Jensen's Alpha

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table13"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table14"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table15"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Alpha</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table16"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="0%" align="left" valign="top"><p class="word_list"> <a href="#Alpha">alpha</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> <td width="100%" align="left" valign="top"><p class="word_list"> <br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" style="border-collapse: collapse" bordercolor="#111111" width="310" align="right" id="table17"> <tr> <td align="right" width="9" height="14"></td> <td width="300" height="14"> <font size="2"> </font><font size="4"> </font></td> </tr> <tr> <td align="right" width="9"> </td> <td width="300">  </td> </tr> </table> <p class="body">In the 1999 comic movie <i>The Spy Who Shagged Me</i>, fictional spy Austin Powers has his "mojo" stolen by a Dr. Evil. With his mojo gone, Powers seems to lose his manliness, decisiveness and vitality. Don't ask what mojo is. The term is as obvious as it is imprecise. In finance, we have a similar term, alpha. <b> <a name="Alpha">Alpha</a></b> is to a portfolio manager what mojo is to Austin Powers. Unlike mojo, we can precisely define alpha. It is a <a href="../articles/RAPM.htm">risk-adjusted performance metric</a> (RAPM) obtainable through a regression of a portfolio manager's <a href="../articles/return.htm">returns</a> against those of a benchmark. But this doesn't capture the significance of alpha to the investment community or the emotions it engenders. There are lots of RAPMs out there, but none are like alpha. You never hear portfolio managers boasting about their <a href="../articles/sharpe_ratio.htm">Sharpe ratio</a> over beers.</p> <p class="body">Alpha was defined by Michael Jensen (<a href="#Jensen">1968</a>). He was investigating the emerging <a href="../articles/efficient_market_hypothesis.htm">efficient market hypothesis</a> and wanted to determine whether mutual fund managers' historical returns indicated an ability for some, at least, to outperform the overall market. A simple approach would have been to compare mutual fund annual returns to annual returns of the market portfolio, which might be represented by some broad index, such as the S&P 500. Such a comparison could be misleading because it doesn't take into account <a href="../articles/risk.htm">risk</a>. Sharpe (<a href="#Sharpe">1964</a>) had recently published his <a href="../articles/capital_asset_pricing_model.htm">capital asset pricing model</a> (CAPM), which indicates that a portfolio's <a href="../articles/mean.htm">expected</a> return will increase with its <a href="../articles/capital_asset_pricing_model.htm">systematic risk</a> (<a href="../articles/beta.htm">beta</a>) according to the formula</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table18"> <tr> <td width="336">  </td> <td width="9"> </td> </tr> </table> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table19"> <tr> <td width="100%" align="center"> <i>E</i>(<i>Z<sub>p</sub></i>) = <i>z<sub>f</sub></i> + <font face="Times New Roman">β[<i>E</i>(<i>Z<sub>m</sub></i>) – <i>z<sub>f</sub></i>]</font></td> <td width="5%" align="right">[1]</td> </tr> </table> <p class="body">That is, a portfolio's expected return <i>E</i>(<i>Z<sub>p</sub></i>) equals the risk-free rate <i>z<sub>f</sub></i> plus the portfolio's beta<font face="Times New Roman"> β</font> multiplied by the expected excess return of the market portfolio <font face="Times New Roman">[<i>E</i>(<i>Z<sub>m</sub></i>) – <i>z<sub>f</sub></i>]</font>. Consistent with this site's <a target="_blank" href="http://www.riskglossary.com/Notation.pdf"> notation system</a>, <i>Z<sub>p</sub></i> and <font face="Times New Roman"><i>Z<sub>m</sub></i></font> are capitalized because they are random variables. The risk-free rate <font face="Times New Roman"> <i>z<sub>f</sub></i></font> is lower-case because it is a known constant. Formula [1] defines the portfolio's expected return as a <a href="../articles/polynomial.htm">linear polynomial</a> of the market expected return. Its graph is a line—the <a href="../articles/capital_market_line.htm">capital market line</a>. Under the assumptions of CAPM, this is the line in risk-return space that passes through the points corresponding to the risk-free asset and the market portfolio. </p> <div align="center"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table20"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Capital Market Line under CAPM</span><br> <span class="exhibit_subheader">Exhibit 1</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex1_alpha.gif" width="300" height="200"></td> </tr> <tr> <td> <p class="exhibit_legend">Under the assumptions of CAPM, the capital market line is the line in risk-return space passing through points for the risk-free asset and the market portfolio.