A Framework for Valuation and Portfolio Optimization of Venture Capital Deals with Contractual Terms

. Venture capitalists invest not only in the business aspect of a deal but also in its contractual terms. Therefore, the selection of deals and the combination of contractual terms pose challenging decisions for them. This paper consists of two main sections. The ﬁ rst section introduces a novel framework for the valuation of venture capital (VC) deals, including startups and their contractual terms. By taking into account risk situations, this section presents the valuation of combined contractual terms, including call options, liquidity preference, and participant rights. In the second section, a new multiobjective mathematical model for VC deals and contractual terms portfolio selection is developed using right-tail probability, strategy alignment, and a utility function. To solve the proposed model, three metaheuristic algorithms — Non-Dominated Sorting Genetic Algorithm (NSGA-II), Multi-Objective Binary Harmony Search Algorithm, and Dynamic Tuning Parameter Binary Harmony Search Algorithm (DTPBHS) — are applied. Based on numerical examples, DTPBHS outperforms other algorithms in the “ Mean Ideal Distance ” index, but NSGA-II demonstrates the best performance in the “ Rate of Achievement of two objectives simultaneously ” index. Furthermore, we demonstrate that the proposed utility function is more robust than the right-tail probability function under default deals conditions.


Introduction
Venture capitalists, a specific type of financial intermediary, identify investees, such as startups, with significant potential for growth and entrepreneurial capacity.They provide these companies with funds, networking capabilities, and business skills to capitalize on market opportunities [1][2][3], playing a crucial role in the survival of new ventures [4].
On the one hand, investee selection involves not only evaluating venture projects but also considering contractual terms for venture capital (VCs).When selecting investees, VCs should assess factors such as the status of technology and the market, competitive position, growth strategy, and customer management [5].This situation introduces a significant level of uncertainty and complexity in the valuation and selection of deals and their contractual terms.Therefore, VCs need to evaluate and maximize the profitability of investments by employing reliable and flexible methods for choosing investees and determining their contractual terms.
On the other hand, the investment strategies of VC firms play a primary role in the survival of new investments [6] and the selection of investees.These strategies may encompass the degree of VC participation in portfolio companies [7], as well as the level of diversification or localization.
In this paper, we will address three questions: How to value deals that combine contractual terms in uncertain conditions?How to model the portfolio selection of deals?What constitutes a robust objective function in the context of deal defaults?
Because the contractual terms of deals have a direct impact on investee selection, we propose a multidimensional method for the valuation of deals and their components, such as call options, liquidity preference, and participant rights.We apply the stochastic real options (ROs) method, stochastic DCF, and default rates to consider uncertainty and risk, offering an alternative to using risk-adjusted discount rates.Moreover, we introduce a new multiobjective mathematical model that incorporates both VC strategic and financial objectives, considering uncertainty and constraints.The model helps in better-assessing risks and portfolio budgeting for deal selection, considering both financial and nonfinancial dimensions.Furthermore, we compare the performance of a utility selection approach to that of a right-tail selection approach when designing an index robust against deal defaults.
The main contributions of this paper include: (i) Introducing a novel method for valuing deals and their contractual terms, incorporating default rates.(ii) Developing a new multiobjective mathematical model for the portfolio selection of deals and contractual terms.(iii) Employing a novel utility function for the selection of a VC portfolio and comparing its efficiency to the right-tail function in the context of a deal default scenario.
The remainder of this paper is organized as follows: In Section 2, we provide an overview of recent research on the valuation of deal contractual terms, VC strategy, and portfolio optimization.Section 3 explores the concepts of contractual terms valuation, VC portfolio selection, and strategic alignment.Additionally, we present a method for estimating the strategic misalignment of a deal with a VC strategy.In Section 4, we develop a meta-heuristic algorithm based on the harmony search (HS) algorithm to address the NP-hard computational complexity of the model for solving real-size problems.Section 5 discusses the applicability and results of the model using a numerical example, evaluating the performance of the proposed algorithms and the utility metric instead of the right-tail metric.Finally, in Section 6, we draw our conclusions and provide a summary to conclude the paper.

