A Different Risk–Return Relationship

Abstract

We challenge the widely accepted premise that the valuation of an early-stage firm is simply the capital invested (USD) divided by the equity received (%). Instead, we argue that this calculation determines the break-even point for the investor; for example, investing USD 1.0 in exchange for a 10% equity sets a firm-level free cash flow target of USD 10.0, resulting in a 0% return for the investor. The design of our study is that of a descriptive analysis of the phenomenon, based on three assumptions: that angel investing is a two-issue negotiation, that negotiation positions are communicated sequentially from capital to equity, and that the capital is fixed to a strategic trajectory. We note that when pausing the negotiation once a strategic trajectory (and thus capital) has been defined, utilizing the break-even point as the main reference point provides a structure that can serve as a guiding barometer for negotiators, as they evaluate their options across the full range of equity greater than 0% and less than 100%. We draw attention to the diminishing benefit of the marginal equity percentage point [diminishing at a rate of (−1/x²)] for the investor to break even on their investment. This relationship tracks to the equation [value = 1/equity], which presents the full option set for any offer, once the capital is determined. Our study provides the practitioner with the subtle benefit of situational awareness and the scholar with a logical foundation for future research.

1. Introduction

The phenomenon of angel investing has captured the attention of households across the globe, popularized by the nail-biting negotiations broadcast into the living room of billions of people, via television shows like the Shark Tank (USA, first aired 2009), Hakrishim (Israel, first aired 2006), Dragons’ Den (United Kingdon, first aired in 2005; Canada, first aired 2006), and Tigers of Money (Japan, first aired 2001). We presume that the appeal is largely due to the inherent risk–return relationship, where an amount of capital is put at risk (typically dollars) in exchange for a partial equity ownership in a firm that serves as a proxy for return expectations. These dramatic interactions arguably revolve around this core exchange.

The entrepreneurial finance literature also centers on this relationship, developing an analytical framework, where the quotient of capital (USD) divided by equity (%) yields firm valuation, or more precisely, post-money valuation (Da Rin and Hellmann 2020, see chap. 4). The buyer’s offer serves as a catalyst to engage the analytical framework, which aligns with the central premise in price theory (i.e., micro-economics), where value only explicitly exists if there is a buyer (i.e., demand). This starting point is reflected in Equation (1), where (INV) represents the angel investment and (F_INV) represents the percentage ownership of the firm’s future cash flow. The analysis then proceeds to derive the pre-money valuation by subtracting the investment dollars from the post-money valuation. If Equation (1) reflects the central premise, Equation (2) represents the base model of entrepreneurial finance.

$$\mathrm{Post-money}\mathrm{Valuation}({\mathrm{V}}_{\mathrm{POST}})={\displaystyle \frac{\mathrm{I}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \left(\mathrm{I}\mathrm{N}\mathrm{V}\right)}{{\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y} (\mathrm{F}}_{\mathrm{I}\mathrm{N}\mathrm{V}})}},$$

(1)

Pre-money Valuation (V_PRE) = Post-money Valuation (V_POST) − Investment (INV),

(2)

The field of entrepreneurial finance emerged as recently as the 1980s, and pertains to only a sliver of firms, referred to as entrepreneurial firms, that are growth oriented, as opposed to small firms that represent the vast majority of entities (~75%, with a median of four employees; Hurst and Pugsley 2011); entrepreneurial firms that are younger than 5 years are referred to as startups (Da Rin and Hellmann 2020). The field is therefore a subset of traditional finance addressing a subset of firms that can accommodate an investment in the space granted by growth expectations. The challenge is to unlock that perceived growth in a variety of ways, beyond just the dollars invested, such as one’s personal networks, strategic acumen, and the working relationship post-investment.

It is noteworthy that an alternative analytical framework is even necessary for the seemingly simple exchange of capital for partial equity ownership in a firm (Da Rin and Hellmann 2020; Oksoy 2020), set against the backdrop of analytical tools available from traditional finance that utilize the basic structure of a discounted cash flow analysis (DCF). Applicable to nearly all other investment decisions, a basic DCF analysis requires two inputs: an investment to initiate the effort and the set of subsequent cash flow over a period of time; the aggregate figure in relation to the duration of the analysis yields return-metrics such as IRR, NPV, etc. (Graham and Harvey 2001). However, this framework breaks down when applied to angel investing because of the limited ability to account for time due to the limited historical information at the firm level, which in turn diminishes one’s ability to utilize a discount rate. Therefore, expectations revert to the aggregate of cash flow upon exit and are typically communicated in terms of multiples of the initial investment (i.e., the cash-on-cash method)1. And the result is a different economic game; a different bet, that requires an alternative framework.

We ultimately agree with Da Rin and Hellmann (2020) on the need for an alternative analytical framework and recognize that Equations (1) and (2) also address this same core relationship. However, we identify limitations in their analytical structure as pertaining to the dichotomy between pre-money and post-money valuation alongside the calculation of pre-money as the nominal difference in post-money valuation and the investment. There are circumstances such as the Shark Tank case that attracted investment dollars even though the product was only an idea, and this example can be further generalized to a patent that has yet to be commercialized; deriving the value of the patent in terms of a post-investment outcome, and assigning only the investment portion of value to the buyer would appear to discount the role of the buyer as the main catalyst. We propose an alternative communication structure that reflects a more direct interpretation of the core transaction, while also accounting for the behavioral implications associated with the structure of this particular transaction while at the negotiation table.

With respect to the mechanics of the exchange, we develop our argument on the existing foundations of the traditional finance literature as embodied in the analytical framework of a DCF analysis. Succinctly, we note that a traditional utilization of DCF to derive the target entity’s valuation includes the acquirer’s profit expectations. This raises the question of why then does a subset of traditional finance utilize the term valuation while only accounting for the period in time ending at the break-even point? The quotient from Equation (1) reflects the amount of free cash flows that the firm must accumulate for the investor to break even on their initial investment, which excludes profit expectations. We therefore suggest that the field of entrepreneurial finance either alter the terminology in Equation (1) away from post-money valuation to the break-even point to reflect that the quotient identifies the firm’s free cash flow objective that enables the spot market investor to break even, or, that the quotient in Equation (1) is multiplied by a factor of 11.0.2 The latter suggestion reflects another nuance of this space, that the high-risk, high-reward, and entirely discretionary nature of the bet would indicate that all bets carry an initial expectation of a 10× return; otherwise, the purely return-motivated buyer would presumably preserve their dollars as afforded by their discretionary stance. That is, the buyer would presumably be unable to self-justify any bet that does not have a 10× return (or higher), because of the risks involved, when ~70% of bets result in a total loss.

Perhaps more important are the behavioral implications for practitioners once we accept that Equation (1) reflects the break-even point. Our position is built on three assumptions: (1) that angel investing is a two-issue negotiation, (2) that negotiation positions are communicated sequentially from capital to equity, and (3) that capital is fixed to a strategic trajectory. Just prior to articulating a position on equity, if we push pause on the buyer’s mental deliberations as they consider their option set, we point out that the break-even point tied to each equity position exhibits an exponential distribution that can be generalized to the function y = 1/x, where x represents equity ranging from 0% to 100%, y represents a cash flow multiplier of the initial investment to break even, and 1 reflects the generalized decision to invest [whatever nominal amount] versus a decision to not invest (0). We note that the buyer is afforded the luxury of directly influencing value due to the subjectivity in growth expectations in relation to their discretionary stance and their inherent power advantage as a buyer (Bazerman et al. 1985; Emerson 1962). Behaviorally, this means that the buyer can untether the logic of the capital position from that of the equity position when communicating an offer, and we envision that those mental deliberations just prior to an offer utilize the ceteris paribus principle holding the investment as a constant (i.e., equal to 1) while increasing or decreasing the equity dial.

We recognize that this calibration exercise will vary according to the nuances encountered with the marginal opportunity, where the strategic trajectory of each investment will differ alongside the capabilities of the entrepreneur. However, we point out that this calibration exercise follows a pattern in the form of an exponential distribution. Furthermore, this is a point reached during the negotiation in every angel investment, and we draw attention to the implications of being aware of the pattern of one’s option set. The exponential nature of the distribution function conveys the key insight: that with respect to the investor’s ability to break even on their initial investment from the accumulation of cash flows from operational activity (i.e., traversing from −100% to a 0% return), the benefit of the marginal equity percentage for the investor diminishes at a rate of [−1/(x²)]. While the break-even point serves as a less sophisticated return metric, as compared to DCF analyses, we note that this informal sliver of the economy significantly revolves around this point; the alternative framework presented in Equations (1) and (2) center on this point, as does the practitioner’s objectives given the poor return outcomes, where ~90% of investments are either a total loss or only slightly exceed break even.

