## Abstract

This article introduces the excess value method for calculating the dollar value that a private market investment generates relative to a benchmark. To the authors’ knowledge, this is the first published method of doing so. It is based on the commonly used direct alpha and Kaplan–Schoar public market equivalent (KS-PME) measures of private market relative performance as a rate and multiple. The article demonstrates that direct alpha and KS-PME cannot be directly translated into dollar terms for all but the simplest cash flow streams. It thus introduces adaptations that enable this translation to take place algorithmically. The authors believe that excess value can give investors and fund managers useful insights into the value creation of private market investments over time and can facilitate an alternative to traditional carried interest compensation. Unlike carried interest, which compensates managers for absolute performance regardless of public market behavior, excess value–based performance fees enable performance compensation to be paid only for outperformance versus a public market benchmark.

**TOPICS:** Real assets/alternative investments/private equity, performance measurement

**Key Findings**

▪ This article introduces the excess value method for calculating the dollar value a private market investment generates above or below a benchmark. It is based on the commonly used direct alpha and Kaplan–Schoar public market equivalent performance measures.

▪ The excess value method introduces adaptations that enable dollar outperformance to be calculated algorithmically even for complicated cash flow streams.

▪ Excess value can enable parties to measure dollar outperformance over time and to develop performance fee agreements that create better alignment by being based on a split of value in excess of an agreed benchmark rather than absolute returns.

In this article, we introduce a method for calculating the dollar value that a private market investment, such as a private equity fund or portfolio company, generates above or below a benchmark. To our knowledge, this is the first published method of doing so. We have developed this method for two primary purposes. The first is as a performance measurement tool to help private market investors (i.e., limited partners [LPs]) and fund managers (i.e., general partners [GPs]) to better understand the sources of value created by their investments and the evolution of these sources over time. The second is to enable investors and fund managers to develop compensation agreements that, unlike traditional carried interest, distinguish between profits attributable to market returns and profits that represent returns above or below market. We call our method *excess value* (abbreviated as EVM for the excess value method).

## ON ALIGNING COMPENSATION WITH OBJECTIVES

One of the major reasons why LPs invest in private markets is to achieve a return in excess of public market alternatives (Kaplan and Schoar 2005; Cambridge Associates 2013; Gredil, Griffiths, and Stucke 2014; Griffiths and Charles 2014; Griffiths and Silva 2016; Bass, Gladstone, and Ang 2017; Korteweg and Sorensen 2017; Phalippou 2020; Polen 2020). In general, return factors that are accessible via public markets are available more cheaply through that path (Bass, Gladstone, and Ang 2017; Phalippou 2020), so a strong argument can be made that LPs should want to pay private market managers only for returns generated above similar public market investments. To make this point more concretely, a private market investment may generate value because of its exposure to macroeconomic conditions or to certain sectors or geographies. Because LPs can generally achieve these exposures more cheaply via public markets, they may not want to compensate private market managers for this performance. The traditional carried interest model is at odds with this objective. By basing compensation on absolute return thresholds, the traditional model can lead LPs to pay high fees for mere market returns.

## A HISTORICAL PERSPECTIVE ON PRIVATE MARKET PERFORMANCE MEASUREMENT

Private market academics and practitioners have worked since the mid-1990s to develop increasingly sophisticated methods to estimate private market alpha (Kaplan and Schoar 2005; Cambridge Associates 2013; Gredil, Griffiths, and Stucke 2014; Griffiths and Charles 2014). By *alpha*, we mean the part of investment performance that is driven by manager skill rather than by systematic sources of return unrelated to skill. Examples of non-alpha sources include the macroeconomic conditions, sector, and geography exposures described previously and certain uses of leverage. (See the “Considerations for Implementing the EVM” section for a discussion of leverage). Random luck also contributes to the performance of any investment and is one reason why alpha can only be estimated but not established precisely (Griffiths and Silva 2016; Korteweg and Sorensen 2017). We do not consider the influence of luck in this article, beyond an implication described in the “Conclusions” section.

In recent years, the academic and practitioner communities seem to have settled on two primary measures of alpha: one a rate-based measure developed at Landmark known as direct alpha, and the other a multiple-based method known as the Kaplan–Schoar public market equivalent (KS-PME) (Kaplan and Schoar 2005; Gredil, Griffiths, and Stucke 2014; Griffiths and Charles 2014). Both measures calculate outperformance precisely rather than heuristically and are consistent with modern portfolio theory (Kaplan and Schoar 2005; Gredil, Griffiths, and Stucke 2014; Griffiths and Charles 2014). Direct alpha measures performance relative to a benchmark as a rate. For example, 5% direct alpha indicates that an investment generated an annualized return 5% above a public market benchmark. KS-PME measures relative performance as a multiple. For example, a 1.15 KS-PME indicates that a private investment outperformed the public benchmark by 15% in total over the life of the investment.

The EVM that we propose builds on the direct alpha and KS-PME methods to measure performance relative to a benchmark in dollars. This method can enable investors and managers to examine the value an investment generates over time or to compare the value generated, in dollars, by different investments. This method can also be used by investors to measure the dollar contribution of their own private market investments to their overall portfolios. These performance measurement uses for excess value are in addition to its compensation use, which is also introduced in this article.

