R-squared, the linearity of investment returns.

December 24, 2016

[updated 12/25/2016: R2 is a useful measure of indexing]

The R-squared (R2) statistic describes a pattern of plotted data with respect to a straight line. R-squared is called the coefficient of determination (ref 1,2).


The black dots in figure 1 represent investment returns that are poorly related to market returns. There is a random distribution of investment returns with respect to market returns. The blue line is an inadequate representation of the relationship simply because there is no relationship. The R2 score for this distribution is 0.03. Conversely, the black dots in figure 2 show the ‘herding’ of data around a straight line.


Figure 2’s investment returns are highly related to market returns with an R2 of 0.997.


The R2 score represents the degree of alignment of data to a best-fit line as determined by regression analysis. The lowest possible score of 0 indicates a random pattern of data with absolutely no alignment. The highest possible score of 1 represents complete alignment.

The product of R2 X 100 represents the percent of variation in investment returns that are related to market returns (ref 1,2). In other words, R2 measures the relavance of the best-fit line to a set of data. Relavance increases as the R2 score varies from 0 to 1.

The lowest score of 0 defies any financial analyst to draw a meaningful line for investment returns as they relate to market returns. In figure 1, the incline (β) and Y-intercept (⍺) of the blue line are unreliable measurements of investment performance.

The highest R2 score of 1.00 identifies a straight line of near-perfect predictions of returns. Any R2 above 0.75 identifies a straight line for making predictions of returns. Lower scores represent increasingly random events. In figure 2, the incline (β) and Y-intercept (⍺) are reliable measurements of investment performance.

R-squared is an excellent measure of index fund performance.  Websites for index mutual funds and ETFs publish R2 as a measure of alignment between fund returns and the market index.   Funds that have an R2 score of nearly 1.00 track the index very closely.


1.  Lain Pardoe, Laura Simon, and Derek Young. STAT 501, Regression Methods. 1.5- The coefficient of determination, r-squared. Pennsylvania State University, Eberly College of Science, Online courses. https://onlinecourses.science.psu.edu/stat501
2.  R-squared. 2016, Investopedia http://www.investopedia.com/terms/r/r-squared.asp?lgl=no-infinite

Lead article: Stock Index Funds

January 16, 2016

The only way an individual investor can quickly invest in hundreds of different stocks is to buy shares of a stock index fund. The tremendous advantage is an immediate ownership of a diversified portfolio in one affordable investment. It’s the surest way of earning the stock market’s returns provided the correct investment is held through a series of ‘bull’ and ‘bear’ markets. Selecting the ‘correct’ fund requires only a few hours of easy research based on the following information:

INDEX. Stock index funds are passively-managed investment funds designed to imitate a stock index. The index measures the investment performance of a hypothetical portfolio of stocks. Some indices are riskier than others by virtue of the underlying securities in the hypothetical portfolio. For example, micro-cap stocks are riskier than all stocks combined by virtue of differences in turnover, liquidity, and diversification.

FUND MANAGEMENT. The investment fund is an actual portfolio of stocks that are managed for the benefit of the fund’s shareholders. Passive management is an investment style that imitates the performance of the selected index. Active management intentionally avoids imitating the index and is a more costly endeavor.

The legal structure of an index fund regulates its style of management. A unit investment trust (UIT) is bound by a trust agreement to manage a portfolio of fixed composition. The UIT has an unmanaged portfolio because there is no allowance for adjustment of composition by the manager. The open-end investment company (OEIC) operates a managed portfolio of adjustable composition. The OEIC is bound by its investment strategy to operate either a passively or actively managed fund. OEIC managers of an index fund are bound to passive management but have leeway to supplement the fund’s income by revising, lending, or borrowing a minor portion of the portfolio. These operations may increase the risk and tax burden of investment.

PRICING. The pricing mechanism of an index fund is closely regulated. Mutual funds are OEICs that trade shares at net asset value (NAV); in other words, they are priced at the fund’s net worth-per-share. The mutual fund’s share price is not quoted until the next day because the NAV is determined after trading hours from closing prices of the underlying stocks. Mutual funds are marketed through an authorized broker and guaranteed to be priced at the NAV. Exchange-traded funds (ETFs) are OEICs or UITs that trade the fund’s shares in the stock market, which means that the share price is quoted by public auction during trading hours. ETFs are traded the same way as stocks. The intraday net asset value (iNAV) and share price are continually updated and reported by the stock market. The fund’s share price is linked to the fund’s iNAV by arbitrage. Individual investors can neither participate in arbitrage nor redeem ETFs at NAV.