Overall, we suggest that this a priori information can assist practitioners to better qualify their equity position, as tailored to the circumstances. We envision a communication structure that simultaneously accounts for the inherent 10× return expectations that presumably motivates all investment decisions, while utilizing the break-even point as the main reference point in the dialogue. From an expectations standpoint, the 10× return point and the break-even point technically fall along the same line. As such, in the event that one’s foresight is accurate (i.e., a 10× return), we contend that focusing on a point closer to the present and one that must be reached for there to be positive returns should serve to facilitate discussions at the negotiation table due to the direct operational implications of each choice from the full option set.

The rest of this paper is structured in the following way: we first address the field of entrepreneurial finance, followed by an elaboration of the risk–return relationship. We then provide a brief overview of negotiation theory and this particular negotiation task, followed by an elaboration of how the structure of the negotiation influences power differentials and dynamics. In the discussion section we provide visualizations of the relationship amongst equity, returns and time. We then conclude with updated metrics on angel investing and venture capital activity through 2023, reflecting significant fluctuations in VC investments in relation to relatively stagnant activity for angel investing; our updated model addresses the evolving landscape and provides a lens to evaluate newer trends in investor behavior and capital distribution. With respect to the reading of the paper, we suggest applying what Raiffa (1982) would consider an asymmetric approach, where focus is placed on one negotiating party while the other remains out of focus but is still accounted for in the analysis; we focus on the investor. Our ultimate objective is to advance the field of entrepreneurial finance.

2. Addressing the Entrepreneurial Finance Literature

It must be noted that that the fundamentals of entrepreneurial finance are conceptually rooted in the traditional financial methodology of a discounted cash flow analysis, as evident in the primary objective of securing a return on investment and in the nature of communication at the negotiation table that revolves around firm value. The need for an alternative analytical framework is due to the inability to crisply associate firm-level projections with time alongside the highly subjective process of forecasting growth, which reduces the negotiation to a tug-of-war between subjective expectations of higher or lower growth. This renders the use of a discount rate as less practical and requires an alternative analytical framework to facilitate communication between the buyer and the seller. Yet, we note that there are limitations in the existing framework as reflected in Equations (1) and (2), and it is with this in mind that we provide constructive criticism across two interrelated points: the first is the practicality of the framework in relation to the behavior of the buyer and the second is the mechanics of the calculation that assumes that the value of the part is a perfect proxy for the value of the whole.

Addressing practicality, we first consider the terminology, logic and behavior of the buyer in a traditional merger/acquisition for 100% equity of the firm. The nature of the communication is less transparent as driven by the structure of the transaction. The buyer only communicates a single figure that accounts for profit expectations, and the details of the two sides of the DCF analysis (the initial investment and subsequent cash flows projections) remains with the buyer. We can also consider this as the buyer making two payments: one is made to the current owner(s) of the firm, and another is made to the firm itself that can range from no additional needs (strong firm) to significant needs (weak firm). During a full-equity purchase, the funds are exchanged directly between the buyer and seller, and any firm-level investment becomes the buyer’s concern; for the astute buyer, these concerns are baked into the derived valuation. As such, the tug-of-war pertains to the seller’s intrinsic satisfaction (i.e., utility) with the valuation, and the buyer’s ability to influence the seller to accept as low a value as possible3. The buyer will not be interested in paying the seller for any of the growth expected to be generated by the buyer’s efforts, and the same is true for any potential synergies that the buyer would be able to extract from the integration of expenses post-acquisition. Overall, we may classify this traditional buyer’s behavior as driven by the logic of existing operations, as they seek to tether growth expectations to the present, whereas the seller wants to connect growth expectations to the future potential made possible by extant operations.

Notably, this process is quite different from how the transaction unfolds with angel investing. First, this transaction can be viewed as making only a single payment directly into the firm to address operational needs; angel investing is a three-part harmony, where there is no direct exchange of dollars between buyers and sellers because the firm fully mediates the exchange4. Further, the buyer’s investment serves as a legitimizing catalyst to capitalize on the envisioned arbitrage opportunity between the firm’s current and future state. Even further, the buyer’s efforts will directly contribute to growth, which complicates negotiations based on the difficultly of allocating future effort to either buyer or seller beyond the stories told of potential value add while at the negotiation table; growth projections are therefore bounded by the softer guardrails of imagination, pragmatism, and the more powerful hand at the table. Additionally, the nature of communication at the negotiation table differs because positions are advanced on the two issues of capital and equity, which derives a value for the firm; while communication may indeed revolve around the firm’s value, negotiation positions are not communicated as a single value but rather expressed by the two issues. And relatedly, this allows the buyer to partially untether the logic of the investment issue from the logic of the equity issue, where the investment is tethered to the strategic trajectory and where equity is tethered to the twin concerns of a return on investment and the incentivization of the entrepreneur to execute the strategy. This type of basic deal structure naturally shifts the governance concern significantly towards market mechanisms via the lever of equity and additional rounds of investment, as opposed to hierarchical governance under full-equity acquisitions.

These factors reflect structural differences in the nature of the transaction that leads to behavioral differences for negotiators. In our assessment of the existing analytical framework (Da Rin and Hellmann 2020), we found that it does not fully address the behavioral implications associated with the structure of the transaction across the full range of possible scenarios. We suggest that this hinders communication at the negotiation table and is one potential contributing factor to the observed stagnation in angel investing activity.

The discussion on pre- and post-money valuation appears to approach matters from a purely mechanical perspective by assigning the difference between post-money valuation and the investment to the firm’s pre-money state. In this scenario, value did not explicitly exist prior to the investment other than in its potential form, and the firm was practically unable to manifest any of that potential without the investment. And indeed, scholars have identified this issue for the initial round, where pre-money valuation is referred to as a numaire, while justifying a purpose for the term during subsequent rounds of investment (Da Rin and Hellmann 2020). As also pointed out via the notion of a numaire, there is a limited practical benefit because one cannot enter a financial institution and utilize the pre-money valuation figure as collateral to secure a loan; this cannot happen before or after an investment is made. Furthermore, there is no such practice in the publicly traded markets—the finance literature has addressed this issue with the term “spot market” to reflect how value is continuously recalibrated and that it can only be derived in the event of a market transaction. And to reflect the similar reality in angel investing, the amount of sunk costs (e.g., sweat equity and/or actual dollars) by the entrepreneur or any former investor holds no influence on the spot market investor other than informing them of the magnitude of those efforts that are now sunk—depending on their power position, the spot buyer can dictate terms.

Succinctly, the simplistic calculation in Equation (2) eliminates the underlying bet in this transaction. While investment dollars are indeed specified towards a strategic trajectory, it is unknown ex ante if that strategic trajectory is the correct path; also unknown are the variety of random events that may or may not contribute to failure or success. Comparing across potential investments is likewise difficult because each typically pertains to completely different strategic trajectories if not also entirely different industries; the subjectivity of the bet complicates things for both buyer and seller and influences market efficiency (Fama 1980).

To further the point, we can consider the mechanistic analogy of a lawnmower that is missing a component and therefore cannot operate. However unlikely, the buyer’s ideal scenario would involve a single pull of the cord once the investment completes the mechanism, after which operations will generate cash flows. We consider this analogy in relation to the ways that we can view the value of the firm before an investment, striking at the difference between judgement (subjective) and opportunity (objective) (Foss and Klein 2012; Zayadin et al. 2023). An opportunity perspective would extend the mechanical analogy of a lawnmower by considering that a mechanism had been developed and an investment merely completes the puzzle; a perspective that correlates to the pre-money and post-money structure (Da Rin and Hellmann 2020). In contrast, a judgement perspective accounts for the initial decision to select this particular lawnmower (i.e., the firm), the diagnostic of what the investment should be (i.e., the firm-level need), and avoids the difficulty of assigning value to a dormant asset prior to any investment. Value, then, is deferred to the realm of subjectivity that we suggest is more fitting of the phenomenon as reflected in both the art and science of developing projections for an entity with limited operational history (Raiffa 1982). Furthermore, we feel that a judgement-based view provides a more direct application of price theory under the central premise that the numerical value paid for an item reflects all marginal utility. It is from this perspective that we’ll introduce the break-even point as a useful tool to navigate the subjectivity associated with judgements.

