# Do Large Banks Have More Leverage?

10 juli 2015 Plaats een reactie

In an interesting IMF Staff Discussion Note, the relationship between bank size and systemic risk is discussed. In the process, it is argued that large banks tend to have a higher leverage: “Large banks tend to have lower capital”, “large banks hold less capital than small banks, as measured either by risk-weighted capital ratios or a simple leverage ratio”, and on p.11 “lower capital” is called a characteristic of large banks.

The IMF substantiates this claim using the graph that is reproduced below in Panel A. The data for this graph is assembled by the IMF and is collected from 370 banks in 52 countries. The data are for the year 2011 and assets are measured in log billions of USD dollars.

The problem with the conclusion that larger banks use more leverage is that it does not follow immediately from the graph. When looking at the graph I see a scatter plot with very much variation. There is also a red regression line and, yes, it has an upwards slope. But p-values and goodness-of-fit statistics are not reported. So what do the data tell us? Given the high level of noise I wouldn’t be inclined to call ‘lower capital’ a characteristic of large banks.

For further evaluation we need the data. Since I am not aware that the IMF has published the data file underlying the analysis, we seem to be stuck.

## Enter Digitization software

Luckily digitization software exists that can reverse engineer the data from a graph. I have used 2 digitization software solutions and Digitizeit turned out to be the better one. The routines for this software have been developed at the CERN lab in Switzerland.

We import the scatter plot picture in Digitizeit, provide scaling information and turn on the digitization process. In Panel B we show how the picture is presented in Digitizeit and how the data is reverse engineered (at the right hand side of the picture).

## Analysis in R

Once we have the data we can use it in a statistical software package such as R. Panel C shows how we have recreated the scatter plot plus regression line in R. The p-values and goodness-of-fit statistics are reported by R as:

Call:lm(formula = Leverage ~ Size, data = data1)Residuals:Min 1Q Median 3Q Max-18.3056 -4.3609 -0.9075 3.5670 18.3760Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept) 10.2154 1.0778 9.478 < 2e-16 ***Size 1.3094 0.2554 5.126 5.48e-07 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Residual standard error: 6.01 on 284 degrees of freedomMultiple R-squared: 0.08469, Adjusted R-squared: 0.08147F-statistic: 26.28 on 1 and 284 DF, p-value: 5.479e-07

The R-squared value is extremely low, 8%. At the same time we have a highly significant coefficient (P<0.1%). This means that the model is a poor fit and that size is not an effective predictor for leverage. At the same time, it does have, on average, a significant positive slope.

So it is true that, *on average*, large banks tend to have a high leverage. But for individual banks, we cannot predict leverage only on the basis of size. There are probably also other factors. The IMF claim that large banks tend to have high leverage is true. However, their claim that “lower capital” is a characteristic of large banks is, in my opinion, not true.

This is a typical case of low R-squared, low P value. When I did my undergraduate studies (more than 20 years ago), econometric professors told us that low R-squared means that we have a specification error since the model lacks important drivers. If we would include those drivers, the coefficients of the current model could look fundamentally different. This implies that the estimators in a poorly fit specification are not reliant and even biased. This is confirmed in the econometric textbooks that we used, e.g. Greene’s *Econometric Analysis*. For the IMF analysis, this implies that we conclude that the variance explained is extremely low and that there must be some omitted variables, and hence that the results for the slope of the size driver for leverage is biased and meaningless.

Today there seems to be some tolerance for low P value, low R-squared models.

## Highly leveraged banks

Going forward, we can show that there is some ground in the IMF’s claim that large banks tend to have high leverage. If we define high leverage as a leverage > 20, we see that the percentage of high leverage banks increases with size. We show this in Panel C, where the dots for the high-leverage banks are red and the rest is blue. (Thanks to the digitization process we have the data so that we can represent the data and highlight the high-leverage observations.)

If we divide the banks into 6 groups based on size (i.e. 2 < Size ≤ 3, …, 7 < Size ≤ 8, we can generate the graph in Panel 4 which is a new view on the data. It shows the increasing trend of the percentage of high-leverage banks with increasing bank size. It confirms the trend identified by the IMF without the noise that is the main feature of their scatter plot.

## Parting thoughts

By reverse engineering the IMF data on bank size and leverage we set up an analysis in R. We confirm the IMF claim that large banks tend to have more leverage. This is true *on average*. We reject their claim that “low capital” is a characteristic large banks.