## REJECTING A STRONG INTUITION—KS-PME NUMERATOR MINUS KS-PME DENOMINATOR

We had a strong early intuition, shared almost universally by early readers of this article, that the dollar value generated by a private market investment above a public market benchmark can be calculated simply by subtracting the KS-PME denominator from the KS-PME numerator using the future value version of KS-PME. Stated differently, the future value version of KS-PME is the future value of the private market cash flows divided by the future value of the public market alternative cash flows (Kaplan and Schoar 2005; Gredil, Griffiths, and Stucke 2014; Griffiths and Charles 2014). Thus, if we take the numerator (i.e., the future value of the private market cash flows) minus the denominator (i.e., the future value of the public market alternative cash flows) at the fund liquidation date, the result should be the dollar value generated by the private market investment in excess of the public market benchmark.

As we show in our Scenario II in the following, this intuition holds only for private market investments with one single distribution. Once a second distribution takes place (e.g., different investments in a fund are sold at different dates, or one investment makes distributions at different times), the intuition no longer works for translating outperformance into dollars. The reason is that distributions that take place prior to fund liquidation are future-valued to the fund liquidation, which leads to the manager being rewarded (or penalized) for market movements when capital is no longer at risk. If an LP wants to compensate a GP for, and only for, value the GP actually creates, then market moves when capital is not at risk must be excluded. The same is the case when using the EVM for performance measurement purposes.

## ANOTHER INTUITION—THE CARRIED INTEREST WITH A MARKET-BASED PREFERRED RETURN

Another intuition shared by several early readers of this article was that the compensation goal of excess value can be accomplished via a method that begins with the traditional carried interest method but sets a market-based compounding preferred return and pays carry only for performance above this preferred return. (In other words, there is no catch-up of profit-sharing back to dollar zero, as is common under traditional carried interest arrangements.) Under such a method, the starting point for a compensation agreement would be the same as under a carried interest agreement, with the manager receiving, say, 20% of profits. However, instead of the manager receiving performance compensation only if the investment outperforms a fixed hurdle (e.g., 8%), the manager would receive compensation only if the investment outperforms a public benchmark. Additionally, instead of the manager receiving compensation back to the first dollar of profit generated by the investment (via a catch-up once the investment return surpasses the preferred return), the manager would receive compensation only for performance that exceeds the preferred return.

It turns out that this method of manager compensation is a special case of our EVM. We discuss why this is so in further depth in the “Scaling Factor” section later in this article. Additionally, when expressed through our EVM equations, this preferred-return method requires an assumption that distributions generate no excess value (i.e., profits above the benchmark) until such time as capital distributed to the investor exceeds capital invested plus the preferred return. (In other words, all excess value profits before this point are attributed to unrealized net asset value [NAV].) Although the parties to a compensation agreement may wish to make such an assumption, there is no reason why this is assumption is necessary, and, more importantly, it is unlikely to be a good representation of empirical reality. Our EVM, in contrast, allows the parties to choose from an infinite array of assumptions regarding the split of excess value at any point in time between realized and unrealized amounts and to aim for splits that represent reality as accurately as possible. For reasons described in the “Scaling Factor” section, the choice of this split affects not just the division of excess value between realized and unrealized components but also the total excess value calculated. As a result, we argue that accuracy should be preferred.

## DEFINING A SOLUTION

As explained earlier, we argue that investors should generally choose private managers based on these managers’ anticipated returns above a benchmark rather than absolute returns and that investors may wish to compensate managers explicitly for their ability to achieve this outperformance. We believe that a framework that best enables both performance measurement and compensation must adhere to the following criteria:

**1.**Sources of return must be expressed in dollar or other currency terms, rather than rates or multiples.**2.**The method must distinguish between alpha value (created by the manager), beta value (created by the market), and the value created via the alpha value growing at the market (or beta) rate while capital is at risk. We refer to this third component as the interaction between alpha and beta and describe it further in the article.**3.**The method must apportion the components of value creation between realized amounts (distributions) and unrealized amounts (remaining NAV). This apportioning is important in enabling investors to track when and where excess value is created and in allowing performance compensation to take place along with distributions, rather than waiting for final liquidation.**4.**As mentioned earlier, the method must consider movements in the chosen benchmark only when capital is at risk. That is, for performance measurement purposes, the method must not show that excess value is created when capital is no longer at risk, and for compensation purposes, it must not lead to a manager benefiting from or being penalized for market adjustments to capital that has not yet been invested or has been previously returned to investors.**5.**The method should be algorithmic and precise (with caveats about NAVs and scaling factors described in the following), not heuristic and imprecise. If compensation is to be based on this measure, it must directly calculate outperformance (as do direct alpha and KS-PME), rather than calculating heuristically (as do some other alpha methods) (Cambridge Associates 2013; Gredil, Griffiths, and Stucke 2014; Griffiths and Charles 2014).

## THE EXCESS VALUE METHOD

In the following, we describe our proposed EVM and demonstrate that it meets all five of our criteria. We show how the value created by a private market investment can be attributed to movements in the broader market (represented by a public benchmark), the alpha that a manager adds relative to this amount, and the interaction between the two. We call these amounts the beta value, alpha value, and interaction value, respectively.

Through examples, we demonstrate how the EVM calculates these return sources in dollar terms. We also describe how the EVM apportions these dollars between realized and unrealized amounts during the life of an investment. As explained earlier, this apportioning is necessary both for accurate performance measurement and for performance compensation to be timed with distributions, rather than waiting until final liquidation. We also elaborate on excess value’s key characteristic of considering movements in the reference benchmark only when capital is at risk. Failure to exclude market movements when capital is no longer at risk is why we rejected the KS-PME intuition described previously.

