Net Stable Funding Ratio and Liquidity Hoarding

Abstract

As a component of the liquidity requirements of Basel III, the Net Stable Funding Ratio (NSFR) seeks to limit the maturity transformation of banks. This paper examines whether the NSFR affects the inefficient precautionary liquidity hoarding of banks and the stability of interbank markets. Based on Acharya and Skeie (2011), the model introduces regulation into a two-period framework with asymmetric information and stochastic credit risk. As a result, due to regulatory costs, the NSFR increases the bid-ask spread on the interbank market. The effects depend strongly on the quality and the forbearance of the regulator. High-quality supervision counters the precautionary liquidity hoarding of banks resulting from asymmetric information, thereby decreasing market failure.

Appendix

1.1 Appendix A: Balance Sheet Considerations

1.1.1 Laissez-faire

Banks of type $L$ fund with time deposits TD and demand deposits DD (Table 2).

Table 2 Bank $L$ in $t_{0}$

Banks of type $L$ use the hoarded liquidity to mitigate the funding gap in $t_{1}$ (Table 3).

Table 3 Bank $L$ in $t_{1}$

Banks of type $B$ use the interbank borrowing $b$ to increase their investment position (Table 4).

Table 4 Bank $B$ in $t_{0}$ and $t_{1}$

1.1.2 NSFR regulation

Long-term lending $l$ to the interbank market requires a bank of type $L$ to adjust its balance sheet. In particular, when the NSFR is a binding constraint, the bank has to raise additional costly equity $c_{E}^{L}$ to hold a liquidity buffer $\textit{Cash}_{\textit{NSFR}}$ (Table 5).

Table 5 Bank $L$ in $t_{0}$

A bank of type $L$ must not use its liquidity $\textit{Cash}_{\textit{NSFR}}$ in $t_{1}$ to mitigate the funding gap in $t_{1}$, which results from withdrawn demand deposits, because otherwise the bank would violate the NSFR (Table 6).

Table 6 Bank $L$ in $t_{1}$

Borrowing $b$ from the interbank market requires a bank of type $B$ to adjust its balance sheet. In particular, when the NSFR is a binding constraint, the bank has to raise additional costly equity $c_{E}^{B}$ and hold a liquidity buffer $\textit{Cash}_{\textit{NSFR}}$. Long-term borrowing $b$ from the interbank market decreases the regulatory pressure, resulting from the difference between the RSF from investing additional units into the investment project and the ASF from borrowing from the interbank market (Table 7).

Table 7 Bank $B$ in $t_{0}$ and $t_{1}$

1.2 Appendix B: Banks’ Objective Function and The Role of Equity

The equity costs in this framework result from agency problems between the bank management and the shareholders. This approach is based on Myers and Majluf (1984) and Pausch and Welzel (2013).

In this model, the bank management acts in the old shareholders’ interest, assuming that those shareholders do not adjust their portfolios in response the banks’ decisions. Thus, maximizing $E(\pi)$ implies acting in the old shareholders’ interest. The high costs of new shareholders $i_{E}> r$ result from asymmetric information in this framework.

By assumption, the bank management has an informational advantage about the value of the long-term investment towards the shareholders. Since the value of the project is private information, the market is not willing to compensate the banks for this information. Thus, issuing new shares would be a “bad” signal concerning the value of a bank’s project. This in turn lowers the price that new shareholders are willing to pay for the issue. Thus, issuing new shares comes at a cost for old shareholders. In the model these costs exceed the positive net present value of the project.

As shown in Appendix A, in order to meet the regulatory constraint of the NSFR, the banks raise costly equity $c_{E}^{B}i_{E}$ and $c_{E}^{L}i_{E}$ from new shareholders to hold additional liquidity.

The regulatory mechanism presented here is very similar to a solvency constraint. Alternative adjustment processes to meet the NSFR would be possible. However, since leaving the optimum due to regulation always decreases $E(\pi)$, alternative adjustments do not change the results qualitatively.

1.3 Appendix C: Incentive Constraint

The bank is not viable when being incited to engage in risk shifting because the demand depositors withdraw their amount of DD in $t_{1}$. In this case, demand depositors are better off when withdrawing in $t_{1}$:

1.3.1 (Expected) pay-off structure without risk shifting:

$$\text{Return per unit}=\begin{cases}1,&\text{if the demand depositor withdraws}\\ \theta{f_{L}}=1,&\text{if the demand depositor prolongs funding}\end{cases}$$

(35)

The banks of type $L$ choose $f_{L}$ (or respectively banks of type $B$ choose $f_{B}$) in $t_{1}$ in a way to make the demand depositors indifferent between withdrawing and prolonging funding in $t_{1}$.

