# Public Market Equivalent

Public Market Equivalent (or PME) designs a set of analyses used in the Private Equity Industry to evaluate the performance of a Private Equity Fund against a public benchmark or index. The analysis is also referred by the acronym ICM for Index Comparison Method. More than a specific calculation method, PME encompasses a number of different analyses sharing the same objective. The first PME analysis was proposed by Long and Nickels in 1996.[1]

## Objective

The two indicators used by the Private Equity industry to assess the performance of an investment are the Internal Rate of Return (IRR) and the Market Multiples. These indicators can allow comparison between private investments, but not with other asset classes which are using time-weighed rate of returns (TWR).[2] The public market equivalent aims to benchmark a private investment against a public benchmark or index.

PME Analyses can be divided into two groups, depending on the general method used to calculate them. Heuristic PMEs (LN-PME, PME+, mPME) simulate the investments of the Private Equity cashflow into a theoretical benchmark investment. The KS-PME, Direct Alpha and PME Alpha start with the definition of alpha in Modern portfolio theory, and extend it to Private Investments.

## Long Nickels PME

The first PME analysis was proposed by Austin M. Long and Craig J. Nickels in 1996.[1]

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):[3]

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

The PME analysis is covered under US patent 7058583[4]

### Methodology

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 :

• When paying a capital call, we assume that the same amount is used to 'buy the index'
• When receiving a distribution, we assume that the same amount of the index is sold.

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.

 Period Cashflows Index Index Performance Theoretical Investment p1 -100 100 0.00% 100 p2 -50 105 5.00% 155 p3 60 115 9.52% 109.76 p4 10 117 1.74% 101.67 Valuation (p5) 110 120 2.56% 104.28 IRR 6.43% PME 5.30%

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.

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.

### Formula

The PME is an IRR on the cashflows of the investment, using as final cashflow an adjusted PME NAV.

${\displaystyle NAV_{PME}=\sum _{s}^{T}C_{s}\times {\cfrac {I_{T}}{I_{s}}}}$

Where :

${\displaystyle C_{s}}$ is the cashflow from the investment at date s, positive for a contribution, negative for a distribution

${\displaystyle I_{s}}$ is the value of the index at date s

then :

${\displaystyle PME=IRR(C_{s},NAV_{PME})}$

### Limitation

As stated in Long and Nickels paper:[1]

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

This can be simulated in the previous example by having a period where the fund distributes a large amount and the index dives :

 Period Cashflows Index Index Performance Theoretical Investment p1 -100 100 0.00% 100 p2 -50 105 5.00% 155 p3 60 115 9.52% 109.76 p4 100 100 -13.04% -4.55 Valuation (p5) 20 120 20% -5.47 IRR 7.77% PME 1.34%

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.[5]

The next methods by Rouvinez, and Kaplan and Schoar are partly designed to address this issue.

## PME+

The PME+ was initially described in 2003 by Christophe Rouvinez[2] in a paper Private Equity Benchmarking with PME+. 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.

### PME+ Methodology

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.

 Period Cashflows Index Theoretical Contributions Discounted Distributions Discounted Cashflows p1 -100 100 100 0 -100 p2 -50 105 50 0 -50 p3 60 115 51.63 51.63 p4 100 100 86.05 86.05 Valuation (p5) 20 120 20 Lambda 0.86 IRR 7.77% PME+ 2.05%

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.

### PME+ Formula

Using Henly Notation in PME Benchmarking Method:[6]

${\displaystyle NAV_{PME+,T}=\sum _{s=0}^{t}(contribution_{s}-\lambda _{T}.distribution_{s}).{\cfrac {I_{t}}{I_{s}}}}$

where

${\displaystyle \lambda _{T}={\cfrac {(S_{c}-NAV_{PE,T})}{S_{d}}}}$

and

${\displaystyle S_{c}=\sum _{s=0}^{T}(contribution_{s}.{\cfrac {I_{T}}{I_{s}}})}$

${\displaystyle S_{d}=\sum _{s=0}^{T}(distribution_{s}.{\cfrac {I_{T}}{I_{s}}})}$

In other words, lambda is chosen so that :

${\displaystyle NAV_{PME+,T}=NAV_{PE,T}}$

The IRR is then calculated using as cashflows :

${\displaystyle PME+_{T}=IRR(Contributions,\lambda _{T}.Distributions,NAV_{PE,T})}$

## Modified PME

The modified PME (or mPME) method was released by Cambridge Associates in October 2013.[7][8] It provides an alternate way to tackle the negative NAV limitation of the LN-PME.