Later in this article, we formalize our logic by introducing the governing equations behind excess value. These equations can be used to calculate sources of return in dollar terms at any point in time, regardless of the complexity of an investment’s stream of cash flows. We conclude with a discussion of points that investors and managers will need to consider in developing compensation agreements based on excess value, as well as several implications that could result from the widespread adoption of excess value–based compensation.

We have also created an Excel template that demonstrates the calculation of excess value for the examples in this article, and we are developing an R package to do the same. The Excel template is available upon request.

### Sources of Value

In the following paragraphs, we explain the intuitions behind excess value and its components. We define excess value as comprising both alpha value and interaction value. We show how to separate these in our first, simple case, but we title their combination excess value under the assumption that LPs and GPs are generally interested in compensating for above-market performance and are less interested in the alpha/interaction distinction.

### Excess Value

Let us illustrate excess value at its highest level with a simple example. Assume that a private market manager invests $100 that grows at 15% for five years. At the end of Year 5, the value of the investment is returned as a distribution of $201. This represents a total gain of $101 (total value minus initial contribution). At the same time, the public benchmark returns 10% per year. Had the manager not invested in this private market investment but instead put the initial $100 in the public benchmark, the value in Year 5 would have been $161. The $40 difference between the private market investment and the hypothetical public investment represents excess value.^{1}

### Alpha Value

One component of this excess value is alpha value. As introductory finance courses teach, the value of any investment at inception is the net present value of the cash flows it is expected to generate. Similarly, net present value can be applied to a private market investment to calculate the value of its alpha above a benchmark. Specifically, instead of using the cost of capital or required rate of return, discount the cash flows by the benchmark return. The resulting number represents the value of the investment above the benchmark, present-valued to the inception of the investment. This number is the realized net present value of the investment; for brevity, we call it alpha value. For those familiar with the direct alpha and KS-PME calculations, this alpha value equates exactly to the sum of the market-adjusted distributions less the sum of the market-adjusted contributions used in the present value versions of both direct alpha and KS-PME (Kaplan and Schoar 2005; Gredil, Griffiths, and Stucke 2014; Griffiths and Charles 2014).

More intuitively, in our example, the component of the $40 excess value that represents alpha value is the present value in Year 0 (i.e., the date of the initial investment) of the benchmark-adjusted cash flows. This calculation is described in Equation 1, where *m*_{0} is the level of the reference benchmark in Year 0 (100) and *m*_{5} is the level in Year 5 (161). The alpha value is computed as $25. This $25 represents the value at Year 0 of the contribution that the manager will make to the return on investment over the investment life (i.e., the return above the benchmark). The remaining $15 of the $40 in excess value is the growth of this alpha value at the benchmark rate through Year 5, or what we call the interaction value. Intuitively, interaction value can be thought of as the dollars generated by investing the alpha value at the benchmark rate over the life of the investment.

Another intuitive way of thinking about interaction value is that the manager contributed $25 of alpha value measured in Year 0 alongside the LP’s initial investment of $100. Both amounts grew at the benchmark rate until distribution. We can measure the alpha value as of each period using the generalized form shown in Equation 2, depicted visually in Exhibit 1. In Exhibit 1, alpha value is as described previously (i.e., the present value at Year 0 of the value created as of each year on the *x*-axis), and *influence of market* includes the remainder of total gain and both beta value and interaction value.

### Beta Value

As described previously, influence of market in Exhibit 1 includes the entire portion of total gain that is not alpha value and includes both beta value and interaction value. In this example, measuring beta value alone is straightforward. Because beta value represents the gain that would have been achieved by investing in the benchmark, beta value equates to the profit on the initial $100 contribution compounded at the 10% per year benchmark rate, resulting in a Year 5 beta value of $61 (Equation 3).^{2}

### Interaction Value

Once we have calculated the alpha value and the beta value, calculating the interaction value is easy. In any period, the gain that remains unattributed after calculating the alpha and beta values represents the combined influence of the manager and the market and is the interaction value. We can find the interaction value in any year by subtracting the alpha value and the beta value from the total gain, as shown in Equation 4.

4In our example, the value of each component increases in each year. Alpha value increases because the private investment’s return exceeds that of the benchmark. Beta value increases in each year because the market generates a positive return. Interaction value increases because both the alpha and beta values do, and interaction value represents their effects on each other (Exhibit 2).

This example can also give us a simple view of the major distinction between compensating based on excess value versus carried interest. The investment in our example generated a $101 total gain but only $40 of excess value. Under a traditional carried interest arrangement, the performance fee to the manager would be calculated (roughly) as 20% of the $101 total gain. Under an excess value arrangement, the performance fee to the manager would be some presumably higher agreed share of the $40 excess value. Sometimes the carried interest arrangement would provide for a higher performance fee, specifically when absolute gains are high but predominantly or entirely market driven. However, sometimes the excess value arrangement would provide for a higher performance fee, such as when market performance is weak but the manager produces high alpha.

Thus far, we have kept our example intentionally simple. All three sources of return are easy to calculate in dollar terms. Disentangling these sources becomes a more involved exercise as the cash flow profile increases in complexity, but the intuition remains the same: Value creation comes from alpha, beta, and interaction.

