Public Market Equivalent

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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#PME+"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>PME+</span> </div> </a> <button aria-controls="toc-PME+-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle PME+ subsection</span> </button> <ul id="toc-PME+-sublist" class="vector-toc-list"> <li id="toc-PME+_Methodology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#PME+_Methodology"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>PME+ Methodology</span> </div> </a> <ul id="toc-PME+_Methodology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-PME+_Formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#PME+_Formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>PME+ Formula</span> </div> </a> <ul id="toc-PME+_Formula-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Modified_PME" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Modified_PME"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Modified PME</span> </div> </a> <button aria-controls="toc-Modified_PME-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Modified PME subsection</span> </button> <ul id="toc-Modified_PME-sublist" class="vector-toc-list"> <li id="toc-Formula_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formula_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Formula</span> </div> </a> <ul id="toc-Formula_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kaplan_Schoar_PME" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kaplan_Schoar_PME"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Kaplan Schoar PME</span> </div> </a> <button aria-controls="toc-Kaplan_Schoar_PME-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Kaplan Schoar PME subsection</span> </button> <ul id="toc-Kaplan_Schoar_PME-sublist" class="vector-toc-list"> <li id="toc-Formula_3" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formula_3"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Formula</span> </div> </a> <ul id="toc-Formula_3-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formula_Simplification" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formula_Simplification"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Formula Simplification</span> </div> </a> <ul id="toc-Formula_Simplification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Usage" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Usage"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Usage</span> </div> </a> <ul id="toc-Usage-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relation_between_LN-PME_and_KS-PME" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_between_LN-PME_and_KS-PME"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Relation between LN-PME and KS-PME</span> </div> </a> <ul id="toc-Relation_between_LN-PME_and_KS-PME-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Direct_Alpha" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Direct_Alpha"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Direct Alpha</span> </div> </a> <button aria-controls="toc-Direct_Alpha-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Direct Alpha subsection</span> </button> <ul id="toc-Direct_Alpha-sublist" class="vector-toc-list"> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Excess_IRR" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Excess_IRR"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Excess IRR</span> </div> </a> <button aria-controls="toc-Excess_IRR-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Excess IRR subsection</span> </button> <ul id="toc-Excess_IRR-sublist" class="vector-toc-list"> <li id="toc-Formula_4" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formula_4"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Formula</span> </div> </a> <ul id="toc-Formula_4-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Methodology_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Methodology_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Methodology</span> </div> </a> <ul id="toc-Methodology_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparison_with_Direct_Alpha" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Comparison_with_Direct_Alpha"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Comparison with Direct Alpha</span> </div> </a> <ul id="toc-Comparison_with_Direct_Alpha-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_PME_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_PME_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Other PME analysis</span> </div> </a> <ul id="toc-Other_PME_analysis-sublist" class="vector-toc-list"> 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>The <b>public market equivalent</b> (<b>PME</b>) is a collection of performance measures developed to assess private equity funds and to overcome the limitations of the <a href="/wiki/Internal_rate_of_return" title="Internal rate of return">internal rate of return</a> and multiple on invested capital measurements. While the calculations differ, they all attempt to measure the return from deploying a private equity fund's cash flows into a stock market index. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Long-Nickels_PME">Long-Nickels PME</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=1" title="Edit section: Long-Nickels PME"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first PME measure was proposed by Austin M. Long and Craig J. Nickels in 1996.<sup id="cite_ref-LS_PME_1-0" class="reference"><a href="#cite_note-LS_PME-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The analysis is referred in the industry as Long Nickels PME, LN-PME, PME, or ICM. Long and Nickels stated that they preferred the acronym ICM (Index Comparison Method):<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p><i>The ICM is also known as the Public Market Equivalent (PME). We prefer the term ICM, because it better describes the methodology, which is not limited to the use of a public market index to calculate its results</i> </p><p>The PME analysis is covered under US patent 7058583<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Methodology">Methodology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=2" title="Edit section: Methodology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Long and Nickels compared the performance of a private equity fund with the S&P500 Index by creating a theoretical investment into the S&P using the Private Equity fund cashflows : </p> <ul><li>When paying a capital call, we assume that the same amount is used to 'buy the index'</li> <li>When receiving a distribution, we assume that the same amount of the index is sold.</li></ul> <p>As the index price evolves, the value of the theoretical amount invested in the index changes. When receiving a valuation for the fund, we can then compare the value of the fund investment to the theoretical value of the index investment. </p> <table class="wikitable"> <tbody><tr> <td style="text-align:left;"><b>Period</b> </td> <td style="text-align:center;"><b>Cashflows</b> </td> <td style="text-align:center;"><b>Index</b> </td> <td style="text-align:center;"><b>Index Performance</b> </td> <td style="text-align:center;"><b>Theoretical Investment</b> </td></tr> <tr> <td>p1 </td> <td style="text-align:right;">-100 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">0.00% </td> <td style="text-align:right;">100 </td></tr> <tr> <td>p2 </td> <td style="text-align:right;">-50 </td> <td style="text-align:right;">105 </td> <td style="text-align:right;">5.00% </td> <td style="text-align:right;">155 </td></tr> <tr> <td>p3 </td> <td style="text-align:right;">60 </td> <td style="text-align:right;">115 </td> <td style="text-align:right;">9.52% </td> <td style="text-align:right;">109.76 </td></tr> <tr> <td>p4 </td> <td style="text-align:right;">10 </td> <td style="text-align:right;">117 </td> <td style="text-align:right;">1.74% </td> <td style="text-align:right;">101.67 </td></tr> <tr> <td> </td> <td style="text-align:right;">  </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td></tr> <tr> <td style="text-align:left; background:#f0f0f0;"><b>Valuation (p5)</b> </td> <td style="text-align:right; background:#f0f0f0;">110 </td> <td style="text-align:right;">120 </td> <td style="text-align:right;">2.56% </td> <td style="text-align:right; background:#f0f0f0;">104.28 </td></tr> <tr> <td style="text-align:left; background:#f0f0f0;"><b>IRR</b> </td> <td style="text-align:right; background:#f0f0f0;">6.43% </td> <td style="text-align:right;"> </td> <td style="text-align:right; background:#f0f0f0;"><b>PME</b> </td> <td style="text-align:right; background:#f0f0f0;">5.30% </td></tr> </tbody></table> <p>Negative cashflows are treated as contributions. On the first period, a $100 call in the fund is matched by a $100 investment into the index. On the second period, the $100 index investment is now worth $105, to which is added $50 of new investment. A positive cashflow is treated by decreasing the index investment by the same value. On the valuation period, we compare the valuation received from the fund to the value of the theoretical investment. The PME IRR is obtained by computing an IRR with the index valuation as the final cashflow. </p><p>The Long Nickels PME tells how an equivalent investment in the public market would have performed. This then needs to be compared to the actual IRR of the fund. In the above example, the IRR is 1.13 percentage points above the PME, which means that the private fund outperformed the public index. The difference between the IRR and the PME is called the IRR spread. </p> <div class="mw-heading mw-heading3"><h3 id="Formula">Formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=3" title="Edit section: Formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The PME is an IRR on the cashflows of the investment, using as final cashflow an adjusted PME NAV. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PME}=\sum _{s}^{T}C_{s}\times {\cfrac {I_{T}}{I_{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>M</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PME}=\sum _{s}^{T}C_{s}\times {\cfrac {I_{T}}{I_{s}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763c083a8301037fa33e8147be6a6f122ee1592c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.205ex; height:7.343ex;" alt="{\displaystyle NAV_{PME}=\sum _{s}^{T}C_{s}\times {\cfrac {I_{T}}{I_{s}}}}"></span> </p><p>Where : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6d160c0ac08478db6e852c204a78cd9bc2e14d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.665ex; height:2.509ex;" alt="{\displaystyle C_{s}}"></span> is the cashflow from the investment at date s, positive for a contribution, negative for a distribution </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da7c52cd07d12dbec808d9b0c5a17ccd346054f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.026ex; height:2.509ex;" alt="{\displaystyle I_{s}}"></span> is the value of the index at date s </p><p>then : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PME=IRR(C_{s},NAV_{PME})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>=</mo> <mi>I</mi> <mi>R</mi> <mi>R</mi> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>,</mo> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>M</mi> <mi>E</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PME=IRR(C_{s},NAV_{PME})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd45e8e59cd31ad98cd84ac54f255a350741d34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.881ex; height:2.843ex;" alt="{\displaystyle PME=IRR(C_{s},NAV_{PME})}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Limitation">Limitation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=4" title="Edit section: Limitation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As stated in Long and Nickels paper:<sup id="cite_ref-LS_PME_1-1" class="reference"><a href="#cite_note-LS_PME-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><i>If a private investment greatly outperforms the index because it makes frequent, large distributions, it is possible for the final value determined by the index comparison to be negative. In effect, the frequent large withdrawals from the index result in a net short position in the index comparison</i> </p><p>This can be simulated in the previous example by having a period where the fund distributes a large amount and the index dives : </p> <table class="wikitable"> <tbody><tr> <td style="text-align:left;"><b>Period</b> </td> <td style="text-align:center;"><b>Cashflows</b> </td> <td style="text-align:center;"><b>Index</b> </td> <td style="text-align:center;"><b>Index Performance</b> </td> <td style="text-align:center;"><b>Theoretical Investment</b> </td></tr> <tr> <td>p1 </td> <td style="text-align:right;">-100 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">0.00% </td> <td style="text-align:right;">100 </td></tr> <tr> <td>p2 </td> <td style="text-align:right;">-50 </td> <td style="text-align:right;">105 </td> <td style="text-align:right;">5.00% </td> <td style="text-align:right;">155 </td></tr> <tr> <td>p3 </td> <td style="text-align:right;">60 </td> <td style="text-align:right;">115 </td> <td style="text-align:right;">9.52% </td> <td style="text-align:right;">109.76 </td></tr> <tr> <td>p4 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">-13.04% </td> <td style="text-align:right;">-4.55 </td></tr> <tr> <td> </td> <td style="text-align:right;">  </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td></tr> <tr> <td style="text-align:left; background:#f0f0f0;"><b>Valuation (p5)</b> </td> <td style="text-align:right; background:#f0f0f0;">20 </td> <td style="text-align:right; background:#f0f0f0;">120 </td> <td style="text-align:right;">20% </td> <td style="text-align:right; background:#f0f0f0;">-5.47 </td></tr> <tr> <td style="text-align:left; background:#f0f0f0;"><b>IRR</b> </td> <td style="text-align:right; background:#f0f0f0;">7.77% </td> <td style="text-align:right;"> </td> <td style="text-align:right; background:#f0f0f0;"><b>PME</b> </td> <td style="text-align:right; background:#f0f0f0;">1.34% </td></tr> </tbody></table> <p>When the final valuation of the theoretical investment is negative, the IRR formula for the PME may not return any results. Even if a PME can be calculated, while the investment stays negative, every increase in the index will be interpreted as a hit in the performance of the theoretical investment : on the above example, the value of the index went back up to 120, which had a negative impact on the value of the theoretical investment. Even if the investment eventually goes back into positive values and a PME can be computed, the time spent under 0 will be improperly taken into account.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The next methods by Rouvinez, and Kaplan and Schoar are partly designed to address this issue. </p> <div class="mw-heading mw-heading2"><h2 id="PME+"><span id="PME.2B"></span>PME+</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=5" title="Edit section: PME+"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The PME+ was initially described in 2003 by Christophe Rouvinez<sup id="cite_ref-capdyn_5-0" class="reference"><a href="#cite_note-capdyn-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> in a paper <i>Private Equity Benchmarking with PME+</i>. It is written to resolve a common issue of the Long Nickels PME : an investment outperforming the index will yield a negative value in the index theoretical investment. </p> <div class="mw-heading mw-heading3"><h3 id="PME+_Methodology"><span id="PME.2B_Methodology"></span>PME+ Methodology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=6" title="Edit section: PME+ Methodology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Instead of modifying the NAV of the investment, the PME+ discount every distribution by a factor computed so that the NAV of the index investment matches the NAV of the fund. </p> <table class="wikitable"> <tbody><tr style="background:#f0f0f0;"> <td style="text-align:left;"><b>Period</b> </td> <td style="text-align:center;"><b>Cashflows</b> </td> <td style="text-align:center;"><b>Index</b> </td> <td style="text-align:center;"><b>Theoretical Contributions</b> </td> <td style="text-align:center;"><b>Discounted Distributions</b> </td> <td style="text-align:center;"><b>Discounted Cashflows</b> </td></tr> <tr> <td>p1 </td> <td style="text-align:right;">-100 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">-100 </td></tr> <tr> <td>p2 </td> <td style="text-align:right;">-50 </td> <td style="text-align:right;">105 </td> <td style="text-align:right;">50 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">-50 </td></tr> <tr> <td>p3 </td> <td style="text-align:right;">60 </td> <td style="text-align:right;">115 </td> <td style="text-align:right;"> </td> <td style="text-align:right;">51.63 </td> <td style="text-align:right;">51.63 </td></tr> <tr> <td>p4 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;"> </td> <td style="text-align:right;">86.05 </td> <td style="text-align:right;">86.05 </td></tr> <tr> <td> </td> <td style="text-align:right;">  </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right;">  </td></tr> <tr> <td style="text-align:left; background:#f0f0f0;"><b>Valuation (p5)</b> </td> <td style="text-align:right; background:#f0f0f0;">20 </td> <td style="text-align:right; background:#f0f0f0;">120 </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right; background:#f0f0f0;">20 </td></tr> <tr> <td> </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right; background:#f0f0f0;"><b>Lambda</b> </td> <td style="text-align:right; background:#f0f0f0;">0.86 </td></tr> <tr> <td style="text-align:left; background:#f0f0f0;"><b>IRR</b> </td> <td style="text-align:right; background:#f0f0f0;">7.77% </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right; background:#f0f0f0;"><b>PME+</b> </td> <td style="text-align:right; background:#f0f0f0;">2.05% </td></tr> </tbody></table> <p>Like the Long Nickels PME, the PME+ needs to be compared to the IRR. An IRR outperforming the PME means that the fund outperformed the public index. </p> <div class="mw-heading mw-heading3"><h3 id="PME+_Formula"><span id="PME.2B_Formula"></span>PME+ Formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=7" title="Edit section: PME+ Formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using Henly Notation in <i>PME Benchmarking Method</i>:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PME+,T}=\sum _{s=0}^{t}(contribution_{s}-\lambda _{T}.distribution_{s}).