Dynamic Banking and the Value of Deposits: Bolton, Li, Wang & Yang (2025)

What this is. The paper’s core model, its key propositions on the value and management of bank deposits under leverage regulation, and the numerical findings: enough to know what it derives and why, without reading all 43 pages. To replicate or extend it, read the full source at the original.

TL;DR

Bolton, Li, Wang, and Yang (2025) propose a continuous-time structural model of a bank that maximizes risk-neutral shareholder value while facing two uncontrollable random processes: asset return shocks and deposit flow shocks. The central state variable is the ratio of equity capital to deposits, $k_t = K_t / X_t$. Because depositors freely move money in and out, the bank cannot perfectly control $X_t$, distinguishing it from nondepository intermediaries and nonfinancial firms. Under equity issuance costs and leverage regulations (the supplementary leverage ratio, SLR), deposit inflows can lower $k_t$ and push it toward a costly equity issuance boundary. As a result, the marginal value of deposits (the deposit marginal $q$) turns sharply negative when $k_t$ is low, the bank reduces its deposit rate toward the zero lower bound (ZLB), and lending falls rather than rises. The model relates to Merton (1969) in casting the bank’s problem as a portfolio choice, extends the dynamic capital structure approach of Leland (1994a) and Brunnermeier and Sannikov (2014) to include stochastic deposit liabilities, and departs from Diamond and Dybvig (1983) by focusing on deposit inflow risk (not runs) under equity issuance costs. Unlike Drechsler, Savov, and Schnabl (2021), who treat deposits as long-duration liabilities without interest-rate risk, the paper incorporates deposit flow risk as the central friction. The model explains why (i) banks did not expand lending during the COVID-19 deposit surge, (ii) the SLR relaxation stimulated lending only in the short run, (iii) a low risk-free rate compresses banks’ deposit management flexibility, and (iv) the deposit rate and loan growth co-move positively with bank capitalization.

Core results

Magnitudes and qualitative characterizations are as reported from the numerical solution. Locators point into the source PDF.

#	Result	Locator	Magnitude
R1	Deposit marginal q is positive for well-capitalized banks but turns sharply negative near the equity issuance boundary	Figure 4 Panel A, p. 2087; Figure 5 Panel A, p. 2088	Deposit marginal q ~0.11 for 80% of stationary distribution of k; drops to ~-0.18 as k approaches the equity issuance lower boundary (~0.052)
R2	Loan-to-capital ratio is procyclical in bank capitalization; capital requirement binds about 7% of the time	Figure 1 Panel B, p. 2084; Figure 3 Panel B, p. 2086	A/K rises from ~0 at the lower boundary to ~14 (capital requirement ceiling) as k increases; capital requirement binds 7% of the time; SLR binds far more frequently
R3	Marginal value of equity capital is sharply elevated near the equity issuance boundary, casting a long shadow	Figure 1 Panel A, p. 2084; Figure 3 Panel A, p. 2086	v’(k) reaches ~7 at k = underbar (0.052); stays between 1.022 and 1.029 for 25% of stationary time; exceeds 1.08 for 5.5% of stationary time
R4	Relaxing the SLR raises lending immediately but reduces long-run risk-taking per unit of equity, contrary to conventional wisdom	Figure 7 Panel A and B, p. 2091	Under 4% SLR, A/K is higher in Panel A (given k) but lower in Panel B (against stationary c.d.f.) than under 5% SLR; tightening SLR generates reach-for-yield over the long run
R5	Relaxing the SLR raises deposit marginal q and deposit rates, stimulating deposit-taking; but deposit marginal q becomes more negative near the new lower equity issuance boundary	Figure 8 Panel A and B, pp. 2092-2093	Deposit rate jumps up under 4% SLR; region where ZLB binds shrinks; deposit marginal q is negative before the regulatory change, turns even more negative near the new lower boundary
R6	Lower risk-free rate reduces bank lending (counterintuitive): bank reduces A/K because it has less room to manage deposit risk via the deposit spread	Figure 9 Panel A and B, p. 2094	At r=1% vs r=2%, the deposit rate is less than 0.8 pp higher under r=2%; A/K is lower at all quantiles of k under r=1% than r=2%

Overall (paper’s conclusion). Deposit-taking is a double-edged sword: it lowers funding costs in normal times, but deposit inflows under leverage regulation drive up leverage and can trigger costly equity issuance. When the bank is close to its equity issuance boundary, it reduces deposit rates to the ZLB, cuts lending, and holds safe assets. Banks in low-interest-rate environments have less flexibility to control deposit flows, amplifying this mechanism and explaining patterns observed after the Global Financial Crisis and the COVID-19 pandemic.

Theory / model

The model features a single bank maximizing risk-neutral shareholder value. Two state variables govern the bank’s balance sheet: the deposit stock $X_t$ and equity capital $K_t$. The bank controls five variables: the risky loan book $A_t$, bond issuance $B_t$, deposit rate $i_t$, dividends $dU_t$, and equity issuance $dF_t$. The resource constraint is $A_t = K_t + X_t + B_t$ (p. 2072, eq. 3).

