# Key Points in Valuing Profits Interests

*Estimated reading time: 12 minutes*

The valuation of profits interests is often an undertaking for private equity (PE) firms that acquire an enterprise. Typically the PE firm organizes the portfolio company in the form of a limited liability company (LLC) and capitalizes it in combination with much smaller equity investments from management or other interested parties, along with a target amount of debt financing. Capital units and incentive units most frequently populate such an equity structure.^{1} Capital units are defined by contributed capital invested at any time, where generally, the most significant investment occurs at formation.^{2} Frequently, these units will accrue a preferred yield over time providing a baseline return to the contributed capital in priority to all other dilutive units. Consider the corporate analog to a capital unit as a share of preferred stock with participation rights.

Incentive units represent profits interests. Their formation is not a reflection of contributed capital. Instead, they are purely dilutive units that number far less than the capital units and frequently represent less than 15% of total equity on a fully diluted basis. An incentive unit’s corporate analog is the stock option.

Authorized by the LLC’s board, these profits interests are defined by their vesting criteria and are eligible to participate in future equity distributions given three significant occurrences:

- The incentive unit meets the vesting criteria,
- The future equity distribution is enough to provide the capital units with a complete return of their contributed capital and the accrued preferred return, and
- Aggregate distributions exceed the defined “participation threshold” (can also be called hurdle amount or distribution amount).

Typically as of the initial sponsor acquisition, items 2 and 3 are one and the same.

### Two-Part Discussion

The valuation of profits interests begins with a two-part discussion. The first part is more familiar, involving the aggregate valuation of the enterprise or LLC through traditional methods, including the income and market approaches frequently applied in valuing corporate entities. An estimate of the total equity value is made after appropriately considering debt. Equity value could also be implied through a back-solve approach based on a recent transaction with sophisticated investors.

The second part involves a separate undertaking that considers the values of the individual capital and profits interests in the context of the total equity value. As an equity class, profits interests are highly customizable and take many forms. However, one general valuation approach is universally applicable—option-pricing models. This is the first critical insight into the valuation of profits interests.

**Complicating Factors. **Knowing the broader valuation approach is undoubtedly an excellent first step. Still, deeper consideration may be required as there are several possible complicating nuances distinguishing a profits interest from a traditional stock option and, therefore, the suitableness of one option-pricing model over another. For instance, it will be seen that a stand-alone Black-Scholes model is not applicable when valuing an incentive unit. This may be surprising given that it is suggested that an option pricing model is an appropriate approach to estimating an incentive unit’s fair value and the fact that the Black-Scholes formulation is the most famous and frequently applied option pricing model. However, this idea will be built on below, as the context, for now, still needs to be completed.

### Black-Scholes Model

The Black-Scholes model is used to value options and is referred to as a closed-form model and an equation or a solution. The model is inflexible and must strictly fit and adhere to certain conditions. However, the principles underlying the concept and approach to valuing an option may be applied in less restrictive environments, using a numerical method, in a broader, more generalized approach to option pricing.

Such a departure from using the Black-Scholes model is not a departure from its underlying pricing fundamentals; it is instead just a relaxation of the strict formulation to allow for other intervening components relevant to the valuation. The nature of profits interests usually requires a more generalized application, but it will be seen that one common approach is a multi-layered application of the Black-Scholes formula, often one that requires a simple external correction or adjustment that is highlighted below.

**Inputs. ** A Black-Scholes model—and by extension, any option pricing model—requires five essential inputs and may include a sixth input if dividends are expected; each input is considered below concerning profits interests:

*Stock price (S).*Typically, a profits interest unit is a derivative of the company’s total equity value vis-à-vis the capital units at any time. The stock price then must be the company’s total equity value, which requires that all the equity classes be considered in the approach.-
*Strike price (X).*In LLC structures, the strike price takes the form of the previously discussed participation threshold, typically set at the total equity level. *Term (T).*Profits interests do not have a contractual life as an option. Instead, the term is the expected timing to a future liquidity event in a sale (M&A) scenario. Typically in a sale scenario, time-vested profits interests receive accelerated vesting for any unvested portion at the sale. At the same time, performance units are vested or forfeited based on the distributable proceeds. The profits interest may survive an initial public offering (IPO) depending on the terms of the agreement.*Volatility (b).*Volatility in the LLC context refers to total equity volatility. The effect of leverage should generally be considered, given that an LLC tends to have a meaningful amount of debt and a greater leverage ratio than other industry participants. Profits interests are the most volatile equity class of units in an LLC since their access to distributions is limited to upside valuations of the total equity.*Risk-free rate (r).*The risk-free rate is the appropriate discount rate in an option pricing model and is measured for a holding period commensurate with the expected term or time to liquidity.*Dividend (y).*Since there is no “stock price,” there also is no dividend yield, at least in the traditional sense of the Black-Scholes model. However, there frequently is a referred return afforded to the capital units. This is explicitly considered and, similar to a dividend yield on a stock, reduces the value of a dilutive security. LLCs may pay significant one-time dividends to provide a return to the capital unitholders. Such dividends reduce the total equity value and lower the participation thresholds for all classes of profits interests outstanding. These one-time dividends are not modeled unless specifically anticipated; any dividends before the valuation date are considered and reflected in the adjusted strike price and performance threshold. Tax distributions may also count toward distribution and performance thresholds and be treated as an advance depending on the terms of the award agreement.

