Assuming that the bank has a monopoly power on the loans

Question

Using the CAPM for Loan Pricing A bank computes the nominal interest rate rL that it charges on a certain type of loan by the following formula:

where δ = the proportion of defaulting loans, assuming the proceeds of a defaulting loan are zero. ϒ_L = the management cost per unit of loan. r = the interbank rate, taken as the riskless rate. π = the risk premium demanded by stockholders. a = the capital coefficient required for this type of loan.

1. Compute the expected return on a loan, and show that the preceding pricing formula is closely related to the CAPM approach.

2. Assuming that the bank has a monopoly power on the loans side but faces a competitive market on the liabilities side, compute what should be the modified pricing formula, for given d and b, as a function of the elasticity of the demand for loans.

3. How should the formula be modified if the bank faces a tax on profits at a rate t?