</td> </tr> </table> </div> <p class="body">According to CAPM, portfolios may randomly outperform or underperform the market from one year to the next. Over many years, the random good years will tend to cancel the random bad years, and the portfolio's long-run performance will fall on the capital market line (if it is optimized under CAPM) or under the capital market line (if it is not). </p> <p class="body">Jensen was interested in whether mutual fund managers add value over the long-term. Could they—through skill, privileged information or intuition—outperform the market reasonably consistently, year after year? This was not about having randomly good or bad years, but about having good years with noticeable consistently. The CAPM formula [1] didn't accommodate this possibility, so Jensen added a term to it that <a name="[2]">did</a>:</p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table22"> <tr> <td width="100%" align="center"> <i>E</i>(<i>Z<sub>p</sub></i>) = <font face="Times New Roman">α +</font> <i>z<sub>f</sub></i> + <font face="Times New Roman">β[<i>E</i>(<i>Z<sub>m</sub></i>) – <i>z<sub>f</sub></i>]</font></td> <td width="5%" align="right">[3]</td> </tr> </table> <p class="body">and so alpha, <font face="Times New Roman">α</font>, was born. This allows for a persistent positive (or negative!) contribution to a portfolio's expected return due to the manager's skill (or incompetence). Formula [2] is illustrated in Exhibit 2.</p> <div align="center"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table23"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Alpha</span><br> <span class="exhibit_subheader">Exhibit 2</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex2_alpha.gif" width="300" height="200"></td> </tr> <tr> <td> <p class="exhibit_legend">By adding alpha to CAPM, Jensen considered the possibility of a portfolio residing above the capital market line due to the skill, privileged information of intuition of the portfolio manager..</td> </tr> </table> </div> <p class="body">Note that the three terms summed on the right side of [<a href="#[2]">2</a>] are stacked up above each other in the exhibit, illustrating how their sum is the portfolio's expected return. You can see how the alpha term lifts the portfolio above the capital market line.</p> <p class="body">Jensen was not saying that some mutual fund managers do consistently outperform the market. His model simply allowed for the possibility in order that he might test whether any do. His next step was to calculate some mutual fund's alphas and see if any were positive. He gathered annual return data for the S&P 500, which he used as a proxy for the market portfolio, and 115 mutual funds. He used fund returns after fees but not including any sales loads. He had complete data for 1955-1964, but some funds had data going back as far as 1945, which he used as well. He performed a regression for each mutual fund to determine its alpha. His estimated alphas for all 115 mutual funds are summarized in Exhibit 3, which is reproduced from his paper.</p> <div align="center"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="350" cellpadding="0" id="table24"> <tr> <td> <p class="exhibit_header"><span class="exhibit_header">Jensen's Results: <br> Estimated Alphas for 115 Mutual Funds</span><br> <span class="exhibit_subheader">Exhibit 3</span></td> </tr> <tr> <td bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td> <img border="0" src="../images/ex3_alpha.gif" width="400" height="294"></td> </tr> <tr> <td> <p class="exhibit_legend">A frequency distribution of the alphas Jensen estimated for 115 mutual funds based on at least ten years of data for each. The vast majority have estimated alphas that are less than zero. The average fund's alpha was –.011, or –1.1%. Results are after fees but not including sales loads. Returns, and hence alphas, are with continuous compounding. Reproduced from Jensen (<a href="#Jensen">1968</a>).</td> </tr> </table> </div> <p class="body">The vast majority of the funds had negative estimated alphas, with the average being –1.1%. This means that, after fees, but not including sales loads, the average fund underperformed the overall market by 110 <a href="../articles/basis_point.htm">basis points</a> a year. Looking at fund returns before fees, the results were only marginally better. A majority still had negative estimated alphas, but with the average being –0.4%. Jensen <a name="return_[1]">noted</a> <sup><font size="2"> <a href="#footnote_[1]">[1]</a></font></sup></p> <p class="quote">An examination [of the data] ... reveals only 3 funds which have performance measures which are significantly positive at the 5% level. But before concluding that these funds are superior we should remember that even if all 115 of these funds had a true <font face="Times New Roman">α equal to zero, we would expect (merely because of random chance) to find 5% of them or about 5 or 6 funds ... at the 5% level.</font></p> <p class="body">Jensen's results leant strong support for the efficient market hypothesis, suggesting that no investment managers have positive alpha.