{\cfrac {I_{t}}{I_{s}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>+</mo> <mo>,</mo> <mi>T</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>b</mi> <mi>u</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>.</mo> <mi>d</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>b</mi> <mi>u</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PME+,T}=\sum _{s=0}^{t}(contribution_{s}-\lambda _{T}.distribution_{s}).{\cfrac {I_{t}}{I_{s}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6acc396cdce4bd9b084c0fd3defbb89bce3ce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.291ex; height:7.176ex;" alt="{\displaystyle NAV_{PME+,T}=\sum _{s=0}^{t}(contribution_{s}-\lambda _{T}.distribution_{s}).{\cfrac {I_{t}}{I_{s}}}}"></span> </p><p>where </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{T}={\cfrac {(S_{c}-NAV_{PE,T})}{S_{d}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>−</mo> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>E</mi> <mo>,</mo> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{T}={\cfrac {(S_{c}-NAV_{PE,T})}{S_{d}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bbc0fb5795f680e05befee2ac94458a4451e3b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.196ex; height:7.176ex;" alt="{\displaystyle \lambda _{T}={\cfrac {(S_{c}-NAV_{PE,T})}{S_{d}}}}"></span> </p><p>and </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{c}=\sum _{s=0}^{T}(contribution_{s}.{\cfrac {I_{T}}{I_{s}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>b</mi> <mi>u</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{c}=\sum _{s=0}^{T}(contribution_{s}.{\cfrac {I_{T}}{I_{s}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/210143cf63a6830bece375420ecfcac0c7d155d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.629ex; height:7.343ex;" alt="{\displaystyle S_{c}=\sum _{s=0}^{T}(contribution_{s}.{\cfrac {I_{T}}{I_{s}}})}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{d}=\sum _{s=0}^{T}(distribution_{s}.{\cfrac {I_{T}}{I_{s}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mi>d</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>b</mi> <mi>u</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>.</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{d}=\sum _{s=0}^{T}(distribution_{s}.{\cfrac {I_{T}}{I_{s}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1496c0b7c1836a137d3a31fc422744f655e90982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.357ex; height:7.343ex;" alt="{\displaystyle S_{d}=\sum _{s=0}^{T}(distribution_{s}.{\cfrac {I_{T}}{I_{s}}})}"></span> </p><p>In other words, lambda is chosen so that : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PME+,T}=NAV_{PE,T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>+</mo> <mo>,</mo> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>E</mi> <mo>,</mo> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PME+,T}=NAV_{PE,T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74793580851a7b71388cc55539f17706bc45b59a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.101ex; height:2.843ex;" alt="{\displaystyle NAV_{PME+,T}=NAV_{PE,T}}"></span> </p><p>The IRR is then calculated using as cashflows : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PME+_{T}=IRR(Contributions,\lambda _{T}.Distributions,NAV_{PE,T})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>M</mi> <mi>E</mi> <msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mi>I</mi> <mi>R</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mi>o</mi> <mi>n</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>b</mi> <mi>u</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>.</mo> <mi>D</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mi>r</mi> <mi>i</mi> <mi>b</mi> <mi>u</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mo>,</mo> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>E</mi> <mo>,</mo> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PME+_{T}=IRR(Contributions,\lambda _{T}.Distributions,NAV_{PE,T})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99843e566e461aaad9b68171c2b0717d50eb9c17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:63.28ex; height:3.009ex;" alt="{\displaystyle PME+_{T}=IRR(Contributions,\lambda _{T}.Distributions,NAV_{PE,T})}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Modified_PME">Modified PME</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=8" title="Edit section: Modified PME"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The modified PME (or mPME) method was released by <a href="/wiki/Cambridge_Associates" title="Cambridge Associates">Cambridge Associates</a> in October 2013.<sup id="cite_ref-DirectAlpha_7-0" class="reference"><a href="#cite_note-DirectAlpha-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-mPME_8-0" class="reference"><a href="#cite_note-mPME-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> It provides an alternate way to tackle the negative NAV limitation of the LN-PME. </p><p>Like the LN-PME and the PME+, the mPME consider an hypothetical public investment whose performance follows the public benchmark. Each contribution in the private investment is matched by an equal contribution in the public investment. However, rather than subtracting the distributed amounts from the public investment, we compute the weight of the distribution in the private investment, and remove the same weight from the public one. </p> <table class="wikitable"> <tbody><tr style="background:#f0f0f0;"> <td style="text-align:left;"><b>Period</b> </td> <td style="text-align:center;"><b>Call</b> </td> <td style="text-align:center;"><b>Dist</b> </td> <td style="text-align:center;"><b>NAV</b> </td> <td style="text-align:center;"><b>Index</b> </td> <td style="text-align:center;"><b>Theoretical Contributions</b> </td> <td style="text-align:center;"><b>Distribution Weight</b> </td> <td style="text-align:center;"><b>Theoretical NAV</b> </td> <td style="text-align:center;"><b>Weighted Distributions</b> </td> <td style="text-align:center;"><b>Net CF</b> </td></tr> <tr> <td>p1 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">-100 </td></tr> <tr> <td>p2 </td> <td style="text-align:right;">50 </td> <td style="text-align:right;"> </td> <td style="text-align:right;">165 </td> <td style="text-align:right;">105 </td> <td style="text-align:right;">50 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">155 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">-50 </td></tr> <tr> <td>p3 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">60 </td> <td style="text-align:right;">125 </td> <td style="text-align:right;">115 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">0.32 </td> <td style="text-align:right;">114.70 </td> <td style="text-align:right;">55 </td> <td style="text-align:right;">55.06 </td></tr> <tr> <td>p4 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">15 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">0.87 </td> <td style="text-align:right;">13.01 </td> <td style="text-align:right;">87 </td> <td style="text-align:right;">86.73 </td></tr> <tr> <td style="text-align:left; background:#f0f0f0;"><b>Valuation (p5)</b> </td> <td style="text-align:right; background:#f0f0f0;"> </td> <td style="text-align:right; background:#f0f0f0;"> </td> <td style="text-align:right; background:#f0f0f0;">20 </td> <td style="text-align:right; background:#f0f0f0;">120 </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right;">15.61 </td> <td style="text-align:right;"> </td> <td style="text-align:right;">15.61 </td></tr> <tr> <td style="text-align:right; background:#f0f0f0;"><b>IRR</b> </td> <td style="text-align:right; background:#f0f0f0;">7.77% </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right;"> </td> <td