Why Do Banks Bear Interest Rate Risk?

Abstract

This paper investigates determinants of banks’ structural exposure to interest rate risk in their banking book. Using bank-level data for German banks, we find evidence that a bank’s exposure to interest rate risk depends on its presumed optimization horizon. The longer the presumed optimization horizon is, the more the bank is exposed to interest rate risk in its banking book. Moreover, there is evidence that banks hedge their earnings risk resulting from falling interest levels with exposure to interest rate risk. The more a bank is exposed to the risk of a decline in the interest rate level, the higher its exposure to interest rate risk.

Appendix

1.1 Appendix 1: Optimal Interest Rate Risk Exposure

Let there be the following relationship between the initial wealth $W_{0}$ and the wealth in time $t,W_{t}$:

$$W_{t}=W_{0}\cdot e^{R\cdot t}$$

(15)

where $R$ is the relevant (annualized) interest rate.

Using the first-order Taylor approximation at $t=0$, we obtain

$$W_{t}=W_{0}\cdot\left(1+R\cdot t\right)\> .$$

(16)

We assume that there is an overnight parallel shift of the term structure at $t=0$. This assumed shock is permanent and no further interest rate shocks will take place. Therefore, the initial wealth $W_{0}$ is exposed to interest rate risk. If the interest rate changes, the present value of the asset changes as well where its modified duration $D$ gives the sensitivity ($\bar{R}$ and $\bar{W}$ are the expectations of $R$ and $W$, respectively):¹¹

$$W_{0}=\bar{W}\cdot\left(1+D\cdot\left(\bar{R}-R\right)\right)$$

(17)

Combining (16) and (17) and neglecting terms of second-order importance, we obtain

$$\begin{aligned}\displaystyle W_{t}&\displaystyle=\bar{W}\cdot\left(1+D\cdot\left(\bar{R}-R\right)+R\cdot t+D\cdot t\cdot\left(\bar{R}-R\right)\cdot R\right)\\ \displaystyle&\displaystyle\approx\bar{W}\cdot\left(1+D\cdot\left(\bar{R}-R\right)+R\cdot t\right)\\ \displaystyle&\displaystyle=\bar{W}\cdot\left(1+D\cdot\bar{R}+R\cdot\left(t-D\right)\right)\> .\end{aligned}$$

(18)

We assume that the relevant interest rate risk $R$ is composed of the interest rate level $r$ and a term premium that is proportional to the duration, i.e.

$$R=r+c\cdot D$$

(19)

where $c$ is the term premium per year of duration $D$; in case of a normal term structure, it is positive.

If we assume, in addition, that the interest rate level $r$ is normally distributed with $E(r)=\bar{r}$ and ${var}(r)=\sigma_{r}^{2}$, then the wealth in $t$, $W_{t}$, is normally distributed as well (see Eq. (18)) and we can, assuming constant absolute risk aversion, state the preferences as follows (see Sect. 3.1), where $t=T_{i}$ is the exogenous optimization horizon for bank $i$:

$$\begin{aligned}\displaystyle\phi_{T_{i}}&\displaystyle=E\left(W_{T_{i}}\right)-\frac{\gamma}{2}\cdot {var}\left(W_{T_{i}}\right)\\ \displaystyle&\displaystyle=\bar{W}\cdot\left(1+T_{i}\cdot(\bar{r}+c\cdot D_{i})\right)-\frac{\gamma}{2}\cdot\bar{W}^{2}\cdot\left(T_{i}-D_{i}\right)^{2}\cdot\sigma_{r}^{2}\end{aligned}$$

(20)

Accordingly, the optimal modified duration $D_{i}^{*}$ for bank $i$ is

$$D_{i}^{*}=\frac{T_{i}\cdot c}{\gamma\cdot\bar{W}\cdot\sigma_{r}^{2}}+T_{i}\> .$$

(21)

If the term structure is normal (and $c$ is therefore positive), the optimal duration for bank $i$ is greater than its optimization horizon $T_{i}$.