**Similar to Options.** A discussion of the inputs to an option pricing approach includes the unique aspects of profits interests and the similarities of this class of equity to an option. Concretely, the payoff mechanism associated with the profits interests makes an option pricing approach correct. Thus, the valuation methodology is a volatility/time model, not a discounted cash flow.

The more significant implication is the irrelevance of the required rate of return, which fits precisely with the nature of these securities having no contributed capital. The risk-free rate serves as the appropriate discount factor and may readily be observed in the market and applied to a term commensurate with the expected time to liquidity. This avoids a potentially significant problem—estimating the risk-adjusted required rate of return.

### Model Framework

Having established option pricing as the correct approach for the valuation of profits interests, the next step is to apply the approach to estimating fair value at any time. To start, think about the profits interests at a high level. Like an option at the “extremes,” meaning having either very low equity values (deep out-of-the-money) or very high equity values (deep-in-the-money), a profits interest will track closely either to a zero value or a fully diluted unit adjusted for the capital units’ contributed capital. This is similar in concept to a traditional stock option which can never be valued less than zero nor greater than its theoretical underlying stock. The equity values in between make the model useful.

**Total Equity Value. **Model selection and construction begin with a consideration of the company’s total equity value. The total equity value serves as the underlying security (S), which necessitates the inclusion of all outstanding equity capital and profits interests in the modeling process. This underscores the theme that profits interests are purely dilutive securities. As a class, their future equity distributions are a percentage of the company’s total equity value at the time of a liquidity event subordinate to the outstanding capital interests’ contributed capital with a preferred return and any vesting requirements.

Thus, a model approach should:

- Incorporate multiple classes of profits interests based on ending equity values.
- Incorporate vesting requirements and strike prices.
- Model the distribution of ending equity values based on the volatility and term inputs.

**Numerical Methods. **The Monte Carlo simulation is well suited to handle all three elements necessary to value profits interests. This method models the price evolution through millions of scenarios from the valuation date to the assumed liquidity date. As the price path and ending equity value are known for each simulation path, a Monte Carlo simulation is used widely to capture vesting conditions that are path dependent. An alternative to a numerical method is the option pricing method (OPM),^{3} which models multiple strike prices to capture the various changes in equity participation until a fully-diluted structure is realized. OPM is often used when the vesting conditions only depend on the “state” of the liquidity event.

**Waterfall. ** In a literal sense, the most direct valuation thought process begins with the liquidity event and works top-down from equity value to security value. A single ending equity value would be allocated across all the units in priority and according to vesting outcomes. This is commonly called a waterfall, where the distribution outcomes are known based on the realized equity value.

Without variability in the ending equity value, each security’s present value equivalent would be measured by applying a discount factor based on a risk-free rate. This would be the entire valuation process if the ending total equity value were known with certainty. But it is not. Without this luxury, basing the valuation on a single-ending equity value or a limited range of potential equity values is insufficient. Instead, the ending equity value is viewed as a distribution of values. Option pricing models all share the common premise that the ending equity value has a log-normal distribution shaped by volatility and time. A distribution of equity values that is sufficiently continuous allows for the full incorporation of the profits interests’ cash flow in a risk-neutral framework, which measures a present value equivalent for the equity interest.

### Vesting

An LLC’s profits interests are an analog of a traditional call option for the fundamental, omnipresent reason that all equity distributions and claims are subordinate to the full recovery of invested capital, represented by the capital units. Beyond this baseline preference, including any preferred yield, the vesting conditions determine the pro rata participation at future equity levels. Vesting—time and performance-based–only allocates equity beyond the capital units’ baseline preference.

**Nuanced Modeling. ** The variety of an LLC’s outstanding profits interests and their vesting requirements influence the valuation model’s construction. As the classes grow, so do the equity participation and valuation complexities. Typically, the longer-lived LLC structures have numerous classes of outstanding profits interests, resulting from various grants that often have their own vesting targets and participation thresholds. As the priority of equity distributions among capital and profits interests continue to evolve, the valuation modeling becomes more nuanced, favoring the selection of a numerical method.

