# RDP 2023-05: The Impact of Interest Rates on Bank Profitability: A Retrospective Assessment Using New Cross-country Bank-level Data 6. Analytical Framework

In this project, 10 central banks estimate the same regressions using bank-level data for their country – that is, Equation (1) below is estimated separately for each country. Our baseline model is a variant of the specification used in Borio et al (2017):

(1) $profi t i,t = α 0 + α 1 profi t i,t−1 + α 2 r t + α 3 sprea d t + α 4 lo w t + α 5 larg e i + α 6 r t *lo w t + α 7 r t *larg e i + α 8 r t *lo w t *larg e i + α 9 sprea d t *lo w t + α 10 sprea d t *larg e i + α 11 sprea d 1 *lo w t *larg e i + γ 1 Y t + γ 2 X i,t−1 + δ i + ε i,t$

Here profiti,t is the profit of bank i in period t. In our baseline specification, profit is measured as the ROA. We also explore specifications in which profit is replaced by one of its underlying components – the NIM, Non-II and LLP.

The variables of interest are the 3-month interest rate, rt, and the spread between the yield on 10-year government bonds and the short-term rate, spreadt, as well as their interactions. The coefficient ${\alpha }_{6}$ on the interaction term rt *lowt indicates the differential impact a change in the short-term rate has on the profitability of smaller banks when interest rates are low. lowt is a dummy equal to 1 if the home country in a specific year t is in a ‘low-rate environment’. In our baseline specification, we consider a country to be in a low-rate environment when its 3-month interest rate is in the first quartile of the country's sample distribution (Figure 1). The coefficient ${\alpha }_{7}$ on the interaction term between rt and the largei dummy indicates the differential impact a change in the policy rate has on profitability for large banks compared to their smaller counterparts in a normal rate environment. The largei dummy is equal to 1 if the bank is in the group of around 80 global banks that made the BIS' main G-SIB assessment sample for the end-2019 G-SIB exercise. Finally, the coefficient ${\alpha }_{8}$ on the triple interaction term rt *lowt *largei indicates the differential impact of a change in short-term rates in a low-rate environment for larger banks compared to their smaller counterparts. All of these interactions are repeated for the spread between the 10-year rate and the 3-month rate, spreadt.

Yt are macroeconomic controls and consist of real GDP growth, housing price growth and CPI inflation. Xi,t–1 are bank-level controls that include deposits over total liabilities, the liquidity ratio and total equity capital over total assets, all lagged one period (definitions are provided in Table 2). These controls remove any correlation interest rates might indirectly have with profitability via their impact on the state of the economy and funding conditions. Bank-level controls are lagged one period as bank profitability could have a contemporaneous impact on these controls. ${\delta }_{i}$ are bank fixed effects and ${\epsilon }_{i,t}$ is an error term. We use robust standard errors, clustered by bank to accommodate within-bank serial correlation.

In addition to the baseline regression given by Equation (1), contributing central banks estimated two additional regressions. The first of these replaces the lowt dummy with a variable that captures for how long interest rates have been low. A longer period of low interest rates could be expected to increase the negative effect of lowering rates on profitability because interest rate hedges become less effective in a protracted low-rate environment. The second of these replaces the lowt dummy with a dummy variable for whether rates are negative. Negative interest rates could have a detrimental effect on banks' profitability because of banks' limited willingness to pass along negative rates to depositors.

Each contributing country estimated this model over their confidential data using a fixed effects (FE) estimator. Because of the lagged dependent variable we have relaxed the strict exogeneity assumption $\left(E\left({\in }_{it}|control{s}_{i1},...,control{s}_{iT},{\delta }_{i}\right)\ne 0\right)$ and our regressors are instead weakly exogenous $\left(E\left({\in }_{it}|{\left\{control{s}_{is}\right\}}_{s\le t},{\delta }_{i}\right)=0\right)$, assuming ${\in }_{it}$ is serially uncorrelated. The implication of non-strictly exogenous regressors is that the FE estimator is downward biased. However, when the time period is reasonably large, as it is here for most jurisdictions, this bias is negligible. This notwithstanding, estimates obtained for Germany, Norway, Sweden and Switzerland should be considered lower bounds.

Finally, the dynamic specification used in Equation (1) allows us to examine the short- and long-run impact of changes in policy rates on profitability. The longer-term impact of a permanent change in interest rates on profitability for small banks is given by the expression ${\alpha }_{2}/\left(1-{\alpha }_{1}\right)$ which is obtained by recursively substituting for profiti,t–1 in Equation (1).

The benefit of the simple baseline specification used here is it could be easily communicated to each central banking expert and commonly estimated over their own confidential bank-level data. However, this simplicity means we necessarily omit a number of other interactions that might have been interesting to explore. First, additional evidence might be obtained by interacting the interest rate level and the slope of the yield curve with bank-specific variables, such as excess reserves and deposit dependence, which both could increase the sensitivity to rates. We also do not control for expected macroeconomic conditions. Others have argued that policy and profitability might share a common association with expected economic conditions, and failing to control for these could result in biased estimates (Altavilla et al 2018). For example, improvement to the economic outlook could give rise to higher rates and profitability by stimulating investment and increasing current loan demand. On the supply side, banks might also increase their profitability by increasing their business lending as the improved economic outlook translates into lower credit risk. While these channels are plausible, finding good controls for expected demand that are consistently available across countries is challenging (beyond controlling for current economic conditions). Finally, our specification assumes that bank profitability does not affect monetary policy decisions. As noted in Borio et al (2017), while aggregate banking conditions might affect the stance of monetary policy, the profitability of any given bank is unlikely to affect central bank decisions. This is a key feature of running our baseline regression at the bank level, rather than aggregating across all banks for a given country and estimating a time series regression.