The second issue relates to the mechanics of the calculation and leads to our main argument: that the value of a portion of an early-stage firm (the part) serves as a weak proxy for the value of the whole. The two issues of capital and equity are imperfect proxies; perfect proxies require that we eliminate the judgement portion of the investment and therefore the underlying bet, and that we eliminate the incentivization concern such that equity would have a 1:1 relationship with returns. However, we know that this is not the reality because the explicit fracturing between ownership and control introduces complexities (Berle and Means 1932), and these influence the behavior of actors as addressed earlier.

We may view angel investing as a bottom-up approach that only derives the value for a portion of the firm that is then extrapolated to the whole, in contrast to traditional investment activity that first calculates a value for the whole to then uniformly assign value across each part. We need to first establish that technically, the mechanics of angel investing only identifies the part; the nominal value of the quotient identifies the free cash flows that the firm must accumulate in order to enable the spot market investor to break even. We may also refer to this as the break-even threshold for the firm’s free cash flow objectives or likewise as firm workload for the spot market investor. The term ’firm workload’ here reflects the cumulative effort required from firm operations to accumulate cash flows in order for the investor to secure their initial investment (break-even) and ideally generate a return as reflected in multiples of that initial investment. We accept that the quotient represents a value, but we point out that the calculation is not the valuation in the traditional meaning of the term because the buyer’s profits are not accounted for. Consider the example of USD 100,000 exchanged for 10% of a firm; per Equation (1) the post-money valuation is USD 1.0 million, and per Equation (2) the pre-money valuation is USD 0.9 million. If immediately after the transaction, a new offer is extended for the full equity ownership at the full valuation of USD 1.0 million, this offer would merely allow the initial investor to recoup their initial investment, yielding a 0% return. Such an offer would presumably be rejected by the investor though it may be accepted by the entrepreneurial team. Furthermore, we would expect a rejection even if we relax the earlier constraint and assume that the firm has a tangible asset worth USD 900,000 and needs the additional investment to enable operations and cash flows5. As such, angel investing is a partial valuation exercise—not only is only a part of the firm valued, but that value stops at the break-even point and does not explicitly account for profit expectations6.

Continuing with the mechanics, we note that this extrapolation of the part to the whole leads to another complication with respect to buyer heterogeneity. Scholarly investigations have long concluded that not all investment dollars are equal and that firm value is a co-created phenomenon during these early stages (e.g., Hsu 2004; Hsu et al. 2014). Yet, the simple subtraction of nominal investment dollars in Equation (2) effectively eliminates the influence of the buyer from the calculation. Their investment war chest, their strategic acumen, and their professional networks are not accounted for in the current terminology of valuation even though it is clear that the buyer plays a significant role in attaining the objective of a 10× return. We feel that buyer heterogeneity is better preserved with a focus on the direct operational requirements of attaining a break-even point, because such a perspective would presumably shift the internal deliberations of both parties to the question of how quickly the firm can generate the cash flow hurdle while operating under the influence of any particular buyer.

In short, we suggest that communication should adhere strictly to the mechanics of the exchange—focused only on the part and only up to the break-even point; it is understood that both parties will extrapolate this point to account for their respective profits. Pursuant to price theory, where marginal utility is captured in the price paid for any item, risk is reflected in the proxy of capital that accounts for strategic concerns, whereas equity accounts for return concerns; naturally this is not a crisp dichotomy due to the interdependence of these two issues. By sidestepping subjective discount rates and assumptions about future cash flow accuracy, our model provides a more adaptable framework, particularly suitable for early-stage firms without extensive historical data. We elaborate on the communication benefits in the next section. These benefits exist even if the break-even framework is utilized as merely an accounting mechanism, similar to the notion of unrealized gains; despite the fact that they could, angel investors typically do not receive periodic cash distributions, and typically cash-out in an exit event.

3. A Different Risk–Return Relationship for the Informal Risk Capital Markets

The scholarly literature has thus far presented two perspectives on the risk–return relationship. One stems from economics and finance measuring risk as return volatility (e.g., Bowman 1980; Fiegenbaum and Thomas 1988), and the other accounts for an ex ante perspective, equating risk to a capital investment (March and Shapira 1987). From the vantage point of returns, these two perspectives can be depicted along a continuum, representing the periods before and after the break-even point, which effectively also reflects the two portions of the discounted cash flow model with the initial investment and subsequent cash flows (Graham and Harvey 2001). Stratifying this space into pre- and post-return sections, we present a third perspective on the risk–return relationship.

Having accepted that the quotient of Equation (1) reflects the break-even point (Section 2), we turn to the buyer’s full choice set and investigate the distribution of the quotient across the full range of equity. Considering the behavioral implications at the negotiation table, we derive a generalized rendering that mirrors the root function [y = ${\displaystyle \frac{1}{x}}$] from three assumptions pertaining to the structure of the transaction with respect to its influence on behavior: (1) that angel investing is a two-issue negotiation, (2) that the negotiation positions are communicated sequentially from capital to equity, and (3) that the capital is fixed to a strategic trajectory. In effect, we back into the function [y = ${\displaystyle \frac{1}{x}}$] by considering the behavior of negotiators as set in this unique economic context: it would not make sense to address the equity issue without first specifying the purpose of the investment dollars, which effectively establishes the intended strategy for the firm. There is a natural sequence of events that reveals the structure of this strategic transaction (Commons 1934; Van de Ven 1993).

When viewed at the portfolio level, or also from the lens of a highly active buyer, each opportunity can be conceptualized as a new hand in poker; we may also view matters as an unequal replicated design, where each investment has a different magnitude, pertains to a different firm-level need, and involves a different industry. We therefore replace the capital investment with 1.0 to reflect a generalization of the buyer’s decision to invest (versus 0 for not investing)7. Logic suggests that at this point, the investor’s mental deliberations then shift to consider their equity option set, ranging from more than 0% and less than 100%. We may further presume that the buyer’s power advantage positively influences their ability to request more equity, thereby expanding the range of practical options available from the full option set; this is, however, subject to the counter tension of incentivization that contracts the practicality of that full option set. It is arguably this equity issue that is the most subjective component of this exchange, which clearly invites the role of power as an influencing factor (see Section 5).

At this point in the negotiation when the investment has been determined, a point reached across all cases, we point out that the benefit of the marginal equity percentage internalized by the investor (even if only in their mental deliberations, while reclining in their chair just prior to communicating an offer) diminishes at a rate of [$-{\displaystyle \frac{1}{x^2}}$] in terms of their ability to break-even on the investment. As the buyer’s equity position increases, the practical result is a smaller reduction in the firm’s workload requirements; that is, the operational burden placed on the firm diminishes at an exponential rate. Further, this a priori information is publicly available to all parties at the negotiation table, and it is clear that the investors’ position has a degree of transparency based on that position’s geolocation on the generalized curvature as reflected in the root function y = 1/x, where the points along this generalized curve can be viewed as a workload multiplier (see Figure 1). This information should facilitate communication as practitioners navigate the negotiation table, and this should level the playing field between buyer and seller due to the ability to better qualify an equity position by allowing participants to ask what if questions as pertaining to the operational requirements associated with each equity position.

With respect to the practical application of this knowledge, we work our way along the curve in Figure 1 (from left to right). If the buyer requests 5% equity in exchange for USD 100,000, the firm must accumulate USD 2.0 million in free cash flows for the investor to recoup their initial investment and break-even, as the 5% yields a 20x multiplier of the initial investment. At 10% equity, the free cash flow requirements lower to USD 1.0 million; a 10x multiplier. If the investor requests 20% equity instead, the firm’s workload multiplier reduces to 5x or USD 500k; at 40%, the requirement is further reduced to USD 250k, and so on. The notion of firm workload may also be viewed as a proxy for time8, which is clearly influenced by the unit of measure of each routine transaction: the sales price, sales cycle, and gross margin. Overall, the buyer should have difficulty justifying equity values that fall into what we’ll refer to as negotiation safe zones for the entrepreneur. As per Figure 1, the cumulative benefit of each marginal equity percentage flattens out at approximately 36.78% and it therefore makes it difficult for the buyer to justify an equity position between this point and 49%; we’ll consider this as safety zone #1. Recognizing that different justifications and argumentations would be utilized for either a 50% or 51% equity position, a second safety zone exists between 52% and 99%. These zones allow the entrepreneur to raise legitimate questions regarding the buyers’ rationale based on the mathematical reality tied to a firm’s work requirements.