Literature Review
2.1.VC Contractual Terms Valuation.From the point of view of the information process perspective for VC decisionmaking, there are four stages: screening, origination, evaluation (due diligence), and negotiation [8].Throughout these stages, qualitative and quantitative decision-making factors are hierarchically evaluated.Decision-making about valuation is one of the important challenges for VC investors because there are many uncertainties about the value of the deals.The valuation does not only include the startup's business of a deal but also contractual terms.Liquidity preference, investment amount, ownership stake, board control, option pool, prorata rights, valuation, antidilution, vesting, ownership stake, investment amount, participation, redemption rights, and dividends are negotiable and flexible contractual terms [9].Some of these contractual terms have a direct effect on the valuation of a deal, such as the vesting situation, while others, like the option pool [10], have an indirect effect on the value of deals [11].These contractual terms collectively influence the value of a deal.Therefore, a method is needed to evaluate both the business and contractual terms of deals under uncertain conditions.
There are several quantitative and qualitative methods for startup deal valuation, such as the Berkus method, Jordan Cooper, risk factor, VC methods, First Chicago, Chepakovich, multiples method, market-oriented approaches, score card, RO, and DCF [12][13][14].Numerous of these methods are too qualitative and provide imprecise outputs (in some cases, unrealistic results) for startups.
RO valuation considers the cash flow value and adds the value of managerial flexibility to assets [15], as discussed by Li et al. [16], who proposed an RO valuation method for financial incentive allocation in infrastructural projects.They examined the effects of ambiguity on RO value based on pessimistic and optimistic approaches and concluded that the RO valuation method is better than the net present value (NPV) method for uncertainty analysis.As argued by Montajabiha et al. [17], the n-fold compound option model was used to evaluate multistages investment projects in R&D pharmaceutical projects.They also applied a robust model for R&D portfolio selection but did not utilize the value distribution function and the information on fat-tail distributions.Nigro et al. [18] assessed pharmaceutical industry projects by RO and categorized the project based on growth phases, and optimized the mathematical model, but they did not consider the uncertainty of valuations.One of the most popular methods for option pricing is the Black-Scholes model, which has been the subject of numerous discussions and examinations in academic literature [19].There are different versions of the Black-Scholes model, such as literatures [20,21], but the main difference in the proposed method compared to the literature is that the proposed valuation method is a combination of RO, DCF, decision tree, and Monte Carlo simulation [14].

