When preferred issuance protects existing holders

Citation

Livingston, Adam. 2026. Digital credit and perpetual preferred stock valuation: Protection boundaries, commitment, and funding policy. 01 October 2026, Strive Working paper.

Analysis disclosure: This assessment was generated by ChatGPT Astra on October 02, 2026. It is machine-generated and should be checked against the source before reliance.

Overview

Succinct summary

Adam Livingston develops a formal model of when issuing additional perpetual preferred shares protects existing treasury company preferred holders. New subscriptions add resources but also competing claims. Starting without cash, sufficiently small issues dilute incumbent preferred value under the baseline funding policy; a sufficiently large issue with binding cash retention can reverse that effect. The gain comes at existing common shareholders’ expense when financing is fairly priced and costless. An alternative model changes who funds coupon shortfalls and permits covered default, reversing the main numerical example. The paper supplies conditional financing benchmarks and protection boundaries, with actual-security valuation requiring additional contractual and accounting detail.

Paper at a glance

• Paper type: Formal corporate-finance working paper, combining analytical results with numerical illustrations and parameter maps.
• Question: When can new preferred issuance protect incumbent preferred holders, and which investor group bears the cost?
• Core claim: Issue size, binding cash allocation, coupon-funding responsibility, and default rights jointly determine the financing effect.
• Evidence: Valuation formulas, fixed-point proofs, investor-cohort budgets, a finite protection certificate, and designed numerical scenarios.
• Principal boundary: Results apply to specified payout and funding arrangements; the paper does not estimate a traded security’s fair value or predict an unconstrained issuer’s choices.
• Reader: Researchers, preferred-security analysts, and Bitcoin treasury observers evaluating financing mechanics, reserve commitments, and the distribution of value among investors.

Placement in the research landscape

Livingston places the analysis within structural credit valuation, preferred-stock pricing, financing without commitment, and the economics of liquidity. Section 2 connects its question to the distinction between asset payouts and security payments, the treatment of existing claims when financing expands, and conflicts between common shareholders and senior claimholders. These are established research problems; the paper claims a narrower contribution in finite issuance comparisons and quantitative protection boundaries.

The valuation formula uses a Brownian probability identity attributed to Matsumoto and Yor. Livingston identifies Jackson Fairbanks’s manuscript on perpetual preferred equity against Bitcoin as motivation, while stating the model’s definitions and results within this paper. Its distinctive analytical combination is endogenous subscription pricing, explicit cash allocation, and exhaustive accounting across investor groups. The funding-policy comparison then asks whether protection survives a change in who must support coupons. This placement does not establish priority over the full financing literature.

Implications for policy and practice

The model makes reserve commitments and funding obligations central to financing analysis. A transaction assessment would need to identify which proceeds enter usable cash, which obligations the new shares create, whether management can reallocate the reserve, and who must cover later shortfalls. A statement about intended issuance prices cannot substitute for a binding allocation policy.

For governance and disclosure, the relevant comparison includes both preferred and common shareholders. Under the paper’s baseline budget, improving one group’s position reduces the other’s value before fees. Claims of wider benefits therefore require an additional mechanism, such as lower distress costs or improved investment opportunities. Applying the framework to an actual security would also require its payment discretion, arrears, liquidation, and financing terms; the model’s calculated thresholds do not themselves justify a reserve requirement or issuance recommendation.

Broader implications

1. Liquidity and wealth distribution. If reserves change which investors receive existing resources, stronger protection for one claim can coexist with unchanged aggregate incumbent wealth. A financing policy’s distributional consequences therefore remain important even when the transaction creates no modeled surplus.
2. Commitment as an economic input. If management can reverse a reserve allocation after subscription, investors must value that discretion. Two arrangements with identical initial cash balances could then carry different claim values because their enforceable future policies differ.
3. Scale can change the mechanism. When claim dilution starts at first order but reserve benefits emerge at higher order, extrapolating from a small transaction can miss a finite-issue reversal. Conversely, mathematical existence at large scale does not establish that investors will supply the required capital.
4. Funding institutions shape asset exposure. If common shareholders fund coupon shortfalls rather than allowing coupons to consume the asset account, the same initial risky resources support different continuation and default decisions. For Bitcoin-backed financing, asset exposure alone would then be insufficient to characterize preferred risk.

