Question

To avoid insolvency, regulators decide to provide the bank with \(27 million in bank capital. Assume that bad news about mortgages is featured in the local newspaper, causing a bank run. As a result, \)40 million in deposits is withdrawn. Show the effects of the capital injection and the bank run on the balance sheet. Was the capital injection enough to stabilize the bank? If the bank regulators decide that the bank needs a capital ratio of 10% to prevent further runs on the bank, how much of an additional capital injection is required to reach a 10% capital ratio?

Short answer

Yes, it would be enough.

The bank capital shouls be $8.89 million.

No additional injection is required.

Step-by-step solution

Step 1. Given Information

The balance sheet on the first day would be:

Assets	Liabilities
Required Reserve - $10 million	Checkable deposits - $120 million
Excess Reserve - $51 million	Bank capital - $11 million
Loans - $70 million

After day 2, the balance sheet would change because of further $35 million investment in mortgage.

Assets	Liabilites
Required Reserve - $10 million	Checkable deposits - $120 million
Excess Reserve - $36 million	Bank capital - -$9 million
Loans - $65 million

Step 2. Explanation 

To avoid insolvency, the regulators provided the bank with $27 million in bank capital. But the spread of news regarding mortgages leads to a bank run. People have withdrawn $40 million of deposits.

Because of this the new balance sheet would look like:

Assets	Liabilities
Required Reserve - $6 million	Checkable deposits - $80 million
Excess Reserve - $27 million	Bank capital - $18 million
Loans - $65 million

With the bank run, the checkable deposits fell to $80 million.

The capital ratio can be obtained as follow:

$\frac{18}{98}\times 100=18.37\%$

Now, if the bank capital is 10% then assume bank capital to be x.

$$\begin{gathered} \frac{x}{80+x}=0.1 \\ x=8+0.1x \\ 0.9x=8 \\ x=8.89 \end{gathered}$$

So, for the capital ratio to be 10% bank capital should be $8.89 million.

Since, the capital ratio is already higher than this amount, no additional capital is required.