style="text-align:right; background:#f0f0f0;"><b>mPME</b> </td> <td style="text-align:right; background:#f0f0f0;">2.02% </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Formula_2">Formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=9" title="Edit section: Formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For each distribution, a distribution weight is calculated </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{weight,t}={\cfrac {D_{t}}{D_{t}+NAV_{t}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>e</mi> <mi>i</mi> <mi>g</mi> <mi>h</mi> <mi>t</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{weight,t}={\cfrac {D_{t}}{D_{t}+NAV_{t}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/883622f2742ad9931c81d26c5b14f27381d01cb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.561ex; height:7.176ex;" alt="{\displaystyle D_{weight,t}={\cfrac {D_{t}}{D_{t}+NAV_{t}}}}"></span> </p><p>The NAV of the theoretical investment is then calculated as : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{mPME,t}=(1-D_{weight,t})\times (NAV_{mPME,t-1}*{\cfrac {I_{t}}{I_{t-1}}}+Call_{t})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>e</mi> <mi>i</mi> <mi>g</mi> <mi>h</mi> <mi>t</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>,</mo> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>C</mi> <mi>a</mi> <mi>l</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{mPME,t}=(1-D_{weight,t})\times (NAV_{mPME,t-1}*{\cfrac {I_{t}}{I_{t-1}}}+Call_{t})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bbead5b9fb0ab8feafde1e5e1a60d51c86093ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.947ex; height:7.176ex;" alt="{\displaystyle NAV_{mPME,t}=(1-D_{weight,t})\times (NAV_{mPME,t-1}*{\cfrac {I_{t}}{I_{t-1}}}+Call_{t})}"></span> </p><p>The weighted Distribution is given by : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Dist_{mPME,t}=(D_{weight,t})\times (NAV_{mPME,t-1}*{\cfrac {I_{t}}{I_{t-1}}}+Call_{t})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>i</mi> <mi>s</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>e</mi> <mi>i</mi> <mi>g</mi> <mi>h</mi> <mi>t</mi> <mo>,</mo> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>,</mo> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>C</mi> <mi>a</mi> <mi>l</mi> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Dist_{mPME,t}=(D_{weight,t})\times (NAV_{mPME,t-1}*{\cfrac {I_{t}}{I_{t-1}}}+Call_{t})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74f391109f4e9806b180e5f9cc07b787eacace08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.439ex; height:7.176ex;" alt="{\displaystyle Dist_{mPME,t}=(D_{weight,t})\times (NAV_{mPME,t-1}*{\cfrac {I_{t}}{I_{t-1}}}+Call_{t})}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IRR_{mPME}=IRR(Call,Dist_{mPME},NAV_{mPME,T})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>R</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>P</mi> <mi>M</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mi>I</mi> <mi>R</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> <mo>,</mo> <mi>D</mi> <mi>i</mi> <mi>s</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>P</mi> <mi>M</mi> <mi>E</mi> </mrow> </msub> <mo>,</mo> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>,</mo> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle IRR_{mPME}=IRR(Call,Dist_{mPME},NAV_{mPME,T})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc20b815a75c56109b4948d3a84ca168c8ad68a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:49.866ex; height:3.009ex;" alt="{\displaystyle IRR_{mPME}=IRR(Call,Dist_{mPME},NAV_{mPME,T})}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Kaplan_Schoar_PME">Kaplan Schoar PME</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=10" title="Edit section: Kaplan Schoar PME"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kaplan Schoar PME was first described in 2005 by <a href="/wiki/Steven_Kaplan_(economist)" title="Steven Kaplan (economist)">Steve Kaplan</a> and <a href="/wiki/Antoinette_Schoar" title="Antoinette Schoar">Antoinette Schoar</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> While the Long Nickels PME returns an IRR, the Kaplan Schoar PME (or KS-PME) returns a market multiple. A simple explanation of its computation is described into Sorensen & Jagannathan paper:<sup id="cite_ref-ssrn_10-0" class="reference"><a href="#cite_note-ssrn-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p><i>Let X(t) denote the cash flow from the fund to the LP at time t. This total cash-flow stream is divided into its positive and negative parts, called distributions (dist(t)) and capital calls (call(t)). Distributions are the cash flows returned to the LP from the PE fund (net of fees) when the fund successfully sells a company. Capital calls are the LP’s investments into the fund, including the payment of ongoing management fees. The distributions and capital calls are then valued by discounting them using the realized market returns over the same time period, and the [KS-]PME is the ratio of the two resulting values:</i> </p> <div class="mw-heading mw-heading3"><h3 id="Formula_3">Formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=11" title="Edit section: Formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When considering an investment at time T. The KS-PME first considers the current valuation of the investment as a distribution at date T. KS-PME is then defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle KS-PME={\cfrac {FV(Dist)}{FV(Call)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mi>S</mi> <mo>−</mo> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle KS-PME={\cfrac {FV(Dist)}{FV(Call)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2c06c2e66e476275384e6dddb2161f94b1a9df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.298ex; height:7.176ex;" alt="{\displaystyle KS-PME={\cfrac {FV(Dist)}{FV(Call)}}}"></span> </p><p>with </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FV(Dist)=\sum _{t}(dist(t)\times {\cfrac {I_{T}}{I_{t}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mi>d</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FV(Dist)=\sum _{t}(dist(t)\times {\cfrac {I_{T}}{I_{t}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2bfc29eeda59294340e05f028fb765d28fed07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.943ex; height:7.176ex;" alt="{\displaystyle FV(Dist)=\sum _{t}(dist(t)\times {\cfrac {I_{T}}{I_{t}}})}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FV(Call)=\sum _{t}(call(t)\times {\cfrac {I_{T}}{I_{t}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mi>c</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FV(Call)=\sum _{t}(call(t)\times {\cfrac {I_{T}}{I_{t}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6882d02ac3856c5203c4f9e7a40249514c978aa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.344ex; height:7.176ex;" alt="{\displaystyle FV(Call)=\sum _{t}(call(t)\times {\cfrac {I_{T}}{I_{t}}})}"></span> </p><p>Using the previous example : </p> <table class="wikitable"> <tbody><tr> <td style="text-align:left;"><b>Period</b> </td> <td style="text-align:center;"><b>Contribution</b> </td> <td style="text-align:center;"><b>Distribution</b> </td> <td style="text-align:center;"><b>Index</b> </td> <td style="text-align:center;"><b>DPI</b> </td> <td style="text-align:center;">  </td> <td style="text-align:center;"><b>Discounted Contribution</b> </td> <td style="text-align:center;"><b>Discounted Distribution</b> </td> <td style="text-align:center;"><b>KS PME</b> </td></tr> <tr> <td>p1 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">0 </td> <td style="text-align:center; background:#f0f0f0;">  </td> <td style="text-align:right;">120 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">0 </td></tr> <tr> <td>p2 </td> <td style="text-align:right;">50 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">105 </td> <td style="text-align:right;">0 </td> <td style="text-align:center; background:#f0f0f0;">  </td> <td style="text-align:right;">57.14 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">0 </td></tr> <tr> <td>p3 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">60 </td> <td style="text-align:right;">115 </td> <td style="text-align:right;">0.40 </td> <td style="text-align:center; background:#f0f0f0;">  </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">62.60 </td> <td style="text-align:right;">0.35 </td></tr> <tr> <td>p4 </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">10 </td> <td style="text-align:right;">117 </td> <td style="text-align:right;">0.47 </td> <td style="text-align:center; background:#f0f0f0;">  </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">10.26 </td> <td