1.2 Appendix 2: Hedging

We approximate bank $i$’s balance sheet as follows. On the asset side, there are two types of assets, namely loans (share: $\theta_{A,i}$) and cash (share: $1-\theta_{A,i}$). The loans are granted on a revolving basis, are free of credit risk, have an original maturity of $M_{A,i}$ (which corresponds to the fixed interest period) and their coupons correspond to the prevailing interest rate at the time the loans were issued. Assume that, at the beginning of the year under consideration, there is a parallel shift of the entire term structure by $\triangle R$. The change in the bank’s net interest margin $\triangle {NIM}_{i}$ of this year is

$$\triangle {NIM}_{i}=\left(\varphi_{A,i}\cdot\theta_{A,i}-\varphi_{L,i}\cdot\theta_{L,i}\right)\cdot\triangle R+\varepsilon_{i}\> ,$$

(22)

where $\varepsilon_{i}$ is some bank-specific noise and $\varphi_{A,i}$ is the fraction of the bank’s loans that have become due in the year after the shock, weighted by the period of time in which they mature, i.e. within the year (see Busch and Memmel (2017)):

$$\varphi_{A,i}=\frac{1}{2\cdot M_{A,i}}$$

(23)

The modified duration of this bank’s loan portfolio is approximately

$$D_{A,i}=\frac{1}{2}M_{A,i}\> .$$

(24)

An example is given to clarify the topic. Assume that bank $i$ hands out loans with a maturity (and a fixed interest maturity) of $M_{A,i}=8$ years. According to Eqs. (23) and (24), the weighted new business in the first year is $\varphi_{A,i}=6.25\%$ and the modified duration of this portfolio is $D_{A,i}=4$.

The variables $\theta_{L,i}$, $\varphi_{L,i}$ and $M_{L,i}$ are the corresponding variables on the bank’s liability side.

Rearranging Eq. (22) gives

$$\triangle {NIM}_{i}=\left(\varphi_{L,i}\cdot\theta_{i}+\left(\varphi_{A,i}-\varphi_{L,i}\right)\cdot\theta_{A,i}\right)\cdot\triangle R+\varepsilon_{i}\> ,$$

(25)

where $\theta_{i}:=\theta_{A,i}-\theta_{L,i}$ is bank $i$’s (net) long-term pass-through.

Combining Eqs. (24) and (23) and incorporating the result into Eq. (25), we obtain

$$\triangle {NIM}_{i}=\left(\varphi_{L,i}\cdot\theta_{i}+\left(\frac{1}{4\cdot D_{A,i}}-\frac{1}{4\cdot D_{L,i}}\right)\cdot\theta_{A,i}\right)\cdot\triangle R+\varepsilon_{i}$$

(26)

With $\beta=\varphi{}_{L,i}> 0$ and $\gamma=12.5\cdot E_{i}/(A{}_{i}\cdot D_{A,i}\cdot D_{L,i})> 0$, where $E_{i}:=A_{i}-L_{i}$ denotes the bank $i$’s equity, Eq. (26) turns into Eq. (11). To obtain the expression for $\gamma$, two additional assumptions need to hold. First, we assume that bank $i$’s interest bearing assets $\theta_{A,i}\cdot A_{i}$ correspond to its interest bearing liabilities $\theta_{L,i}\cdot L_{i}$,¹² so that the duration $D_{i}$ of the bank’s equity $E_{i}$ can be written as

$$D_{i}=\frac{A_{i}\cdot\theta_{A,i}}{E_{i}}\cdot\left(D_{A,i}-D_{L,i}\right)\> .$$

(27)

Second, we assume that of the two scenarios for the Basel interest rate coefficient, the rise in the interest level is the relevant one meaning that ${IRR}_{i}=D_{i}\cdot 0.02$.