A bank offers a loan to be repaid with a single payment after 10 years. The bank requires an annual interest rate of 3.5% compounded continuously assuming no default. It is given that the percentage of borrowers that will default is 15%, and it is expected that the bank will be able to recover 38%. Find the annual interest rate compounded continuously that the bank should charge taking into account defaults and recoveries.
Given that,
A bank offers a loan to be repaid with a single payment after 10 years
interest rate = 3.5% compounded continuously,
So, Loan value to be repaid after 10 year is e^(r*t) times
So, Value at year 10 = e^(0.035*10) = 1.419 times
Now let interest rate with default be r%
recovery rate RR = 38%
Probability of default PD = 15%
So, with default and recovery rate, final payment after 10 years is
Value = e^(r*t)*(1-PD) + e^(r*t)*PD*RR
=> Value = e^(10r)*(1-0.15) + e^(10r)*0.15*0.38 = e^(10r)*0.907
Since this value must be equal to value at risk free rate, we get
e^(10r)*907 = 1.419
=> e^(10r) = 1.5646
=> 10r = ln(1.5646)
=> r = 4.48%
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