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## Bank capital and solvency risks

Here is a simplified balance sheet for a large bank: (£ billions)
Assets: Liabilities:
Loans: £1050 Equity £60
Financial assets £425retail deposits £950
Physical assets £15bonds and other long term debt £290
Reserves at the central bank £25 short term wholesale borrowing £215

The annual interest rates the bank offers on its liabilities are:
Retail deposits: 3%
Bonds and other long term loans: 5%
Short term wholesale loans: 4%
These are the promised interest rates – if there is a chance of insolvency the expected rates of return on these forms of debt finance will be lower. Shareholders have limited liability (their equity can be worth no less than zero).
The expected value of the return on the assets is 6%. Suppose that the total value of the assets a year ahead is normally distributed around its expected value and has a standard deviation of £60 billion.
1. What is the probability that the bank is insolvent a year ahead?
2. What is the expected – or average ‐ return on the equity over the year? (remember the shareholders have limited liability so this calculation is not straightforward and you definitely cannot do it with a pocket calculator!).
3. Devise a fair scheme to pay the holders of all the debt of the bank if the bank is insolvent. (Take note of the fact that there are three types of debt and each has been promised a different interest rate and each has provided a different amount of debt).
4. Assuming that there was no change in the return promised on all types of funding or in the returns on assets, how much of the deposits of the bank would need to switched to equity funding for the chances of insolvency to be halved?
Could the situation described in question 4 be an equilibrium? If it is not an equilibrium what might change?
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