Deposit dynamics (p. 2070, eq. 1): The deposit stock evolves as a diffusion process partially controlled by the bank via the deposit rate:

$$dX_t = -X_t(\delta_X \, dt - \sigma_X \, d\mathcal{W}_t^X) + X_t \, n(i_t) \, dt \tag{1}$$

where $\mathcal{W}_t^X$ is a standard Brownian motion, $\delta_X$ is the drift of payment flows out of the bank, $\sigma_X$ is the deposit flow volatility, and $n(i_t) = \omega_0 + \omega_1(i_t - r)$ is the deposit demand function (eq. 21, p. 2079): raising the deposit rate above the risk-free rate $r$ attracts deposits, lowering it repels them. The deposit rate is bounded below by zero: $i_t \geq 0$.

Bank equity dynamics (p. 2072, eq. 2): Bank equity evolves as

$$dK_t = A_t \left[(r + \alpha_A) \, dt + \sigma_A \, d\mathcal{W}_t^A\right] - B_t r \, dt - X_t i_t \, dt - C(n(i_t), X_t) \, dt - dU_t + dF_t \tag{2}$$

where $\alpha_A$ is the bank’s excess return on lending, $\sigma_A$ is asset return volatility, $\phi \, dt$ is the instantaneous covariance between deposit and asset shocks ($d\mathcal{W}^X$ and $d\mathcal{W}^A$), and $C(n(i_t), X_t) = c(n(i_t)) X_t$ is the cost of maintaining the deposit franchise.

Homogeneity and the HJB equation. The functional forms imply the shareholder value function is homogeneous of degree one: $V(X, K) = v(k) X$ where $k \equiv K/X$ (eq. 9, p. 2075). Within the dividend/issuance boundaries $[\underline{k}, \overline{k}]$ the HJB equation for the scaled value function $v(k)$ is (eq. 10, p. 2075):

$$\rho v(k) = \max_{\pi^A, i} \left\{ [v(k) - v'(k)k]\left[-\delta_X + n(i)\right] + \tfrac{1}{2} v''(k) k^2 \sigma_X^2 + v'(k)(1+k)\left(r + \pi^A \alpha_A\right) + \tfrac{1}{2} v''(k)(1+k)^2 (\pi^A \sigma_A)^2 - v'(k)[i + c(n(i))] - v''(k) k(1+k) \pi^A \sigma_A \sigma_X \phi \right\} \tag{10}$$

where $\pi^A = A/(X+K)$ is the portfolio weight on risky assets, $\rho > r$ is the shareholders’ discount rate. The deposit marginal $q$ equals $V_X(X,K) = v(k) - v'(k)k$ and the equity marginal $q$ equals $V_K(X,K) = v'(k)$.

Regulatory constraints. The capital requirement (eq. 6, p. 2073) restricts the risky asset-to-equity ratio: $A_t / K_t \leq \xi_K$ (baseline $\xi_K = 14.3$). The supplementary leverage ratio (SLR, eq. 7-8, pp. 2073-2076) imposes a lower bound on $k$:

$$k \geq \underline{k} \equiv \frac{1}{1 - \xi_L^{-1}} - 1 \tag{13}$$

with $\xi_L = 20$ in the baseline (implying $\underline{k} \approx 0.05$). This is the equity issuance boundary; hitting it requires the bank to raise costly external equity.

Equity issuance costs. Equity issuance costs are $dH_t = \psi_1 dF_t + \psi_0 X_t \, dt$, where $\psi_1 = 5\%$ is the proportional cost per dollar issued and $\psi_0 = 0.14\%$ is a fixed flow cost proportional to the deposit stock size (governing how frequently equity issuance is triggered). The bank maximizes (eq. 5, p. 2073):

$$V_0 = \max_{\{A,B,i,U,F\}} \mathbb{E}\left[\int_{t=0}^{\tau} e^{-\rho t}(dU_t - dF_t - dH_t)\right] \tag{5}$$

where $\tau$ is the stochastic closing time (when regulatory constraints are violated).

Method

The model is solved numerically as an ODE boundary value problem. The homogeneity reduction to the one-dimensional $v(k)$ transforms the two-dimensional HJB into the ordinary differential equation (10), which is solved over the interval $[\underline{k}, \overline{k}]$ using shooting/iteration.

Optimal risky asset allocation. The first-order condition from (10) for $\pi^A$ yields the optimal loan-to-capital ratio (eq. 18, p. 2077):

$$\frac{A}{K} = \frac{\alpha_A}{\gamma(k) \sigma_A^2} + \frac{\sigma_X}{\sigma_A} \phi \tag{18}$$

where $\gamma(k) \equiv -v''(k)k / v'(k)$ is the bank’s endogenous relative risk-aversion (eq. 19, p. 2078), derived from the curvature of the value function. Even though shareholders are risk-neutral, $\gamma(k) > 0$ because equity issuance costs make the bank endogenously risk-averse. The hedging term $(\sigma_X / \sigma_A)\phi$ reflects the deposit risk as a natural hedge for the asset-side shock when $\phi > 0$.