## ILLUSTRATED CALCULATION OF EXCESS VALUE IN MORE COMPLICATED CASES

In the preceding example, the calculations of excess value, alpha value, beta value, and interaction value are straightforward. However, the real world presents practical challenges beyond this simple form. First, real-world cash flows are much more complex. With added complexity, and specifically the introduction of multiple distributions, the market-driven components of total gain (i.e., beta value and interaction value) require a more robust calculation framework. Second, the total gain created by an investment is not monolithic in terms of timing and liquidity. If LPs and GPs wish to compensate along with distributions rather than waiting for liquidation, total gain, as well as its components, measured at any point prior to the liquidation of the investment must be attributed between distributions (realized value) and NAV (unrealized value).

The EVM solves these challenges by providing a language for calculating the various components of total gain and a framework in which excess value (which in our definition includes both alpha value and interaction value, and so all returns above the benchmark) can be apportioned between distributions and remaining NAV over time. Although in practice we can mathematically separate alpha value and interaction value individually into their realized and unrealized components, we combine both in our definition of excess value, as explained earlier. Specifically, we assume that LPs and GPs are generally interested in compensating for above-market performance and are less interested in the alpha versus interaction distinction.

### Scenario I

Let us return to our simple example to illustrate where these challenges arise and how the EVM provides the tools to solve them.

The parameters of our simple example are described in Exhibit 3. The manager invests $100 that grows at 15% for five years and is then distributed as a bullet payment. During this time, the public benchmark returns 10% per year. At the end of Year 5, the private market manager returns approximately $40 more than an investment in the benchmark would have. This amount is excess value. The rest of the total gain is attributable to the benchmark alone. Algebraically, excess value in this simple case can be expressed through Equation 5 as the terminal distribution less the initial contribution scaled by the change in benchmark level.

5In this equation, *Excess value _{Year }*

_{5}is the excess value calculated at the end of Year 5,

*Dist*

_{Year }_{5}is the bullet distribution received at the end of Year 5,

*Cont*

_{Year }_{0}is the single contribution made at the end of Year 0,

*m*

_{5}is the benchmark value at the end of Year 5, and

*m*

_{0}is the benchmark level at the end of Year 0. Thus, excess value is the dollar gain above what would have been achieved through an investment in the public benchmark. It is also the future value of the alpha value grown at the benchmark rate.

### Scenario II

Scenario II modifies Scenario I so that the fund distributes 50% of remaining NAV in each year, beginning in Year 3, and then liquidates at the end of Year 5. Matters become more complicated in this case (see Exhibit 4).

With this cash flow profile, the process for computing excess value is no longer as simple as finding the future value of the alpha value at the benchmark rate. Instead, distributions prior to Year 5 (liquidation) cause a complication. To the extent that these distributions are future-valued to fund liquidation, the excess value measured will be affected by changes in the benchmark when capital is no longer at risk. Our Criterion 5 prohibits this, as described previously.

To avoid rewarding or penalizing a manager when capital is no longer at risk, the EVM introduces the concept of *cost bases*. We can think about cost bases as the portion of contributions, future-valued to the time of each distribution, that is attributable to remaining NAV and to each distribution. The cost basis of each distribution is calculated by future-valuing the contribution(s) using the change in the public market benchmark up to, and only to, the time of that distribution.

In Scenario I, the cost basis of the liquidating distribution in Year 5 is the initial contribution future-valued by the return of the market. This cost basis must not remain $100 because the benchmark increased over the time capital was at risk. Therefore, the actual value of the distribution above the benchmark growth represents the excess value of that distribution. However, in Scenario II, there are multiple distributions, and each must receive its own cost basis, which only recognizes market returns with respect to the corresponding distribution. Thus, the EVM takes the total value (NAV plus distributions) at any point in time, separates out the value of each distribution and NAV, and separately deducts the cost basis of each distribution and the cost basis of the remaining NAV (Equation 6). By doing so, the EVM apportions the cumulative excess value between realized distributions and unrealized NAV.

6In this equation, *Excess value _{t}* is the cumulative excess value at time

*t*,

*NAV*is the remaining NAV at time

_{t}*t*,

*CB*is the cost basis of this NAV (the proportion of contributions attributable to this NAV, future-valued to time

_{NAV,t}*t*),

*Dist*is the value of each distribution at the date of the distribution (not future-valued to time

_{k}*t*but frozen at time

*k*of the distribution, so the manager does not benefit from or incur a penalty for later market moves), and

*CB*is the cost basis of each distribution (the proportion of contributions attributable to each distribution, future-valued only to the date of the distribution at time

_{Dist,k}*k*).

Importantly, this cost basis freezing is necessary not only for calculating excess value but also for calculating beta value. Specifically, beta value is the sum of all cost bases (i.e., contributions future-valued by the benchmark up to the date of each distribution) minus total nominal (i.e., not future-valued) contributions (Equation 7). Intuitively, beta value is the return on invested capital attributed only to the benchmark and only when the capital under consideration is at risk.