style="text-align:right;">0.41 </td></tr> <tr> <td>Valuation (p5) </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">110 </td> <td style="text-align:right;">120 </td> <td style="text-align:right;">1.20 </td> <td style="text-align:center; background:#f0f0f0;">  </td> <td style="text-align:right;">0 </td> <td style="text-align:right;">110 </td> <td style="text-align:right;">1.03 </td></tr></tbody></table> <p>While the Long Nickels PME needs to be compared to the actual IRR, the KS PME gives a direct indication of the performance of the fund compared to the performance of the index. A KS PME above 1 indicates that the fund overperformed the index. A KS PME below 1 indicates that the public index was a better investment than the fund. </p> <div class="mw-heading mw-heading3"><h3 id="Formula_Simplification">Formula Simplification</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=12" title="Edit section: Formula Simplification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The KS-PME formula can be simplified by removing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0555645749867b84164d65d1ac50ee8087220601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle I_{T}}"></span> from the sums : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle KS-PME={\frac {\sum _{t}{\frac {dist(t)}{I_{t}}}}{\sum _{t}{\frac {call(t)}{I_{t}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mi>S</mi> <mo>−</mo> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle KS-PME={\frac {\sum _{t}{\frac {dist(t)}{I_{t}}}}{\sum _{t}{\frac {call(t)}{I_{t}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00bd7961248e97a481c944e901ca80f70978cbd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:25.471ex; height:9.676ex;" alt="{\displaystyle KS-PME={\frac {\sum _{t}{\frac {dist(t)}{I_{t}}}}{\sum _{t}{\frac {call(t)}{I_{t}}}}}}"></span> </p><p>The Kaplan Schoar formula is independent of the time period used to forecast or discount the cashflows. This is an advantage over PME formulas that use an IRR calculations, whose final value will decrease over time. </p> <div class="mw-heading mw-heading3"><h3 id="Usage">Usage</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=13" title="Edit section: Usage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The KS PME is the subject of a paper from the <a href="/wiki/Columbia_Business_School" title="Columbia Business School">Columbia Business School</a><sup id="cite_ref-ssrn_10-1" class="reference"><a href="#cite_note-ssrn-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> assessing that <i>The [Kaplan Schoar] PME provides a valid economic performance measure when the investor ("LP") has log-utility preferences and the return on the LP’s total wealth equals the market return.</i> </p> <div class="mw-heading mw-heading3"><h3 id="Relation_between_LN-PME_and_KS-PME">Relation between LN-PME and KS-PME</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=14" title="Edit section: Relation between LN-PME and KS-PME"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a 2008 paper <i>The common Mathematical Foundation of ACG's ICM and AICM and the K&S PME</i>,<sup id="cite_ref-alignmentcapital1_11-0" class="reference"><a href="#cite_note-alignmentcapital1-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Austin Long studies the mathematical link between LN PME and KS PME. </p><p>Starting with KS PME formula : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle KS-PME={\cfrac {FV(Dist)}{FV(Call)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mi>S</mi> <mo>−</mo> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle KS-PME={\cfrac {FV(Dist)}{FV(Call)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2c06c2e66e476275384e6dddb2161f94b1a9df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.298ex; height:7.176ex;" alt="{\displaystyle KS-PME={\cfrac {FV(Dist)}{FV(Call)}}}"></span> </p><p>From the LN-PME formula : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PME}=\sum _{t}^{T}C_{t}{\cfrac {I_{T}}{I_{t}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>M</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </munderover> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PME}=\sum _{t}^{T}C_{t}{\cfrac {I_{T}}{I_{t}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55b6fc36ff5473e8a87dad5aeb3a678ffee464e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.188ex; height:7.343ex;" alt="{\displaystyle NAV_{PME}=\sum _{t}^{T}C_{t}{\cfrac {I_{T}}{I_{t}}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PME}=\sum _{t}^{T}call(t){\cfrac {I_{T}}{I_{t}}}-\sum _{t}^{T}dist(t){\cfrac {I_{T}}{I_{t}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>M</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </munderover> <mi>c</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </munderover> <mi>d</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PME}=\sum _{t}^{T}call(t){\cfrac {I_{T}}{I_{t}}}-\sum _{t}^{T}dist(t){\cfrac {I_{T}}{I_{t}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0240eb035e9542a1f814cf2937db35b3bcd6683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.4ex; height:7.343ex;" alt="{\displaystyle NAV_{PME}=\sum _{t}^{T}call(t){\cfrac {I_{T}}{I_{t}}}-\sum _{t}^{T}dist(t){\cfrac {I_{T}}{I_{t}}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PME}=FV(Call)-FV(Dist)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>M</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mi>F</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mi>i</mi> <mi>s</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PME}=FV(Call)-FV(Dist)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85f331ad2752028b59a7a462bb4b54748d5b5151" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.264ex; height:2.843ex;" alt="{\displaystyle NAV_{PME}=FV(Call)-FV(Dist)}"></span> </p><p>By merging the two formulas : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle KS-PME=1-{\cfrac {NAV_{PME}}{FV(Call)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mi>S</mi> <mo>−</mo> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>M</mi> <mi>E</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mi>a</mi> <mi>l</mi> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle KS-PME=1-{\cfrac {NAV_{PME}}{FV(Call)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f8be4f56525c2836ccf469402be2de66ab38fa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.026ex; height:7.176ex;" alt="{\displaystyle KS-PME=1-{\cfrac {NAV_{PME}}{FV(Call)}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Direct_Alpha">Direct Alpha</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=15" title="Edit section: Direct Alpha"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Direct Alpha was introduced on March 6, 2014, in a paper by Gredil, Griffiths, and Stucke.<sup id="cite_ref-DirectAlpha_7-1" class="reference"><a href="#cite_note-DirectAlpha-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>It is deduced from the KS-PME calculation by computing an IRR using the discounted contributions and distributions, and take its natural logarithm. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=IRR(FV(C),FV(D),NAV_{PE})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>I</mi> <mi>R</mi> <mi>R</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>F</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>E</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=IRR(FV(C),FV(D),NAV_{PE})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a268ca73ac1df5de07e0e6cc0202e7431c9d272c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.154ex; height:2.843ex;" alt="{\displaystyle a=IRR(FV(C),FV(D),NAV_{PE})}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\cfrac {ln(1+a)}{\Delta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\cfrac {ln(1+a)}{\Delta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2013a6d439318611c1925f49d749fa8156cf71a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.552ex; height:7.176ex;" alt="{\displaystyle \alpha ={\cfrac {ln(1+a)}{\Delta }}}"></span> </p><p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span> being the time interval for which alpha is computed (usually in years)<sup id="cite_ref-DirectAlpha_7-2" class="reference"><a href="#cite_note-DirectAlpha-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <td style="text-align:left;"><b>Period</b> </td> <td style="text-align:center;"><b>Cashflows</b> </td> <td style="text-align:center;"><b>Index</b> </td> <td style="text-align:center;"><b>Index Performance</b> </td> <td style="text-align:center;">  </td> <td style="text-align:center;"><b>Discounted Cashflows</b> </td></tr> <tr> <td>p1 </td> <td style="text-align:right;">-100 </td> <td style="text-align:right;">100 </td> <td style="text-align:right;">1.20 </td> <td