# Key Points in Valuing Profits Interests

As seen in Valuation Strategies: Key Points in Valuing Profits Interests

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. Such an equity structure is most frequently populated by capital units and incentive units.^{1} Capital units are defined by contributed capital invested at any point in time; where generally the largest investment occurs at formation.^{2} Frequently, these units will accrue a preferred yield over time that provides a baseline return to the contributed capital in priority to all other classes of dilutive units. Consider the corporate analog to a capital unit as being a share of preferred stock with participation rights.

Incentive units represent profits interests. Their formation is not a reflection of contributed capital. Rather, 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 two significant occurrences: (1) the incentive unit meets the vesting criteria, and (2) the future equity distribution is enough to provide the capital units with a complete return of their contributed capital and the accrued preferred return.

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

**Complicating Factors. **Knowing the broader valuation approach is certainly a good first step, but 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 the 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, is incomplete.

#### Black-Scholes Model

The Black-Scholes model is referred to as a closed-form model, and also as an equation or a solution. Given a certain set of criteria and defined relevant inputs, the Black-Scholes model is a solution to the desired outcome, that of pricing an option.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, through the use of a numerical method, in simply a broader, more generalized approach to option pricing. Thus, such a departure from the use of the Black-Scholes model is not a departure from its underlying pricing fundamentals; it is rather 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 basic inputs, and may include a sixth input if dividends are expected; each input is considered below with respect to incentive units:

1. *Stock price (S).* There is not one; this is the big reason a stand-alone use of Black-Scholes is not considered to be an applicable model. An incentive unit is not a derivative of another class of stock the way a stock option is a derivative of a share of common stock. Instead, an incentive unit is a derivative of the company’s total equity value vis-a-vis the capital units at any point in time. The stock price then must be the company’s total equity value which by extension requires that all the equity classes be considered in the approach.

2. *Strike price (X).* There is not one, at least explicitly. Rather, there are implied strike prices that allow the option pricing approach. These generally consist of the capital units’ contributed capital plus preferred return, as well as various participation thresholds associated with equity returns to the capital units. These thresholds represent performance vesting conditions, also referred to as a market condition, for the incentive unit classes.

3. *Term (T).* Incentive units do not have a contractual life as an option. Instead, the term is the expected timing to a future liquidity event, generally a sale (M&A) or an initial public offering (IPO). Typically, time-vested incentive units receive accelerated vesting for any portion that is unvested at the time of a liquidity event, while performance units are vested or forfeited based on the ending equity value at the time of a liquidity event.

4. *Volatility (b).*Volatility here refers to the total equity volatility. The effect of leverage should generally be considered given the tendency for an LLC 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.

5. *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.

6. *Dividend (y).* Since there is not a “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 serves to reduce the value of a dilutive security. LLCs may pay large one-time dividends to provide a return to the capital unitholders. Such dividends serve to reduce the total equity value, but also lower the participation thresholds for all classes of incentive units. These one-time dividends are not modeled unless specifically anticipated; any dividends prior to the valuation date are included in the threshold measures.

**Similar to Options.** A discussion of the inputs to an option pricing approach includes both 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 the correct one. Thus, the valuation methodology is a volatility/time model, not a discounted cash flow. The larger implication is the irrelevance of the required rate of return, which fits exactly 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—that of estimating the risk-adjusted required rate of return.

#### Model Framework

Having established option-pricing as the correct approach for valuation of profits interests, the next step is to apply the approach to estimating fair value at any point in time. To start, think about the incentive units 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 to 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 the underlying stock. The equity values in between make the model useful.

**Total Equity Value. **Model selection and construction begins with consideration of the company’s total equity value.The total equity value serves as the underlying asset (S), which necessitates the inclusion of all outstanding capital and incentive units in the modeling process.This underscores the theme that profits interests are purely dilutive securities without a link to the value of another class of equity. As a class, their future equity distributions are a function 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 incentive units based on ending equity values.
- Incorporate vesting requirements through implied strike prices.
- Model the distribution of ending equity values based on the volatility and term inputs.

**Numerical Methods. **Numerical methods—primarily Monte Carlo simulation and lattice (e.g. binomial) models—are well suited to handle all three of these elements necessary to value profits interests. These methods have the advantage of beginning with the end in mind; that is they start with an ending equity value and can tailor the equity participation accordingly. An alternative to using 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. But, the fact that the OPM is based on the Black-Scholes option pricing model means that this model cannot tailor the equity participation to a singular ending equity value. Capturing the underlying participation of securities relative to an ending equity value is a necessity when valuing profits interests. That is the primary strength of a numerical method, relative to the use of the Black-Scholes formula.

**Waterfall. ** In a literal sense, the most direct valuation thought process begins with the liquidity event and works backward in time to a present value. A single ending equity value would be allocated across all the units in priority and according to vesting outcomes. This is commonly referred to as 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, of course, 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 even 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 serve to determine the pro rata participation at future equity levels. Vesting—both time and performance-based serve only to allocate equity beyond the capital units’ baseline preference. As such, all profits interests are subject to zero participation.

**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 in number, so does the complexity of the equity participation and the valuation. Typically, the longer-lived LLC structures have numerous classes of outstanding profits interests, the result of various grants that often have their own vesting targets. As the priority of equity distributions among capital and incentive units continues to evolve, the valuation modeling becomes more nuanced favoring the selection of a numerical method.

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

#### Digital Option Correction

Consider a portfolio company with $200 million of invested equity capital and a 10% preferred yield. In one year, a liquidity event results in a $270 million equity value. 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.
- All time-vested units would become vested based on the liquidity event and be eligible for 5% pro rata participation.
- The capital units need to receive $30 million in order 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 incentive units, and the conditional dilution of performance units. The advantage of a numerical method is made clear 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 then 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, the reason a time-vested unit will always be valued more than a performance-vest unit is 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 Blind Spot

The term “digital option” refers to a single cash flow that serves as a catchup to correct for the OPM’s blind spot. Since the total equity value gap between preferred return 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:

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

This dollar amount is 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 simple. 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 are applicable to valuing a class of incentive units. Valuing profits interests simply 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, recognize that the valuation complexity is driven more by detail than a 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 with respect to the capital units and performance vesting.

In brief summary, the valuation of profits interests involves three critical components: (1) assessment of the total equity value, (2) recognition of profits interests as options, and (3) 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 the context of option math that adjusts for the cumulative probability of exceeding the threshold N(d2) and present value using a risk-free rate.**Disaggregate refers to the process of 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.*