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="345" align="left" id="table25"> <tr> <td width="336">  </td> <td width="9"> </td> </tr> </table> <p class="body">Today, few practitioners recall Jensen's paper, but they all know what alpha is. There are many <a href="../articles/RAPM.htm"> RAPMs</a> at investment managers' disposal, including the <a href="../articles/sharpe_ratio.htm">Sharpe ratio</a> and the <a href="../articles/sharpe_ratio.htm">Treynor ratio</a>, but none is as popular as alpha. Alpha isn't actually calculated that often. This may be partly due to the fact that it requires many years of performance data. Another reason, no doubt, is that Jensen's conclusion has been reaffirmed many times. Investment managers' empirical alphas tend to be embarrassingly negative. </p> <p class="body">Instead, alpha has become a symbol. It is a one-word moniker for investment managers' belief they can outperform the market. Alpha is out-performance, and it is the job of an active manager to produce alpha. There are investment strategies with names like "<a href="../articles/market_neutral_strategy.htm">alpha transport</a>" and books with titles like <i> <a target="_blank" href="http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http://www.amazon.com/Searching-Alpha-Exceptional-Investment-Performance/dp/0471348228?ie=UTF8&s=books&qid=1187203033&sr=1-5&tag=contingencyanaly&linkCode=ur2&camp=1789&creative=9325">Searching for Alpha</a><img src="http://www.assoc-amazon.com/e/ir?t=contingencyanaly&l=ur2&o=1" width="1" height="1" border="0" alt="" style="border:none !important; margin:0px !important;" /></i>. If active investment management were a religion, alpha would be its god.</p> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table26"> <tr> <td height="30" valign="bottom"> <p class="header2_black">Related Internal Links</td> <td align="right" valign="bottom"> </td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table29"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/arbitrage.htm">arbitrage</a> <strong style="font-weight: 400"> A transaction which generates a risk-free profit</strong>.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/beta.htm">beta</a>—a metric of the systematic risk of a portfolio.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><span class="verdana10"><strong style="font-weight: 400"><a href="../articles/capital_asset_pricing_model.htm">c</a></strong><a href="../articles/capital_asset_pricing_model.htm"><strong style="font-weight: 400">apital asset pricing model</strong></a></span><strong style="font-weight: 400"> A model for valuing financial assets based upon their systematic risk.</strong><p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/directional_strategy.htm">directional strategy</a> A trading or investment strategy that entails taking net long or short positions in a market.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/efficient_market_hypothesis.htm">efficient market hypothesis</a> A financial theory that markets are efficient in the sense that <strong style="font-weight: 400"> prices </strong>reflect all available information.<p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/event_driven_strategy.htm">event driven strategy</a> Speculative trading strategy that seeks to exploit relative mispricings between securities whose issuers are involved in mergers, divestures, restructurings or other corporate events.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/law_of_one_price.htm">law of one price</a> The notion that, if two assets have identical cash flows, they should have the same market value.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/market_neutral_strategy.htm">market neutral strategy</a> Speculative trading strategy that seeks to exploit relative mispricings between instruments while avoiding systematic risk.<p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/portfolio_theory.htm">portfolio theory</a> A body of theory relating to how investors optimize portfolio selections.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/random_walk_hypothesis.htm">random walk hypothesis</a> Financial model based on the empirical observation that stock and commodity prices behave like a random walk.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/RAPM.htm">risk-adjusted performance metric</a> Any metric of performance that balances reward against risk.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/sharpe_ratio.htm">Sharpe ratio, Treynor ratio</a> Two risk-adjusted performance metrics developed for testing the efficient market hypothesis and widely used by investment managers since.</strong></td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom"> <p class="header2_black">References</td> <td align="right" valign="bottom"> </td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table30"> <tr> <td width="100%"> <p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Jensen">Jensen</a>, Michael (1968). The performance of mutual funds in the period 1945-1964, <i>Journal of Finance</i>, 23 (2), 389-416<p class="explanatory_text"> <img border="0" src="../images/bullet_paper_glossary.gif" width="21" height="15"><a name="Sharpe">Sharpe</a>, William F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk, <i>Journal of Finance</i>, 19 (3), 425-442.</td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom"> <p class="header2_black">Footnotes</td> <td align="right" valign="bottom"> </td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table31"> <tr> <td width="100%"> <p class="explanatory_text"> <a name="footnote_[1]">[1]</a> Jensen (1968), p. 410. 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