1.3.2 (Expected) pay-off structure with risk shifting:

However, when the banks are incited to engage in risk shifting, the actual (expected) pay-off structure for the demand depositors changes as follows:

$$\text{Return per unit}=\begin{cases}1,&\text{if the demand depositor withdraws}\\ \theta_{R}{f_{L}}<1,&\text{if the demand depositor prolongs funding}\end{cases}$$

(36)

Hence, demand depositors are strictly better off by withdrawing in $t_{1}$ instead of leaving their money in the bank until $t_{2}$ because the demand depositors are liable but not participate in case of success.

Consequently, a bank faces insolvency in $t_{1}$ because it is not possible to prolong funding via demand deposits to close the bank’s funding gap. Moreover, equity it is too expensive due to asymmetric information as discussed in Appendix B.

1.4 Appendix D: Comparative Statics

1.4.1 Maximization problem of bank $L$ in case b):

$$\begin{array}[]{l}\max_{l}E(\pi)=\int_{\hat{\theta}^{L}}^{\overline{\theta}}\bigg[\theta\left(y+lr-\dfrac{\rho_{L_{\textit{TD}}}}{1+i_{\textit{TD}}}i_{\textit{TD}}-i_{E}\left(\gamma\dfrac{y}{1+i_{I}}+\delta{l}-\alpha\textit{DD}-\beta\dfrac{\rho_{L_{\textit{TD}}}}{1+i_{\textit{TD}}}\right)\right)\\ \quad\quad-\big(DD-(1-l)\big)\bigg]g(\theta)\mathrm{d}{\theta}\end{array}$$

(37)

The necessary and sufficient conditions are given by the following:

$$\begin{aligned}\displaystyle\begin{array}[]{rl}\dfrac{\partial{E(\pi)}}{\partial{l}}=&\int_{\hat{\theta}^{L}}^{\overline{\theta}}[\theta(r-i_{E}\delta)-1]g(\theta)\mathrm{d}{\theta}\\ &-\left[\hat{\theta}^{L}lr+k^{L}\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g(\hat{\theta}^{L})\dfrac{\partial\hat{\theta}^{L}}{\partial l}=0\\ {\dfrac{\partial^{2}{E({\pi})}}{\partial{l}^{2}}}=&-\left[\hat{\theta}^{L}r+\dfrac{\partial\hat{\theta}^{L}}{\partial l}lr\right]g(\hat{\theta}^{L})\dfrac{\partial\hat{\theta}^{L}}{\partial l}\\ &-\left[\hat{\theta}^{L}lr+k^{L}\left(1-\dfrac{\gamma_{R}}{1+i_{I}}i_{E}\right)\right]g^{\prime}(\hat{\theta}^{L})\left(\dfrac{\partial\hat{\theta}^{L}}{\partial l}\right)^{2}\\ &-\left[\hat{\theta}^{L}lr+k^{L}\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g(\hat{\theta}^{L})\dfrac{\partial^{2}\hat{\theta}^{L}}{\partial l^{2}}<0\end{array}\end{aligned}$$

1.4.2 Comparative statics:

$$\dfrac{d{l}}{{dk^{L}}}=-\dfrac{{\dfrac{\partial^{2}{E({\pi})}}{\partial{l}\partial{k^{L}}}}}{{\dfrac{\partial^{2}{E({\pi})}}{\partial{l}^{2}}}}\leq 0$$

(38)

The partial derivative of the first-order condition with respect to $k^{L}$ is given by the following:

$$\begin{array}[]{rl}{\dfrac{\partial^{2}{E({\pi})}}{\partial{l}\partial{k^{L}}}}=&{-(\hat{\theta}^{L}(r-i_{E}\delta)-1)g(\hat{\theta}^{L})\dfrac{\partial\hat{\theta}^{L}}{\partial k^{L}}}\\ &-\left[\dfrac{\partial\hat{\theta}^{L}}{\partial k^{L}}lr+\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g(\hat{\theta}^{L})\dfrac{\partial\hat{\theta}^{L}}{\partial k^{L}}\leq 0\end{array}$$

(39)

Moral hazard mitigates interbank lending $l$, ceteris paribus.