VC Portfolio.
Operation research problems usually answer only one of the "what," "where," and "when" questions [22].This paper aims to address the combination of these questions for the VCs portfolio.Generally, VCs aim to select the best deals for success, but various risks can undermine profitability.As discussed in the review paper by Chaparro et al. [23], portfolio selection methods can be classified into 12 sectors: financial methods, probabilistic methods, option pricing theory, strategic methods, scoring methods, combinatorial optimization, behavioral methods, mapping approaches, ROs, integrated methods, information gap theory, and scenariobased approach.The paper suggests that the portfolio selection method should be based on the innovation level.When there is radical innovation, qualitative methods are preferred.As a result, the priority is behavioral and information gap theory.The second priority is integrated methods and ROs.In this paper, we consider that VCs do not invest in extreme radical innovation investees.Therefore, integrated methods and ROs are assumed to be appropriate.
According to Guo et al.'s research [24], different stakeholders have different tendencies toward portfolio selection goals, which are assumed in the proposed mathematical model.At the first level, top managers prefer to select a portfolio aligned with organizational strategies.At the second level, portfolio managers want to choose the projects that maximize portfolio return.They pay great attention to the interrelation and synergies of projects.At the third level, the project managers should meet resource limitations and risk tolerance.In this paper, the proposed model belongs to the first and second levels of stakeholders.Aouni et al. [25] used the stochastic goal programing model and the concept of utility function in the case of 15 deals in order to design the decision-making process of VCs under three different scenarios.In another article published by Aouni et al. [26], a goal programing model based on fuzzy logic VC decision was proposed.The effect of the entry of a new deal on the existence of a portfolio and the effect of mutual interaction between new deals are considered by Zhao et al. [27].Moreover, VCs can choose their deal portfolio based on the risk-taking or risk-aversion of investors by using a tuning parameter [27].
One of the most important criteria for VC portfolio optimization is portfolio size.According to Cumming [28], four categories of factors affect the size of a portfolio: (1) Characteristics of VC funds, including the type of fund (such as public or private VC), fund duration, fundraising, and the number of VC fund managers.(2) Characteristics of entrepreneurial companies, including development stage, technology, and geographical location.(3) The character of financing, including staging, syndication, and deal size.(4) Market conditions.In an area where the number of VCs is limited, entrepreneurs are likely not in a position to select among VC financiers [29].Overall, the number of investees in a portfolio depends on many factors, such as funding stage, VC budget, VC type, the level of diversification, geographical distance, VC experience, financed entrepreneurial firms number, VC firm age, fundraising number and successful exits number [29,30].For example, Huntsman and Hoban [31] have demonstrated that ten investments may not be enough to reach a reasonably steady portfolio return for a VC portfolio, and portfolio diversity is not proportional to the fund size.Also, VC fund contractual conditions usually deny each investment from investing in more than a specific percentage (usually 10%) of the fund, and it is critical to maintain the appropriate diversity level [29].In addition, the fund availability established by each investment company is seriously and directly associated with the fund size.As the investable companies grow, they are more likely to enter new funds.So, the small fund size restricts the VCs from accessing the advantages of investment results [29].In the proposed model, the number and amount of investments are considered, and it is assumed that the stock offering of each deal is fixed for a certain amount of money because there are limited stock volume choices available for VCs to select.This assumption reflects the fact that investees, such as startup founders, prefer to keep a certain amount of equity to preserve their decision-making power in corporate governance.
Diversification criteria are another factor for portfolio selection.There are three principal categories of diversification (specialization): industry, stage of venture, and geography.Organizational learning theory discusses that specialized investor perseverance relies on extraordinary success in the field.However, diversified VCs are more likely to sustain due to less competition in a particular industry.Portfolio diversification is often beneficial in the early stages/high-risk stages.Informal venture capital (IVC) (including lesser-known individuals, entrepreneurial friends, and angel investors forming groups of IVCs) is less profited than formal venture capital (FVC) in terms of industry and stage diversity.Insufficient investment for IVC can push them to participate in fewer investment rounds, despite successful investment selections.It may prevent them from thoroughly enjoying the diversity benefits.The previous investment experience is suggested to provide a more robust alternative for IVC diversification needs.As a result, IVC can profit more by concentrating on the typical industry by achieving an expressive investment [32].The small funds can protect forthcoming assets under management by adopting specialized strategies with relatively small investments in a restricted number of companies with losing diversification.In contrast, small funds may invest in many portfolio companies and choose a diversification strategy [29].Therefore, in this article, according to the importance of this concept, the diversification index is used.
As discussed by Treville et al. [33], the worth of a VC portfolio is determined by the value of the small number of top deals that are available for assessment.Therefore, it can be acknowledged that the value of a portfolio relies on the presence of fat right-tail distribution deals rather than averages (IVC or FVC investor returns are extremely skewed, high probability of low-expected NPV and low probability of high-expected NPV, and existing literature confirms it [32]).The combination of the right tail of deals for the portfolio deal's probability distribution of NPV (we call it PDNPV) is different from the summation of deal value because this problem is not always subadditivity, no smooth, and nonconvex [34][35][36].
The "strategy" is the key to reaching profit for VCs [5].As a result, the alignment of investee characteristics with the VC strategies is an important issue in selecting deals.Consequently, VC investors have strategies for reaching their goals.Ayob and Dana [37] proposed three strategies for VC products in developing countries, consisting of producing lowcost, differentiated, and specialized products.Park and Bae [38] proposed a 3D integrative framework of VC strategy in developing countries.The framework consists of three dimensions: target market, product/market maturity, and technological capability.They have also proposed seven practical strategies based on the main three strategies.
Various models exist for VC portfolio selection, including stochastic goal programing [25], fuzzy goal programing [26], modern portfolio theory [39], data envelopment analysis [40], and fuzzy inference system-agent [41].Many research studies have focused on assessing the weight of selection criteria and integrating multicriteria decision-making techniques with fuzzy theory.However, only a limited number of studies have applied mathematical models specifically to the selection of VCs' investees (as opposed to deals).This limited usage underscores the untapped potential of mathematical models in the field of deal portfolio selection.