7With these principles in mind, let us return to Scenario II and examine the period in this hypothetical fund life that most closely aligns with the simple bullet payment example (Scenario I). During Years 1 and 2, prior to any distribution, total value has a single component: NAV at the end of each year. The EVM in each of these years follows the same procedure as for Scenario I. In this case, we can take Equation 6 and cancel out its second half because there have been no distributions. We can then restate the cost basis of NAV, or *CB _{NAV,t}*, as the initial contribution future-valued to time

*t*via our market adjustment. This term becomes , where

*Cont*

_{0}equals the contribution made in Year 0, and

*m*and

_{t}*m*

_{0}are the benchmark levels in Year

*t*and Year 0, respectively. The result is Equation 8, which is functionally the same as Equation 5 in Scenario I.

Exhibit 5 illustrates the value components in Scenario II through Year 2. As we should expect, excess value is an increasing positive amount, consistent with NAV growth being in excess of benchmark growth by 5% each year. Likewise, beta value (represented in Exhibit 5 as the difference between the cost basis of NAV and nominal contributions) increases each year, driven by a positive benchmark return.

As Exhibit 6 shows, in Year 3, the investment begins to distribute. Up to this point, we have only considered examples in which total value has a single component: unrealized NAV or a liquidating distribution. However, at Year 3, as the investment begins to distribute while unrealized NAV remains, we must account for the excess value attributable to both.

Using the concept of cost bases, the EVM apportions the cost basis (i.e., future-valued contribution) between the distribution and remaining NAV. For now, we will apportion the cost basis proportionally; that is, the cost basis of the distribution in Year 3 is the percentage of total remaining value that the distribution represents, , multiplied by the future value of the contribution to Year 3 (Equation 9). The same process is applied to the remaining NAV (Equation 10) using the parallel scaling factor, . Exhibit 6 illustrates these cost bases and the excess value through Year 3.

9 10In the year of this first distribution, the scaling factor used to do this apportioning (recall that we used a scaling factor that apportions proportionately) does not affect the excess value measurement because the total value is simply the NAV plus that distribution. How we split the cost basis between them does not affect the cumulative measure of excess value. However, the assignment of cost basis to a distribution becomes critical when we calculate the cost basis of the next distribution and every one that follows. We will elaborate later on the choice of scaling factor and its implications for excess value calculations.

Let us now consider the application of the EVM in periods following the first distribution. Through Year 3, excess value amounts simply to the total value in each year minus its cost basis (i.e., the future-valued contribution). However, starting in Year 4, this relationship no longer follows this simple form. When a distribution is returned to the investor, that value is no longer at risk. Per our Criterion 5, the measurement of value created by the manager must be independent of market returns that take place after that capital is returned to investors. Subsequent benchmark movements should not benefit or penalize the manager.

Thus, in each period following the first distribution, we must exclude the change in value of the cost bases attributable to past distributions, just as we froze the distributions themselves in Equation 6. As Equation 11 shows, the cost basis of Year 4’s distribution is calculated as follows: We begin with the total future-valued contribution to Year 4 . We then subtract from this the cost basis of Year 3’s distribution measured at Year 3, adjusted for the benchmark return between Year 3 and Year 4 . We future-value the cost basis of the Year 3 distribution in this way because we need to account for the entire future value of the contribution as of Year 4. However, the excess value attributable to the Year 3 distribution remains frozen at the value calculated in Year 3. We then multiply this result (i.e., the term in brackets in Equation 11) by the scaling factor that apportions the remaining cost basis between the distribution and remaining NAV . The cost basis of NAV is calculated similarly, differing only in the numerator of the scaling factor.

11 12In effect, we freeze the cost basis for each distribution (and each distribution itself) at the time of said distribution. By doing so, we remove a phantom excess value that would result from market-adjusting distributions and their cost bases beyond the distribution dates.

Equations 13 and 14 demonstrate how the excess value cost bases are calculated in Year 5.

13 14To illustrate the evolution of cost bases and cumulative excess value, Exhibit 7 shows the cost basis of the remaining NAV, the cost basis of the current distribution, the cost basis of previous distributions, and excess value over the life of our hypothetical Scenario II investment.

Exhibit 8 shows the same Scenario II total value evolution but expressed in terms of the sources of this value.

Exhibit 9 shows the calculated cumulative excess value and beta value in each year using the calculations and scaling factor used in this example.

We can also use our Scenario II to examine how compensation to the GP would differ between a traditional carried interest arrangement and an excess value arrangement. Under a traditional carried interest arrangement, the manager would receive (roughly) 20% of the $70 total gain of the investment ($43 cumulative beta value plus $27 cumulative excess value). Under an excess value arrangement, the manager would receive a presumably higher share of the $27 excess value. Unlike our simpler Scenario I, in this case, the scaling factor matters because it determines the division of excess value into realized and unrealized portions at the time of each distribution and it affects the total excess value calculated in periods after the first distribution.

### The Scaling Factor

In this example, the scaling factor we used to apportion cost basis between distributions and remaining NAV was the ratio between each and the sum of the two. Note that this is the same scaling factor used by Cambridge Associates in their mPME method of calculating alpha heuristically (Cambridge Associates 2013). In other words, the cost basis of a distribution and of NAV were determined proportionally by and , respectively.

