International Banking and Liquidity Allocation

Abstract

This paper explores the comparative advantage of multinational banking over cross-border financial services in terms of capitalizing on a global access to funding sources. We argue that this advantage depends on the benefit and the cost of multinational banks’ intimacy with local markets. The benefit is that it allows multinational banks to create more liquidity. The cost is that it causes inefficiencies in internal capital markets, on which a bank relies to allocate liquidity across countries. We analyze the conditions under which multinational banking is then likely to arise and show that capital requirements have an effect as they influence the degree of inefficiency in internal capital markets for alternative organization structures differently.

Appendix

Proof of Lemma 1

To solve program (9) consider first the sum of rents

$$ \begin{array}{lll} R_{1}(\hat{\chi})+R_{2}(\hat{\chi}) & = & \frac{3}{8}\beta_{mb}C+T_{1}-\frac{1}{4}\hat{\chi}\left(\beta_{mb}C-L\right) \\ & & -\frac{1}{8}L-\frac{1}{2}d_{0}-\min\left\{ T_{1},\frac{1}{4}\left(1-\hat{\chi}\right) \frac{\beta_{mb}C}{2} \right\} , \end{array} $$

(24)

which has the following property

$$ \frac{d}{d\hat{\chi}}\left( R_{1}(\hat{\chi})+R_{2}(\hat{\chi})\right) =\left\{ \begin{array}{ccc} \displaystyle\frac{1}{4}(L-\beta_{mb}C)<0 & if & \hat{\chi}<\chi ^{crit} \\[5pt] \displaystyle\frac{1}{4}(L-\frac{\beta_{mb}C}{2})>0 & if & \hat{\chi}\geq \chi ^{crit} \end{array} \right. \label{equ18} $$

(25)

where $\chi ^{crit}:=\frac{\beta_{mb}C-8T_{1}}{\beta_{mb}C}$. We proceed by distinguishing between four cases. □

Case 1

If deposits d ₀ are too large while the transfer T ₁ is too small, depositors will hold such a high claim on the banker that no liquidation rate smaller than 1 allows the banker to raise enough funds in order to deter investors from enforcing the liquidation of all late loans. Formally, there is no $\hat{\chi}<1$ satisfying Eq. 6, i. e.

$$ \frac{\frac{1}{2}(L-\beta_{mb}C)+2d_{0}-4T_{1}}{L}\geq 1 $$

(26)

holds true. Rewriting this condition yields

$$ T_{1}\leq \frac{1}{2}\left[ d_{0}-\frac{1}{4}\left( L+\beta_{mb}C\right)\right] . \label{equ30} $$

(27)

Hence χ ^∗ is equal to 1 for any combination of d ₀ and T ₁ satisfying Eq. 27.

Case 2

If χ ^crit is smaller than 0, i. e. if $T_{1}>\frac{1}{4}\frac{\beta_{mb}C}{2}$ holds, it follows for all $\hat{\chi}>0$ that $\frac{d}{d\hat{\chi}}\left( R_{1}(\hat{\chi})+R_{2}(\hat{\chi})\right) >0$. It is thus optimal for the bank manager to set χ ^∗ = 1, even though a $\hat{\chi}$, which satisfies constraint (6) with equality, is smaller than 1. The reason here is that T ₁ is too large, so the banker has an incentive to liquidate all late loans for strategic reasons: She simply pockets the high transfer T ₁ at t = 1, but she is not inclined to repay it at t = 2.

For the remaining two cases, neither the condition for case 1 nor that for case 2 holds. Hence we finally consider those cases, where

$$ \frac{1}{2}\left[ d_{0}-\frac{1}{4}\left( L+\beta_{mb}C\right) \right] <T_{1}\leq \frac{1}{4}\frac{\beta_{mb}C}{2} \label{equ7} $$

(28)

holds.

Case 3

Apparently, the bank manager will set χ ^∗ = 0 if the constraint (6) is slack for $\hat{\chi}=0$ and if

$$ R_{1}(0)+R_{2}(0)\geq R_{1}(1)+R_{2}(1) \label{equ11} $$

(29)

is additionally fulfilled. Constraint (6) will not be binding for $\hat{\chi}=0$ if

$$ \frac{1}{4}\beta_{mb}C+T_{1}\geq \frac{1}{2}\left[ \frac{1}{4}\left(\beta_{mb}C+L\right) +d_{0}\right] \label{004} $$