Mathematical Problems in Engineering
To the best of our knowledge, no research has been conducted on the combined valuation and selection of startup deals and their contractual terms (such as call options, liquidity preference, and participant rights) for VC.Therefore, we propose a new method for valuation and a multiobjective portfolio selection mathematical model for VC, considering both financial and nonfinancial objectives.Various factors, including strategy alignment, the level of investee innovation, diversification, deal options, fund limitations, and the number of investments, are taken into account.

Problem Description and Formulation
The framework of this paper consists of two main concepts: valuation and selection of VC portfolio (Figure 1).
In Figure 1, we apply valuation and portfolio selection concepts to design a new model for deal contractual terms valuation and portfolio optimization.For valuation, the input data involve free cash flows (FCFs) probability function, default rates in different periods, the data related to ROs, and contractual terms.The inserted data for portfolio selection includes VC strategies, diversification index, the minimum NPV required for VC, valuation of deals and their contractual terms, budget constraints, and the maximum number of investees.Then, a solver method like a Meta-heuristic algorithm will determine the portfolio's compound.
3.1.The Valuation of VC Contractual Terms.In this paper, it is assumed that there are several deals with a variety of contractual terms.Therefore, we not only evaluate the NPV of a deal but also assess the effect of deals with contractual terms between the founder and the investors.We consider all kinds of deal risks into three groups: hard factors, soft factors, and scenario factors.
The factors with sufficient data about future scenarios and probabilities are classified as hard factors.In this group, these factors will be estimated using decision trees and probability distribution functions.The second group consists of scenario factors, where the scenarios are identifiable, but their probabilities cannot be determined directly.Here, we employ the RO method to account for them.In the third group, soft factors are typically examined qualitatively, including individual and managerial attributes of the investor, the motivation and experience of the founders, the team members' ability to recover in case of partial failures, and their resilience in the face of economic, political, social, and legal conditions [42].The evaluation of these soft factors is carried out using expert-elicited Bayesian network (EBN) analysis, as described by Valaei and Khodakarami [42].The output of EBNs is the default rate in each period of a deal's life.
In the following, By using an example, we will present how to calculate the value of a stochastic European call option by using a Monte Carlo Simulation for a startup deal.Then, the formula for the valuation of contractual terms, specifically for liquidity preference and participant rights, is proposed.
In this paper, the price of a European call option is calculated based on the Black-Scholes model (Equation ( 1)) (it is evident that various alternative methods exist for RO pricing that can be used, including Binomial Models) [43].The inputs of the model consist of the current underlying asset price, the strike underlying asset price, the time to maturity, the riskfree rate, and the volatility of the underlying asset [43].
The parameters of the valuation model are described as follows: (O t ) Call option premium at time t * -the value of a call option with expiration time "t"; * If there is a sign of "′" such as O′ t , it shows the present value of each parameter.
For instance, there is a probability distribution of net present value (PDNPV) for a deal.Because the NPV of a deal is not a single value, the NPV of derivative tools, such as a call option, will not be a singular value either.Therefore, in this paper, we employ the Monte Carlo simulation method to generate PDNPV for deals and their options, reflecting the premium cost in the cash flow of a deal.
In this paper, the value of the current underlying asset is a probability distribution function.Assuming that the NPV motion follows a random walk based on Geometric Brownian Motion, as described in Equation ( 2) [44], We calculate the standard NPV of a new deal (without considering the default rate) with a call option using Equation (3).
Based on Equation ( 3), if the price of the deal is higher than the strike price, the call option will be exercised.Otherwise, the investment will not continue, and the deal will be abandoned, resulting in the NPV of the deal will be negative, equal to the option premium.For instance, Figure 2 shows the PDNPV of a deal with a European call option and a time to maturity of 4 years, demonstrated using a Monte Carlo simulation (The Monte Carlo simulation is done by Crystal Ball software).If there is a failure probability in each period (1 − P t ), the NPV of the startup is shown in Figure 2.
The NPV of a startup deal with a call option (with expiration g year) and the default rate is shown in Equation (4). ...