style="text-align:center; background:#f0f0f0;">  </td> <td style="text-align:right;">-120 </td></tr> <tr> <td>p2 </td> <td style="text-align:right;">-50 </td> <td style="text-align:right;">105 </td> <td style="text-align:right;">1.14 </td> <td style="text-align:center; background:#f0f0f0;">  </td> <td style="text-align:right;">-57.14 </td></tr> <tr> <td>p3 </td> <td style="text-align:right;">60 </td> <td style="text-align:right;">115 </td> <td style="text-align:right;">1.04 </td> <td style="text-align:center; background:#f0f0f0;">  </td> <td style="text-align:right;">62.60 </td></tr> <tr> <td>p4 </td> <td style="text-align:right;">10 </td> <td style="text-align:right;">117 </td> <td style="text-align:right;">1.03 </td> <td style="text-align:center; background:#f0f0f0;">  </td> <td style="text-align:right;">10.26 </td></tr> <tr> <td>Valuation(p5) </td> <td style="text-align:right;">110 </td> <td style="text-align:right;">120 </td> <td style="text-align:right;">1 </td> <td style="text-align:center; background:#f0f0f0;">  </td> <td style="text-align:right;">110 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td>a (IRR)  : </td> <td>1.09% </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td>Direct Alpha </td> <td>1.08% </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Derivation">Derivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=16" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As an introduction, it is reminded that the computation of an IRR for the set of cashflows <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{0\ldots n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>…</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{0\ldots n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b50bf4ef8e95bd701d5cf9bac351ef3268e2c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.628ex; height:2.509ex;" alt="{\displaystyle C_{0\ldots n}}"></span> and final value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea72bd790a9c616a989d1abeb58d78bdff04968" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.594ex; height:2.176ex;" alt="{\displaystyle NAV}"></span> is done by solving <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> for : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot (1+r)^{n-i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot (1+r)^{n-i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0872d31a5ed64c2c03f53c21fdffe7c9ac2974dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.135ex; height:6.843ex;" alt="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot (1+r)^{n-i}}}"></span> </p><p>The direct alpha formula is derived from the definition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> in <a href="/wiki/Modern_portfolio_theory" title="Modern portfolio theory">Modern portfolio theory</a>. We define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, the rate of return, as the sum of a market return plus an alpha : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(t)=b(t)+\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(t)=b(t)+\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55f414196fbfa9a54284b5c2f9cbfa999842a2fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.771ex; height:2.843ex;" alt="{\displaystyle r(t)=b(t)+\alpha }"></span> </p><p>in the scope of direct alpha, we consider that r(t) and b(t) are continuous rate. Hence, the value of a cashflow <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}"></span> at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271566db7e8ca8616a4dc3efb6c5982a2d987ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.058ex; height:2.343ex;" alt="{\displaystyle t_{n}}"></span> is : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}(t_{n})=c_{i}\cdot e^{\int _{t_{i}}^{t_{n}}{(b(t)+\alpha )dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}(t_{n})=c_{i}\cdot e^{\int _{t_{i}}^{t_{n}}{(b(t)+\alpha )dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7beb3e4369f2349b8adb9c2604c928adf712b79c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.374ex; height:4.009ex;" alt="{\displaystyle v_{i}(t_{n})=c_{i}\cdot e^{\int _{t_{i}}^{t_{n}}{(b(t)+\alpha )dt}}}"></span> </p><p>using the benchmark values, we know that : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {I_{n}}{I_{i}}}=e^{\int _{t_{i}}^{t_{n}}{b(t)dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {I_{n}}{I_{i}}}=e^{\int _{t_{i}}^{t_{n}}{b(t)dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a49f81ce419bb5e97705c845d60a3a17cdfec5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.562ex; height:5.676ex;" alt="{\displaystyle {\frac {I_{n}}{I_{i}}}=e^{\int _{t_{i}}^{t_{n}}{b(t)dt}}}"></span> </p><p>Hence : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}(t_{n})=c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\int _{t_{i}}^{t_{n}}{\alpha dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>d</mi> <mi>t</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}(t_{n})=c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\int _{t_{i}}^{t_{n}}{\alpha dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2bfa84c11218062fdf6c35c29409ca5bbed179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.995ex; height:5.676ex;" alt="{\displaystyle v_{i}(t_{n})=c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\int _{t_{i}}^{t_{n}}{\alpha dt}}}"></span> </p><p>by resolving the integral, and discretizing the time variable such as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}=i\Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}=i\Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4ac68c583645bd1e64c8d3da43a3c6f820f650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.476ex; height:2.509ex;" alt="{\displaystyle t_{i}=i\Delta }"></span> : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}(t_{n})=c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\alpha \cdot (n-i)\Delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}(t_{n})=c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\alpha \cdot (n-i)\Delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e078c5ca3ffe22cd30ca76e6d891e837bb133e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.441ex; height:5.676ex;" alt="{\displaystyle v_{i}(t_{n})=c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\alpha \cdot (n-i)\Delta }}"></span> </p><p>We use this formula for every contribution in the private investment : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\alpha \cdot (n-i)\Delta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\alpha \cdot (n-i)\Delta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d03705e714b57b97a7087e74b4e86252627c8cce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.272ex; height:6.843ex;" alt="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\alpha \cdot (n-i)\Delta }}}"></span> </p><p>Finally, we define a as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+a=e^{\alpha \cdot \Delta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>⋅</mo> <mi mathvariant="normal">Δ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+a=e^{\alpha \cdot \Delta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9181795f67f9c3866080600975a2a3bc505dbbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.525ex; height:2.843ex;" alt="{\displaystyle 1+a=e^{\alpha \cdot \Delta }}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot (1+a)^{n-i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot (1+a)^{n-i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21b1d7ebcd4091f09847c620ddbe0b23db3413b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.073ex; height:6.843ex;" alt="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot (1+a)^{n-i}}}"></span> </p><p>This brings us back to a typical IRR formula. In other words, the direct alpha is calculated by computing an IRR with the benchmark discounted cashflows, and then computing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\cfrac {ln(1+a)}{\Delta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\cfrac {ln(1+a)}{\Delta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2013a6d439318611c1925f49d749fa8156cf71a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.552ex; height:7.176ex;" alt="{\displaystyle \alpha ={\cfrac {ln(1+a)}{\Delta }}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Excess_IRR">Excess IRR</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=17" title="Edit section: Excess IRR"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Different names for this methodology includes alpha, excess IRR, Implied Private Premium ("IPP") and PME Alpha.