tracking error.
  • Active management involves designing loading factors and portfolio assets to outperform the fund’s predicted returns.

Copyright © 2014 Douglas R. Knight


April 21, 2013

(updated 7/31/2013)

The Small Trades Portfolio Designer is used to test model portfolios that hold 1-9 sectors of financial market returns plus a cash supply of U.S. dollars.  The program is pre-loaded with monthly returns computed from broad market indices during the 15 year period of 1997 to 2011.  You create the model portfolio by entering an allocation plan, investment amount, and rebalancing strategy.  The results are displayed in tables and charts on the same worksheet. You have the option of assessing the impact of trading costs and investment fund fees on portfolio returns.  The program can be downloaded for free by clicking here: SmallTradesPortfolioDESIGNER.

Allocation plan

At the top of the worksheet, each class of securities is labeled according to a unique combination of market region, market sector, and asset class.


Consider, for example, funding a portfolio that is 54% invested in large-cap U.S. stocks, 36% invested in U.S. bonds and 10% stored in cash.   For every $100 invested, $54 are allocated to U.S. large-cap stocks, $36 to U.S. bonds, and $10 to a cash reserve.  The allocation plan consists of weighting factors 0.54/0.36/0.10 [the article designing a buy-and-hold portfolio offers advice on creating allocation plans relevant to your investment goal].  The following entries are made next to the appropriate labels:


Rebalancing strategy

designer3Suppose the portfolio is funded with $10,000 [comment: lower payments might be less efficient investments when factoring in the costs of trading fees and expense ratios].  Two methods of rebalancing the portfolio are scheduled (e.g., every year) and signaled.  Suppose you wish to test the signaled method by choosing “no schedule” from the pull down list of the “Rebalance schedule” cell and “signal 3” from the pull down list of the “Rebalance signal” cell.   “Signal 3” is a command to rebalance the portfolio when market forces unbalance the portfolio to an unacceptable degree of error.  The result is an intermittent series of rebalancing episodes that modify the historical returns.  “Signal 1” and “signal 2” evoke a different number of rebalancing episodes by modifying the boundary for unacceptable allocation error.  It’s an empirical process for finding the best result.

Investment costs


The “risk-free bond rate” is used to calculate the Sharpe ratio.  I recommend a rate that estimates the risk free return for the holding period of the test portfolio (e.g., a 10 year Treasury Note at inception of the portfolio).  The default bond rate is 2.98% for the 15 year period of this program.  Trading fees and annual expense ratios always reduce investment returns, sometimes by a considerable amount.  Assess these by entering the typical trading fee charged by your broker and an estimated annual expense ratio derived from investment funds and advisor’s fees.  Or consider testing the default costs of $10 for trading and 1% for an annual expense ratio.  These entries are left blank for this tutorial.


The historical returns are summarized by statistics and charts for the  “Unbalanced” (“buy-and-hold”) and “Rebalanced” portfolio. The outcomes of the “Unbalanced” and “Rebalanced” portfolio would be identical without a rebalancing strategy [furthermore, a portfolio of one asset cannot be rebalanced].  In the following table, “CAGR” is the annualized growth rate of the portfolio’s accumulated returns.  “Sharpe ratio” is the average annual investment return adjusted for market fluctuations.   A negative Sharpe ratio implies that risk-free U.S. government bonds are better investments.  Higher values of CAGR and Sharpe are preferred.  The “final value” is the portfolio’s market value at the end of the investment period.


Chart 1 shows the returns based on test conditions.  The market fluctuations ultimately reach the final values shown in the table.  An effective rebalancing strategy creates a gap between both curves.


Rebalanced portfolio

Rebalancing may not improve the investment performance of a portfolio.  However in this example, the signaled rebalancing strategy outperformed the unbalanced portfolio (CAGR 6.59% is better than CAGR 5.65%).  Not shown is that scheduled rebalancing “every 3 years” also outperformed the unbalanced portfolio (CAGR 6.38% vs CAGR 5.65%).  In this tutorial, the result of selecting “signal 3” for the signaled strategy generated a 37.7% boundary error labeled as the “rebalance signal” in the program.


“Signal 3” also triggered 4 rebalance episodes over 15 years (chart 2) when there were no trading costs at inception or rebalance.