FEES. Managers of investment funds are compensated by charging an annual expense ratio that diminishes the NAV. Competition has decreased the expense ratio of stock index funds to only a few basis points (1 basis point = 0.01%), but beware that the expense ratios of bond index funds and actively managed mutual funds are typically higher; read the prospectus. Mutual funds are notorious for adding special fees to trades and imposing minimal holding periods; check with the broker and read the prospectus. New, small index funds are at risk for early termination when the NAV fails to grow above an estimated fifty million dollars. The expense ratios of small funds generate insufficient compensation for the fund sponsors, so they close shop.

TAXES. OEICs and UITs are registered Investment companies (RICs) that pass all income taxes to the shareholders. The amount of tax depends on dividends and capital gains earned by the fund. Managed portfolios incur a higher tax burden due to the more frequent turnover of portfolio securities. Consequently, mutual fund shareholders pay taxes on unrealized capital gains that ETF shareholders don’t have to pay. In theory, UITs are more tax efficient than OEICs.

INVESTMENT PERFORMANCE. During the 10 year period of 2006-2015, the compound annual growth rate of Standard and Poor’s 500 Total Return Index was 7.2%. In comparison, the growth rates of an index ETF (ticker: SPY) and an index mutual fund (ticker: VFINX) were 7.1% and 7.0% respectively. The slight differences in performance were due to an expense ratio, tracking error, and pricing error of the investment funds compared to the index.

OTHER INDEX FUNDS. There are indices to measure the investment performance of bonds, commodities, precious metals, and other assets. Likewise, there are mutual funds and ETFs that track the various indices. Bond index funds are managed by OEICs and require frequent turnover of the underlying bonds. The index funds for commodities, precious metals, and other assets are structured as grantor trusts, partnerships, or debt instruments. Stock index funds are generally less expensive, taxed at lower rates, and less risky than other index funds. Leveraged ETFs are exceptionally risky investments designed for same-day trading.

CONCLUSION. A broad-market stock index fund is the correct investment for earning returns from the entire stock market or a sector of the stock market. Simply choose an established, reputable index for the particular market that interests you. Then choose an established, reputable mutual fund or ETF that imitates the index. Use screeners or reputable fund families to select appropriate funds. Verify the fund’s expense ratio, extra fees (if any), NAV, longevity, and passive management by reading the prospectus and/or research reports. XTF.com is a free and excellent rating service for screening and assessing ETFs. Cross check your research with a trusted broker.

Investment strategy of the SmallTrades ETF Portfolio

February 14, 2014

An index ETF is designed to capture the investment returns from a financial market.  The SmallTrades ETF Portfolio (“Portfolio”) uses index ETFs to invest in several financial markets.  The goal of the Portfolio is to earn returns at a faster rate than possible by investing in risk-free bonds or the broad market of U.S. stocks, thereby ensuring that the accumulation of returns outpaces the inflation of prices in the American economy.  Success is measured by the following benchmark indices:

Investment strategy

The Portfolio is a high-risk, high-return investment in ETFs that duplicate well-established market indices for global stocks, U.S. bonds, U.S. real estate investment trusts, and gold bullion.  Twenty five percent of the portfolio’s market value is allocated to each index.  The ETFs generate at least 99% of the portfolio’s value and any remaining value is stored in a money market fund.  The ETFs will be held indefinitely except when faced with the advantage of replacing one with a more suitable ETF for the same index.

Table of holdings

ETF trading symbol Market Allocation
 AGG   U.S. bonds 25%
 GLD   Gold bullion 12.5%
 SGOL     Gold bullion 12.5%
 VNQ     U.S. real estate investment trusts 25%
 VT  Global stocks 25%

Expected return

Unfortunately there is no 50-100 year history of ETF performance that enables the forecast of an expected return.  To compensate for this limitation, two models were used to test the allocation plan shown in the table of holdings.  In one model of the 15-year recovery from the 1997 Asian Financial Crisis, the allocation plan outperformed the U.S. stock market.  In the other model of the 5-year recovery from the 2008 Global Financial Crisis, the allocation plan underperformed the U.S. stock market.  Among both time periods, the lowest return of the model portfolio was 8.5%.