1.5 Appendix E: Comparative Statics

1.5.1 Maximization problem of bank $L$ in case c):

$$\max_{l}E(\pi)=\int_{\hat{\theta}^{L}}^{\overline{\theta}}\left[\theta\left(y+lr-\dfrac{\rho_{L_{\textit{TD}}}}{1+i_{\textit{TD}}}i_{\textit{TD}}\right)-(DD-(1-l))\right]g(\theta)\mathrm{d}{\theta}$$

(40)

The necessary and sufficient conditions are as follows:

$$\begin{aligned}\displaystyle\begin{array}[]{rl}\dfrac{\partial{E(\pi)}}{\partial{l}}&=\int_{\hat{\theta}^{L}}^{\overline{\theta}}[\theta r-1]g(\theta)\mathrm{d}{\theta}\\ &\quad-\left[\hat{\theta}^{L}lr+k^{L}\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g(\hat{\theta}^{L})\dfrac{\partial\hat{\theta}^{L}}{\partial l}=0\\ {\dfrac{\partial^{2}{E({\pi})}}{\partial{l}^{2}}}&=-\left[\hat{\theta}^{L}r+\dfrac{\partial\hat{\theta}^{L}}{\partial l}lr\right]g(\hat{\theta}^{L})\dfrac{\partial\hat{\theta}^{L}}{\partial l}\\ &\quad-\left[\hat{\theta}^{L}lr+k^{L}\left(1-\dfrac{\gamma_{R}}{1+i_{I}}i_{E}\right)\right]g^{\prime}(\hat{\theta}^{L})\left(\dfrac{\partial\hat{\theta}^{L}}{\partial l}\right)^{2}<0\end{array}\end{aligned}$$

1.5.2 Comparative statics:

$$\dfrac{d{l}}{{dk^{L}}}=-\dfrac{{\dfrac{\partial^{2}{E({\pi})}}{\partial{l}\partial{k^{L}}}}}{{\dfrac{\partial^{2}{E({\pi})}}{\partial{l}^{2}}}}\leq 0$$

(41)

The partial derivative of the first-order condition with respect to $k^{L}$ is given by the following:

$$\begin{array}[]{rl}{\dfrac{\partial^{2}{E({\pi})}}{\partial{l}\partial{k^{L}}}}=&{-(\hat{\theta}^{L}r-1)g(\hat{\theta}^{L})\dfrac{\partial\hat{\theta}^{L}}{\partial k^{L}}}\\ &-\left[\dfrac{\partial\hat{\theta}^{L}}{\partial k^{L}}lr+\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g(\hat{\theta}^{L})\dfrac{\partial\hat{\theta}^{L}}{\partial k^{L}}\leq 0\end{array}$$

(42)

Moral hazard mitigates interbank lending $l$, ceteris paribus.

The results of the comparative statics for case b) in Eq. (38) and for case c) in Eq. (41) are valid if and only if banks are regulated by an informed regulator. The NSFR mitigates the moral hazard problem because it imposes regulatory costs $\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}$ on risk shifting for the banks.

1.6 Appendix F: Comparative Statics

1.6.1 Maximization problem of bank $B$ in case b):

$$\begin{array}[]{rl}\max_{b}E(\pi)=&\int_{\hat{\theta}^{B}}^{\overline{\theta}}\Bigg[\theta\left(y+b(y-r)-i_{E}\left(\gamma(1+b)-\delta{b}-\alpha\textit{DD}-\beta\dfrac{\rho_{B_{\textit{TD}}}}{1+i_{\textit{TD}}}\right)\right)\\ &-\textit{DD}\Bigg]g(\theta)\mathrm{d}{\theta}\end{array}$$

(43)

The necessary and sufficient conditions with $g^{\prime}(\hat{\theta}^{B})=0$ are as follows:

$$\begin{array}[]{rl}{\dfrac{\partial{E({\pi})}}{\partial{b}}}=&\int_{\hat{\theta}^{B}}^{\overline{\theta}}\left[\theta(y-r-i_{E}(\gamma-\delta))\right]g(\theta)\mathrm{d}{\theta}\\ &-\left[k^{B}(1+b)\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g({\hat{\theta}}^{B})\dfrac{\partial\hat{\theta}^{B}}{\partial b}=0\\ {\dfrac{\partial^{2}{E({\pi})}}{\partial{b^{2}}}}=&-[{\hat{\theta}}^{B}(y-r-i_{E}(\gamma-\delta))]g({\hat{\theta}}^{B})\dfrac{\partial\hat{\theta}^{B}}{\partial b}\\ &-\left[k^{B}\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g({\hat{\theta}}^{B})\dfrac{\partial\hat{\theta}^{B}}{\partial b}\\ &-\left[k^{B}(1+b)\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g({\hat{\theta}}^{B})\dfrac{\partial^{2}\hat{\theta}^{B}}{\partial b^{2}}<0\end{array}$$