Warning messages

The next chart uses red arrows to show the location of warning messages.  These disappear when satisfactory entries are made in the program.  Be aware that the “asset allocations” must total 100% or else the blue-lettered message “Allocations are incomplete” reminds you to check the entries.



The investable securities of the program’s market sectors are index funds, stocks, bonds, real estate investment trusts (REITs), and commodities futures. Index funds are particularly good substitutions for market sectors of the model portfolio.

Test other model portfolios.  The 60/40 Stock-Bond Portfolio, exclusive of a 10% cash holding, is a favorite of many investors.  The 60/40 unbalanced portfolio’s 6.07% “CAGR” and 0.29 “Sharpe ratio” provides a standard for comparison with other allocation plans.  Try creating higher returns by experimenting with different allocations.  Consult the article designing a buy-and-hold portfolio for advice on creating allocation plans relevant to your investment goal.

Apply the rebalancing strategy.  Either the scheduled or signaled strategy can be used to rebalance a portfolio of index ETFs that match the allocation plan of a model portfolio.  The scheduled strategy is straight forward.  Simply rebalance the ETFs according the best schedule determined by this program.  The signaled strategy is not straight forward.  It requires transcribing  data from this program to the Small Trades Portfolio REBALANCER program in the following way:

1. Enter the “Rebalance signal” from the Results of this Designer program into step 1 of the Rebalancer program.  In this example, the correct entry would be 37.7%.

2. Result 1 of the Rebalancer program will display a “Rebalance” message when any of the portfolio’s ETFs satisfies criteria for correction.


A leap of faith is needed to apply the model portfolio to your investment goals.  This program is based on recent 15-year returns and your best bet is to assume that the next 15 years will provide a different investment performance due to market uncertainty.  Even so, I don’t know any investor who completely ignores history.

This program tests strategies for rebalancing a model portfolio.  I know of no other program that provides such information!

The potential impact of trading fees and fund expense ratios is considerable when many portfolio holdings are rebalanced frequently and the expense ratios are high [that’s why respected authors recommend minimizing costs by seeking high-quality, no-fee, no-load investments].  A good rebalancing strategy should augment the expected return of the unbalanced portfolio.

You can download this program free of charge by clicking on SmallTradesPortfolioDESIGNER.  If the program inspires your investing for the betterment of self and society, consider giving a tax-deductible contribution to your favorite charity or my favorite charity.

Copyright © 2013 Douglas R. Knight  


March 24, 2012

[Revised on 9/4/2013 by addition of a sample problem]

The Small Trades Portfolio Rebalancer is a clone of existing calculators (1-3)  that is uniquely designed to rebalance a portfolio of 10 or fewer holdings.  DOWNLOAD the Excel file program by clicking on rebalance_calculator, then work from left to right, top to bottom of the spreadsheet.  A sample problem is provided in the appendix.


DATA ENTRY.  Two worksheets offer the option of entering every holding in the portfolio (i.e., REBALANCER price) or groupings of holdings(i.e., REBALANCER volume).  A grouping of holdings is made by entering the sums of asset values and asset volumes.  Both worksheets require 6 steps of data-entry.

  • Step 1 is optional, for use in timing the rebalance based on the “rebalance signal” published in the Portfolio Designer program.
  • Step 2 is a listing of the portfolio’s holdings.  Each holding is a unique investment referred to as an “asset”.
  • Step 3’s allocations were assigned at the inception of the portfolio.
  • Step 4’s Value and step 5’s Price (REBALANCER price worksheet) are current market data. “Today” refers to the day that the portfolio is rebalanced.
  • Step 5’s Volume (REBALANCER volume worksheet) is the number of asset units currently held in the portfolio.  This is typically the number of shares of stock, investment fund, or other exchange traded security.
  • Step 6 requires the desired portfolio value.  It can be: 1) the current value when the only goal is to rebalance assets, 2) a higher value when more money is being invested, or 3) a lower value when money is being withdrawn.
  • Both worksheets have 4 error messages that appear in yellow script when Steps 3-6 are incomplete.

RESULTS.  Both worksheets offer 5 results.