  • MARKETS portfolio of financial-market returns from 1997 to 2011: The global-stocks market was simulated by a mixture of 75% U.S. large capitalization stocks and 25% emerging markets stocks.  Trading and management fees were excluded from the model.  The annualized return of the portfolio was 8.5% in comparison to the 5.7% annualized return of U.S. large capitalization stocks.
  • ETF portfolio of historical prices from 2008-2013: Trading fees, but not management fees, were included in the calculations (– management fees are charged in the primary market before ETFs are listed in the stock market).  The annualized return of the portfolio was 10.9% in comparison to the 17.8% annualized return of SPY, an ETF that tracks the Standard & Poor’s 500 Total Return.

Risk management

The holding period will be at least 5 years.  Fluctuation in market prices is the main risk of investing in index ETFs.  The likelihood of incurring a loss from a declining market decreases as the length of the holding period increases (– e.g., the risk of loss from stocks and bonds declines by 50% as the length of the holding period increases from 1 to 5 years; and, the risk declines by 80% when the holding period is extended to 10 years (1)).

The Portfolio will be rebalanced as needed to maintain the allocation plan within an acceptable limit of 28% error.  The Portfolio is concentrated in 4 markets and losses may occur when one or several markets decline.  The 25% allocation plan assigns equal weightings to each financial market in order to smooth the effect of market declines.  After accounting for trading fees, the strategy of rebalancing a large allocation error is more cost-effective than using a rebalancing schedule.

The Portfolio holdings are investable, have established reputations, charge low management fees, and are safely structured.  Although there’s no guarantee that the index ETFs will sustain their historical performance, the stock market, bond market, and real estate market ETFs provide diversified investments in underlying assets.  The risk of investing in these ETFs is lower than the risk of investing in an underlying asset.  Gold bullion ETFs are non-diversified investments in the volatile gold market.  Gold bullion is theoretically susceptible to physical damage by theft or fire.  This risk is diminished by investing in two funds, GLD and SGOL, that store the bullion in separate vaults located in London and Lucerne.

The investor’s tax burden can be reduced by holding these index ETFs in a tax-deferred retirement account.

Copyright © 2013 Douglas R. Knight


1.           James B. Cloonan, A lifetime strategy for investing.  American Association of Individual Investors, Chicago, 201

#ETF Scorecard, Vanguard Total World Stock ETF (VT: nyse)

January 31, 2014

The Vanguard Total World Stock ETF (VT) is an index fund that holds stocks and REITs issued by companies in the emerging and developed markets. Stocks from the frontier markets are excluded by the Fund. The following chart shows that VT is an established fund which efficiently manages its stock portfolio with the good prospect of sustaining its operations:


The following scorecard rates VT as a wealthy, tax efficient fund with several risks: 1) market volatility. 2) potential management error. 3) fluctuating exchange rates for foreign currency.



The FTSE All-World Index is derived from the FTSE Global Equity Index Series (GEIS), which covers 98% of the world’s investable market capitalizations. The FTSE All-World Index is a market-capitalization weighted index of the large- and mid-cap stocks listed in developed and emerging markets; frontier markets are excluded. The stocks are selected and weighted to ensure that the GEIS is an ‘investable’ index.

Fund operations

VT operated a $3 billion, diversified portfolio of 800 stocks in the year 2013. Most of the stocks had large capitalizations (77% large cap, 13% mid cap, 10% small cap) and most were issued in developed markets. The regional distribution of stock issuers was greatest in North America & Europe (78%) followed by Asia (19%), Latin America (2%), and Africa (1%).

According to the ETF.com website (1), VT’s major competitors in 2013 were ACWI, ACIM, ACWV, HECO, and ONEF. Among these, VT offered the lowest annual expense ratio, 0.19%. VT, ACWI, and ACIM operated in very similar ways to match the performance of their market indices; their returns were tax-efficient. In comparison, ACWV, HECO, and ONEF sought to outperform the global equities market.

