PCSY-D-24-00123

Complex economics of simple systems

PLOS Complex Systems

Dear Dr. Kärenlampi,

Thank you for submitting your manuscript to PLOS Complex Systems. After careful consideration, we feel that it has merit but does not fully meet PLOS Complex Systems's publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

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Bohui Wang

Academic Editor

PLOS Complex Systems

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Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Does this manuscript meet PLOS Complex Systems’s publication criteria ? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.

Reviewer #1: Partly

Reviewer #2: Yes

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: No

Reviewer #2: Yes

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3. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

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4. Is the manuscript presented in an intelligible fashion and written in standard English?<br/><br/>PLOS Complex Systems does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

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5. Review Comments to the Author<br/><br/>Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: Title and abstract

Title and abstract should be revised to point to complex systems analysis of simple financial investment processes

It is not complex economics which is addressed, but the temporal and probability structures of financial returns and the analysis of main indicators commonly used to determine those returns

Introduction

Your literature review on capital budgeting techniques is insufficient. You should delve into existing literature reviews (such as: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1700154) to summarise existing criteria. NPV and IRR belong to the same family, when your summary currently ignores accounting-based indicators and cash-based indicators (without discounting).

Moreover, you ignore a renewal of the capital budgeting debate triggered by a critique of compound return structure (see https://doi.org/10.1080/09672560600875281 for an historical reconstruction and https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1330587 for analytical development).

From this viewpoint, NPV includes an implicit hypothesis of re-investment of proceeds at compound return (at the same rate as the initial capital), while modified internal rate of return relaxes this hypothesis.

Your introduction did not define return on investment (ROI). And you refer only on a peculiar reformulation of it by Magni et al. ROI has been an accounting-based measurement which is not based on multiplicative process but an additive one (see https://doi.org/10.1515/ael-2023-0132). Moreover, ROI – being the ration of operational earnings over invested capital / total assets - takes earnings and invested capital from accrual-based accounting figures, which are not cash-based.

All the introduction goes on referring to variables and indicators without defining them analytically. Quite the contrary, you should define them according to the current understanding, before applying your understanding and analysis to them.

Section 2.1

Without a clear analytical formulation, claims such as “absence of intermediate harvest” are meaningless. You implicitly refer here to the compound return structure which is assumed by the only family of measure you consider. Moreover, your comparative choice between ROI, RROI and RROC is meaningless and obscure for readers.

Formula 1 is introduced without defining the variables.

Supposedly, k is a flow of earnings which accrue through each time period [b; b+t]; p(t) is the probability of their occurrence.

I understand the your variable k is net operating earnings, not net earnings. Amortisations are capital withdrawals. You are surely referring here to ‘depreciations’ (capital consumption) .

Again, formula 2 does not define K. Is it shareholder equity or total assets?

Formula 3 points to a ratio between two averaging formulas of operational earnings and invested capital (which one?). It is then a prospective measure taken at time b, I guess

It is your definition of RROC, three line equality symbol should be applied

In formula 4, your variable I is ill-defined. You should introduce an explicit accounting system with an elementary income statement to define earnings (k) and an elementary balance sheet to define invested capital (K) and investments/disinvestments (I). And you should make clear that NPV and IRR do not point to those accrual-based financial statements, but to cash-based financial budget statements.

In formula 5, it is not clear which one is an independent variable.

In formula 6, IRR is ill-labelled as “o”. It should be labelled as IRR. In this formula, you introduce again another variable C without definition and explanation, and a continuous compound rate of return ‘o’. How does C relate to k and K and especially I?

In formula 7, you should acknowledge that IRR is the ‘d’ rate which makes NPV = 0

At line 143 (page 5) you wrongly assume that ROI is computed on cash flow basis, while is it an accrual-based measure dividing operational earnings over invested capital (total assets)

At line 148 ff. (ibidem), you refer to accrual-based increments of capital as ‘negative cash flow’ while they are positive accretions to it. Please check and amend or clarify.

In formula 8, your growth model clearly adopts a compound return structure based on r(t)

In formula 9, I am not convinced by your normalisation strategy. If k is an independent flow of earnings, it is not constant over time and its averaging value between the final moment t and the initial moment 0 is not valid to define its flowing.

You never clarified how k and K related to each other, but you assume here that k is the compound return on K under certainty, notwithstanding your probability p(t) which mysteriously disappears in formula 9.

In formula 10, you introduce the same rate r as in formula 9. You should decide if K depends on k or vice-versa.

In formula 11, you introduce a periodic structure of r(t). This structure was never discussed before. How your structure relates and reshapes the traditional compound interest structure which is applied by NPV and IRR?

