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.

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

5. Investing Answers. Alpha Definition & Example. 2016.

6. Professor Lasse H. Pederson. The capital asset pricing model (CAPM). New York University Stern School of Finance. undated.

7. MoneyChimp. Regression, Alpha, R-Squared. 2016.

8. Invest Excel. Calculate Jensen’s Alpha with Excel. undated.

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

ETF risk, investment strategy

June 20, 2012


ETFs rely on a market index for the development of an investment strategy and standard of performance.  The standard of performance is measured as an annual percentage change in index value.  The ETF investment goal is a statement of the desired performance of the fund’s portfolio relative to the performance of the market index.  Investment strategy is the fund’s stated plan for achieving its investment goal.  Most ETFs use the investment strategy of physical replication.

Physical replication (passive management)

Most ETFs are index funds that seek to match the performance of a securities market index by the process of passive management.  Passive management attempts to earn a return equal to the index performance, less fund expenses, by investing in equities or bonds.  Tracking error is used to grade how closely the fund matches its index.  The ideal tracking error is 0, but an acceptable tracking error is 0.25%/year.  One risk of index funds is the volatility of share prices in parallel with fluctuations of the market index1,2.

Physical replication involves SECURITY REPLICATION or SECURITY SAMPLING of the market index.

  • SECURITY REPLICATION is used to acquire every security listed in the market index in proportion to a weighting factor3.  The fund’s tracking error is expected to be minimal.
  • SECURITY SAMPLING does not acquire every security listed in the market index; instead, managers choose a representative sample of the market index.  Sampling is used when thousands of different securities in the index (some of which are likely illiquid) make replication impractical if not impossible.  Managers must adjust the sampling process to minimize tracking error2,3.

Derivative replication

DERIVATIVE REPLICATION (‘synthetic’ replication) involves investing in derivatives to match the performance of a commodities index.  Managers of synthetic ETFs use a proprietary model to invest in futures, asset swaps, options, and other derivatives.  Beware of synthetic ETFs (called ETNs) that purchase contracts or notes which guarantee payment of the total return of an index from a counterparty bank.  The bank supplies collateral that might be illiquid assets unrelated to the investment goal of the fund.  The documents of the synthetic ETF should disclose this strategy and the type of collateral supplied by the counterparty bank.  DERIVATIVE REPLICATION incurs the risks of counterparty default and acquisition of illiquid collateral assets1,4,5,7.

Inverse strategy

The INVERSE STRATEGY seeks the opposite performance of a market index.  For example, the successful inverse fund achieves a 10% increase in value when its market index declines by 10%, or a 10% decrease in value when the index rises by 10%4.  Purchasing an inverse index fund can yield profits from a declining market by shorting the market or buying derivatives.  Shorting the market means to sell borrowed stocks and repurchase them later at a lower price.  The profit is the price difference between sale and purchase2.

Leverage strategy

The LEVERAGE STRATEGY uses debt to increase investment performance.  Leveraged ETFs (so-called geared funds) seek to multiply the performance of a market index by a stated numerical factor.  For example, the successful 2X leveraged fund achieves a 20% increase in value when its index increases by 10%.  Likewise, the 2X leveraged-inverse fund seeks a 20% increase in value when its index decreases by 10%.  The borrowed money is used to buy financial assets in anticipation that the return from purchased assets exceeds the cost of the loan.  Otherwise, the fund will underperform the market.  The best case for a leveraged fund is when interest rates decline (i.e., lower cost of borrowing) and the stock market rises (i.e., higher return).  The worst case for a leveraged fund is when the stock market declines and interest rates rise.  The fund manager of a leveraged-inverse fund borrows money to purchase extra assets when the market is declining2.

Leveraged funds are risky and leveraged-inverse funds are extremely risky2,4,6,7 for two reasons:  1) the investor risks a magnified loss.  2) Leveraged ETFs are designed for day trading, not for longer holding periods.  The risk of a compounding loss increases with holding a leveraged ETF beyond one day.

Active management

Only a few ETFs (e.g., some bond funds) seek to outperform the market index by process of ACTIVE MANAGEMENT.  The active manager neither replicates nor samples an index to match its performance, but instead creates a unique mix of investments to satisfy the investment objective8.  The risk of ACTIVE MANAGEMENT is a failure to outperform the market based on inherent inability to accurately predict the future.  Poor predictions and selections of assets lead to under-performance of the market2.

Copyright © 2012 Douglas R. Knight


1.  BIS Working Papers No 343: Market structures and systemic risks of Exchange-traded funds. Srichander Ramaswamy, Monetary and Economic Department of the Bank for International Settlements, April, 2011.

2.  All About Index Funds, second edition.  Richard A. Ferri, CFA. McGraw Hill, 2007

3.  Investment Company Factbook, 50th Edition, A Review of Trends and Activity in the Investment Company Industry. Investment Company Institute, 2010.

4.  Potential financial stability issues arising from recent trends in Exchange-Traded Funds (ETFs).  Financial Stability Board, 12 April 2011.

5.  Exchange-traded funds: A good idea in danger of going bad. The reckless expansion of “synthetic” funds requires a few new rules, June 23rd 2011, The Economist.

6.  SEC Looks Into Effect of ETFs on Market, Scott Patterson, Wall Street Journal, September 7, 2011.

7.  Exchange-Traded Funds: Too much of a good thing. The risks created by complicating a simple idea, June 23rd 2011, The Economist .

8.  2012 Investment Company Factbook, 52nd Edition. Investment Company Institute, 2012.

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