- What is the minimum proportion of good borrowers in the economy required for b
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Question
- What is the minimum proportion of good borrowers in the economy required for banks to be willing to lend?
- What is the interest rate that banks will charge at this proportion?
Now suppose that these is a business cycle upswing which doubles the proportion of good borrowers (the proportion of bad borrowers drops correspondingly), and that this change is expected to last for the duration of the investment project. Further assume that the project runs over 5 years. Everything else is unchanged.
- What is the annual interest rate charged by banks?
Explanation / Answer
What is the minimum proportion of good borrowers in the economy required for banks to be willing to lend?
Chance of success in the economy = chance of success of good firms * percentage of good firms + chance of success of basd fimrs * percentage of basd firms = 0.7 * a + 0.4 * (1-a) = 0.4 + 0.3 * a
Pay-off from success = 100
PAy off in the economy = Pay-off from success * Chance of success in the economy =100 *(0.4 + 0.3 * a)
=40+30a
a should be such that Pay off in the economy>cost of investment 50
40+30a>50
a>1/3. At least one thirds the firms should be good firms.
What is the interest rate that banks will charge at this proportion?
Pay-off from good firms = a * 0.7 *100 = 70a
Pay-of from bad firms = (1-a)*0.4*100 = 40-40a
total payoff = Pay-off from good firms + Pay-of from bad firms - investment
= 70a + 40-40a - 50 = 30a -10
return on investment = return/investment = (30a-10)/50 = interest rate charged
interest rate will be in proportion of returns,
(interest on good firms)/total interest = (return of good firms)/(return of all firms) = 70a/(40-40a+70a)
=70a/(40+30a)
hence, interest on good firms = 70a/(40+30a) * (30a-10)/50 =(28a+21a2)/(3a-1)
where a=1/3 no interest will be charge since no net return
What is the annual interest rate charged by banks?
If there is twice as many good borrowers, a =2/3
(30a-10)/50 = interest rate charged = (20-10)/50 = 20%
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