Consider the example of a sweater and jacket company called ProfessorSweaters.com, selling each unit of apparel for USD 150 and generating USD 90 in gross profit (60% gross margin) via the use of a direct-to-consumer business model with a boutique presence online. In

Table 1. Reflects the operational implications associated with each equity percentage point as it relates to the specific business case of Professor Sweaters.

Equity Choice and Operational Implications
Equity	1%	5%	10%	20%	25%	30%	35%	36.78%	40%	50%	55%	60%	…	99%
B/E Multiplier	100.0	20.0	10.0	5.0	4.0	3.3	2.9	2.718	2.5	2.0	1.8	1.7	…	1.01
Capital (USDM)	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	…	0.10
Break-Even Point (USDM)	10.00	2.00	1.00	0.50	0.40	0.33	0.29	0.27	0.25	0.20	0.18	0.17	…	0.10
Units sold @ B/E (000s)	133.3	26.7	13.3	6.7	5.3	4.4	3.8	3.6	3.3	2.7	2.4	2.2	…	1.3
Firm Value @ 10× Return (USDM)	110.0	22.00	11.00	5.50	4.40	3.67	3.14	2.99	2.75	2.20	2.00	1.83	…	1.11
Units Sold @ 10× Return (000s)	1466.7	293.3	146.7	73.3	58.7	48.9	41.9	39.9	36.7	29.3	26.7	24.4	…	14.8

, we present the operational implications of each equity option, utilizing the gross profit as the equivalent of cash flows after subcontracting away all manufacturing and shipping logistics (contained in the 40% of the cost of goods sold). We assume potential sales will be generated from organic digital activity (clicks, likes, shares); we further assume that start-up costs of USD 100,000 are required to transition from the pure-idea stage to generating cash flow from operations.

Having explained both the behavioral and practical implications of utilizing the break-even point as the primary reference point, we present Equation (3), which mirrors Equation (1) but with modified terminology, and Equation (4), which presents the generalized rendering in the form of a multiplier, that once multiplied by the nominal investment, reverts to Equation (3).

$$\mathrm{y}={\displaystyle \frac{\mathrm{I}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \left(\mathrm{I}\mathrm{N}\mathrm{V}\right)}{{\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y} (\mathrm{F}}_{\mathrm{I}\mathrm{N}\mathrm{V}})}} = \mathrm{Firm} {\mathrm{Workload}}_{(\mathrm{Break-Even} \mathrm{Point})} = 0\% {\mathrm{Return}}_{(\mathrm{Spot} \mathrm{Investor})},$$

(3)

$$\mathrm{y}={\displaystyle \frac{1}{{\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y} (\mathrm{F}}_{\mathrm{I}\mathrm{N}\mathrm{V}})}} = \mathrm{Firm} \mathrm{Workload} {\mathrm{Multiplier}}_{(\mathrm{Break-Even} \mathrm{Point})} = 0\% {\mathrm{Return}}_{(\mathrm{Spot} \mathrm{Investor})},$$

(4)

One may be inclined to dismiss the above as an alternative framework that returns the same results as the entrepreneurial finance literature and that merely alters the terminology of Equation (1) by replacing the term post-money valuation with the break-even point. The applicability of this alternative communication structure becomes more apparent when we consider that the investor has two broad avenues to generate a return: (1) through a buy-hold strategy that pertains to the firm’s ability to accumulate free cash flow from operations, and/or (2) through a flip strategy that pertains to a secondary strategic transaction (Oksoy et al. 2024). We consider the buy-hold strategy as the default assumption that must underlie all investment decisions, regardless of one’s expectations of future potential buyers; that is, one would not invest only on the expectation of future buyers and would rather do so based on the merits of the firm. While in reality the vast majority of investors seek to flip their shares, we realize that a strong buy-hold strategy yields a strong flip strategy, but that the opposite is not necessarily true.

Considering the implications, one perspective suggests that this sliver of the economy is undervalued because of the exclusion of the investors’ profit. Assuming that all bets carry a uniform expectation of at least a 10× return (else, why invest), the valuation of this market in terms of the initial expectations should be multiplied by a factor of 11.0, or that expectations need to be recalibrated to reflect the realities of the calculation. And yet, to temper the potential impact of this discussion on the larger economy, the described risk-return relationship only exists for those firms that cross the threshold of investment-ability by exhibiting significant growth potential, and is not applicable for those firms that do not want an investor or cannot justify an investment. This model pertains to those circumstances when DCF breaks down.

Finally, it is worth mentioning that while we have built our argumentation around a single-period game in order to condense our argumentation, our analysis can be extended to an n-period game; the theoretically ideal scenario of a single pull of the lawnmower cord was merely a convenient way to simplify the problem space. In Appendix A and Appendix B we depict scenarios of a non-dilutive (1a) and a proportionately dilutive (1b) structure across multiple rounds. From the perspective of the marginal buyer, the same risk–return relationship applies, but under more bound circumstances given the existing shares of equity already allocated. We expect that the negotiation dynamics conform to the new realities at this 2nd (or 3rd, 4th, …, nth) period investment event; that is, the structure in Equation (3) and (4) remains the same, while there may or may not be greater constraints on the equity issue after each successive investment period.

4. Negotiation Theory

Research on the subject of decision making is itself relatively nascent –– it was only in the 1940s and 1950s that research focused directly on demystifying how humans generally solve problems and reach a decision (e.g., Simon 1955; Newell et al. 1958; Newell and Simon 1972). Newell and Simon (1972, p. 145) reflect how problem “…solving was regarded by many, at that time, as a mystical, almost magical, human activity—as though the preservation of human dignity depended on man’s remaining inscrutable to himself, on the magic-making processes remaining unexplained”. These decision scientists approached the challenge from a psychology and cognition perspective, laying the foundation for future research on the observed deviances from the theoretical notion of the rational economic man, documenting a wide range of mechanisms that influence errors from what we may generalize as an individual’s subjective assessment of probabilities when reaching a judgement (e.g., Tversky and Kahneman 1971, 1974; Thaler 1986; Kahneman 1991).

A negotiation is a joint decision-making process, and we may view negotiation theory as largely providing navigational tools for the practitioner; conceptual frameworks that help make sense of one’s encountered circumstances. A discussion on negotiation is therefore interwoven with the literature on cognition, as evident in the heavy reliance on reference points that reflect various tactics that can be deployed at the negotiation table by one party against the other. Two well-known mechanisms that influence judgement are the framing effect and the anchoring effect; it has been repeatedly shown that choices can be altered based on how the problem is framed (Bazerman 1983; Kahneman 1992; Kristensen and Gärling 1997) and can be significantly influenced by the initial framing during the negotiation (Brown 1953; Furnham and Boo 2011). These foundational theories provide a backdrop for understanding behavioral tendencies in early-stage investments, where uncertainty and power dynamics are often heightened, which creates conditions that are ripe for errors in judgement as evidenced further by the dismal return metrics for angel investors. By situating angel investing within these conceptual frameworks, we acknowledge how high-stakes decision making, cognitive framing, and power differentials uniquely shape investor and entrepreneur interactions.

However, the circumstances of angel investing reduce the benefits of nearly all the prevailing navigational tools as supplied by negotiation theory due directly to the subjectivity associated with growth projections and the post-investment future. Setting aside the structure of the table and the related discussion on power (Section 5), in this section we focus on the applicability of these available navigational tools, contending that they provide little guidance, leading us to ultimately advocate for the use of the break-even point as the primary reference point. We suggest that by anchoring the conversation around breaking even rather than total valuation, investors can manage the negotiation dynamics to mitigate potential power imbalances (Brown 1953; Oksoy 2020). This focus on the break-even point allows investors to prioritize organic growth objectives, while entrepreneurs retain realistic performance incentives—a critical balance in high-risk, early-stage environments.