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.

 Period Call Dist NAV Index Theoretical Contributions Distribution Weight Theoretical NAV Weighted Distributions Net CF p1 100 0 100 100 100 0 100 0 -100 p2 50 165 105 50 0 155 0 -50 p3 0 60 125 115 0 0.32 114.70 55 55.06 p4 0 100 15 100 0 0.87 13.01 87 86.73 Valuation (p5) 20 120 15.61 15.61 IRR 7.77% mPME 2.02%

### Formula

For each distribution, a distribution weight is calculated

${\displaystyle D_{weight,t}={\cfrac {D_{t}}{D_{t}+NAV_{t}}}}$

The NAV of the theoretical investment is then calculated as :

${\displaystyle NAV_{mPME,t}=(1-D_{weight,t})\times (NAV_{mPME,t-1}*{\cfrac {I_{t}}{I_{t-1}}}+Call_{t})}$

The weighted Distribution is given by :

${\displaystyle Dist_{mPME,t}=(D_{weight,t})\times (NAV_{mPME,t-1}*{\cfrac {I_{t}}{I_{t-1}}}+Call_{t})}$

${\displaystyle IRR_{mPME}=IRR(Call,Dist_{mPME},NAV_{mPME,T})}$

## Kaplan Schoar PME

Kaplan Schoar PME was first described in 2005 by Steve Kaplan and Antoinette Schoar.[9] in the paper : Private Equity Performance : Returns, Persistence and Capital Flows. 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:[10]

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:

### Formula

When considering an investment at time T. The KS-PME first considers the current valuation of the investment as an distribution at date T. KS-PME is then defined as

${\displaystyle KS-PME={\cfrac {FV(Dist)}{FV(Call)}}}$

with

${\displaystyle FV(Dist)=\sum _{t}(dist(t)\times {\cfrac {I_{T}}{I_{t}}})}$

${\displaystyle FV(Call)=\sum _{t}(call(t)\times {\cfrac {I_{T}}{I_{t}}})}$

Using the previous example :

 Period Contribution Distribution Index DPI Discounted Contribution Discounted Distribution KS PME p1 100 0 100 0 120 0 0 p2 50 0 105 0 57.14 0 0 p3 0 60 115 0.40 0 62.60 0.35 p4 0 10 117 0.47 0 10.26 0.41 Valuation (p5) 0 110 120 1.20 0 110 1.03

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.

### Formula Simplification

The KS-PME formula can be simplified by removing ${\displaystyle I_{T}}$ from the sums :

${\displaystyle KS-PME={\frac {\sum _{t}{\frac {dist(t)}{I_{t}}}}{\sum _{t}{\frac {call(t)}{I_{t}}}}}}$

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.

### Usage

The KS PME is the subject of a paper from the Columbia Business School [10] assessing that 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.

### Relation between LN-PME and KS-PME

In a 2008 paper The common Mathematical Foundation of ACG's ICM and AICM and the K&S PME,[11] Austin Long studies the mathematical link between LN PME and KS PME.

Starting with KS PME formula :

${\displaystyle KS-PME={\cfrac {FV(Dist)}{FV(Call)}}}$

From the LN-PME formula :

${\displaystyle NAV_{PME}=\sum _{t}^{T}C_{t}{\cfrac {I_{T}}{I_{t}}}}$

${\displaystyle NAV_{PME}=\sum _{t}^{T}call(t){\cfrac {I_{T}}{I_{t}}}-\sum _{t}^{T}dist(t){\cfrac {I_{T}}{I_{t}}}}$

${\displaystyle NAV_{PME}=FV(Call)-FV(Dist)}$

By merging the two formulas :

${\displaystyle KS-PME=1-{\cfrac {NAV_{PME}}{FV(Call)}}}$

## Direct Alpha

The Direct Alpha was introduced in March 6, 2014 in a paper by Gredil, Oleg and Griffiths.[7]

It is deduced from the KS-PME calculation by computing an IRR using the discounted contributions and distributions, and take its natural logarithm.