Mathematical Problems in Engineering
The NPV of a startup with a call option (with expiration g), default rate, and liquidity preference is shown in Equation (5).
The NPV of a startup with a call option (with expiration g), default rate, liquidity preference, and participant right is shown in Equation (6).To assess the impact of certain independent contractual terms, please refer to reference [11].
3.2.VC Portfolio Selection.The mathematical model in this paper is based on the multidimensional multiple-choice knapsack problem (MMKP) [45] and the portfolio selection problem under the value-at-risk measure [36,46], which does not have the subadditivity property.MMKP is a generalization of the ordinary knapsack problem [47], and it is a more complex variant of the binary knapsack problem and computationally is an NP-Hard problem [45,48,49].Moreover, the NP-Hard complexity of portfolio selection problem with Value at Risk objective (such as right tail probability) is proved by Benati and Rizzi [46].Since the real-world investment problems are composed of different objectives, it is proposed a multiobjective stochastic version of the multidimensional multichoice knapsack problem.Thus, the proposed model is a multiobjective binary model to select a set of deals and their RO, maximizing the right tail of a cumulative probability of the portfolio NPV and minimizing the strategic misalignment of the selected deals.
In this section, the proposed mathematical model is introduced.The following assumptions are made: Mathematical Problems in Engineering In the following, decision variables are determined, and the abbreviations and parameters used in the mathematical model are defined.

Model Indices. Investee: i;
Industry: j; Deals: u (all kinds of contractual terms for each deal are considered independent variables).

Model Parameters. Z 1 :
The first objective function is designed for calculating the right tail of the portfolio NPV distribution; Z 2 : The second objective function aims to quantify the strategy misalignment; MNPV: The minimum attractive NPV for the VC portfolio; D ji : The total amount of strategy misalignment for investee i in industry j; d y ji : The amount of strategy misalignment for investee i in industry j regarding strategic dimension y; U: The maximum number of company strategy dimensions; C jiu : The investment cost for the investee i in industry j with deal u; P jiu : The present value of future cost for the investee i in industry j with deal u; B: The maximum available budget; E: The maximum reserved available budget; N: The maximum number of the deals that VC can manage; L: The minimum amount of Herfindahl diversification index for VC portfolio; H: The maximum amount of Herfindahl diversification index for VC portfolio; k: The maximum number of industries; n j : The maximum number of investee in industry j; T ji : The maximum number of deals for the investee i in industry j; NPV jiu : NPV of the investee i in industry j with deal u; Φ jiu ðMNPVÞ: The complementary cumulative distribution function of the NPV of investee i in industry j with deal u for the MNPV; α jiu : Auxiliary binary variable for selecting the deal i in industry j.
To evaluate the strategic misalignment between investees and VCs, a distinct version of the index, influenced by literatures [50,51] and incorporating the Herfindahl index [52,53], is employed.This index measures the Euclidean Distance for each dimension.The formulations are as follows (Equation ( 7)): where D ji represents the overall strategic misalignment of deal i in industry j, and d y ji is the strategic misalignment of deal i in industry j at dimension y.
As given in Table 1, For instance, if a VC with an "import substitution" strategy intends to invest in a deal with an "early market entry" strategy, conflicts may arise in strategic alignment [38].
In order to show the formulas, in Equations ( 8)-( 17), we illustrate the complete mathematical model as follows: Min Z 2 : Mathematical Problems in Engineering  High:3 High:3 For example, if the assumed strategy of a VC is Import Substitution and the strategy of an investee is Early Market Entry, the misalignment strategy for the selection is calculated in the last column.Bold columns are examples of strategies and are relevant.
8 Mathematical Problems in Engineering As we aim to capitalize on the asymmetric returns of deals, the first objective (8) employs the right tail of PDNPV at the point of MNPV.This aligns with the opportunistic strategy of seeking the highest investment return.Proceeding from the left side of the distribution, we utilize numerical integration with a trapezoidal base and calculate the cumulative probabilistic distribution functions until reaching the value of MNPV (its proof is presented in the Appendix [54]).The second objective (9) focuses on estimating the strategic misalignment of a portfolio, resulting in a value of one or greater.
Constraints (10) and (11) are related to budget considerations.Constraint (10) pertains to the maximum budget available for selecting deals right now, while Constraint (11) entails reserving a budget for the future cost of the selected deal, including the exercise of options in subsequent years.
Additionally, Constraint (12) establishes boundaries for both the maximum and minimum number of investees that can be managed, resembling a cardinality constraint [55].In this research, industrial diversification is considered due to the importance of the industrial type of investees that operate.A diversification index called the Herfindahl index [52,53] is applied in Constraint (13) on the portfolio selection model.It is proposed that the upper and lower bounds of the constraint be determined based on successful and similar VC practices (see [56]).
Since VCs can choose only one deal for each investee, we use Constraint (14).Constraint (15) is intended to decide between the primary and auxiliary variable choices.In the first part of the primary objective function, you must select one of the deals.Therefore, we add Constraint (16) to the model.Constraint (17) represents binary variables.Each variable shows the value of a deal with individual contractual terms (consisting of options).

