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Bank runs

Toxic Limited is an investment bank with the following balance sheet:

The CMOs can be sold at $t=1$ for a price of $P_1=160,$ but if held until $t=2,$ their

value rises to $P_2=210.$ Short term debt is in the form of demand deposits, and is held

by $200$ investors, each of whom has $$1.$ At $t=1,$ Toxic promises a $5\%$ net return to

each of these investors if they keep their money in the bank until $t=2.$ Assume that

$5\%$ is high enough that if an investor is confident that the bank will remain solvent,

then it is optimal to keep the money in the bank.

$($a$)$ Suppose that all investors keep their money in the bank until $t=2.$ Will they

get the promised $5\%$ return? If you are one of the $200$ investors, and know that

the other $199$ keep their money in the bank, should you withdraw at $t=1?$ Will

a bank run take place in this scenario?

$($b$)$ Now suppose you know for certain that $180$ other investors want to withdraw at

$t=1.$ Should you keep your dollar in the bank in this case? $[$If withdrawals

exceed available funds, money is shared evenly by the withdrawing depositors.$]$

Will there be a bank run now? Explain.

$($c$)$ Now suppose that a wealthy investor injects $$250$ cash in the bank, in exchange

for new equity. Suppose that, as in $($b$),$ you know that $180$ other investors want

to take their money out at $t=1.$ Should you keep the money in the bank now?

Given this result, would the other $180$ investors find it optimal to withdraw? Do

you expect a bank run now?

$($d$)$ Now assume that due to a change in market conditions, the value of CMOs at

$t=2$ drops to $P_2=175.$ Suppose that, as in $($a$),$ all investors keep their money

in the bank until $t=2.$ Do they get their money back? If you know that all other

investors keep their money in the bank, should you withdraw at $t=1?$ How

should the other investors think? What outcome do you expect? Do you think a

wealthy investor will want to step in in this scenario?