(30)

which is equivalent to require

$$ T_{1}\geq \frac{1}{2}\left[ d_{0}-\frac{1}{4}\left( \beta_{mb}C-L\right)\right] . \label{equ10} $$

(31)

Condition (29) holds true if

$$ T_{1}\leq \frac{1}{4}\left( \beta_{mb}C-L\right) . \label{equ8} $$

(32)

Hence, it follows that χ ^∗ = 0 if T ₁ meets conditions (28), (31) and (32) simultaneously, i. e. if

$$ \max \left\{ \frac{1}{2}\left[ d_{0}-\frac{1}{4}\left( \beta _{mb}C-L\right) \right] ,0\right\} \leq T_{1}\leq \frac{1}{4}\min \left\{ \beta_{mb}C-L,\frac{\beta_{mb}C}{2}\right\} , \label{equ9} $$

(33)

where—owing to assumption (1)—we have $\beta_{mb}C-L<\frac{\beta_{mb}C}{2}$. Hence, Eq. 33 further simplifies to

$$ \max \left\{ \frac{1}{2}\left[ d_{0}-\frac{1}{4}\left( \beta _{mb}C-L\right) \right] ,0\right\} \leq T_{1}\leq \frac{1}{4}\left( \beta _{mb}C-L\right) . \label{equb} $$

(34)

Case 4

Consider the case where $T_{1}\leq \frac{1}{4}\frac{\beta_{mb}C}{2}$ holds true but constraint (6) is violated for χ = 0. Investors then require the banker to choose $\chi \geq \hat{\chi}$ where $\hat{\chi}$ is implicitly defined by Eq. 6, i. e.

$$ \hat{\chi}=\frac{\frac{1}{2}(L-\beta_{mb}C)+2d_{0}-4T_{1}}{L}. $$

(35)

Hence, as a consequence of the property (25) of R ₁(χ) + R ₂(χ), the banker will set $\chi ^{\ast }=\hat{\chi}$ only if

$$ R_{1}(\hat{\chi})+R_{2}(\hat{\chi})\geq R_{1}(1)+R_{2}(1) \label{equ13} $$

(36)

holds and sets χ ^∗ = 1 otherwise. Since $\hat{\chi}$ is defined as that χ for which available liquidity at t = 1 exactly equals what is demanded by investors (i. e. for which condition (6) holds with equality), we have $R_{1}(\hat{\chi})=0$ and condition (36) can be rewritten as

$$ \frac{1}{4}\left( 1-\hat{\chi}\right) \beta_{mb}C-\min \left\{ T_{1},\frac{1}{4}\left( 1-\hat{\chi}\right) \frac{\beta_{mb}C}{2}\right\} \geq \frac{1}{8}\left( \beta_{mb}C+L\right) +T_{1}-\frac{1}{2}d_{0}. \label{equ14} $$

(37)

We know from Eq. 25 that $R_{1}(\hat{\chi})+R_{2}(\hat{\chi})<R_{1}(1)+R_{2}(1)$ if $\hat{\chi}\geq \chi ^{crit}$ or, equivalently, if

$$ T_{1}\geq \frac{1}{8}\frac{(L+\beta_{mb}C)-4d_{0}}{2L-\beta_{mb}C}\beta _{mb}C. \label{equ19} $$

(38)

We therefore have χ ^∗ = 1 if T ₁ satisfies Eq. 38. But if this condition is violated, we have $T_{1}\leq\frac{1}{4}\left( 1-\hat{\chi} \right) \frac{\beta_{mb}C}{2}$. In this case Eq. 37 reads as

$$ T_{1}\leq \frac{1}{8}\frac{(L+\beta_{mb}C)-4d_{0}}{2L-\beta_{mb}C}(\beta_{mb}C-L). \label{015} $$

(39)

Its RHS is even smaller than the expression on the RHS in Eq. 38.

We conclude

$$ \chi ^{\ast }= \left\{ \begin{array}{lll} 1 & if & \displaystyle T_{1}>\frac{1}{8}\frac{(L+\beta_{mb}C)-4d_{0}}{2L-\beta_{mb}C}(\beta_{mb}C-L) \\ \hat{\chi}=\displaystyle\frac{\frac{1}{2}(L-\beta_{mb}C)+2d_{0}-4T_{1}}{L} & if & T_{1}\leq \displaystyle\frac{1}{8}\frac{(L+\beta_{mb}C)-4d_{0}}{2L-\beta_{mb}C}(\beta_{mb}C-L) . \end{array} \right. $$

(40)

Comparing the parameter ranges in Eqs. 27, 28, 34 and 40 yields that Eq. 27 is redundant. Hence, three cases are left which yields Eq. 10.