For example, perhaps the best-known dichotomy in negotiation theory is an integrative (win-win) versus distributive (win-lose) negotiated outcome (Walton and McKersie 1965). Distributive bargaining takes the view of a zero-sum game, where one party benefits is at the other’s expense–Bazerman (1983) will later reference a fixed-pie bias that influences this type of negotiation behavior. And traditionally, distributive behavior has been studied via pay-off matrices that reflect a designed experiment centered around a commodity as the negotiation object such as a refrigerator (Bazerman et al. 1985; Northcraft et al. 1994), a used car (Carroll et al. 1988), the supply of metals (Mannix et al. 1995), or a job contract (Brett et al. 1996), among others. In contrast, integrative bargaining relates more broadly to the logic of Simon’s early work on human problem solving, where a problem is first defined and/or reconceptualized, followed by the identification of potential solutions. Broadly speaking, the difference between these two generic approaches to a negotiation can be associated with the available conceptual freedoms, where a distributive approach suggests less freedoms, whereas an integrative approach suggests more freedoms.

The circumstances surrounding angel investing complicate the application of this conceptual framework. With respect to the equity issue, there is no escape from the distributive reality that one party’s improved equity position is at the expense of the other party, but this changes if we frame equity as a proxy for returns. The equity issue addresses returns both directly, via the mathematical computation, and indirectly, via its role as a market-based governance mechanism to incentivize the entrepreneur with retained equity to generate said returns. Therefore, the relationship between equity and returns exhibit both structural and behavioral curvilinearity; structural in the domain of negative returns, and behavioral in the domain of positive returns; both must be accounted for simultaneously when extending an offer.

Behaviorally, equity pertains to the post-investment working relationship. However, the inherent tension between intended strategy and emergent strategy (Mintzberg 1978), combined with the fact that investors are typically one-step removed from operations, complicates not only the negotiation, but also the nature of the contract between buyer and seller. While all contracts are inherently incomplete (Hart 1988; Hart and Moore 1999), this is especially the case with angel investing, leading to a limited number of available safeguards such as ratchets that provide protection from future dilutionary practices, royalties that extract dollars directly from sales prior to their conversion into profits, contingencies placed on the capital investment, such as activity-centric stages, and board seats that position the angel investor with greater managerial control. These contractual limitations almost require the contracts to be unsophisticated, justifying even the most informal structure of a handshake (i.e., no written contract) when considering that safeguards are ultimately rooted in trust (Knight 1921; Hart 1988; Davidson 1996; Hellmann 1998; Ibrahim 2008). Set against these difficulties, maintaining a strategic alignment between the investor and entrepreneur emerges as one non-contractual safeguard to reduce risks as long as both parties embrace the generally emergent nature of entrepreneurship and of strategy at large (Mintzberg 1978). And the strategic management literature does provide some justification for this as a viable mechanism of behavioral control, with the notion of indirect inducement, where attention is directed towards an objectively observable point (such as a strategy) in order to induce behavior in tangential areas that cannot be targeted directly (such as strategy-related operational decisions) (Makadok and Coff 2009; Oksoy 2020)—the specificity of capital deployment and multiple rounds of investing would be two mechanisms that indirectly induce behavior.

The capital issue introduces a number of additional possible outcomes, stemming from the ability to define strategy while at the negotiation table and the existence of more than one correct answer when it comes to selecting a good strategy. Therefore, and considering the impact of the investor’s capabilities, it is entirely possible that a superior offer is reflected in less capital and a higher equity request9. Pay-off matrices are therefore difficult to construct because of the difficulty in assessing a good versus bad choice from the ex-ante vantage point.

Relatedly, BATNA is one highly utilized reference point within negotiation theory as introduced by Fisher and Ury (1981), reflecting one’s best alternative to a negotiated agreement if one should walk away and not reach an agreement. BATNA can serve as a useful conceptual framework when considering the earlier-mentioned negotiation objects that are well-defined commodities where payoff matrices can be constructed (i.e., refrigerator, etc.), and this concept can be grouped with other similar reference points such as an aspiration price (Pruitt 1983) or a reservation price (Raiffa 1982). However, each of these reference points require information to be of practical use, information that is a luxury for the early-stage investor; there are limited benefits when accounting for the heightened subjectivity because they arguably become more elastic due to the greater conceptual freedoms available to the buyer and seller when it comes to projecting growth expectations. The inability to discern across what may appear to be equally likely possibilities from an ex ante perspective (reflecting equifinality of securing a lucrative return), renders BATNA as a less effective conceptual framework.

So then, how should one navigate these circumstances, and what tools are available? One tool is a negotiator’s conceptual clarity, and we suggest that the discussed risk-return relationship can serve as a guiding light through the fog. In light of all of these cognitive pitfalls in an economic domain that is already an extreme sliver of high-risk and high-return investment activity, we place even more importance on the accuracy of the terminology deployed at the negotiation table. The dialogue at the table revolving around the valuation of the firm introduces an additional degree of subjectivity that is unnecessary, and as we earlier argued, is incorrect in regard to the mechanics of the calculation. Yet, we suppose that the free-market mentality that gave birth to angel investing would enable buyers and sellers to communicate using either reference point of a valuation or a break-even point. In consequence, we articulate an alternative communication framework using the same alternative analytical framework as reflected in Equation (1) alongside our modification of terminology in Equation (3). And with respect to communication, it is important to realize that the dialogue at the negotiation table foreshadows the post-investment relationship. A value-centric dialogue would signal the buyer’s intentions to pursue a flip strategy versus a focus on the break-even point that would signal a buy-hold strategy. These different return strategies will certainly impact post-investment behavior. Yet, because communications revolve around the same two issues for either of the return strategies, that can be further masked via communication that utilizes the quotient (i.e., value); all of this retains a bit of a fog.

5. Power Differentials Across a Varied Landscape

Power differentials are a fact in any negotiation, inherent in the roles of either buyer or seller (Bazerman et al. 1985). While the buyer holds an inherent power advantage due to their ability to buy, the overall advantage can be held by either party, and we make note of two immediately identifiable factors that influence power differentials. One is the influence of the firm, where strong firm-level indicators would improve the buyer’s motivation and presumably decrease the seller’s (and vice versa with weak firm-level indicators). The other moderating factor is set against the backdrop of this three-part medley––the structure of the negotiation table among the buyer, the firm, and the seller (Oksoy et al. 2022). In this section, we present how structure can influence power across three observable contexts of angel investing: (1) angel investor syndicates, (2) Shark Tank, and (3) the traditionally dyadic negotiations that primarily remain hidden due to the sensitive nature of the communication.

The results of a recent survey of 1659 angel investors hint at the importance of accounting for power (Huang et al. 2017): deals sourced via independent channels represent 70% of the deal pipeline for those seasoned investors (15+ years of experience) vs. only accounting for 44% of the deal pipeline for novice investors (0–3 years of experience). We could speculate that this difference suggests that seasoned veterans have greater comfort with the elevated power status that a 1v1 negotiation structure affords, though it comes with the risk of operating solo; relatedly, the results also suggest that novice investors are likely to work in groups (i.e., syndicates; see Gregson et al. 2013; Agrawal et al. 2016), which again would influence power differentials as elaborated below.

The inverse relationship between power and dependence allows us to view power differentials as a function of the varied option set available to each negotiator (Emerson 1962; Oksoy et al. 2022). Each of the discussed contexts presents the buyer and seller with different options, to therefore influence how power dynamics can or cannot unfold. However, it is noteworthy that the impact that the structure has on either the buyer’s or seller’s overall power has been greatly reduced by modern technology due to the reduction in search costs as afforded by the greater connectivity via the internet (Dahlman 1979; Dyer 1997). This reduction can also be observed in the aforementioned survey, finding that investment is no longer limited to large metropolitan centers (Huang et al. 2017) and that regions like the Mid-Atlantic in particular are showing strong signs of growth in not only deal flow but also deal size10. Because other negotiation tables are more easily discoverable, all actors enjoy an additional potential option and therefore greater potential power walking into a negotiation; incidentally, this would reflect an improved BATNA position, but only in a potential form because the next best alternative is the potential for a hoped-for-better table, which raises the question of whether the grass is greener on the other side.