As it turns out, this split of cost basis is not the only way to apportion. Although there are practical advantages to this choice of scaling factor, an infinite number of mathematically viable scaling factors can be used. An argument can be made that for a very transparent, simple fund that consists of several discrete underlying investments (e.g., portfolio companies in a private equity fund), each with just one distribution, excess value should be calculated separately for each investment. When each investment is realized, its excess value is crystallized, and the total excess value of the fund then equals the sum of the excess value of the underlying investments. Unfortunately, in our experience few real-world funds are this simple. Investments typically result in multiple distributions, managers often report aggregate cash flows that represent multiple underlying investments, and even contributions can sometimes be difficult to attribute cleanly to individual investments. The proportional scaling factor that we propose in this article is intended to reasonably and simply approximate the result of this correct scaling factor for more realistic funds whose construction is not this simple.

Other alternatives are possible as well. For example, if the parties wish to reach the carried interest-like agreement with a market-based preferred return, as described at the beginning of this article, they can accomplish this goal by using a scaling factor that sets the cost basis of each distribution equal to the minimum of the distribution and the remaining cost basis and then attributes the remaining cost basis to NAV. Alternatively, if they wish to treat the incremental excess value created as of the time of each distribution as being realized in full, they can accomplish this goal by setting the cost basis of NAV in each period equal to NAV and attributing the remaining cost basis to the distribution. There are drawbacks to these scaling factors, which are beyond the scope of this article.

Although we do not delve deeply into these scaling factors here, we do wish to note that the scaling factor is a decision that must be agreed upon by both the GP and LP in any compensation arrangement because it can influence both the dollar amount of excess value that is attributable to each distribution and the cumulative excess value amount itself. We should also note that although there may be alternative ways to produce calculations that achieve identical compensation results for specific scaling factors (e.g., there are other calculation methods that produce identical results for preferred return–based compensation or compensation that attributes all excess value to distributions), the EVM is the only method that we know of that accommodates all of these different methods (via the scaling factor) and facilitates robust performance assessment by dividing the sources of return into alpha, beta, interaction, realized, and unrealized.

### A Generalized Form

Scenarios I and II highlighted the mechanics and intuitions behind the EVM. Thus far, we have demonstrated the concept of cost basis and how it provides a framework to calculate the alpha, beta, and interaction values in dollar terms. We also described how excess value algorithmically apportions these values to realized and unrealized amounts, which is necessary for enabling real-time compensation arrangements.

Up to this point, our examples have had short fund lives and unrealistically simple cash flow profiles. However, we can expand the EVM to a generalized form that can be applied to any private market investment, regardless of the complexity of its cash flow stream. Together, Equations 15–18 describe this generalized form.

15 16 17 18Our examples use the scaling factor described earlier. In this generalized form, the EVM can use any scaling factor that satisfies Equation 18, which simply means that the scaling factor must attribute 100% of the remaining cost basis at the time of a distribution between the distribution and remaining NAV.

### Scenario III

To demonstrate the EVM with a cash flow and performance profile closer to the real world, Exhibit 10 presents an investment that moves between out- and underperformance relative to its benchmark.

Scenario III has the private investment outpacing the public benchmark in Years 1–3, falling below in Years 4–6, and finally returning to outperformance in Years 7–10. The investment begins distributing capital in Year 2, so the EVM must exclude the effects of market movements on each cost basis after its corresponding distribution takes place. In all years, the benchmark returns 10% per annum. Thus, beta value should be an increasing, positive amount.

Exhibit 11 illustrates each of the return components of this hypothetical investment over time, using our equations and the proportional scaling factor. The top of the chart shows the investment’s direct alpha at each point in time.

In this scenario (as well as the earlier examples), direct alpha and excess value move directionally in tandem. In each year in which direct alpha is negative, indicating underperformance relative to the benchmark, excess value is accordingly negative. Likewise, in Year 8, when direct alpha moves from negative back to positive, excess value follows suit. In real-world situations, this will not always hold true for the cumulative excess value amount. Although alpha value will always match the direction of direct alpha, the other component of excess value, interaction value, can be driven negative if alpha is positive and the benchmark declines, or vice versa.

## CONSIDERATIONS FOR IMPLEMENTING THE EVM

### Benchmarking and Beta

As is the case for any comparison of private and public markets, benchmark selection is an important step in measuring a private investment’s out- or underperformance. This applies to excess value in the same way that it does to direct alpha and KS-PME. A growing body of literature is examining the factor exposures of private market asset classes and how they compare to public market benchmarks. Two recent examples from within the pages of *The Journal of Portfolio Management* are by Bass, Gladstone, and Ang (2017), who proposed a factor-based approach to portfolio construction across public and private asset classes, and Arnold, Ling, and Naranjo (2019), who examined the risk exposures of private equity real estate. Outside of this journal, Goetzmann, Gourier, and Phalippou (2019) examined the factor exposures of several types of private market funds, and Blanc-Brude and Tran (2019) examined the factor exposures of private infrastructure.

In line with this literature, when using excess value for performance measurement purposes, the benchmark should capture systematic factors that theory and/or empirical evidence indicate contribute to investment performance but are outside of the manager’s skill and therefore are not related to alpha. For a private equity fund, we believe that appropriate benchmarks should capture broad equity exposures as well as sector and geography exposures that match the fund’s portfolio companies. Similar benchmarking can be done for private real estate, private infrastructure, private credit and other private asset classes.

Admittedly, some discretion applies when selecting these factors. For example, a specialized manager investing exclusively in a single industry, say manufacturing, would probably be most appropriately benchmarked against the performance of comparable public manufacturing companies. In contrast, when benchmarking a broadly diversified private equity manager, the calculating party would need to consider whether it considers sector rotation to be a potential source of skill. If so, a broad benchmark such as the Russell 2000 or MSCI World may be most appropriate. If not, then a benchmark that tracks each company against its own sector (or other factor) comparables may be most appropriate.