${\displaystyle a=IRR(FV(C),FV(D),NAV_{PE})}$

${\displaystyle \alpha ={\cfrac {ln(1+a)}{\Delta }}}$

with ${\displaystyle \Delta }$ being the time interval for which alpha is computed (usually in years[7])

 Period Cashflows Index Index Performance Discounted Cashflows p1 -100 100 1.20 -120 p2 -50 105 1.14 -57.14 p3 60 115 1.04 62.60 p4 10 117 1.03 10.26 Valuation(p5) 110 120 1 110 a (IRR) : 1.09% Direct Alpha 1.08%

### Derivation

As an introduction, it is reminded that the computation of an IRR for the set of cashflows ${\displaystyle C_{0\ldots n}}$ and final value ${\displaystyle NAV}$ is done by solving ${\displaystyle r}$ for :

${\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot (1+r)^{n-i}}}$

The direct alpha formula is derived from the definition of ${\displaystyle \alpha }$ in Modern portfolio theory. We define ${\displaystyle r}$, the rate of return, as the sum of a market return plus an alpha :

${\displaystyle r(t)=b(t)+\alpha }$

in the scope of direct alpha, we consider that r(t) and b(t) are continuous rate. Hence, the value of a cashflow ${\displaystyle c_{i}}$ at time ${\displaystyle t_{n}}$ is :

${\displaystyle v_{i}(t_{n})=c_{i}\cdot e^{\int _{t_{i}}^{t_{n}}{(b(t)+\alpha )dt}}}$

using the benchmark values, we know that :

${\displaystyle {\frac {I_{n}}{I_{i}}}=e^{\int _{t_{i}}^{t_{n}}{b(t)dt}}}$

Hence :

${\displaystyle v_{i}(t_{n})=c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\int _{t_{i}}^{t_{n}}{\alpha dt}}}$

by resolving the integral, and discretizing the time variable such as ${\displaystyle t_{i}=i\Delta }$ :

${\displaystyle v_{i}(t_{n})=c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\alpha \cdot (n-i)\Delta }}$

We use this formula for every contribution in the private investment :

${\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot e^{\alpha \cdot (n-i)\Delta }}}$

Finally, we define a as ${\displaystyle 1+a=e^{\alpha \cdot \Delta }}$

${\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}\cdot {\frac {I_{n}}{I_{i}}}\cdot (1+a)^{n-i}}}$

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 ${\displaystyle \alpha }$ with ${\displaystyle \alpha ={\cfrac {ln(1+a)}{\Delta }}}$

## Excess IRR

Different names for this methodology includes alpha, excess IRR, Implied Private Premium ("IPP") and PME Alpha.[12][13][14]

The first reference of the alpha was in a 2005 paper from Phalippou and Gottschalg[12] 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[15]

### Formula

The excess IRR is calculated by resolving ${\displaystyle \alpha }$ in the following equation :

${\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.((1+b_{i})^{\frac {1}{t_{n}-t_{i}}}+\alpha )^{t_{n}-t_{i}}}}$

with ${\displaystyle b_{i}={\frac {I_{n}}{I_{i}}}-1}$

### Methodology

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.

More specifically, the Implied Private Premium is solved numerically from

${\displaystyle 1={\frac {\sum _{i=1}^{n}d_{i}(1+b_{T_{i},T_{N}}+r_{pp})^{T_{N}-T_{i}}}{\sum _{i=1}^{n}c_{j}(1+b_{T_{j},T_{N}}+r_{pp})^{T_{N}-T_{j}}}},}$

where ${\displaystyle c_{i}}$ and ${\displaystyle d_{j}}$ are contributions and distributions at time ${\displaystyle T_{i}}$ and ${\displaystyle T_{j}}$, respectively; ${\displaystyle b_{T_{i},T_{N}}}$ is the annualized benchmark return from time ${\displaystyle T_{i}}$ to ${\displaystyle T_{N}}$, and ${\displaystyle r_{pp}}$ is the IPP we are solving for.