The concept of Ansatz should be defined and clarified in this context. Formula 11 does not contain tau but t

New variables in formula 11 are not defined and properly explained

Figure 1 cannot clarify anything since the various variables are all defined according to your framework which is not properly compared with the traditional one (based on compound return structure under certainty and without disinvestments between 0 and time t)

Variable IRRs through time occur when a modified IRR is applied, that is, one that consider multiple return through time (similar to your r(t)). But your analysis did not consider that subset of capital budgeting criteria (see previous remark on literature review)

Formula 13 may be potentially valuable as a comparison between traditional NPV and your modified version. But that comparison was never properly clarified. Among other things, you redefine here the discount rate d, its analytical structure and its interpretation. All this has to be explained!

I recommend a function study for that formula. Figure 2 is insufficient (it does not even define the x-axis!)

Discussion of leverage should be set apart in a new section

Formula 14 introduces new variable L and E without defining them. Leverage is traditionally defined as:

L = Liabilities over Equity = (Total Assets – Equity) / Equity = (Total Assets / Equity) – 1

This would imply that you implicitly define K as total assets. Consequently, invested capital was before E (with L = 0) and is now K = E + L

This elementary balance sheet structure should be properly introduced, given its important for a theory of the firm which considers accounting (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=774764)

The proper analytical deduction of formula 15 should be disclosed. The market discount rate was previously labelled d, not u.

Figure 3 does not properly label its axes. Moreover, L=1 has no clear meaning here. Is it an absolute value or a relative value? I recommend taking K=1 and therefore L = [0;1). It cannot be 1 I suppose.

Moreover, L as a cost of debt which should be properly introduced and included in your elementary income statement. This cost element (perhaps embedded in u) was not properly introduced.

The peculiar notion of negative leverage (line 271 ff. page 10) is introduced without explanation. It seems to violate accounting conventions in which numbers cannot be negative

Negative leverage does not imply lending. If by negative leverage you mean capital injections, they should be accounted as E. In fact, L (as total liability) implies a capital injection which has a cost which is different from E (which has another implicit cost, see https://doi.org/10.1515/2152-2820.1068)

The analysis of leverage impact on IRR (line 284 ff. page 11) should be reconsidered and rewritten once the notion of leverage has been clarified

Assumptions underlying equation 16 are introduced too briefly only at line 295 page 11. This is unacceptable for such an important extension

Lines 300 – 305 page 11 point to numerical results which were never properly introduced and explained.

Equation 18 should be reconsidered when the notion and analysis of leverage is clarified

Equation 18 introduced three rates, in line with modified IRR approach. It has been postulated that the three rates must be equal, this is why the traditional NPV and IRR are based upon compound return structure under certainty at that rate

It is unclear how equation 20 is inferred from your conceptual framework or depends on 19

Lines 363-366 page 13 introduced explicitly the concept of financial accumulation (another name for the compound return structure under certainty). It should be introduced before. For a dynamic systems analysis of financial accumulation, see: https://doi.org/10.1007/s11403-020-00281-7

(page 14) your notion of capitalisation is ambiguous here. Either earnings depend on capital, or capital depends on earnings. What do you mean by “growth-induced changes in capitalisation”? growth is implicitly included in your flow of earnings, or earning process you introduced.

(same page) your comparison between a cash-based IRR and an accrual-based profit rate is interesting, but that distinction was never properly explained before and never introduced analytically. Perhaps your analysis is trying to generalise the accounting measure of performance called ROI (operating earnings over total assets both taken under accrual-basis of accounting). Please expand and clarify. This should be done since the beginning and properly modelled

A literature exists comparing accounting measures with cash-based measures (generally trying to prefer the latter against the former). I may agree with that a production process oriented measurement may prefer and build upon the former, but your analysis is not sufficiently clear and developed to uphold this implicit claim

(pages 14-15) conclusions on leverage are alleged and should be reconsidered when your notion and analysis of leverage is clarified and further developed

(page 15) your discussion of ‘intermediate capitalisation’ goes far beyond the simple framework of analysis and modelling strategy you introduced. It is alleged and should be removed

(same page) your formula 20 introduces a completely new issue (spatial distribution of capital investments in production processes). This cannot be treated in this way and deserves a fully-fledged analysis and literature review. It should be removed

(Page 16) It is unclear what do you mean by expected value of the return rate on capital, and all the references are self-references.

(same page) your reference to risk is astonishing since you introduced a probability density over return which is supposed to cover for risk. Please clarify and amend

To deal with uncertainty/risk, dynamical systems analysis by simulation should be developed. Alternatively, you must clarify that your analysis is applied to financial systems under certainty and remove the reference to probability density which is introduced but not treated

Reviewer #2: While the theoretical insights are valuable, the paper could benefit from a clearer explanation of how these findings can be applied in real-world financial decision-making. Providing some concrete examples or case studies could make the implications more tangible. The paper touches on the important topic of leverage effects, but the discussion seems relatively brief. Given the significant impact leverage can have on financial metrics, a more in-depth analysis and illustrative examples could strengthen this section. Some of the mathematical derivations and equations may be challenging for readers without a strong quantitative background. Adding more intuitive explanations or graphical illustrations alongside the equations could make the key concepts more accessible to a broader audience.Clearly stating the key assumptions made in the analysis (e.g., the absence of intermediate divestments) and discussing potential limitations of the approach could help readers better interpret and apply the findings.While the paper focuses on simple periodic growth processes, discussing how the findings might extend to or inform the analysis of more complex systems could broaden the impact and relevance of the work.