  1. The current portfolio value is the sum of all values entered in Step 4.  Note: If the current portfolio meets the calculator’s criteria for a rebalance, based on data from Step 1, a “Rebalance” notification will appear underneath the portfolio value.
  2. The percent error reflects how much each of the holdings deviated from their assigned allocation.
  3. The rebalanced asset value is a realignment of the holdings to their assigned allocation.
  4. The recommended action is “buy” or “sell”.  Caution: It’s impractical to take the recommended action on small volumes of assets when trading fees must be paid.
  5. The volume is the number of asset units to buy or sell when rebalancing the portfolio.

RECOMMENDATIONS.  You can download this program free of charge by clicking on this link, rebalance_calculator.  If the program inspires your investing to support the betterment of society, consider making a tax-deductible contribution to your favorite charity or my favorite charityA sample problem is appended to the references of this post.

Other calculators

There are rebalancing calculators available on the internet at no charge to the reader.  Here are a few of them:

Copyright © 2013 Douglas R. Knight


1.  Provident Planning, Personal Finance for Life in the Kingdom.  Free portfolio rebalancing calculator, by Paul Williams.  December 29, 2009.  ©2009-2011 Provident Planning Inc.  http://www.providentplan.com/1044/free-portfolio-rebalancing-calculator/

2.  Invest-it-Yourself.com.  Portfolio rebalancing.  Copyright ©2010 Invest-it-Yourself.com.  http://invest-it-yourself.com/rebalancing.php#

3.  Portfolio rebalance calculator, by Bob Beeman. May 25, 2007. ©2006 Bob Beeman.  www . bee-man.us  http://www.bee-man.us/finance/portfolio_rebalance.htm

4.  Invest-it-Yourself.com.  Portfolio rebalancing.  Copyright ©2010 Invest-it-Yourself.com.  http://invest-it-yourself.com/rebalancing.php#

Appendix: Sample problem

Suppose your allocation plan calls for 10% cash and 90% securities, with the securities split 40/60 between a bond fund and stock fund.  The first chart illustrates a hypothetical portfolio.  The rebalance signal was obtained by testing a set of historical data in the Portfolio_Designer program [no rebalance signal is required for rebalancing the portfolio according to schedule].  Notice that the bond and stock funds are equally valued at $3,333.33 rather than the planned 40/60 split, clearly indicating that the portfolio is unbalanced.


In the second chart (below), Result 1 shows that today’s unbalanced portfolio value is $10,000.  The program automatically recommends a rebalance of the portfolio based on the Rebalance signal in step 1.   In step 6 the choices for a desired value  are between retaining today’s portfolio value, increasing the portfolio value by adding cash, or decreasing the portfolio value by removing cash.


In the third chart (below), Result 2 illustrates the extent to which each holding deviates from the allocation plan [notice that the absolute values for 2 of 3 errors in result 2, –233.3% & -38.3%–, exceeded the Rebalance signal in step 1, –37%–, which prompted the recommendation to rebalance the portfolio in result 1].  Result 3 shows that if you choose to rebalance the portfolio at $10,000 with a 10% weighting for cash, the rebalanced portfolio will hold $1,000 of cash.  Likewise, the bond fund’s 36% weighting would rebalance the bond fund to $3,600 and the stock fund’s 54% weighting would rebalance the stock fund to $5,400.  Results 4 and 5 show the trading needed to rebalance the portfolio.


Rebalancing an investment portfolio

March 23, 2012

[Updates:  10/21/2014, the inclusion of 2 useful references at the conclusion of this article.  9/4/2013, a new discussion of the rebalancing process and transfer of a sample problem to the #SmallTradesRebalancer ]


Your investment portfolio is balanced when its assets are funded according to your allocation plan.  Suppose the plan assigns weightings of 0.6 (i.e., 60%) to stocks and 0.4 (i.e. 40%) to bonds.  The portfolio is balanced when for every $100 of portfolio value there is $60 of stock value and $40 of bond value.  Otherwise, the portfolio is out of balance and may need to be re-balanced.  There’s no universally accepted time or signal for rebalancing the portfolio.

There are a number of reasons why portfolios drift out of balance.  The commonest is daily fluctuation of market prices during trading hours.  Another is the purchase or sale of assets.  For whatever reason, an unchecked drift could defeat your plan for seeking risk-adjusted returns from a balanced portfolio.  The purpose of this article is to explain the rebalancing process.

Allocation plan

Investment portfolios hold one or more financial assets that are traded in financial markets.  At the time of inception, the market value of the portfolio (P) is the total cash payment for all assets.  The original portions of assets are called the portfolio’s allocation plan.