When using the EVM for compensation purposes, we believe that similar considerations should be taken into account, but with an eye toward providing the manager with the desired incentives rather than toward calculating truer alpha per se. For example, the parties to a compensation agreement would need to consider whether the GP should be compensated for successful sector rotation, regardless of whether this constitutes true skill. Such may be the case if an investor chooses a generalist private equity fund as an alternative to a broad benchmark but wants to give the manager leeway to choose its sector weights.

Leverage is another source of systematic risk that can influence an investment’s return and should be considered when selecting a benchmark. Debt financing is an important feature of many private market investments and may not necessarily constitute manager skill if this return can be mirrored by levering an index of public securities. Therefore, for performance measurement purposes, we argue that the calculating party should consider whether leverage used by the manager can be easily replicated by an investor in public markets (and thus does not represent manager skill) or whether it represents a manager’s ability to use its insight into its value-creation process to strategically lever its alpha or to otherwise use leverage in ways unavailable to public market investors. In the former case, the benchmark should be adjusted to match the leverage used by the manager. In the latter case, perhaps it should not be. As in the case of benchmark selection, the treatment of leverage may also be different for compensation purposes than for performance measurement purposes. For example, an investor may want to apply a lower beta for compensation purposes than for performance measurement purposes to provide incentive for a manager to use more aggressive leverage when it believes doing so will create alpha.

We do not cover in any depth within this article the topic of the appropriate values of beta to use when comparing different types of private market investments to different public benchmarks. However, a body of literature addresses this question, and we are confident that both academics and practitioners can draw from this literature to make useful calculations of excess value, just as they do for KS-PME and direct alpha, in which the same benchmarking considerations apply. Though the betas of private market assets classes cannot be calculated directly because private assets are infrequently traded (Kaplan and Schoar 2005), the literature does offer plausible ranges (Korteweg 2019). In general, betas above 1.0 have been estimated for both venture capital and buyout relative to broad public equity benchmarks. However, the range of estimates is wide; studies suggests betas as high as 2.7 and as low as 1.0 and 0.7 for venture and buyout, respectively (Korteweg 2019). When using direct alpha, KS-PME, and excess value ourselves, we generally apply a beta of 1.0 to 1.5 to the relevant public market benchmarks, increasing the beta for managers who use more leverage or who operate in asset classes in which the impact of leverage on risk is clearer or more pronounced. We believe that, as a rule, investors in private markets should incorporate their own evidence-based views on private market risks into portfolio construction and manager selection and that the selection of appropriate betas is achievable, even if they cannot be estimated with ideal precision.

**NAV smoothing.** Many of the practical scaling factors, including the proportional scaling factor used in Scenarios I–III, take interim NAVs as inputs to differentiate between realized and unrealized excess value. Although we expect that the proportional scaling factor most closely approximates the theoretically correct apportioning of excess value between distributions and remaining NAV, reliance on interim reported NAVs does mean that these estimates are partially governed by unrealized marks and the paths they take over time. In illiquid asset classes, these reported valuations have been shown to be smoothed (Jenkinson, Sousa, and Stucke 2013) and may be slower to incorporate information than do their public benchmarks.

To the extent that NAVs are smoothed and do not reflect a true trading value of the private assets, EVM estimates may over- or understate the amount of excess value created in a given period. Over longer periods of time, this disconnect should lessen as investments are realized and returns made concrete. However, the path dependency of the EVM with the proportional scaling factor means that users should pay attention to how close reported valuations are to their true values to ensure the highest degree of the EVM accuracy. Special care should be taken for periods of increased market volatility when this smoothing effect can have a more pronounced impact on price differences between the benchmark and the private market investments. In these cases, longer windows of time can be used to more accurately reflect sustaining excess value.

When the EVM is used for compensation purposes, there are additional ways to address NAV smoothing, including spreading out incentive payments over time or conducting deal-by-deal excess value calculations. Additionally, there are alternative scaling factors to the proportional that do not require NAV estimates to be useful. One of these is the scaling factor described earlier that leads realized excess value to reflect a European-style waterfall with a compounding preferred return and no catch up. When using this scaling factor, the cost basis associated with a distribution is independent from any calculation of unrealized NAV. Thus, the realized component of excess value (cost basis associated with distributions) can be found without knowing the NAV time series. This scaling factor may not be ideal for performance measurement purposes but may find applications in EVM-based compensation arrangements.

## CONCLUSIONS

In this article, we introduced a framework to compute the value, in dollars, of the gain in a private market investment attributable to alpha, beta, and their interaction (what we call alpha value, beta value, and interaction value, respectively). We also introduced the intuition behind these values. We defined the combination of alpha value and interaction value as excess value, and we showed how to disaggregate it into realized and unrealized amounts over the life of an investment.

We demonstrated that this method meets all five of our criteria for validating a solution. It expresses alpha, beta, and their interaction (Criterion 2) in dollars (Criterion 1). It apportions the components of value between realized and unrealized amounts (Criterion 3). (As mentioned earlier, we combine alpha value and interaction value into one excess value, but we can easily separate them.) The EVM explicitly excludes the effects of market movement on capital that is not at risk (Criterion 4). Finally, it is directly linked to the established measures of alpha in rate (direct alpha) and multiple (KS-PME) forms and represents a precise algorithm, rather than a heuristic (Criterion 5).