### Derivation

Starting with the definition of the IRR, which is computed by resolving ${\displaystyle r}$ in

${\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.(1+r)^{t_{n}-t_{i}}}}$

we consider r as the sum of two components : ${\displaystyle r(t_{i})=\beta _{i,n}+\alpha }$, with ${\displaystyle \beta _{i,n}}$ being the annually compounded benchmark performance between ${\displaystyle t_{i}}$ and ${\displaystyle t_{n}}$.

${\displaystyle \beta _{i,n}=({\frac {I_{n}}{I_{i}}})^{\frac {1}{t_{n}-t_{i}}}-1}$

${\displaystyle \beta _{i,n}=(1+b_{i})^{\frac {1}{t_{n}-t_{i}}}-1}$

by replacing in the original equation :

${\displaystyle NAV_{PE}=\sum _{i=0}^{n}{c_{i}.((1+b_{i})^{\frac {1}{t_{n}-t_{i}}}+\alpha )^{t_{n}-t_{i}}}}$

### Comparison with Direct Alpha

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.

## Other PME analysis

Other less common PME analyses exists, usually as variation from either the Long Nickels PME or the Kaplan Schoar PME.

Alignment Capital defines the Alternative ICM, or AICM[11] as a variation from the Long Nickels PME :

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:

${\displaystyle Value_{Index_{Ending}}=FV_{Returned}-FV_{Invested}}$

In Valuing Private Equity, December 13, 2011,[16] Sorensen, Wang and Yang defines an alternate PME based on the KS PME :

There are three concerns with the standard PME measure. First, the denominator combines two types of cash flows, the investment ${\displaystyle I_{0}}$ 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. To address these concerns, we define the adjusted PME as follows :

${\displaystyle Adj.PME={\cfrac {E[e^{-rT}(LP_{1}(A_{t},T)+LP_{2}(A_{T},T)+LP_{3}(A_{T},T))]}{I_{0}+E[\int _{0}^{T}e^{-rs}mX_{0}ds]}}}$

## References

1. ^ a b c "A Private Investment Benchmark" (PDF). Retrieved 2014-03-05. Cite error: Invalid <ref> tag; name "LS PME" defined multiple times with different content (see the help page).
2. ^ a b "Private Equity Benchmarking with PME+" (PDF). Retrieved 2014-03-05. Cite error: Invalid <ref> tag; name "capdyn" defined multiple times with different content (see the help page).
3. ^ Inside Private Equity : The professional Investor Handbook by Kocis, Bachman, Long and Nickels, page 157
4. ^ "Patent US7058583 - Method for calculating portfolio scaled IRR". Retrieved 2014-03-05.
5. ^ Jost, Philippe; Stoll, Philipp. "Quantifying the shortness issue of PME" (PDF).
6. ^ Samuel Henly (2013-08-12). "PME Benchmarking Methods" (PDF). Retrieved 2014-03-05.
7. ^ a b c "Benchmarking Private Equity: The Direct Alpha Method". SSRN 2403521. Missing or empty |url= (help)
8. ^
9. ^ "Private Equity Performance:Returns, Persistence and Capital Flows" (PDF). University of Chicago Graduate School of Business. Retrieved 2014-03-05.
10. ^ a b "The Public Market Equivalent and Private Equity Performance by Morten Sorensen, Ravi Jagannathan :: SSRN". Papers.ssrn.com. doi:10.2139/ssrn.2347972. SSRN 2347972.
11. ^ a b "A Method for Quantifying Concentration of Returns in Private Equity Portfolios" (PDF). Retrieved 2014-03-05.
12. ^ a b "Performance of Private Equity Funds, page 17". Ludovic Phalippou & Olivier Gottschalg. SSRN 473221. Missing or empty |url= (help)
13. ^ "Evolution of MIRR: PME Alpha". Retrieved 11 December 2014.
14. ^ "The GEM Implied Private Premium (IPP) Private Equity Benchmark" (PDF). Global Endowment Management, LP. Retrieved 2014-11-21.
15. ^
16. ^ "Valuing private equity" (PDF). 2011-12-13. Retrieved 2014-03-05.
• Exposed to the J-Curve: Understanding and Managing Private Equity Fund Investments, Ulrich Grabenwarter & Tom Weidig, Chapter 5