Question

Pacific Shores Bank makes a vacation loan to Steve at an interest rate of 8.50 percent. The one year Tbill rate is 4%. The LGD on a vacation is 100%. When Steve asks for a second loan, this time to buy a car, Pacific Shores Bank offers him a rate of 5.68%. What recovery rate does Pacific Shores Bank use to price the auto loan?

Answer 1

Let loan amount be x
x*(1+Tbill rate)=probability of default*Recovery Rate+(1-probability of default)*x*(1+interest rate)
From vacation:
(1+4%)=p*(1-100%)+(1-p)*(1+8.5%)
=>p=1-(1+4%)/(1+8.5%)

From auto:
(1+4%)=(1-(1+4%)/(1+8.5%))*R+(1-(1-(1+4%)/(1+8.5%)))*(1+5.68%)
=>R=((1+4%)-(1-(1-(1+4%)/(1+8.5%)))*(1+5.68%))/(1-(1+4%)/(1+8.5%))
=>R=65.1733%