Algorithm Design
The number of variables in the proposed problem could increase significantly when a VC has several investees with different contractual terms.The exact algorithms, such as branch and bond algorithms and dynamic programing algorithms, cannot handle large-scale problems, and solving them requires a significant amount of CPU time due to their NP-hardness [57].Given these challenging, several meta-heuristic algorithms have been developed for optimization in the last few decades [58].As discussed by Zou et al. [59], the HS algorithm is a well-organized method for solving knapsack problems, and it can find better solutions compared to other metaheuristics, such as genetic algorithm in a stochastic multiobjective problem under the same situation [60,61].The HS method is an emerging metaheuristic optimization algorithm employed to cope with numerous challenging tasks during the past decade.The HS algorithm is inspired by the underlying principles of musicians' improvization of harmony, which can be seen as a search process for the global optimum in optimization problems that are evaluated by an objective function [62].The musical harmony in HS is similar to a variable vector, and the best harmony achieved in the end can be seen as the optimal global solution [58].
4.1.Dynamic Parameters Tuning Multiobjective HS.A variety of harmony algorithms have been proposed, including an efficient HS algorithm [63], an improved HS algorithm [64], multiobjective binary differential evolution HS (MOBDEHS) [65], and multiobjective HS algorithm [66].A new version of the multiobjective binary HS (MBHS) algorithm [66] is the dynamic parameters tuning multiobjective harmony search algorithm (DPTMOHS), which uses dynamically tuned parameters to create nondominant solutions.The algorithm for dynamically tuning the parameters has been designed based on reference [60], and the Pseudocode of the proposed algorithm (DPTMOHS) is shown in the coding done by MATLAB 2013a software on a Core i7, 8 GB.Before delving into the detailed explanation of the algorithm, it is necessary to present the coding method of the problem, as shown in Table 2.
As depicted in Figure 3, a deal is selected from the harmony memory (based on the elitism parameter) if a new random number is less than the harmony memory consideration rate (HMCR).Subsequently, at the next level, if a new random number is less than the pitch adjustment rate (PAR) (the mutation parameter), a deal is randomly selected from the neighborhood of the last deal position (with all deals sorted in ascending order).If the algorithm does not enter the process of HMCR and PAR, it utilizes a random mechanism, as shown in Figure 3.

Mathematical Problems in Engineering
To handle the constraint mentioned in the model within the algorithm, a pruning approach is employed for newly generated solutions.In essence, if an infeasible solution is generated, the algorithm promptly discards it.