Optimal deposit rate. The first-order condition for $i$ from (10) yields the q-theory formula for the optimal deposit rate (eq. 23, p. 2079):

$$i = r + \frac{V_X(X,K)/V_K(X,K) - 1/\omega_1}{\omega_1 \theta} - \frac{\omega_0}{\omega_1} = r + \frac{(v(k) - v'(k)k)/v'(k) - 1/\omega_1}{\omega_1 \theta} - \frac{\omega_0}{\omega_1} \tag{23}$$

The deposit rate rises in the ratio of deposit marginal q to equity marginal q. When $V_X / V_K$ is high (deposits are more valuable than equity), the bank sets a high $i_t$ to attract deposits. When $V_X$ falls near $\underline{k}$, the bank reduces $i_t$ toward zero.

Boundary conditions. At the equity issuance boundary $\underline{k}$: value-matching $v(\underline{k} + m) = 1 + \psi_1$ (eq. 14) and smooth-pasting $v'(\underline{k}) = 1 + \psi_1$. At the dividend boundary $\overline{k}$: $v'(\overline{k}) = 1$ (eq. 16) and supercontact condition $v''(\overline{k}) = 0$ (eq. 17, p. 2077). The paper builds on hjb-optimal-stopping techniques from Leland (1994a) and value-function-iteration for the numerical ODE solution.

Empirical specifications

This is a pure-theory paper with a calibrated numerical solution; there are no econometric regressions or estimated equations. The authors calibrate the model to match empirical moments of U.S. banking data (Table I, p. 2082), including:

• Average return on assets of 1.04% (matching $\alpha_A = 0.2\%$ at baseline parameters, FRED data).
• Average deposit-to-assets ratio of 92% (matching $\phi = 0.8$, consistent with Drechsler, Savov, and Schnabl (2017)).
• Average equity issuance frequency of once every four years (matching $\psi_0 = 0.14\%$, consistent with Baron (2020)).
• Average bank deposit growth rate of 1.9% per quarter (matching $\omega_0 = 0.06$, consistent with Lin (2019)).
• Average return on equity of 11% (matching $\theta = 0.5$).

Comparative statics and applications. The paper analyzes two applications numerically:

1. Leverage regulation (Section IV.A): Compares model solutions under SLR of 5% vs. 4%, tracing the immediate (given $k$) and long-run (stationary distribution of $k$) effects on $A/K$, deposit marginal $q$, deposit rate, and equity issuance frequency (Figures 7 and 8, pp. 2091-2093).

2. Low-interest-rate environment (Section IV.B): Compares model solutions under $r = 1\%$ vs. $r = 2\%$ (adjusting $\rho$ by 1% to control for the wedge $\rho - r$), tracing effects on deposit rate and $A/K$ against the stationary c.d.f. of $k$ (Figure 9, p. 2094). Drechsler, Savov, and Schnabl (2017) document that a lower $r$ compresses the deposit spread $r - i$ and constrains banks; this paper shows the mechanism via imperfect deposit flow control.

Datasets used

Dataset	Role in paper	Wiki page
Federal Reserve Economic Data (FRED)	Calibration targets: average Fed funds rate, return on assets of U.S. banks	FRED

The paper is theoretical with a calibrated numerical solution. All key moments (average ROA 1.04%, deposit-to-asset ratio 92%, equity issuance frequency, deposit growth 1.9%/quarter) are cited from published empirical papers (Baron (2020), Lin (2019), Drechsler, Savov, and Schnabl (2017)) and FRED aggregate statistics; no proprietary microdata are used.

When to read the full paper

Read the original if you are: (i) building a model of bank balance-sheet management where deposit risk matters alongside equity issuance costs; (ii) analyzing the short-run vs. long-run effects of leverage regulation (SLR) on bank lending and risk-taking; (iii) studying why low interest rates reduce bank lending (the deposit-management-flexibility channel); (iv) studying the COVID-19 episode of massive deposit inflows and the SLR exemption. The Internet Appendix contains the extension with reserve requirements, the jump-risk model, and the negative deposit rate case.

Attribution and rights

Source: peer-reviewed, The Journal of Finance 80(4), August 2025. Published by Wiley on behalf of the American Finance Association. This distillation was extracted by an LLM on 2026-06-05 and is not human-verified or independently reproduced. The publisher licence is Wiley VOR terms (paywalled); extract-only reproduction applies.

Bolton, Patrick, Ye Li, Neng Wang, and Jinqiang Yang. “Dynamic Banking and the Value of Deposits.” The Journal of Finance 80, no. 4 (August 2025): 2063–2105. DOI: 10.1111/jofi.13454. © 2025 the American Finance Association. This page is an extract-only distillation by the Institute for Automated Research; no reproduction of the full text.