<sup id="cite_ref-excessIRR_12-0" class="reference"><a href="#cite_note-excessIRR-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-fundveil_13-0" class="reference"><a href="#cite_note-fundveil-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-gems_14-0" class="reference"><a href="#cite_note-gems-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>The first reference of the alpha was in a 2005 paper from <a href="/wiki/Ludovic_Phalippou" title="Ludovic Phalippou">Phalippou</a> and Gottschalg<sup id="cite_ref-excessIRR_12-1" class="reference"><a href="#cite_note-excessIRR-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and is simply named alpha, or excess IRR. The analysis is also explained in detail and named GEM Implied Private Premium (or "IPP") by Global Endowment Management<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Formula_4">Formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=18" title="Edit section: Formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The excess IRR is calculated by resolving <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> in the following equation : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.((1+b_{i})^{\frac {1}{t_{n}-t_{i}}}+\alpha )^{t_{n}-t_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </msup> <mo>+</mo> <mi>α</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.((1+b_{i})^{\frac {1}{t_{n}-t_{i}}}+\alpha )^{t_{n}-t_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b1b7dba3564e8e53d1b88124f7eacbf4a3793ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.198ex; height:6.843ex;" alt="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.((1+b_{i})^{\frac {1}{t_{n}-t_{i}}}+\alpha )^{t_{n}-t_{i}}}}"></span> </p><p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i}={\frac {I_{n}}{I_{i}}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{i}={\frac {I_{n}}{I_{i}}}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ee3162c35b6b452c436c193dcc8915de55c42c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.976ex; height:5.676ex;" alt="{\displaystyle b_{i}={\frac {I_{n}}{I_{i}}}-1}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Methodology_2">Methodology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=19" title="Edit section: Methodology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To calculate the Implied Private Premium, we compute the future values of a private investment's historical distributions and contributions. Each cash flow is compounded at a rate of return equaling the benchmark's annualized return plus the IPP. We then solve for the required IPP such that the PME ratio is set to one. IPP uses annual compounding to be consistent with other reporting methodologies and comparable to IRR. </p><p>More specifically, the Implied Private Premium is solved numerically from </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1={\frac {\sum _{i=1}^{n}d_{i}(1+b_{T_{i},T_{N}}+r_{pp})^{T_{N}-T_{i}}}{\sum _{i=1}^{n}c_{j}(1+b_{T_{j},T_{N}}+r_{pp})^{T_{N}-T_{j}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>p</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mrow> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>p</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1={\frac {\sum _{i=1}^{n}d_{i}(1+b_{T_{i},T_{N}}+r_{pp})^{T_{N}-T_{i}}}{\sum _{i=1}^{n}c_{j}(1+b_{T_{j},T_{N}}+r_{pp})^{T_{N}-T_{j}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c520fa51fa668e07a6fa23f99cf4be61096e4ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.153ex; height:7.343ex;" alt="{\displaystyle 1={\frac {\sum _{i=1}^{n}d_{i}(1+b_{T_{i},T_{N}}+r_{pp})^{T_{N}-T_{i}}}{\sum _{i=1}^{n}c_{j}(1+b_{T_{j},T_{N}}+r_{pp})^{T_{N}-T_{j}}}},}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fa3426b07cfa37c76382ddbecfb4c880889657f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.119ex; height:2.843ex;" alt="{\displaystyle d_{j}}"></span> are contributions and distributions at time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8dd1c50cb9436474f83624c3f679ccf3eebbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.157ex; height:2.509ex;" alt="{\displaystyle T_{i}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27cdb9041c8aa769beb9153a48f41002297faacc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.267ex; height:2.843ex;" alt="{\displaystyle T_{j}}"></span>, respectively; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{T_{i},T_{N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{T_{i},T_{N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e683c7e71af9afc67b3a1e564b507adee1d4eb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.581ex; height:2.843ex;" alt="{\displaystyle b_{T_{i},T_{N}}}"></span> is the annualized benchmark return from time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8dd1c50cb9436474f83624c3f679ccf3eebbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.157ex; height:2.509ex;" alt="{\displaystyle T_{i}}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e92de0cbe77750cce632b1c75de66e558ad297" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.049ex; height:2.509ex;" alt="{\displaystyle T_{N}}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{pp}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{pp}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c40533a9809b62affe30d832224190723dbdf917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.935ex; height:2.343ex;" alt="{\displaystyle r_{pp}}"></span> is the IPP we are solving for. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_2">Derivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=20" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Starting with the definition of the IRR, which is computed by resolving <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> in </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.(1+r)^{t_{n}-t_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>r</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.(1+r)^{t_{n}-t_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad11ce28c633e809c2842591f97b63fd8ca43c62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.714ex; height:6.843ex;" alt="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.(1+r)^{t_{n}-t_{i}}}}"></span> </p><p>we consider r as the sum of two components : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(t_{i})=\beta _{i,n}+\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(t_{i})=\beta _{i,n}+\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32bbc2d2c2226f76742fda23741084b3c85b397" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.483ex; height:3.009ex;" alt="{\displaystyle r(t_{i})=\beta _{i,n}+\alpha }"></span>, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{i,n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{i,n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a742ea57b61a26f6f4235091e2998ce941ff0f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.559ex; height:2.843ex;" alt="{\displaystyle \beta _{i,n}}"></span> being the annually compounded benchmark performance between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/271566db7e8ca8616a4dc3efb6c5982a2d987ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.058ex; height:2.343ex;" alt="{\displaystyle t_{n}}"></span>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{i,n}=({\frac {I_{n}}{I_{i}}})^{\frac {1}{t_{n}-t_{i}}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{i,n}=({\frac {I_{n}}{I_{i}}})^{\frac {1}{t_{n}-t_{i}}}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d58f782db12f4f7e618181342177e372a28d9905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.146ex; height:5.843ex;" alt="{\displaystyle \beta _{i,n}=({\frac {I_{n}}{I_{i}}})^{\frac {1}{t_{n}-t_{i}}}-1}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta _{i,n}=(1+b_{i})^{\frac {1}{t_{n}-t_{i}}}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{i,n}=(1+b_{i})^{\frac {1}{t_{n}-t_{i}}}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cecc5e01f876f1a76d25db442fafef7a9562426" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.868ex; height:4.509ex;" alt="{\displaystyle \beta _{i,n}=(1+b_{i})^{\frac {1}{t_{n}-t_{i}}}-1}"></span> </p><p>by replacing in the original equation : </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.((1+b_{i})^{\frac {1}{t_{n}-t_{i}}}+\alpha )^{t_{n}-t_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>A</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mi>E</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </msup> <mo>+</mo> <mi>α</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.((1+b_{i})^{\frac {1}{t_{n}-t_{i}}}+\alpha )^{t_{n}-t_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b1b7dba3564e8e53d1b88124f7eacbf4a3793ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.198ex; height:6.843ex;" alt="{\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.