Comparing Metric.
There are various metrics for comparing approximate algorithms when accuracy is unmeasurable.In this research, we employ three metrics to compare the three algorithms.These metrics include the rate of achievement of two objectives simultaneously.The relative achieved spread (RAS), which measures the rate of achieving the two best objectives simultaneously, the mean ideal distance (MID, which quantifies the distance between Pareto solutions and the best answer), and the spread of nondominated solutions (SDS, indicating the range of nonexhaustive solutions) [67].In Equations ( 18)- (20), Z 1s and Z 2s are normalized using the method described in literature [67], and n shows the number of solutions in Pareto solutions.
5. Computational Tests 5.1.Generating Test Problems.To test and verify the algorithms in the proposed model, we generate practical testing by using 10 deals with 40 premium call options (Table 3).The strategy misalignment of each candidate deal is randomly generated from a uniform distribution (1,5), as demonstrated in Table 4.The cumulative probability distributions of NPV are summarized in Table 5.

Comparing Results.
To assess the performance of the DPTMOHS algorithm, we utilize two series of test problems: one for small size and another for medium size.
5.2.1.Small-Size Test Problem.For the small-sized problem, new test problems were designed to evaluate the capability of the DPTMOHS algorithm in finding Pareto solutions.Three groups of problems, with sizes 8, 10, and 12 and each having three levels of MNPV (with values of 50, 75, and 100), were created.The results, based on quality metrics (the proportion of Pareto solutions [68]), are presented for 1,000 iterations of the algorithm in Table 6.The table illustrates that the DPTMOHS algorithm can effectively find Pareto solutions, demonstrating the efficacy of the proposed algorithm in identifying optimal points for the problem (the data are provided in Table 7).

Medium-Size Test
Problem.The algorithm of Nondominated Sorting Genetic Algorithm II (NSGA-II) [69,70] and MBHS algorithm [66] are used for comparison with the DPTMOHS algorithm.These algorithms need to adjust their parameters (DPTMOHS does not need to parameter tuning).In this paper, the NSGA-II (the maximum number of iteration parameters is not considered because the termination condition is determined based on the number of objective evaluations) and MBHS (the harmony memory size (HMS) parameter in the classic harmony search algorithm has been omitted in this study.This omission is based on the assumption that what is crucial for VCs is to generate a range of dominant solutions rather than focusing on a single solution.Furthermore, due to the utilization of dynamic parameter tuning, the count of nondominated solutions may fluctuate throughout the optimization process) parameters [66] are tuned by the Taguchi method for tuning parameters [60].The tables are selected for standard orthogonal array L 16 (4 2 ) and L 9 (3 3 ) experiments, in which three repetitions are performed (the data are provided in Tables 8 and 9).The levels of experiments were conducted using the Taguchi method, and their results were obtained based on the signal-to-noise (S/N) ratio.Then, the best-tuned parameters of the algorithm were calculated using Minitab 17 software [60].Tables 10 and 11 show the best parameters of the MBHS and NSGA-II algorithms.After determining the parameters for both the MBHS and NSGA-II algorithms, their performance will be assessed against that of the DPTMOHS algorithm, utilizing RAS, MID, and SDS metrics (see Table 12).
To determine whether there exists a statistically significant difference in the performance of the algorithms (Table 12), we employ the Kruskal-Wallis test (Figure 4) [67] and a paired ttest (Table 13) [68] with a confidence interval of 0.95.
The results of the Kruskal-Wallis test indicate no significant difference in the SDS index (P-value = 0.321).However, there are notable distinctions in the RAS (P-value = 0.0001) and MID (P-value = 0.0002) indices.Subsequent paired t-test results demonstrate that the MID of the DPTMOHS algorithm outperforms that of NSGA-II.Conversely, the RAS of the NSGA-II algorithm surpasses that of the DPTMOHS algorithm.
As outlined, the probability of failure for any VC deal is very high.Therefore, we propose a new approach for the first   Mathematical Problems in Engineering objective of portfolio selection, which involves replacing the right tail of the probability function.This involves a utility function based on the probability of the right tail of PDNPV relative to the probability of negative PDNPV (see Equation ( 21)).
To facilitate a comparison between the two approaches, we have designed 20 test problems (refer to Table 14).After implementing the DPTMOHS algorithm for each of these test problems, the efficiency of each approach is assessed through the proposed stress test, known as the deal failure rate.In this test, we assume that the NPV of each selected deal in the portfolio fails randomly (using Monte Carlo simulation).The failure is determined by selecting from the negative area of the NPV distribution function for each deal.
To assess whether there is a significant difference between the two approaches, the Paired Wilcoxon Rank test (in this test, there is no assumption of normal data distribution) is employed with a confidence interval of 0.95 (P-value = 0.00001) [67].The results show a significant difference between the two methods, as illustrated in Table 9.