(44)

1.6.2 Comparative statics:

$$\dfrac{d{b}}{{dk^{B}}}=-\dfrac{{\dfrac{\partial^{2}{E({\pi})}}{\partial{b}\partial{k^{B}}}}}{{\dfrac{\partial^{2}{E({\pi})}}{\partial{b}^{2}}}}\leq 0$$

(45)

The partial derivative of the first-order condition with respect to $k^{B}$ is given by the following:

$$\begin{array}[]{rl}{\dfrac{\partial^{2}{E({\pi})}}{\partial{b}\partial{k^{B}}}}=&-[{\hat{\theta}}^{B}(y-r-i_{E}(\gamma-\delta))]g({\hat{\theta}}^{B})\dfrac{\partial\hat{\theta}^{B}}{\partial k^{B}}\\ &-\left[(1+b)\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g({\hat{\theta}}^{B})\dfrac{\partial\hat{\theta}^{B}}{\partial b}\\ &-\left[k^{B}(1+b)\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g({\hat{\theta}}^{B})\dfrac{\partial^{2}\hat{\theta}^{B}}{\partial b\partial k^{B}}\leq 0\end{array}$$

(46)

Moral hazard mitigates interbank borrowing $b$, ceteris paribus.

1.7 Appendix G: Comparative Statics

1.7.1 Maximization problem of bank $B$ in case c):

$$\max_{b}E(\pi)=\int_{\hat{\theta}^{B}}^{\overline{\theta}}\big[\theta\left(y+b(y-r)\right)-\textit{DD}\big]g(\theta)\mathrm{d}{\theta}$$

(47)

The necessary and sufficient conditions with $g^{\prime}(\hat{\theta}^{B})=0$ are given by the following:

$$\begin{array}[]{rl}{\dfrac{\partial{E({\pi})}}{\partial{b}}}=&\int_{\hat{\theta}^{B}}^{\overline{\theta}}\left[\theta(y-r)\right]g(\theta)\mathrm{d}{\theta}\\ &-\left[k^{B}(1+b)\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g({\hat{\theta}}^{B})\dfrac{\partial\hat{\theta}^{B}}{\partial b}=0\\ {\dfrac{\partial^{2}{E({\pi})}}{\partial{b^{2}}}}=&-[{\hat{\theta}}^{B}(y-r)]g({\hat{\theta}}^{B})\dfrac{\partial\hat{\theta}^{B}}{\partial b}\\ &-\left[k^{B}\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g({\hat{\theta}}^{B})\dfrac{\partial\hat{\theta}^{B}}{\partial b}\\ &-\left[k^{B}(1+b)\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g({\hat{\theta}}^{B})\dfrac{\partial^{2}\hat{\theta}^{B}}{\partial b^{2}}<0\end{array}$$

(48)

1.7.2 Comparative statics:

$$\dfrac{d{b}}{{dk^{B}}}=-\dfrac{{\dfrac{\partial^{2}{E({\pi})}}{\partial{b}\partial{k^{B}}}}}{{\dfrac{\partial^{2}{E({\pi})}}{\partial{b}^{2}}}}\leq 0$$

(49)

The partial derivative of the first-order condition with respect to $k^{B}$ is as follows:

$$\begin{array}[]{rl}{\dfrac{\partial^{2}{E({\pi})}}{\partial{b}\partial{k^{B}}}}=&-[{\hat{\theta}}^{B}(y-r)]g({\hat{\theta}}^{B})\dfrac{\partial\hat{\theta}^{B}}{\partial k^{B}}\\ &-\left[(1+b)\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g({\hat{\theta}}^{B})\dfrac{\partial\hat{\theta}^{B}}{\partial b}\\ &-\left[k^{B}(1+b)\left(1-\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}\right)\right]g({\hat{\theta}}^{B})\dfrac{\partial^{2}\hat{\theta}^{B}}{\partial b\partial k^{B}}\leq 0\end{array}$$

(50)

Moral hazard mitigates interbank borrowing $b$, ceteris paribus.

The results of the comparative statics for case b) in Eq. (45) and for case c) in Eq. (49) are valid if and only if banks are regulated by an informed regulator. The NSFR mitigates the moral hazard problem because it imposes regulatory costs $\dfrac{\gamma_{R}}{1+i_{I_{R}}}i_{E}$ on risk shifting for the banks.