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Some statistical aspects of precautionary reserves in banking

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Abstract

We examine the volatility of overall withdrawal demands in banking using a model allowing for correlation between individual withdrawal demands. We derive various volatility and elasticity results which show the effects of correlated withdrawals in setting precautionary reserves. We discuss the implications of this analysis for fractional reserve banking.

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Notes

  1. Note that all summations in the equations are taken over the whole range of values for i and j. We do not need to impose any inequality restriction on the sums in the above working

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Acknowledgement

The author would like to thank two anonymous referees who provided valuable feedback and advice on an earlier draft of this paper.

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Correspondence to Ben O’Neill.

Appendix: Elasticity calculations

Appendix: Elasticity calculations

In order for us to further understand the reserve requirements we can look at the elasticity of the volatility of the overall withdrawal amount with respect to the number of depositors. Since this involves consideration of the change in volatility with respect to the number of depositors, this requires some specification of the change in the structure of deposits as we gain more depositors.

With n depositors we have total deposited funds equal to \( T=n\overline{X} \) and total withdrawal demands equal to \( T\theta =n\overline{X}\theta \). We let \( {\sigma}_{**}^2 \) be the variance of the total withdrawal demands so that \( {\sigma}_{**}=T{\sigma}_{*}=n\overline{X}{\sigma}_{*} \). To allow us to measure elasticity with respect to the number of depositors or the total deposits we will treat n as a real quantity and treat other quantities as continuous differentiable functions of this quantity, notwithstanding that it is actually discrete. On this basis we then define the quantities:

$$ {\varepsilon}_X\equiv \kern0.4em \frac{\partial \overline{X}}{\partial n}\cdot \frac{n}{\overline{X}}\kern1em {\varepsilon}_{*}\equiv \kern0.4em \frac{\partial {\sigma}_{*}}{\partial n}\cdot \frac{n}{\sigma_{*}}\kern1em {\varepsilon}_{**}\equiv \kern0.4em \frac{\partial {\sigma}_{**}}{\partial T}\cdot \frac{T}{\sigma_{**}}. $$
(10)

The first of these quantities is the elasticity of the mean deposit with respect to the number of depositors. The second is the elasticity of the volatility of the overall withdrawal proportion with respect to the number of depositors. The third is the elasticity of the volatility of the total withdrawals with respect to the total deposited funds.

These three elasticity quantities are related via a simple equation. To see this we apply the product rule for derivatives to obtain:

$$ \begin{array}{c}\hfill \frac{\partial \left(n\kern0.2em \overline{X}\right)}{\partial n}=\frac{\partial n}{\partial n}\kern0.2em \overline{X}+\frac{\partial }{\partial n}n=\overline{X}\left(1+\frac{\partial \overline{X}}{\partial n}\bullet \frac{n}{\overline{X}}\right)=\overline{X}\left(1+{\varepsilon}_X\right),\hfill \\ {}\hfill \frac{\partial \left(n\kern0.2em \overline{X}{\sigma}_{*}\right)}{\partial n}=\frac{\partial \left(n\kern0.2em \overline{X}\right)}{\partial n}{\sigma}_{*}+\frac{\partial {\sigma}_{*}}{\partial n}\kern0.2em n\kern0.2em \overline{X}=\overline{X}{\sigma}_{*}\left(1+{\varepsilon}_X+\frac{\partial {\sigma}_{*}}{\partial n}\bullet \frac{n}{\sigma_{*}}\right)=\overline{X}{\sigma}_{*}\left(1+{\varepsilon}_X+{\varepsilon}_{*}\right).\hfill \end{array} $$

We therefore have:

$$ {\varepsilon}_{**}=\frac{\partial {\sigma}_{**}}{\partial T}\bullet \frac{T}{\sigma_{**}}=\frac{\partial \left(n\kern0.2em \overline{X}{\sigma}_{*}\right)}{\partial \left(n\kern0.2em \overline{X}\right)}\bullet \frac{1}{\sigma_{*}}=\frac{\partial \left(n\kern0.2em \overline{X}{\sigma}_{*}\right)}{\partial n}/\frac{\partial \left(n\kern0.2em \overline{X}\right)}{\partial n}\bullet \frac{1}{\sigma_{*}}=\frac{1+{\varepsilon}_X+{\varepsilon}_{*}}{1+{\varepsilon}_X}. $$
(11)

One important special case is where we have equal deposits given by \( \overline{X}={X}_1=\dots ={X}_n \). This structure of deposits is unchanging in the sense that the average deposit remains constant so that ε X  = 0 and ε * * = 1 + ε *.

To obtain more specific elasticity results we need to specify the correlation structure. We will again take the simple case where the correlations between individual withdrawal proportions are all equal. Using our previous volatility results from Eq. (8) we have:

$$ \frac{\partial {\sigma}_{*}}{\partial n}=-\frac{1}{2}\cdotp \frac{1-\rho }{\sigma_{*}}\cdotp \frac{\sigma^2}{n^2}, $$

so that:

$$ {\varepsilon}_{*}=\frac{\partial {\sigma}_{*}}{\partial n}\cdotp \frac{n}{\sigma_{*}}=-\frac{1}{2}\cdotp \frac{1-\rho }{n}/\frac{\sigma_{*}^2}{\sigma^2} $$
$$ =-\frac{1}{2}\cdotp \frac{1-\rho }{n}/\frac{1+\left(n-1\right)\rho }{n} $$
$$ =-\frac{1}{2}\cdot \frac{{1-\rho }}{{1+\left( {n-1} \right)\rho }}. $$

With equal deposits we then have:

$$ {\varepsilon}_{**}=1+{\varepsilon}_{*}=1-\frac{1}{2}\cdotp \frac{1-\rho }{1+\left(n-1\right)\rho }=\frac{1}{2}\cdotp \frac{1+\left(2n-1\right)\rho }{1+\left(n-1\right)\rho }. $$
(12)

In the special case where the individual withdrawals are uncorrelated we have ρ = 0 so that we obtain elasticity ε * * = 1/2. In the special case where the individual withdrawals are perfectly positively correlated we have ρ = 1 so that we obtain elasticity ε * * = 1. Since the elasticity function is monotonically increasing in ρ we will generally obtain an elasticity that is somewhere between these two extremes. In other words, the combination of the two parts of the variance function in cases where we have some positive but not perfect correlation between withdrawals means that the actual elasticity between the volatility of the overall demand and the scale of the deposits is likely to be more than one-half but less than one.

We can also consider what happens to the elasticity as n → . If \( \underset{n\to \infty }{ \lim}\rho >0 \) we have

$$ \underset{n\to \infty }{ \lim }{\varepsilon}_{**}=1. $$

If \( \underset{n\to \infty }{ \lim}\rho =0 \) the elasticity will depend on the relative limiting rates of the number of depositors and the correlation in withdrawals. Taking \( \underset{n\to \infty }{ \lim } n\rho =a \) we have:

$$ \underset{n\to \infty }{ \lim }{\varepsilon}_{**}=\frac{1+2a}{2+2a}. $$

In the special case where \( \underset{n\to \infty }{ \lim } n\rho =0 \) we have \( \underset{n\to \infty }{ \lim }{\varepsilon}_{**}=1/2 \) so that the elasticity approaches the lower bound.

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O’Neill, B. Some statistical aspects of precautionary reserves in banking. Rev Austrian Econ 28, 179–193 (2015). https://doi.org/10.1007/s11138-014-0264-x

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