Consider the model portfolio that holds 2 assets.  The market value of the first asset (A) plus the market value of the second asset (B) are equal to P.  The portfolio equation is written P = A + B.  Furthermore, the value of A is denoted aP; and likewise for the value of B (bP).  Written as an equation, P = aP + bP.  Constants a and b are called the assigned weightings; they are the portfolio’s allocation plan.   The allocation plan is written 1 = a + b to emphasize that the sum of assigned weightings is always 1.0.

Portfolio drift

Market forces typically cause the values of portfolio assets to shrink or grow with the passage of time.  In other words, the effect of time revises the original portions of portfolio assets.  The effect of time on the portfolio equation is written PT = AT + BT.  The effect of time on the portions of portfolio assets is written (AT/PT) and (BT/PT).  The observed weightings (aT & bT) represent portions of the portfolio; (aT=AT/PT) and (bT=BT/PT).  The difference between observed and assigned weightings is called portfolio drift.  The portfolio drift equation is written 0 = (aT-a) + (bT-b).

Allocation error

An asset’s allocation error is determined by calculating the quotient of its drift and assigned weighting.  The allocation errors for both assets are written (aT-a)/a and (bT-b)/b; they represent the percentage difference from the allocation plan.

Rebalancing signal

The portfolio is rebalanced according to a schedule or signal.  One way of generating a signal is to compare allocation errors to an assigned boundary [for example, the #SmallTradesDesigner].  The best chance for assigning an effective boundary is to perform trial-and-error tests on a historical portfolio of financial assets.  In preparation for testing, the historical allocation errors are converted to absolute values and searched for the maximum allocation error (see example in next paragraph).  Then the test boundary is arbitrarily set between 0% and 100% of the maximum error.  The test portfolio is rebalanced when any allocation error exceeds the test boundary.  In order for a rebalancing signal to work on future returns, all allocation errors must be converted to absolute values.

Example: Suppose the test portfolio’s greatest allocation error is -82%, which converts to an absolute value of 82%.  Also suppose that the test boundary is arbitrarily set to 50% of the greatest allocation error.  Then 50% of 82% is 41%.  In this way, the test boundary represents a rebalancing signal of 41%.  Any allocation error with an absolute value exceeding 41% will incite a test rebalance of the historical portfolio.

The best rebalancing signal produces the highest total return of the historical portfolio.  Lower rebalancing signals increase the number of rebalancing episodes and higher signals decrease the number of episodes.  Consider what happens when testing the extreme rebalancing signals of 0% and 100%.  The 0% signal would allow every allocation error to trigger a rebalance and the trading costs would be prohibitive.  The 100% signal is restrictive by preventing all chances for a rebalance.

Correcting the cash-equivalent drift

The goal of rebalancing a portfolio is to restore drifting assets to their allocation plan.  The rebalancing process involves calculating a cash-equivalent drift for every asset.  Asset AT’s cash-equivalent drift is the product of (aT-a) and PT.  Asset A is rebalanced by purchasing additional units (‘shares’) when the cash-equivalent drift is a negative value and selling surplus units when the cash-equivalent drift is a positive value.  The number of units traded is equal to the cash-equivalent drift divided by unit price.  Asset A’s cash-equivalent drift is corrected when (aT-a) is restored to 0.  Asset BT is rebalanced in the same way.


An investment portfolio is always balanced at inception when all assets are simultaneously purchased according to an allocation plan.  Market forces inevitably cause the assets to drift from the allocation plan in which case the rebalancing process restores the allocation plan.  The rebalancing process requires an allocation plan, rebalancing strategy, and correction of the cash-equivalent drift.

Two financial assets were used to illustrate the basic principles of rebalancing an investment portfolio.  The same process applies to portfolios with a greater number of assets but not to the portfolio with one asset.  The portfolio with one asset can’t be rebalanced.  Market forces may change the single asset’s market value, but can never change its assigned weighting of 100%.

October, 2014, update:  I recently discovered (for the first time) two very excellent papers that explain better than I the concepts of allocating and rebalancing investment portfolios. Please read them in order to gain a better insight into portfolio management.

Copyright © 2013 Douglas R. Knight

Book review: What Works on Wall Street, by James P O’Shaughnessy.