A numerical method is best suited to value the progressive dilution of profits interests regardless of the equity structure’s complexity. However, an OPM model can work quite well in simple situations where a single class of profits interests is outstanding, or they are 100% time-vested, or there are only one or two performance targets. The reason is that the altered participation based on ending equity value can be captured using a “barrier option” correction to the OPM result.^{4} An example and application of this correction are explored below.

### Barrier Option Correction

Consider a portfolio company with $200 million invested equity capital and a 10% preferred yield. A liquidity event results in a $270 million equity value in one year. Suppose a single outstanding class of profits interests represents 10% of the fully diluted equity, and half is time-vested and half is performance-vested subject to a 25% internal rate of return (IRR). The following allocation would result:

- The first $220 million would be distributed 100% to the capital units, leaving $50 million distributable equity.
- Based on the liquidity event, all time-vested units would become vested (assuming accelerated vesting) and be eligible for 5% pro rata participation.
- The capital units must receive an additional $30 million for the performance units to vest based on IRR. Because the capital units are assured of receiving at least 90% (up to 95%) of the $50 million, the lower bound participation leads to a $45 million distribution, meeting the IRR condition. The time-vested and performance-vested units would each be distributed $2.5 million.

**Comparison with OPM. **This simple example highlights the immediately dilutive nature of time-vested profits interests and the conditional dilution of performance units. The advantage of a numerical method is clarified in the example by working backward from the ending equity value through the distribution waterfall. The OPM does not have this ability. Instead, in an OPM, the performance units are limited to participation at equity levels equal to or greater than the vesting targets.

That is generally not the intent of the vesting condition. Generally, vesting determines whether a unit is “in or out.” If the unit is in, meaning part of the capital structure, it has full pro-rata participation subordinated only by the capital units’ preferred return. Thus, if the equity level is sufficient, then a time-vested unit is no different than a performance-vested incentive unit. Of course, a time-vested unit will always be valued more than a performance-vest unit in the ability to participate at lower equity values, where a performance unit would be zero (out).

So, rather than being economically equivalent to a time-vesting unit, given that the ending equity value meets the performance condition, the OPM limits the performance incentive unit participation to the equity value required to vest rather than the capital units’ preferred return. This is an artifact of the model and its inability to “begin with the end.”

### Correction for Weakness

The term “barrier option” refers to a single cash flow that is an immediate catchup to correct for the OPM’s weakness. Since the equity value gap between the preferred return (exercise price) and vesting target is known, it is possible to capture the pro rata portion of this gap subject to the probability of meeting the vesting target, which is already calculated as N(d2) in the OPM for that specific target equity value. Thus, the basic correction looks like this:

*Digital Option = ($Target Equity – $ Capital Units’ Preferred Return) * N (d2) * e -rfT * pro rata share*

This dollar amount is a value missed by the OPM but captured in a numerical method. Such a correction puts the two approaches on equal footing, provided the capital structure is reasonably straightforward. As the variety of profits interests and their performance vesting grows, this form of correction becomes unwieldy and mistake-prone.

### Conclusion

The key insight to understanding profits interests is that they are options by another name; therefore, the concepts applicable to valuing an option apply to valuing a class of profits interests. Valuing profits interests requires that the valuator apply what is already known about option valuation but in a parallel framework that looks at all the traditional option inputs in a similar but slightly altered manner.

Also, the valuation complexity is driven more by detail than concept. In other words, the more there is to disaggregate, the more there is to model. And the more there is to model, the more the model selection favors a numerical method.^{5} The equity value-dependent nature of the classes of profits interests is best suited to a numerical method that employs established option pricing principles and incorporates the numerous participation thresholds concerning the capital units and performance vesting.

In summary, the valuation of profits interests involves three critical components:

- Assessment of the total equity value,
- Recognition of profits interests as options, and
- Inclusion of all the various applicable features and conditions imposed to estimate the fair value.

*Footnotes:*

*Units are the “shares” of an LLC.**Some larger LLCs may have several significant investors and multiple classes of capital units, with various per-share contributions and preferred returns.**The OPM is one of the most common equity allocation methods used to value the common stock of a private company with a complex equity capital structure where several classes of preferred stock are outstanding. The OPM is detailed extensively in the AICPA Practice Aid, Valuation of Privately-Held-Company Equity Securities Issued as Compensation.**A digital option is coined as such because it is either applicable or not and, if applicable, is linked to a fixed equity value that exceeds a specific threshold. The fixed equity value is manipulated in option math, which adjusts for the cumulative probability of exceeding the threshold N(d2) and present value using a risk-free rate.**Disaggregate refers to starting with the total equity value and working down through all outstanding classes of equity to imply a value for each where the sum of the parts equals the whole.*