Much of this regional growth is credited to the rise of angel investor syndicates that have provided structure and formality to deal making (Gregson et al. 2013; Agrawal et al. 2016). These modern syndicates typically have a coordinating body that conducts due diligence on the opportunities prior to their presentation to a panel of potential investors who merely have to decide if they should or should not invest. We note the difference in the nature of the decision per the aforementioned survey, where while the median investment is USD 25,000 and the average invested into a firm is USD 375,000. In other words, the median (syndicated) investor is dealing with 1/15th of the risk of the solo investor who must account for the totality of strategic and operational concerns. To extend an analogy, modern angel syndicates resemble initial public offerings that are private, invite-only events––an initial private offering (an IpO). Missing from such interactions is any explicit negotiation and, therefore, any explicit competition among the pool of buyers––they can all participate. The “die has been cast”, so-to-speak, and the investment decision reduces to a choice between participating in the deal or keeping one’s money. And from a power perspective, we note that this prior due diligence process conveys a degree of legitimacy to the entrepreneur, because there would presumably be no presentation if there was a negative outcome.

We can compare this multiple-buyer vs. single-seller context to Shark Tank, which also involves multiple buyers. On the TV show, the due diligence process is conducted after the curtain falls. To extend a partial analogy, this transaction is similar to the purchase of a residential real estate, where there are multiple buyers competing against one another to first settle on the price for a house, to then have the buyer conduct a home inspection (i.e., due diligence) after a contract is signed. It is from this perspective that Smith and Viceisza (2018) refer to the final outcomes of the TV show as an ‘intention-to-fund’; representing an initial negotiation that extends the seller the benefit of the doubt that all is as represented. We note that the lack of a due diligence process prior to the negotiation removes any initial legitimacy, therefore strengthening the hand of the buyers (Sharks) compared to the circumstances found in syndicated investing. Furthermore, Sharks have the option to either compete or cooperate, which expands the option set for both the buyer and seller, to counterintuitively give the seller an advantage at the start of the negotiation –– their option set is reflected in $[y=2^{\left(n\right)}-1$], whereas the individual buyer’s option set is reflected in $[y=2^{(n-1)}$] (Oksoy et al. 2022), where ‘n’ represents the number of buyers in a market.

The third context is a single-buyer versus single-seller negotiation structure; arguably reflecting the most informal context in relation to the other two. Extending the informality logic to the legal contract, we should expect that any due diligence process is likewise informal, occurring concurrently as the relationship develops organically over time. Constrained to the confines of the negotiation table, the investor has the power advantage as per their role as a buyer, but in light of the greater number of tables available for the entrepreneur, this increases the requirements of navigating that table solo. The entrepreneur faces the reflection of this challenge to discern across investors.

We summarize the power differentials across these contexts in

Table 2. Reflects how different negotiation structures influence power dynamics and power differentials.11

Negotiation Context	Negotiation Structure	Generalizable Conditions	Seller’s Initial Power	Buyer’s Initial Power	Occurrence	Due Diligence	Investment Decision	Analogous Transaction
Shark Tank	(n) buyers vs. (1) seller	Competition; Cooperation;	Higher, but potential to fall	Medium	Staged Publicly	Post-Negotiation	Negotiate, Compete, Collaborate, or Abstain	International Relations
Syndicates	(n) buyers vs. (1) seller	Cooperation	High	Low	Staged Privately	Prior to a Negotiation	Participate or Abstain	Initial Private Offering
Independent Angel Investors	(1) buyer vs. (1) seller	Competition	Low	High	Private	Concurrent with a Negotiation	Negotiate or Abstain	Pure Angel Investing
(Straw Man) Residential Real Estate	(n) buyers vs. (1) seller	Competition	Higher	Lower	Private	Post-Negotiation	Negotiate or Abstain	N/a

. The Sharks appear to enjoy the greatest option set, due to the ability to compete and/or collaborate, but risk overbidding for the business due to competition; each individual buyer’s options approximate to 50% as the number of buyers in a market increase, and counterintuitively the entrepreneur initially enjoys more options over each individual buyer. Within syndicated investing (i.e., ’IpO’), the entrepreneur enjoys a tacit agreement on the legitimacy of the venture before the negotiation begins (and even perhaps a pre-determined valuation), which improves their power position and effectively reduces the act of angel investing to a form of window shopping. And within the more informal and traditional structure, the entrepreneur has the weakest hand because neither is there a due diligence process to bestow legitimacy nor is there any competition to allow for a change in one’s power position at the current table. In this 1v1 context, both parties leverage the firm’s current and/or future state to influence shifts in power.

Finally, when we expand the scope to consider the different options that an entrepreneur has to reduce their dependency on capital, crowdfunding is a viable alternative that comes with its own risks (Mollick and Robb 2016). Crowdfunding is effectively securing pre-commitments on the firm offering, enabling the construction of the firm to then deliver the promised offering (e.g., Tesla). While the entrepreneur is able to retain their equity, there is the risk that the market does not respond positively, which places the firm in a difficult situation to explain to a future investor just what went wrong. At the market level, the greater transparency associated with crowdfunding increases the entrepreneur’s exposure to the demand function (for better or for worse), which may be welcome information to investors that will presumably then yield more accurate firm valuations (see also Di Pietro et al. 2018; Dyer et al. 2020).

6. Discussion

The title of Professor Knight’s (1921) influential book progresses from risk to uncertainty to profit; after all, this is the natural order of things. We elaborate upon the relationship between risk and return for the early-stage investor who is operating under conditions of heightened uncertainty and subjectivity, a domain where contracts can be as informal as a handshake and where even the most robust contract offers little recourse in the event of an unsuccessful venture. Such environments are ripe for manipulation at the negotiation table, and a negotiation ultimately boils down to a clash of perceptions where the winning hand enjoys a power advantage that is a combination of actual reality and mere perception. We investigate the clash of realities between the rigid mathematical structure of the strategic transaction (Commons 1934) that exists against the backdrop of this subjectivity. We suggest that the depicted logical structure can serve as a useful tool, serving as a barometer of rationality to gauge negotiation behavior and bestowing the immeasurable value of situational awareness for such foggy conditions for both practitioners and academics.

To recap our position, the framing effect plays a prominent role in research on negotiation, and the alternative communication structure that we propose is one example of this effect: a focus on valuation shifts the mind to the 10x objective requiring that one envisions a more distant future (positive return domain), whereas the break-even point brings the immediate time back into greater focus (negative return domain). Either party can choose either framing and these choices are not mutually exclusive; one may choose to focus only on valuation, only on the break-even point, or weave both together since both reference points lay along the same line of foresight. Regardless of this choice, we note that a valuation framing is more useful for firms that have an operational track record, whereas a break-even framing is more useful in the absence of such historical activity (i.e., commercializing a patent). Incidentally, any operational track record would enable a discounted cash flow analysis; we therefore focus on circumstances where DCF is not as applicable. In this section we provide visualizations of the risk–return relationship reflecting both the structural and behavioral curvilinearity.

Manifesting the power of the framing effect once more, we utilize the break-even point to reframe the risk–return relationship as risk mitigation via the lever of equity, and returns; or succinctly, the equity-return relationship. Greater equity reduces the firm’s workload requirements, thereby shortening the period of time the investor remains at risk prior to breaking-even. This risk mitigation tool, however, has a diminishing benefit with respect to reducing the firm’s workload, both in terms of the additional governance concern with respect to incentivizing the entrepreneur with retained equity and the inherently exponential nature of the risk–return relationship. The weakening of the marginal equity percentage point is thus due to both mathematical and behavioral considerations. Due to this tension, equity at times acts as a strong proxy for returns and at times acts as a weak proxy for returns; we articulate this by conceptually graphing equity onto returns in Figure 2. Momentarily suspending the need to incentivize the entrepreneur via retained equity, in the domain of negative returns we observe how the first equity percentages points have a significant impact on return expectations, but that this weakens as equity increases beyond ~36.78% (2.71828× multiplier). We consider this equity zone between 0% and 36.78% as a strong proxy for returns and we reflect the same section in the positive return domain as a 1:1 relationship; we consider points beyond 36.78% as a weak proxy for returns.