By meeting these criteria, the EVM enables investors and managers to better understand where and when value relative to a benchmark is created. It also enables parties who wish to compensate based on value relative to a benchmark to form such agreements.

As should be clear, compensating based on excess value is substantively different from traditional carried interest. Under traditional carried interest, GPs are compensated for absolute returns regardless of market moves. If a fund (or other investment) generates a large profit because of a strong market, the GP is paid a performance fee regardless of whether they added alpha. If a fund fails to hit the preferred return solely because of a weak market, the GP is not paid a performance fee, again regardless of whether they added alpha. The EVM enables LPs to compensate only alpha-producing GPs and only for value created in excess of a benchmark.

As to how excess value–based compensation frameworks may be structured, these may take any number of forms. One can imagine compensation waterfalls based on splits of excess value amounts and/or preferred return hurdles negotiated on its basis. One can also imagine agreements starting from the point that beta value properly belongs to the LP, alpha value to the GP, and interaction value to them both. The specifics of how these compensation arrangements could be structured is outside the scope of our discussion. However, we note that proper incentives and alignment will have to be part of any well-structured agreement.

Should investors and managers wish to develop compensation agreements based on excess value as opposed to carried interest, they will need to take several considerations into account:

▪ The choice of benchmark will be very important. The parties will need to choose a benchmark that reflects the desired systematic exposures of the private investment as accurately as possible. Sector and geography exposures, as well as other factors, should be considerations.

▪ The parties will also need to account for the amount of leverage a manager uses and how this may increase the beta applied to any benchmark.

▪ Subscription lines are an additional consideration because they lead to timing differences between cash flows into investments and cash flows from LPs. Subscription lines can be handled in several ways under the EVM. For performance measurement, excess value can be calculated excluding or including the effects of the line. For compensation, the parties should agree on how they would like to treat it.

▪ The portion of excess value that the manager keeps as a fee will need to be negotiated.

▪ If investors and managers agree that compensation should not wait until final liquidation of the investment, the timing of payments and the potential establishment of high-water marks and clawbacks will need to be negotiated. Additionally, investors and managers will need to agree on a scaling factor to determine the share of excess value that is realized in each period.

▪ When using a scaling factor that is partially governed by interim NAVs, such as the proportional scaling factor, incremental excess value estimates may see heighted volatility when there are dramatic moves in the benchmark. We recommend that compensation arrangements be constructed with this in mind by some combination of spreading incentive payments out over time, conducting deal-by-deal excess value calculations, using a NAV-independent scaling factor, or using other means to counterbalance the potential short-term effects this point may have on excess value.

▪ Increased attention will also need to be paid to the accuracy of reported NAVs because they might become not just indications of interim investment performance but also drivers of compensation (Brown, Gredil, and Kaplan 2019).

▪ Investors may need to compensate managers for excess value, even if absolute performance is negative.

▪ Compensation agreements will likely need to continue to maintain a management fee payment, perhaps as a draw against future excess value.

▪ LPs and managers will need to consider whether they want to minimize payments for luck (nontrue alpha outperformance) and how they may do so. Applying the EVM to more diversified portfolios, less prone to outlier impacts, may be one way to do this.

Overall, the EVM would add some complexity when initially implemented but could add better incentive alignment to compensation agreements and meaningful richness to diligence processes.

On a more macro level, we believe that the widespread adoption of compensation agreements based on the EVM would have the following notable implications:

▪ Consistently high-alpha managers would likely be paid more, and more consistently, under an EVM than under a traditional carried interest arrangement because their excess value production would be spotlighted. They potentially (depending on the negotiated agreement) would be paid even when market returns, and their absolute returns, are low or negative.

▪ If positive-alpha producers manage some pools of assets under traditional carried interest arrangements and others under EVM-based fee arrangements, these managers would diversify their revenue streams through this combination of a stream that is more closely tied to market performance (carried interest) and a stream that is more closely tied to outperformance versus the market and thus is more likely to pay out in weaker markets (EVM). This diversification could give these managers an advantage in the important quest to attract and retain quality talent by giving more predictability and consistency to performance fee payouts.

▪ Consistently negative-alpha managers would likely see their performance fees meaningfully reduced or eliminated, regardless of their absolute returns.

## ACKNOWLEDGMENTS

The authors would like to thank Professors Oleg Gredil of Tulane University, David Robinson of Duke University, Gregory Brown of the University of North Carolina, and an anonymous reviewer from *The Journal of Portfolio Management*, who provided insightful comments as we developed this article.

## ENDNOTES

↵

^{1}In this simple case with just one distribution, our KS-PME intuition would produce an identical result.↵

^{2}Our examples assume an environment akin to the real world, in which the manager controls the timing of contributions and distributions. Thus, our beta value does not necessarily equal a market return over the entire contractual life of the investment but, rather, only the market impact on capital at risk for the time period during which the manager decides to put it at risk. This distinction will become more apparent in our Scenarios II and III. For the purpose of designing a compensation plan, we assume that LPs will balance the timing of these exposures via their public market portfolio or other assets and thus are not interested in rewarding or penalizing the manager for these timing decisions beyond the fact that the interaction value grows more in up markets.

- © 2021 Pageant Media Ltd