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  

Sharpe ratio (‘historic’)

December 16, 2012

The Sharpe ratio is useful for assessing risk-adjusted returns of an investment.  Higher ratios are better outcomes.

Calculation of the Sharpe ratio

Consider the historic series of returns from an investment portfolio where ri represents any single return during one interval of time.  The corresponding individual return from a benchmark security or portfolio is bi [the typical risk-free benchmark security is a U.S. Treasury bond of appropriate maturity date].  The differential return, di, reflects the difference between the investment portfolio’s return and the benchmark’s return at the same interval of time.

  • di = ribi

D is the arithmetic average of all differential returns.

  • D = sum of all di‘s divided by the number of di’s in the series

σdi is the standard deviation of all differential returns.

  • single deviation = diD at one interval of time in the series
  • single squared deviation = (diD)2
  • N is the number of time intervals in the series
  • σdi = (sum of single squared deviations/(N-1))½

The historic Sharpe ratio is the average difference between investment returns and benchmark returns relative to the variability of differences in returns.

  • Sharpe = D/σdi.

The Sharpe ratio is useful in at least two ways:

  1. A negative Sharpe ratio indicates that the benchmark yields a higher return than the investment portfolio.  For example, the benchmark Treasury bond would outperform the investment portfolio.
  2. A comparatively high Sharpe ratio indicates that the investment returns are comparatively high in relationship to investment risk.  In this analysis, D is an index of investment returns and σdi is an index of investment risk.


William F. Sharpe.  The Sharpe Ratio.  Stanford University.  Reprinted fromThe Journal of Portfolio Management, Fall 1994.  www.stanford.edu/~wfsharpe/art/sr/sr.htm

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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