Quantifying the components of the banks’ net interest margin

Abstract

Using unique data sets on German banks, we decompose their net interest margin and quantify the different components by estimating the costs of the various functions they perform. We investigate three major functions: liquidity and payment management for customers, bearing credit risk, and term transformation. For 2013, the costs of liquidity and payment management correspond, in the median, to 47% of the net interest margin, with bearing of credit risk and earnings from term transformation accounting for 12 and 37%, respectively.

Appendix

In this appendix, the weighting function $h(\cdot )$ is derived, i.e., the function that shows how the past and current expected credit loss rates for an industry are to be weighted for the contribution to the net interest margin that covers the expected losses in a bank’s credit portfolios (see Eq. (1)). The expected loss rates $\mu \left( {t_m ,j,m} \right)$ in $t_m$ in industryj for newly granted loans with maturity m are relevant for today, i.e., at time t, for loans that were granted at time $t_m$ and have not matured yet, i.e., $m>v:=t-t_m$. We make two assumptions. First, the expected loss rate at time $t_m$ is the same for all the loans in the initial maturity bracket under consideration, i.e., $\mu \left( {t_m ,j,m} \right) =\mu _{t_m ,j,k}$. Second, the loan volume for a certain initial maturity m is spread equally across the maturity bracket k, i.e., $g(m,a_k ,b_k )=1/(b_k -a_k )$, where $g(\cdot )$ is the density, and $a_k$ and $b_k$ are the lower and upper bounds of the maturity bracket k, respectively. In a given time span $\Delta t$, the fraction $1/m\cdot \Delta t$ of loans with initial maturity m must be renewed to have a constant balance sheet. For $1\le a_k <b_k$ and $0\le v:=t-t_m <a_k$, the expression for the weighting function $h(\cdot )$ is therefore

$$\begin{aligned} h\left( {v,a_k ,b_k } \right)= & {} \frac{1}{b_k -a_k }\cdot \int \limits _{a_k }^{b_k } {\frac{1}{m}\mathrm{d}m} \\= & {} \frac{1}{b_k -a_k }\cdot \left( {\ln \left( {b_k } \right) -\ln \left( {a_k } \right) } \right) ,\\ \end{aligned}$$

where $v:=t-t_m$ is the time difference between today t and the point in time $t_m$ in the past.

In the event that $a_k \le v<b_k$, we can express $h(\cdot )$ as

$$\begin{aligned} h\left( {v,a_k ,b_k } \right)= & {} \frac{1}{b_k -a_k }\cdot \int \limits _v^{b_k } {\frac{1}{m}\mathrm{d}m} \\= & {} \frac{1}{b_k -a_k }\cdot \left( {\ln \left( {b_k } \right) -\ln \left( v \right) } \right) . \end{aligned}$$

As we have only yearly data for the expected credit loss rates, we calculate the yearly average of the weighting function, i.e.,

$$\begin{aligned} h_{l,k} =\int \limits _l^{l+1} {h\left( {v,a_k ,b_k } \right) } \mathrm{d}v. \end{aligned}$$

For $l\le a_k -1$, we obtain

$$\begin{aligned} h_{l,k} =\frac{1}{b_k -a_k }\cdot \left( {\ln (b_k )-\ln (a_k )} \right) , \end{aligned}$$

and for $a_k -1<l\le b_k -1$, the corresponding expression is

$$\begin{aligned} h_{l,k} =\frac{1}{b_k -a_k }\cdot \left( {\ln (b_k )-(l+1)\cdot \ln (l+1)+l\cdot \ln (l)+1} \right) . \end{aligned}$$

This can be seen using the formula $\int {\ln (x)\mathrm{d}x=x\cdot \ln (x)-x+C}$. For the first maturity bracket, i.e., $a_1 =0$ and $b_1 =1$, the function $h(\cdot )$ and its average $h_{0,1}$ are set to 1. The values of the weighting function $h_{l,k}$ are shown in Table 1.