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Reviewer #1: No

Reviewer #2: No

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PCSY-D-24-00123R1

Complex economics of simple periodic systems

PLOS Complex Systems

Dear Dr. Kärenlampi,

Thank you for submitting your manuscript to PLOS Complex Systems. After careful consideration, we feel that it has merit but does not fully meet PLOS Complex Systems's publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript within 60 days May 18 2025 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at complexsystems@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pcsy/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:

* A rebuttal letter that responds to each point raised by the editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'. This file does not need to include responses to any formatting updates and technical items listed in the 'Journal Requirements' section below.

* A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

* An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, competing interests statement, or data availability statement, please make these updates within the submission form at the time of resubmission. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

We look forward to receiving your revised manuscript.

Kind regards,

Hocine Cherifi

Editor-In-Chief

PLOS Complex Systems

Bohui Wang

%CORR_ED_EDITOR_ROLE%

PLOS Complex Systems

Hocine Cherifi

Editor-in-Chief

PLOS Complex Systems

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Additional Editor Comments (if provided):

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Reviewers' Comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: (No Response)

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2. Does this manuscript meet PLOS Complex Systems's publication criteria ? Is the manuscript technically sound, and do the data support the conclusions? The manuscript must describe methodologically and ethically rigorous research with conclusions that are appropriately drawn based on the data presented.

Reviewer #1: Partly

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3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: N/A

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4. Have the authors made all data underlying the findings in their manuscript fully available (please refer to the Data Availability Statement at the start of the manuscript PDF file)?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception. The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

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5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS Complex Systems does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: No

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6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The author developed minor amendments to comply with my comments and suggestions.

However, these amendments are not sufficient to clarify the conducted analysis, its novelty and findings (which should be clearly stated preferably in the form of lemmas), and its implications for theory and practice of dynamic systems analysis as applied to financial performance measurements

The author should revise his manuscript to make it more clear-cut and robust on all these points. Moreover:

- the notion of 'operating profits' is not consistent with accounting practice and standards, although your analysis claims it to be accrual-based;

- From a theoretical viewpoint, the recourse to a probability density function is problematic since your analysis is conducted under certainty. The probability function under examination is not clarified or discussed either;

- Equation N. 5 introduces r(t) without defining it and showing it deduction from previous equations in a coherent system;

- Equations N. 6 and 7 shifts toward a cash-basis but the relationship of them with an accrual-basis of accounting is not elucidated;

- Equation N. 9 appears to introduce back a compound-discounting mechanism, although the profit rate <r> or may be made compatible with a simple-discounting mechanism;

- How does the illustrative definition of r(t) in equation N. 11 relate, analytically, with the previous development and system of equations?

Furthermore, it is unclear how the section 2.3 relates to the previous development and system, since this section draws upon and discusses the NPV rather than the profit rate (the supposedly main finding and novelty). The risk here is to try discussing too much in just one paper, losing analytical robustness and a consistent and convincing flow of argument.

Therefore, I recommend removing section 2.3 while replacing it with a new section better explaining and illustrating the features and properties of the profit rate just introduced. This new section should made clear which are the original contribution of your own analysis and how they align or differ from previous literature. Section 3 should be consequently shortened and amended.</r>

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Reviewer #1: No

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Complex economics of simple periodic systems

PCSY-D-24-00123R2

Dear Prof. Dos. Kärenlampi,

We are pleased to inform you that your manuscript 'Complex economics of simple periodic systems' has been provisionally accepted for publication in PLOS Complex Systems.

Before your manuscript can be formally accepted you will need to complete some formatting changes, which you will receive in a follow-up email from a member of our team. 

Please note that your manuscript will not be scheduled for publication until you have made the required changes, so a swift response is appreciated.

IMPORTANT: The editorial review process is now complete. PLOS will only permit corrections to spelling, formatting or significant scientific errors from this point onwards. Requests for major changes, or any which affect the scientific understanding of your work, will cause delays to the publication date of your manuscript.

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Thank you again for supporting Open Access publishing; we are looking forward to publishing your work in PLOS Complex Systems.

Best regards,

Hocine Cherifi

Editor-In-Chief

PLOS Complex Systems

Hocine Cherifi

Editor-in-Chief

PLOS Complex Systems

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Reviewer Comments (if any, and for reference):