Detailed summary

Argument architecture

• Specify delivered payments. The baseline separates a cash account from risky resources. Cash earns the discount rate and pays preferred coupons first; after cash exhaustion, coupons consume risky resources, which also support proportional common payouts. This specification determines what each investor actually receives.
• Price resources and claims together. New preferred shares add subscription proceeds and increase the number of claims sharing the resource pool. Their fair price must equal the value supported by the post-issue state, although that state itself depends on the subscription price.
• Distinguish small and finite issues. From zero cash, cash and risky resources have the same initial marginal preferred value. Small issues therefore impose immediate dilution before the funding-allocation benefit becomes large enough to offset it. Finite cash retention can change the sign.
• Identify the payer. Fair subscribers receive value equal to their contribution. With no financing fees or additional value-creation channel, incumbent preferred gains consequently equal incumbent common losses. Common shareholders favor the lowest permitted cash allocation, making commitment essential.
• Change the funding rules. An added default option is inactive when common receives only nonnegative payments and need not contribute. Requiring common to fund shortfalls makes default meaningful and can overturn protection. The warranted conclusion concerns dependence on funding and enforcement rules, rather than a universal benefit from issuing preferred shares.

Assumptions and scope conditions

The baseline fixes a positive discount rate, volatility, coupon, and proportional common payout under a declared valuation law. Cash must remain below the amount that could fund coupons indefinitely, and the specified continuation excludes later financing, an operating franchise, arrears, calls, rate resets, and terminal principal recovery at resource exhaustion (pp. 4–7). Binding allocation determines how proceeds support payments. The positive common payout closes the perpetual resource budget; it represents neither Bitcoin yield nor an observed Strive dividend policy. Removing it requires an explicit alternative way for common shareholders to receive value. The alternative funding model instead requires common contributions and covered default, while the future-financing extension permits one dated, capped common-equity round. Each change defines a different economic problem.

Evidence base and method

• Analytical valuation. The paper derives preferred coupon value from an established probability identity, then establishes increasing, concave claim value and marginal resource values bounded by one. These properties support the cash-allocation comparison and make the issuance-price mapping contractive, yielding a unique feasible fair price under its fee condition.
• Numerical illustration. The reference state uses $60 in risky resources per existing preferred share, no cash, a $13 annual coupon, a 4.4% discount rate, 50% volatility, and a 1% common payout rate. With two new shares per existing share and no fees, all-cash issuance raises incumbent value by about $0.426; all-risky issuance lowers it by about $6.218 (Table 1).
• Parameter exploration and verification. The baseline maps cover 96,720 designed parameter cells. Section 9 reports quadrature refinement and alternative numerical checks, while Appendix G supplies a separate arithmetic certificate for a narrow finite region. The grid proportions have no estimated frequency interpretation for actual issuers.
• Can establish. Formal results identify conditional price closure, dilution, transfers, and protection boundaries. Numerical examples demonstrate their operation at specified states.
• Outside its reach. No fitted issuer sample identifies market prevalence, issuance capacity, or observed pricing performance. The bounded future-equity exercise cannot establish robustness to unrestricted recurring finance, and the theoretical model alone cannot price all contractual rights of a traded preferred security.

Claim ledger

1. Claim: A feasible issue has a unique fair subscription price under the stated continuation, and protection can be determined by evaluating the pricing map at the no-issue value. Locus: pp. 8–9, Theorems 6.1 and 7.1. Basis: formal derivation. Support: supported within design. Safe use: Cite endogenous pricing that includes both proceeds and additional claims. Boundary: Feasibility requires the exact fee ceiling; uniqueness concerns a specified policy, not equilibrium selection among competing issuer strategies.
2. Claim: Every sufficiently small feasible issue from positive risky resources and zero cash dilutes incumbent preferred, uniformly across proceeds allocations. Locus: pp. 10–11, Theorem 8.1; Appendix D, p. 32. Basis: formal derivation. Support: supported within design. Safe use: Explain why fair pricing does not prevent local dilution. Boundary: Positive common payouts, the resource-withdrawal continuation, and a fixed fee below the no-issue preferred value control the result.
3. Claim: Finite committed-cash issues can protect incumbents, including throughout a specified parameter box. Locus: pp. 12–14, Theorem 8.2, Corollary 8.2, and Table 1; Appendix G, pp. 35–36. Basis: formal derivation. Support: conditionally supported. Safe use: Cite existence and sufficient protection conditions. Boundary: The certificate uses 16–20 new shares per existing share and 99.5–100% cash retention; it neither identifies a minimum feasible transaction nor establishes market capacity.
4. Claim: With fair subscription and zero fees, incumbent preferred gains exactly equal incumbent common losses. Locus: p. 9, Proposition 7.1, equations (27)–(31). Basis: formal derivation. Support: supported within design. Safe use: Separate protection of a claim from aggregate incumbent wealth creation. Boundary: The identity assumes exhaustive resource delivery and excludes investment synergies, distress-cost savings, and favorable mispricing; those channels require additional modeling.
5. Claim: Common-funded coupons with covered default reverse the reference cash-protection example, although protection remains possible elsewhere. Locus: pp. 18–22, Lemma 11.1, Propositions 11.2–11.4, and Table 3. Basis: formal derivation. Support: conditionally supported. Safe use: Demonstrate dependence on funding responsibility and recovery rights. Boundary: The liquidation preference must cover the derived default boundary; the comparison changes both funding and stopping, so it cannot isolate a default-option effect.
6. Claim: A sufficiently small, bounded future common-equity round preserves the sign of the baseline issuance effect. Locus: p. 24; Appendix H, p. 39, Corollary H.2 and equation (94). Basis: formal derivation. Support: supported within design. Safe use: Cite a quantified limited robustness result. Boundary: The discounted net funding cap per original preferred share must be smaller than the absolute baseline margin; arbitrary future financing and freely chosen positive fundraising remain outside the guarantee.

