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I have found this exercise in some open bank of problems. The task is as follows:

The bank gives out credits of 100000$ to people. But it has the following strategies:

  1. Give the loan and receive 8% profit with 95% probability
  2. Not give the loan and buy bonds worth 100k for 4% profit (100% probability)
  3. Check the customer for 1000$ and base the decision on the result:

a) In case of favourable recommendation, the customer will not default with the probability of 80%

b) The probability of a favourable recommendation for a customer who will default is 0.15.

The question is what strategy will be the best for the bank...

In all my attempts I was facing either negative expected returns or ones worse than just buying bonds. This does not feel right to me, since there would be no reason for the bank to exist then :). How would you solve this?

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I believe you should build a decision tree to solve this problem. By first knot you will have three splits: Immediately approve, Immediately decline and Do investigation. The first will have a profit expectation of 2600, the second of 4000. The last one will have more splits with new probabilities of default, but all combinations will lead to negative profit. So, the best strategy is Immediately decline and buy bonds.

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