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You are seeking a bank loan for $12000 and go to Prosper Bank and to Skyline...

You are seeking a bank loan for $12000 and go to Prosper Bank and to Skyline Bank to see which loan has a lower rate. Your plan is to open a restaurant, which is quite risky. Prosper Bank expects that it could recover $10,000 if you defaulted while Skyline thinks it would only recoup $9000. However, Skyline puts your probability of repayment at 97% while Prosper only has it at 96%. Which loan has the lower interest rate? Assume both banks are aiming to earn 6%. Please show all work.

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Answer #1

Loan*(1+required earning rate)=probability of repayment*Loan*(1+interest rate charged)+(1-probability of repayment)*Recovery in case of default

Let the interest rate be r
SKYLINE:
12000*(1+6%)=97%*12000*(1+r)+(1-97%)*9000
=>r=(12000*(1+6%)-(1-97%)*9000)/(97%*12000)-1=6.96%

PROSPER:
12000*(1+6%)=96%*12000*(1+r)+(1-96%)*10000
=>r=(12000*(1+6%)-(1-96%)*10000)/(96%*12000)-1=6.94%

PROSPER has lower interest rate

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