((1+b_{i})^{\frac {1}{t_{n}-t_{i}}}+\alpha )^{t_{n}-t_{i}}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Comparison_with_Direct_Alpha">Comparison with Direct Alpha</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=21" title="Edit section: Comparison with Direct Alpha"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The theoretical foundation of IPP is similar to that of Direct Alpha; however, the implementation details differ. The advantage of IPP is that it's an annually compounded, arithmetic excess return. This allows IPP to be directly comparable to generally accepted performance metrics such as IRR (also an annually compounded quantity). By contrast, the continuous direct alpha is not measured in the same unit as IRR, while the discrete direct alpha is a geometric excess return. </p> <div class="mw-heading mw-heading2"><h2 id="Other_PME_analysis">Other PME analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=22" title="Edit section: Other PME analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Other less common PME analyses exists, usually as variation from either the Long Nickels PME or the Kaplan Schoar PME. </p><p>Alignment Capital defines the Alternative ICM, or AICM<sup id="cite_ref-alignmentcapital1_11-1" class="reference"><a href="#cite_note-alignmentcapital1-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> as a variation from the Long Nickels PME : </p><p><i>While ACG’s ICM calculation assumes that the capital invested into the index is a long position, the alternative index comparison method (AICM) assumes the opposite – that is, the cash used to invest in the private market investment results, not from a source external to both the private market investment and the index, but from a short position in (i.e., a sale of) the index. Expressed in the same terms, the AICM calculation of the ending value of the index (the ending value used to calculate the AICM) is as follows: </i> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Value_{Index_{Ending}}=FV_{Returned}-FV_{Invested}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mi>a</mi> <mi>l</mi> <mi>u</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>n</mi> <mi>d</mi> <mi>e</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> <mi>n</mi> <mi>d</mi> <mi>i</mi> <mi>n</mi> <mi>g</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mi>F</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>e</mi> <mi>t</mi> <mi>u</mi> <mi>r</mi> <mi>n</mi> <mi>e</mi> <mi>d</mi> </mrow> </msub> <mo>−</mo> <mi>F</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mi>n</mi> <mi>v</mi> <mi>e</mi> <mi>s</mi> <mi>t</mi> <mi>e</mi> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Value_{Index_{Ending}}=FV_{Returned}-FV_{Invested}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df9e00c5480e11a2848f44064335119fce38ffec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:41.186ex; height:3.009ex;" alt="{\displaystyle Value_{Index_{Ending}}=FV_{Returned}-FV_{Invested}}"></span> </p><p>In <i>Valuing Private Equity</i>, December 13, 2011,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Sorensen, Wang and Yang defines an alternate PME based on the KS PME : </p><p><i>There are three concerns with the standard PME measure. First, the denominator combines two types of cash flows, the investment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/893d08e90ea73781dc133414d661529d0651ca80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.077ex; height:2.509ex;" alt="{\displaystyle I_{0}}"></span> and the management fees. Management fees are effectively a risk-free claim and should be discounted at a rate close to the risk-free rate. Second, the numerator contains the total proceeds net of carried interest. The carried interest is effectively a call option, making the LP's total payoff at maturity less risky than the underlying asset. Hence, it should be discounted at a lower rate than the underlying PE investment. Finally, the beta of the PE investment may not equal one.</i> <i>To address these concerns, we define the adjusted PME as follows : </i> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Adj.PME={\cfrac {E[e^{-rT}(LP_{1}(A_{t},T)+LP_{2}(A_{T},T)+LP_{3}(A_{T},T))]}{I_{0}+E[\int _{0}^{T}e^{-rs}mX_{0}ds]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>d</mi> <mi>j</mi> <mo>.</mo> <mi>P</mi> <mi>M</mi> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> <mo stretchy="false">[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>r</mi> <mi>T</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>L</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>L</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>L</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> </mstyle> </mrow> <mrow> <mpadded width="0" height="8.6pt" depth="3pt"> <mrow /> </mpadded> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>E</mi> <mo stretchy="false">[</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>r</mi> <mi>s</mi> </mrow> </msup> <mi>m</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>d</mi> <mi>s</mi> <mo stretchy="false">]</mo> </mrow> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Adj.PME={\cfrac {E[e^{-rT}(LP_{1}(A_{t},T)+LP_{2}(A_{T},T)+LP_{3}(A_{T},T))]}{I_{0}+E[\int _{0}^{T}e^{-rs}mX_{0}ds]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b46e095b9c94b025b51407db873b6692e337b5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:64.561ex; height:7.676ex;" alt="{\displaystyle Adj.PME={\cfrac {E[e^{-rT}(LP_{1}(A_{t},T)+LP_{2}(A_{T},T)+LP_{3}(A_{T},T))]}{I_{0}+E[\int _{0}^{T}e^{-rs}mX_{0}ds]}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Public_Market_Equivalent&action=edit&section=23" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-LS_PME-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-LS_PME_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-LS_PME_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://alignmentcapital.com/pdfs/research/icm_aimr_benchmark_1996.pdf">"A Private Investment Benchmark"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. 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Retrieved <span class="nowrap">11 December</span> 2014</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Evolution+of+MIRR%3A+PME+Alpha&rft_id=http%3A%2F%2Fblog.fundveil.com%2Fblog%2F2014%2F8%2F12%2Fevolution-of-mirr-pmea&rfr_id=info%3Asid%2Fen.wikipedia.org%3APublic+Market+Equivalent" class="Z3988"></span></span> </li> <li id="cite_note-gems-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-gems_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://globalendowment.com/File_Store/GEM_FactSheet_Fin.pdf">"The GEM Implied Private Premium (IPP) Private Equity Benchmark"</a> <span class="cs1-format">(PDF)</span>. Global Endowment Management, LP<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-11-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+GEM+Implied+Private+Premium+%28IPP%29+Private+Equity+Benchmark&rft.pub=Global+Endowment+Management%2C+LP&rft_id=http%3A%2F%2Fglobalendowment.com%2FFile_Store%2FGEM_FactSheet_Fin.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APublic+Market+Equivalent" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://uspto.report/TM/86436078">"Trademark GEM implied Private Premium"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Trademark+GEM+implied+Private+Premium&rft_id=https%3A%2F%2Fuspto.report%2FTM%2F86436078&rfr_id=info%3Asid%2Fen.wikipedia.org%3APublic+Market+Equivalent" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www0.gsb.columbia.edu/faculty/nwang/papers/swy44.pdf">"Valuing private equity"</a> <span class="cs1-format">(PDF)</span>. 2011-12-13<span class="reference-accessdate">. Retrieved <span class="nowrap">2014-03-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Valuing+private+equity&rft.date=2011-12-13&rft_id=http%3A%2F%2Fwww0.gsb.columbia.edu%2Ffaculty%2Fnwang%2Fpapers%2Fswy44.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APublic+Market+Equivalent" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><i>Exposed to the J-Curve: Understanding and Managing Private Equity Fund Investments</i>, Ulrich Grabenwarter & Tom Weidig, Chapter 5</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Public_Market_Equivalent&oldid=1227948421">https://en.wikipedia.org/w/index.php?title=Public_Market_Equivalent&oldid=1227948421</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Private_equity" title="Category:Private equity">Private equity</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden category: <ul><li><a href="/wiki/Category:CS1_errors:_missing_periodical" title="Category:CS1 errors: missing periodical">CS1 errors: missing periodical</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 8 June 2024, at 17:16<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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