Proof of Proposition 2

We have to distinguish two cases: □

Case 1

When $\hat{k}\leq \alpha$, deposits $d_{0}^{\ast }$ are equal to $\frac{\beta_{mb}C+L}{4}$. Thus, according to Eq. 15, there will be no transfer and it follows from Lemma 1 that χ ^∗ = 1.

Case 2

When $\hat{k}>\alpha$, deposits are smaller than $\frac{\beta _{mb}C+L}{4}$ and monotonically decreasing in $\hat{k}$.

Since there are no deposits for $\hat{k}=1$, and since

$$ \min \left\{ \frac{\beta_{mb}C-L}{4},\frac{(\beta_{mb}C+L-4d_{0}^{\ast })(\beta_{mb}C-L)}{8(2L-\beta_{mb}C)}\right\} =\frac{\beta_{mb}C-L}{4} $$

(41)

if $d_{0}^{\ast }=0$, the intermediate value theorem implies that there is a critical capital-to-asset ratio, denoted by η, such that

$$ \min \left\{ \frac{\beta_{mb}C-L}{4},\frac{(\beta_{mb}C+L-4d_{0}^{\ast })(\beta_{mb}C-L)}{8(2L-\beta_{mb}C)}\right\} =\frac{\beta_{mb}C-L}{4} $$

(42)

holds for any $\hat{k}\geq \eta$ and

$$ \min \left\{ \frac{\beta_{mb}C-L}{4},\frac{(\beta_{mb}C+L-4d_{0}^{\ast })(\beta_{mb}C-L)}{8(2L-\beta_{mb}C)}\right\} <\frac{\beta_{mb}C-L}{4} $$

(43)

for any $\hat{k}<\eta$. There is thus no need to liquidate loans at all if $\hat{k}\geq \eta$, i. e. χ ^∗ = 0. For intermediate capital-to-asset ratios satisfying $\hat{k}\in (\alpha ,\eta )$ we have χ ^∗ ∈ (0,1). In this case, $d_{0}^{\ast }$ decreases as $\hat{k}$ increases, which allows for higher transfers. Both, lower deposits and higher transfers, lead to a lower χ ^∗ as shown in Lemma 1.

Proof of Proposition 3

Expected repayments to initial investors depend on $\hat{k}$ (Table 1). The proof will show that a necessary condition for repayments associated with cross-border financial services being higher than those associated with multinational banking is that $\hat{k}$ is sufficiently small. Moreover, an increasing $\hat{k}$ will make the provision of cross-border financial services less favorable. Once multinational banking dominates for some $\hat{k}$, there will be no $\hat{k}$ beyond that ratio where cross-border financial services will dominate again.

Table 1 Liquidity provision by internationally active banks

We distinguish two cases, depending on whether capital requirements are binding. □

Case 1

We start with capital adequacy ratios satisfying $0<\hat{k}\leq \alpha$. Cross-border financial services are then associated with higher expected payments than multinational banking if

$$ \Delta _{0\leq \hat{k}\leq \alpha }=\frac{W\left( 0\right) }{1+\hat{k}}-\frac{3V\left( 1,0\right) +U\left( 0,0\right) }{2}>0. \label{equ44} $$

(44)

In what follows we show that Eq. 44 reaches its maximum at $\hat{k}=0$ and that this maximum is strictly positive if and only if Eq. 23 is met. Differentiating Eq. 44 yields

$$ \frac{d}{d\hat{k}}\Delta _{0<\hat{k}\leq \alpha }=-\frac{W(0)}{(1+\hat{k})^2}<0. \label{045} $$

(45)

For $\hat{k}=0$, rearranging Eq. 44 yields that $\Delta _{0\leq\hat{k}\leq \alpha }>0$ if Eq. 23 is satisfied. It can easily be checked that this condition holds for at least some parameters, for instance for β _mb and β _cfs being very close to each other.

Case 2

When $\alpha <\hat{k}\leq 1$, cross-border financial services still dominate multinational banking if

$$ \Delta _{\alpha <\hat{k}\leq 1}=\frac{W\left( 0\right) }{1+\hat{k}}-\frac{V\left( 1,0\right) +U\left( 0,0\right) }{1+\hat{k}}>0 $$

(46)

that is if

$$ \frac{\beta_{cfs}C-L}{(\beta_{mb}-\beta_{cfs})C}>3. $$

(47)