The following chart was found in the ETF.com website (1). Visual inspection of the chart gives an impression that the net asset value (NAV) of VT’s portfolio closely tracked the performance of its index (the underlying FTSE index) and the market segment’s index (MSCI All World Investable Markets + Frontier).


According to the ETF.com website (1), VT’s operations were less transparent than the competitors’ due to Vanguard’s practice of reporting holdings on a monthly rather than daily basis. Similar to other competitors, VT lent its portfolio holdings to help offset the Fund’s expenses. Institutional investors incurred a higher cost for buying creation units from VT than from other Funds.


VT is a good long-term investment.

#01. http://www.etf.com/, © ETF.com 2014

#ETFscorecard_SPY, SPDR S&P 500 ETF

July 12, 2013

SPY was the first exchange-traded index fund sold in the U.S. Stock Market.   Today’s investment goal remains the same from inception: to earn returns, before expenses, that correspond to the price and yield performance of the S&P 500 Total Return Index.  The S&P 500 index measures the intraday value of the 500 largest stocks listed in U.S. stock exchanges.  SPY’s annual expense ratio is 0.09% and the Fund is considered to be tax-efficient.   It is structured as a unit investment trust (UIT) to protect the Fund from management error.  The following profile and scorecard support a long-term investment in SPY.





Copyright © 2013 Douglas R. Knight

#ETFscorecard_VWO, Vanguard FTSE Emerging Markets ETF

June 28, 2013

VWO is an index fund that invests in stocks issued by companies in the emerging markets.  The following profile shows that VWO is an established fund that operates at a low rate of turnover.


The VWO scorecard (below) reveals a wealthy, experienced index fund that offers tax-efficient returns.  The risks to underperformance are 1) market uncertainty and volatility, 2) erroneous judgment in management of the portfolio holdings, and 3) unfavorable exchange rates for foreign currency.


Fund operations

The Fund is a registered investment company for tax purposes.  The annual expense ratio is 0.18% and the net assets are about $46 billion.  The Fund’s strategy is to invest 95% of assets in stocks of the index diversified across industrial sectors (financial 27%, energy 13%, technology 11%, basic materials 10%, etc.) and regions of emerging markets (China 19%, Taiwan 13%, Brazil 12%, India 8%, South Africa 8%, Russia, 6%, other).  Dividends are distributed quarterly.

The FTSE Transition Index currently tracks ~800 stocks from emerging markets, including South Korea.  In 2014, the “transition index” will be replaced by an FTSE Index that excludes South Korea.

RISKS:  Emerging market stocks may be more volatile and less liquid than domestic stocks.  Emerging markets are less regulated than developed markets.  Foreign countries have different regulatory mechanisms and risks of disaster.  Foreign currency may decline in value, lowering the stock values.

VWO is a good investment for diversifying your ETF portfolio.

#ETFscorecard_VNQ, Vanguard REIT ETF

June 28, 2013

VNQ is an index fund that invests in U.S. real estate properties by purchasing shares of real estate investment trusts (REITs).  The following profile shows that VNQ is an established fund that operates at a low rate of turnover.


The VNQ scorecard (below) reveals a wealthy, experienced index fund that invests in a well-developed market for REITs.  The fund may be tax inefficient when cash distributions are taxed as ordinary income.  The risks to underperformance are 1) erroneous judgment in management of the portfolio holdings, and 2) generic risks of investing in the real estate market.



Fund operations

The Fund is a registered investment company for tax purposes and issues quarterly dividends.  The annual expense ratio is 0.1% and the net assets are valued at approximately $17 billion dollars.

The benchmark Index measures the performance of 85% of the U.S. Equity REIT market (~116 REITs) and is rebalanced quarterly.  The equity REITs are diversified among residential and commercial properties.

Real estate is an illiquid asset, but equity REITs are easily traded in the stock market.  Equity REITs own and manage real estate.  They must receive 75% of income from rents and sales and they must distribute 90% of taxable income to shareholders.  Equity REITs earn income by attracting tenants, renewing leases, and financing property purchases/improvements.  REIT stocks are typically small and mid-cap sized.

The main investment risks are: 1) concentrated investment in the REITs/real estate industries, and 2) competing interest rates may attract investors away from REITs.

VNQ is a good investment for diversifying your ETF portfolio.

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