Discussion.
As pointed out earlier, both the value of a business's cash flows and the contractual terms have a significant impact on the benefit to an investor when considering a startup.In this research, we illustrate how to evaluate a combination of specific contractual terms.Utilizing RO theory, decision trees, and PDNPV, we propose a practical formulation for valuing call options, liquidity preferences, and participant rights in VC contracts.We then present a practical mathematical model for selecting a combination of a deal portfolio based on the right tail of PDNPV and strategy alignment.To solve the model, we compare the performance of NSGA-II, DPTMOHS, and MBHS algorithms.The results of statistical hypothesis tests indicate no significant difference Mathematical Problems in Engineering     Mathematical Problems in Engineering in the SDS index (P-value = 0.321).However, the RAS (Pvalue = 0.0001) and MID (P-value = 0.0002) indexes show distinctions.According to the results of the paired t-test, the MID of the DPTMOHS algorithm is superior to that of NSGA-II, while the RAS of NSGA-II surpasses that of the DPTMOHS algorithm.
The proposed approach can be beneficial for VCs aiming to choose investees with potentially higher returns, given certain probability conditions [32].Furthermore, we introduce a utility function that is more robust in the right tail of the PDNPV approach for portfolio selection optimization.All these methods prove highly useful for VC decision-making and for private equity investors.

Conclusion
VCs employ various quantitative and qualitative models for deal valuation and portfolio selection.This study introduces a novel framework for valuing and selecting deals, incorporating a combination of contractual terms, such as call options, liquidity preferences, and participant rights, within VC portfolios.VCs aim to select investee portfolios that not only yield the highest possible returns but also align with the strategic goals of their portfolio.Thus, we have proposed a multiobjective mathematical model that considers both financial and nonfinancial dimensions in VC portfolio selection.In this context, we utilize a numerical integrated method where the NPV of each deal is stochastic, determining the probability of the right tail side of the stochastic NPV portfolio.Given the NP-hard computational complexity of the model, we employ the DPTMOHS algorithm-a metaheuristic based on harmony search-to address and solve the proposed multiobjective model, followed by a thorough assessment of the algorithm's performance.Furthermore, we introduce a new robust utility function tailored for VC firms.This function demonstrates superior resilience compared to the right tail side of the NPV probability function approach when dealing with default.The proposed function empowers VC decision-makers to make more informed and resilient decisions in the face of potential failures.We encourage future researchers to explore exact algorithms to discover the best and optimum Pareto solutions.Furthermore, the integration of various types of contractual terms would significantly contribute to the advancement of this field.Moreover, we propose the development of bi-level mathematical models for deal selection, incorporating diversification indices across various investment stages and geographical locations. ðA:1Þ

FIGURE 2 :
FIGURE 2: A simple NPV decision tree of a deal.
strategy index in dimantion of y − VC strategy index in dimantion of y Maximun strategy index in dimantion of y − Minimum strategy index in dimantion of y ;

TABLE 2 :
Coding of solutions.
(0.1724, 0.3958) T-test of mean difference = 0 (vs.≠ 0): T-value = 5.2 P-value = 0.00 u¼1 χ jiu ðhÞ: f ðhÞ is a joint probability distribution function for the sum of the probability distribution function of ∑ k j¼1 ∑ Decision Variables.χ jiu : Binary variable for selecting the deal i in industry j;

TABLE 3 :
Estimation of startup cash flow.

TABLE 4 :
Strategy misalignment of each candidate.

TABLE 5 :
Cumulative probability distribution of NPV.

TABLE 6 :
The percentage of finding Pareto solutions in algorithms.This algorithm solves the problem by counting the entire feasible space.

TABLE 10 :
The best level of MBHS algorithm.

TABLE 11 :
The best level of NSGA-II algorithm.

TABLE 14 :
Comparison of right-tail and utility function approach.