In Figure 3, we introduce a temporal dimension and depict the three-way relationship amongst equity (i.e., risk mitigation, x-axis), returns (y-axis), and time (z-axis). Note that time and returns are in many ways inseparable, especially when viewed from an operational perspective focusing on the throughput and cash flows; and while the ideal outcome is a 10× return in the shortest time possible, that temporal dimension is the most subjective and reflects an aspect of the bet that is most prone to recalibration. Strategy (i.e., capital) is effectively fixed and only periodically recalibrated, and the same can be said about return expectations that are tethered to a 10x objective. In effect, the buyer’s view is that attaining return expectations is only a matter of time. We therefore treat the time dimension as an asymptote that reflects a distant point in the future from one’s current vantage point of making an investment. And because of the close relationship between time and returns, we expect the elasticity of time to also influence elasticity in return expectations that are (re)calibrated as time passes. And as such, we expect that both dimensions are prone to a phenomenon called response shift bias that refers to “a change in the meaning of one’s self-evaluation of a target construct as a result of: (1) recalibration, that is, a change in the respondent’s internal standards of measurement; (2) reprioritization, that is, a change in the respondent’s values (i.e., reevaluation of the importance of component domains constituting the target construct); or (3) reconceptualization, that is, a redefinition of the target construct” (Oort et al. 2009, p. 1127; see also Ewert and Sibthorp 2009). We may presume that an initial offer reflects an ideal future state for the temporal and return dimensions, and that this offer lies within a range of offers that would each satisfice (Simon 1955; see also Winter 2000). We expect this to continue post-investment as the passage of time and new events leads to recalibrations of foresight; analogous to chasing after a rainbow, “or to the phenomenon seen in the treatment of hyperthermia where the body core temperature suddenly drops during the rewarming phase” (Ewert and Sibthorp 2009, p. 382).

Frame #1 in Figure 3 is a direct corollary to Figure 2. When viewing the domain of negative returns in isolation, we note that the most an investor can lose is 100% of their initial investment, which yields a lower bound at (−1.0×) return. We presume that because the investor has a uniform expectation of a 10x return across all bets, that the ‘distance’ between a −100% return and breaking-even (a 0% return) is also a uniform expectation, albeit a shorter distance. The temporal dimension in this negative domain is represented in terms of firm workload, because lower equity positions, that burden the firm with a larger workload, will presumably take more time to accumulate under a buy–hold strategy (i.e., a 2.5% equity position in exchange for USD 100,000 represents a 40× workload to attain a break-even point), as opposed to a higher equity position (i.e., a 36.78% position that only places a 2.71828× burden on the firm). Even though we can crisply identify the burden placed upon the firm, identify its exponential distribution that can be generalized to the function y = 1/x, the subjectivity resides in assessing how long it will take for firm operations to complete the work.

Overall, Figure 3 is an attempt to visualize the relationships that reflect the structural curvilinearity in the domain of negative returns, and the behavioral curvilinearity in the domain of positive returns as driven by the need to incentivize the entrepreneur with retained equity. Frame #1 reflects the structural consideration, whereas Frames #2–#4 reflect our extrapolations of the baseline relationship in Frame #1. We view these frames in pairs, where each of Frames #2–#4 are individually coupled with Frame #1, which represents the common denominator. In so considering, Frame #2–#4 depict what we expect to be the baseline behavioral relationship prior to equity parity (Frame #2), at equity parity (Frame #3), and at equity dominance (Frame #4). Under the simplified assumption that the entrepreneur is incentivized by the retained equity, we anticipate that there is a negative direct relationship between equity and returns when equity is greater than 51% (Frame #4). Similarly, we anticipate a positive direct relationship between equity and returns when equity is less than 49% (Frame #2).

While we do state that all negotiation positions related to the exchange of capital for partial equity must take a position on the generalized curvature of y = 1/x, we are unable to predict ex ante which position on the curve is better for the marginal event. We draw on an analogy from Kellert (1993): the overall system of a leaky faucet may be viewed as quite stable, while simultaneously, the pattern of the individual leaking droplets may itself present extreme complexity; that is, complexity is a matter of perspective (Simon 1962; Jensen 1983). Similarly, we show that the risk–return relationship as related to the break-even point presents a pattern that all drops must follow, even though the specifics of the marginal/incremental drop remain unknowable (Knight 1921; Arrow 1974). This comment points to the difference between possibility and probability–we address the former.

7. Conclusions

This theoretical discussion aside, angel investing activity has lagged significantly behind venture capital activity, even though both involve an exchange of capital for partial equity ownership in the firm. This is the larger puzzle that we endeavor to solve.

The key tensions pertain to the level of available information and the discretion to justify an investment decision; as a wealthy individual, the angel investor’s discretion requires only self-justification, in contrast to VCs, who require a greater scale of operations (i.e., information) to justify spending third-party capital and benefit from the ability to utilize a DCF analysis to secure approvals. As a result, VC bets are larger, with a loftier exit goal of an initial public offering, while angel investing is estimated to impact ~20 times the number of firms impacted by venture capitalists (Wetzel 1983; Freear et al. 2002; Sohl 2003; Shane 2008; Bernthal 2018; Hellmann et al. 2021). In light of these differences, we find it interesting that these two significant pistons of the entrepreneurial engine propelling the economy have exhibited different growth patterns over the past 18 years (Figure 4). Both peaked in 2021 (angel investing reached USD 29.1 billion, while venture capital reached USD 345.4 billion), yet venture capital investments have exhibited significant fluctuation, when considering that the 5-year average from 2006 to 2010 was USD 33.1 billion, and 2023 activity was down ~50% to USD 170.6 billion from its peak in 2021. In contrast, angel investing has remained more stable across this period, even though the 2006 to 2010 average activity was at a relatively similar scale (USD 21.7 billion).13 Why has angel investing activity not exhibited similar growth?

There have been significant developments in this domain since Gaston’s (1989) analysis, yet the peak of USD 29.1 billion, experienced in 2021, falls short of his extrapolations of activity, at USD 32.7 billion, 36 years ago. It is therefore noteworthy that activity in this space has effectively plateaued at a level below Gaston’s extrapolation; this remarkable divergence in activity is made starker when considering that both shared similarly modest beginnings in the recent past, despite the importance of this domain that “ranks as the largest source of external equity funds available to U.S. small business corporations…[and that] is more than double the next largest external equity source and eight times larger than professional venture capital commitments.” (Gaston 1989, pp. 228–29). This growth differential is even more surprising given the more recent reports that the role of venture capitalists is shrinking in favor of angel investors who are willing to spend the time and attention that larger organizations are unable to spend: “Why go to a big firm when a business angel can cover financing and give more personalized attention?” Rouzies and Ghalbouni (2010, p. 23); as well as the reported shift in mentality in our modern “age of capital superabundance”, which no longer requires that we view investment decisions from the traditional mentality of a business executive who is dealing with capital constraints; the authors suggest moving “away from making a few big bets…and start making numerous small and varied investments” (Mankins et al. 2017, p. 4).

We find these differences in growth and scale to be interesting in light of the greater freedoms an angel investor has due to discretion. Historical differences are easier explained when considering a pre-digitalized economic environment with limited connect-ability between buyer and seller, reflecting the private nature of angel investing that led William Wetzel (1983) to refer to this domain as the informal risk capital markets (Prowse 1998). Furthermore, envisioning two boundedly rational individuals negotiating in isolation about growth expectations, the resulting quotient of capital divided by equity is, indeed, an uncalibrated figure, an informal one. Yet, despite significant advances in not only our communication infrastructure but also in angel investment syndicates that have added a degree of formality to the space (Wirtz et al. 2017; Bonini et al. 2018; Mason et al. 2019; Fisher and Neubert 2023; Neubert 2022), it is important to note that we are still orbiting around informality as driven by the underlying bet. And extant return metrics do not shed a favorable light on the collective judgement of angel investing, where ~70% of bets result in a total loss and ~20% only break-even, noting, further, that these outcome metrics pertain to the 3 to 5% of ideas that actually receive funding (Riding et al. 1995; Maxwell et al. 2011).

