Dear all, after working on this problem for several hours I still can’t wrap my head around how I would determine the lending volume taking account probability of a solvency of 95%. I know how to calculate maximum lending volume L =< E/(1-(1-x)R)”, but as said not with all variables considered.
I am quite sure you need to mainly consider downturn and the fact that outcome needs to be more or equal to 95%. (Leverage I can calculate)
Here is the question:
“There are two dates, date t and t+1. A bank at date t chooses its lending volume so as to maximize its expected profits, that is the profits it will have at t+1 as a result of the lending it makes at date t, conditionally upon being solvent (able to repay its debt at date t+1) with a probability not smaller than 95%.
The environment is as follows:
The bank has a net worth (equity), E, and can raise funding by borrowing at a zero net interest rate, that is for every unit, Euro, it borrows at date t, the repayment due at date t+1 is one Euro.
A loan that the bank makes at date t, at date t+1, either performs, in which case the bank gets R > 1 , or it does not perform, in which case the bank gets nothing (zero).
There is aggregate risk: at date t+1, with probability 2/3 the economy will be in an upturn and all loans made at date t perform. With the residual probability of 1/3 the economy will be in a downturn and the percentage x of the loans do not perform.
Assume that (R-1)/R < x.
Derive the size of the lending volume, L, that the bank chooses at date t, and the bank’s leverage.
As a numerical example, work out the solution for x=10% , R= 1.05 , E= 20”
I am quite sure you need to mainly consider downturn and the fact that outcome needs to be more or equal to 95%. (Leverage I can calculate)
Here is the question:
“There are two dates, date t and t+1. A bank at date t chooses its lending volume so as to maximize its expected profits, that is the profits it will have at t+1 as a result of the lending it makes at date t, conditionally upon being solvent (able to repay its debt at date t+1) with a probability not smaller than 95%.
The environment is as follows:
The bank has a net worth (equity), E, and can raise funding by borrowing at a zero net interest rate, that is for every unit, Euro, it borrows at date t, the repayment due at date t+1 is one Euro.
A loan that the bank makes at date t, at date t+1, either performs, in which case the bank gets R > 1 , or it does not perform, in which case the bank gets nothing (zero).
There is aggregate risk: at date t+1, with probability 2/3 the economy will be in an upturn and all loans made at date t perform. With the residual probability of 1/3 the economy will be in a downturn and the percentage x of the loans do not perform.
Assume that (R-1)/R < x.
Derive the size of the lending volume, L, that the bank chooses at date t, and the bank’s leverage.
As a numerical example, work out the solution for x=10% , R= 1.05 , E= 20”