December 29, 2011

(9/29/2013 Update:  The American Association of Individual Investors created test portfolios of the Cornerstone Growth and Value strategy described in this book and of several best strategies from O’Shaughnessy’s newest book on formula investing (entitled Predicting the Markets of Tomorrow: A Contrarian Investment Strategy for the Next Twenty Years).  The test-portfolio returns are published free of charge in the AAII.com stock screens web site.)


Author James O’Shaughnessy tested a variety of strategies for investing in stocks with the use of numerical models.  His winning strategies outperformed both the broad U.S. stock market and Standard & Poor’s 500 Stock Index by wide margins.


Mr. O’Shaughnessy cited publications from the scientific and financial literature to support the policy of investment-by-formulation rather than investment-by-intuition.  Formulation involves the application of stock data to a quantitative model and intuition depends on human judgment.  He formulated numerous single-factor and multifactor models of investment, then back-tested the models by analyzing historical returns over 40- or 52-year time periods.  The benchmark of performance was one of several stock universes that the author obtained from Compustat’s large database.   The universes were categorized according to levels of market capitalization among stocks.  The risks and returns of his test portfolios were compared to the appropriate universe.

Winning strategies


The PERFORMANCE TABLE presents a selection of the author’s investment strategies that yielded exceptional returns.  Column headings are the labels of 7 investment strategies that were tested over 40 years (white columns) and 52 years (blue columns), both periods ending on 12/31/2003.  Notice that the cornerstone and S&P500 strategies were tested in both periods.  Row headings are the labels of 4 statistics commonly used to describe the risk-return performance of investment portfolios.  Cells contain numerical spreads.  Each spread is the difference in statistical results between an investment strategy and the All Stocks universe (described in the Appendix).  For example, a spread of 0 would mean that the outcomes of the strategy and universe are identical.

The spreads in the PERFORMANCE TABLE provide a comparison of exceptional strategies to the All Stocks UniverseCAGR spread: Compound annual growth rate (CAGR or geometric mean) is a statistic for the annualized growth rate of the portfolio’s market value.  Positive spreads show the desired result, namely that the strategy outperformed the universe.  All strategies outperformed the universe except the S&P500, which performed worse than the universe.  Std Deviation spread: The standard (Std) deviation is used to evaluate an investment’s risk, which is the chance that an investment unexpectedly increases or decreases in value.   A larger standard deviation implies a greater scatter of portfolio values over the time period of analysis.  In the performance table, a positive std deviation spread infers that investing according to strategy is riskier than investing in a representative sample of the universe.  All strategies except the S&P500 were riskier than the universe.   Downside risk spread:  Downside risk is the chance that the investment’s market value will decline.  In the performance table, the desired result is a negative downside risk spread.  The S&P500 and mending values(tri-ratio) had lower downside risks than the universe.  Investors who are risk averse might consider using these strategies.  Sharpe Ratio spread: Sharpe ratio is a statistic that relates investment return (numerator) to investment risk (denominator).  In the performance table, the desired result is a positive Sharpe ratio spread.  All strategies except the S&P500 outperformed the universe.

Here’s a description of exceptional strategies listed in the performance table:

  • Cornerstone improved (book table 20-7), a strategy tested over 40 years while the portfolio is rebalanced at monthly intervals to account for stocks with a monthly depreciation of price.   The strategy selected 50 stocks from the All Stocks Universe (described in the Appendix of this article) with the best 1-year price appreciation among stocks with market capitalizations exceeding the deflated $200 million value, having a Price-to-Sales ratio (P/S) below 1.5, showing 3- and 6-month price appreciations above average, and showing a 12-month increase in earnings-per-share (EPS).  The selected stocks were equally weighted.  [notes: Multifactor strategies might reduce risk and increase return.  Betting on price momentum supports the theory that stock prices have “memory” and opposes the claim that past price performance cannot predict future price performance.]    
  • Mending small value (book table 18-3), a strategy tested over 40 years while the portfolio was rebalanced at monthly intervals.   The strategy selected 50 stocks from the All Stocks Universe with the best 3-, 6-, & 12-month price appreciations coupled with a low P/S from the sub-universe of small stocks.  The small stocks had market capitalizations above the inflation-adjusted value of $185 million USD and below the database average.  The selected stocks were equally weighted.
  • Cornerstone, a strategy tested over 52 years (book table 20-1) and 40 years (book table 20-7) while the portfolio was rebalanced at yearly or monthly intervals.  The strategy selected 50 stocks from the All Stocks Universe with the best 1-year price appreciation among stocks with market-capitalizations above $200 million USD, P/S ratios below 1, and 12-month increase in EPS.  The selected stocks were equally weighted.  [note: A side-benefit of annual rebalancing is the lower tax rate on annual capital gains compared to monthly capital gains.]
  • S&P500 (book tables 4-1, 17-2), an index of 500 U.S. stocks with the largest market capitalizations exclusive of foreign stocks traded in U.S. stock exchanges.  The test portfolio was weighted according to the market capitalization of the stocks.
  • Mending value(tri-ratio) (book table 16-4), a strategy tested over 52 years while the portfolio was rebalanced at yearly intervals.  The strategy selected 50 stocks from the All Stocks Universe with the best 1-year price appreciation among stocks pre-screened for desired ranges of low price-to-earnings ratio (P/E), low price-to-book ratio (P/B), and low P/S.  The selected stocks were equally weighted.  [notes: The author found that investing in bargain, single-value factors (i.e., low P/E, low P/B, low P/S, or low P/C) provided superior returns among several universes (i.e., all stocks, large stocks, small stocks, market leaders) whether using monthly or annually rebalanced test portfolios.  The disadvantage of using single-value factors was volatility, which makes it difficult for “jittery investors” to sustain the strategy in real time with real money.  “Jittery investors” tend to prefer index funds.]
  • Mending value(P/B) (book table 16-1), a strategy tested over 52 years while the portfolio was rebalanced at yearly intervals.  The strategy selected 50 stocks from the All Stocks Universe with the best 1-year price appreciation among stocks screened for P/B below 1.  The selected stocks were equally weighted.
  • Mending value(P/S) (book table 16-2), a strategy tested over 52 years while the portfolio was rebalanced at yearly intervals.  The strategy selected 50 stocks from the All Stocks Universe with the best 1-year price appreciation among stocks screened for P/S below 1.  The selected stocks were equally weighted.

Summary of the author’s strategies

The data published by author James O’Shaughnessy are re-plotted in the following chart to show that the Sharpe ratio is a predictor of long-term return.  The Sharpe ratio is the difference between a portfolio’s rate of return and that of a risk-free investment, such as the 10-year U.S. Treasury bond, divided by the standard deviation of the portfolio’s return.  The result is an expression of the portfolio’s risk-adjusted return, in which a high ratio is the desired value.

Chart.  Outcomes of the back-tests.

The chart’s X axis, labeled relative Sharpe Ratio, displays values for the quotient of a test portfolio’s Sharpe ratio divided by the Sharpe ratio of the benchmark universe.  X >1 is the domain for portfolios with Sharpe ratios exceeding (better than) the universe.  The Y axis, labeled relative Return, displays values for the quotient of the final market value of the test portfolio divided by the final market value of the universe.  Y >1 is the range for test portfolios with higher (better) investment outcomes than the universe.   The dashed line in the chart represents the best fit of all data to the exponential equation Y = aebX.  A regression analysis provided the values of a = 0.0144 and b = 4.309 for the equation, and R2 = 80.2% for the ‘predictability’ of the equation.  The data-point markers are black triangles for all back-tested portfolios except blue dots for the winning portfolios and yellow squares for the S&P500 Index.  The winning portfolios and S&P500 Index were discussed in the preceding table of this article


This is a book about picking stocks that yield high returns.  It was written to provide useful information for household and institutional investors.  Due to the book’s vast number of statistics, the more appropriate audience is the institutional investor who manages stock portfolios for clients.  The author’s winning strategies are based on historical data reviewed over 40-52 year time periods.  Readers should be cautioned that applying the winning strategies to 5-10 year time periods might not achieve the same fantastic results.

What Works on Wall Street, A Guide to the Best Performing Investment Strategies of All Time.  Third Edition.  James P. O’Shaughnessy.  McGraw-Hill, New York, 2005.


All Stocks Universe table

Legend:  The All Stocks Universe (book tables 16-2, 17-2) was comprised of stocks in the Standard & Poor’s Computstat database with market capitalizations above $185 million USD.  Smaller market-capitalized stocks were excluded due to the high risk of illiquidity.  Compustat is the largest database for the U.S. Stock market

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