There is no reason why angel investment activity cannot reach the same dollar volume as venture capital activity, and the relative stagnation of angel investing warrants an investigation. One barrier is the dismal return metrics that hinder the recycling of prior investment dollars from a prior win, given that only 30% break-even or better. Another is clearly psychological, given the high-risk and high-reward atmosphere leading to the anecdotal advice by angel investors to not commit more than 5 to 10% of one’s total wealth to the effort. Yet another barrier pertains to the scale of engagement, with anecdotal advice to only engage with a minimum of 10 bets, with some scholars suggesting that risk mitigation efforts may even require 50 bets (Gregson et al. 2017). Yet this guidance presents a significant barrier to entry when considering that the median (syndicated) investor puts at risk USD 25,000, whereas the average firm receives USD 375,000 (Huang et al. 2017), and the barrier of entry for the syndicated investor ranges from USD 250,000 for a 10-bet portfolio to USD 1.25 million for a 50-bet portfolio, whereas for the solo investor this presents a range of USD 3.75 million to USD 18.75 million. This has no doubt led to the recent growth of angel syndicates, but syndication does not necessarily resolve the underlying issue with dismal return outcomes; it rather serves as a mechanism to mitigate risk through a brute-force, volume-centric approach. There is arguably also a need to address the underlying risk associated with the marginal bet, which motivates our research on the risk–return relationship.

Our paper depicts an embedded negotiation tool unique to the negotiation task of capital exchanged for partial equity ownership in a nascent firm. This tool can serve both the practitioner and the scholar in making sense of the behavior of negotiating counterparties, perhaps facilitating even more transactions due to the improved conceptual clarity. This effectively amounts to a reduction in bargaining costs. Beyond angel investing, the break-even point framework may also provide strategic value in assessing equity exchanges in corporate joint ventures and other partial ownership arrangements. This approach could facilitate more balanced negotiations in scenarios where equity is exchanged for strategic contributions rather than strictly financial returns, thus broadening its application to diverse investment contexts. For the practitioner we suggest that the gained understanding provides a subtle self-awareness of the landscape allowing for legitimate questions to be raised. In turn, this forces the counterparty to justify their positions against the inescapable logic encountering all parties at the table; these questions may give the negotiators enough room to gain the upper hand or at least walk away from the table with greater confidence if those answers are found to be unsatisfactory. For the scholar, we add to the conversation on the risk–return relationship, noting that for the distance between the point of −100% and 0% return (i.e., the negative return domain), the relationship is curvilinear—positive but with diminishing benefit for the investor (Bowman 1980; Fiegenbaum and Thomas 1988; McNamara and Bromiley 1999).

Our analysis depicts the operational implications of choice along the full range of equity; our analysis is therefore descriptive of possibilities as opposed to addressing probability (prescriptive). As such, the notion of uncertainty is with respect to the firm’s ability to generate cash flow from operations; our model is intended to help reduce subjectivity by crisply identifying the operational implications associated with any given choice along the full range of equity; this may partially reduce uncertainty at the negotiation table with respect to knowing the operational requirements associated with one’s choice of an equity position (see Table 1), but not with the operational uncertainty inherent in the proposed new venture. Our model is, thus, intended to improve the economic intuition and forecasting ability and allow practitioners to better incorporate downside risks. We anticipate that market activity will recalibrate once this alternative communication structure is embraced by practitioners, and we expect this to be of primary interest to policymakers as well as other financial institutions (e.g., banks, insurance, financial services authorities, regulators, et cetera).

As mentioned, a lower equity position places a greater operational burden on the firm, dictating the higher scale of operations that introduces additional risks that need to be considered in contrast to a smaller boutique operation. We may therefore suggest that low-equity positions promote a growth agenda, whereas moderately higher equity positions can establish a slower pace to operations and a smaller scale for the firm that is less concerned about growth. We clarify this point using the earlier example of Professor Sweaters selling apparel for USD 150 per unit at a 60% gross margin, to specify the operational burden for each equity percentage point. If the buyer invests USD 100,000 for a 36.78% equity stake, the operational burden is ~3.6k units sold. This operationally means 12.86 units per day in a 280-day work year. This figure only reduces to 9.64 units sold per day if equity was 50%, but increases to 47.5 units per day at 10% equity. We therefore connect the choice of equity directly to the firm’s strategy and therefore competitive positioning. Further, this perspective may also help to better qualify the capital position because if USD 100,000 is considered in relation to a boutique operation (small scale), then larger scale requirements of a lower equity position may also require a larger capital investment. As such, these two issues of capital and equity may help calibrate each other when positioned in relation to firm strategy. On a similar note, we suggest that future research investigate the firm value metrics in relation to strategy in order to better understand any impact on negotiation behavior.

Policymakers may want to consider developing an options market to serve as a de-risking mechanism for the lead angel investor14. This would present a different negotiation structure than those depicted in Table 2, reflecting a (n) buyer vs. (n) seller marketplace, where the single buyer has the opportunity to de-risk their initial investment in a secondary market of buyers. Beyond this conceptualization, we defer a deeper integration of this hypothetical options market to the relevant literature for future research, while articulating some appropriate comments from Merton’s 1997 Nobel Lecture (Merton 1998), stating how option pricing was still a “relatively new discipline…which relates mathematical finance theory and finance practice [as a] special sphere of finance [to study the] allocation and deployment of economic resources, both spatially and across time, in an uncertain environment” (pg. 323). Merton continues: “paradoxically, the mathematical model was developed entirely in theory, with essentially no reference to empirical option-pricing data as motivation for its formulation…[and] provided a launching pad for refinements of the theory” (pg. 324). We wish the same for our proposed communication structure, sharing a similar developmental story based entirely in theory, deriving our model from an analysis of behavior as influenced by the inherent structure of this strategic transaction.

When considering angel investing as a global phenomenon, and how different countries utilize different schemas to mediate relations between buyers and sellers, we make note of the impact that our generalized rendering may have to serve as the baseline relationship to facilitate the development of more efficient schemas. Our research further carries implications for cross-border investments and the mathematical structure of the investor-investee relationship that we have identified may help offset language and cultural barriers. We also present the implications of our research on the development of regional economies, as reflected in the extensive literature on agglomerations. Marshall (1920) noted the tendency of similar firms to agglomerate, and research has since found a wide range of ways that clusters form (Porter 1998; McCann and Folta 2008; Mathias et al. 2021). Some include homogenous clusters, such as the positioning of auto dealerships, others include a diverse set of firms that may (and often do) or may not have a common thread such as a core industry focus, a natural resource, and/or a core technological focus. We raise this point only to suggest that the angel investor’s geographic travel patterns (as an autonomous economic actor) as a function of their natural life may serve as a facilitating mechanism of how regional economies develop, where an angel investor can populate one or more angel syndicates that are typically regional in focus and where the syndicate serves as a force multiplier.

Additionally, we suggest that future research focuses on buyer heterogeneity, and the nature of how information is conveyed post-investment; we might presume that individual buyers are more likely to impart intellectual talent on the entrepreneur(s)15. Perhaps most importantly, we suggest that future research considers any potential shifts in market behavior before and after the awareness of our proposed alternative communication structure; we suggest designed experiment structures are used to better understand how individuals recalibrate judgement and longitudinal studies that investigates any market-wide recalibrations.

For these reasons and more, we call attention to the importance of the informal risk capital markets. Overall, we find it surprising that such a rigid mathematical structure exists in a negotiation that is seemingly at an extreme end of subjectivity, where extant negotiation theory offers little guidance to assess the rationality of decisions made by either party. We argue that the intersection of subjectivity and mathematical structure is the core intrigue, and we suggest that the depicted structural reality can serve as a barometer to navigate the inherent subjectivity. However, it is important to note that the structure cannot be used to prescribe a rational decision because of the ex ante nature of the decision itself; it merely affords negotiators with situational awareness. Our intention is to partially dispel this fog, created from the conflicting realities that we presume contributes to the much-romanticized mysticism surrounding angel investment, mysticism that is in many ways well-justified in light of all the challenges. We expect our model to serve as a counterweight against unchecked subjectivity, mitigating the role of artificial power that feeds off such circumstances; as such, we anticipate more pragmatic firm valuations that should translate to improved return metrics and greater market participation. With respect to the application of this paper to this issue, we note that the scope of risk management “involves the practices of identifying, addressing, and monitoring the risks associated with investment and business operations decisions” (Call for Papers: Special Issue: Risk Management for Capital Markets). For the informal risk capital markets, we contend that the subjective nature of this economic domain makes it imperative that one properly understands the circumstances associated with the decision point.