Falsification and testability

The formal claims can fail through a proof error or a counterexample satisfying their stated assumptions; market disagreement alone would instead question their applicability. Computational verification should distinguish the certificate’s arithmetic enclosures from ordinary convergence checks and reproduce financing prices across protection boundaries. Empirical application requires a separate test: identify the pre-transaction resource state, binding allocation, contractual payments, and financing expectations, then compare predicted incumbent-value changes with observed outcomes. A pre-announcement market price may already incorporate expected issuance and therefore need not equal the model’s no-issue counterfactual (p. 9). Evidence that cash allocations remain discretionary, or that coupon support differs from the assumed funding law, would materially qualify a security-level application without refuting the conditional theorem.

Open questions and information gaps

1. Which enforceable reserve and payment provisions make an actual preferred security comparable to either funding economy? Establishing control over proceeds, payment discretion, recovery, and shortfall responsibility would determine whether the model’s protection mechanism applies to the security being evaluated.
2. How much of the computed protective region survives realistic issuance fees, transaction sizes, and initial cash balances? Mapping these constraints jointly would distinguish mathematical existence from financing opportunities available at economically relevant resource states.
3. What additional benefit would induce common shareholders to accept a binding protective allocation? Quantifying franchise preservation, avoided distress, or improved financing access would identify whether those benefits can offset the modeled transfer to incumbent preferred holders.
4. How do repeated financing opportunities and discretionary payment policies change the protection boundary? Extending beyond one committed future round would reveal whether recurrent funding preserves the local-versus-finite distinction or creates materially different incentives and claim values.
5. Do contract-matched predictions explain incumbent preferred repricing around financing events? Separating anticipated issuance from new information about resources, allocation, and rights would provide a meaningful test of empirical applicability without confusing market reactions with the model’s no-issue counterfactual.

Assessment

High confidence: I regard the explicit investor-cohort budget and the separation of funding responsibility from default discretion as the paper’s strongest contributions. They explain why fair issuance can dilute incumbents, why protecting preferred can cost common shareholders, and why a reserve balance has no policy-independent credit meaning. The paper also distinguishes its imported probability identity from the financing results it develops, and its own counterexamples limit the headline interpretation.

Moderate confidence: I would use the protection results as a formal benchmark, with the stated assumptions attached. The proofs, sufficient bounds, and numerical comparisons support that use, but the supplied material does not permit a full executable audit of the source package or certificate implementation. The reference example also has a narrow margin: it requires about 96.72% cash retention, and a fee of approximately $0.259 per new share removes protection (pp. 14–15). Neither sensitivity invalidates the result; both constrain its practical interpretation.

Low confidence: I would place little confidence in a direct actual-security valuation inferred from this paper alone. Its principal application gap is the connection between the specified payout economy and enforceable security rights, issuer incentives, and financing access. Livingston acknowledges that gap and discloses employment and financial interests in Strive (p. 27); Appendix K also states that independent human mathematical review has not yet been obtained. These disclosures warrant attention to verification without constituting evidence of error.

Cite this for conditional preferred-issuance valuation, investor-cohort transfers, and the dependence of reserve protection on commitment and funding rules. Do not cite this for a calibrated SATA price or evidence that actual preferred issuance generally benefits incumbent holders.