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Suppose you have the opportunity to invest in a second-hand machine for $100M, and you are unsure of its quality. There is a 75 percent chance it will be high quality, and a 25 percent chance it will be low quality. If it is high quality, its production will yield you $180M with 60 percent probability, and $140M with 40 percent probability. If it is low quality, its production will yield you $80M or $40M, with equal probability. The machine stops working at t=1, regardless of the quality. You observe the output of the machine and thus you will know the quality of the machine at t=1. Similar projects of similar risk have a required rate of return of 10 percent.

a) What is the expected NPV from this project?

Suppose you also get a maintenance warranty with your purchase. At t=1, you can pay another $50M to keep the machine going for another period. At t=2, the machine will yield you $90M with 60 percent probability, and $70M with 40 percent probability if the machine is high quality. The machine will yield you $40M or $20M with equal probability if the machine is low quality.

b) At t=1, would you want to pay to keep the machine going for another period if the machine was high quality? What about if the machine was low quality?

c) What is the NPV of your investment project given your choices in part (b)?

d) Suppose you had to pay for the maintenance warranty at time t=0. What is the maximum you would be willing to pay for it?

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Keith Leannon
Keith LeannonLv2
28 Sep 2019

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Suppose that you are a bank regulator who knows that there are two types of bank’s under your jurisdiction, high quality (H) and low quality (L), who differ in their ability to screen loan applications for credit worthy borrowers. Using past data you estimate that roughly 70% of banks are high quality and 30% are low quality. High quality bank’s generate a cashflow per loan of 100k with probability .8 and 0 with probability .2. A low quality bank on the other hand generates a cashflow per loan of 100k with probability .5 and 0 with probability .5.

These banks primarily make small loans to local businesses which creates a social benefit of 4% per dollar lent (i.e. if the bank lends $10 then the social benefit is 4%*$10). When a bank fails though, it generates a social cost of 15% per dollar of losses to the banks creditors (i.e. if creditors lose $10 then the social costs is 15%*$10).

As a regulator you must choose a “capital requirement”, dictating how much the banks can borrow to make loans, in order to maximize the expected social value. Each loan costs $50k and generates the returns described above. Additionally, the bank’s start with $50k of their own money to invest. You, as the regulator, are debating whether to let the banks borrow money to issue 0, 1, or 2 additional loans. We assume the bank’s find it optimal to issue as many loans as possible but can only raise additional funds by borrowing. This means banks will always make at least 1 loan with their initial $50k and will make an additional 0, 1, or 2 loans with borrowed money depending on how much the regulator allows them to borrow. All loans generate the same cash flow (e.g. if the H-type makes 3 loans then with prob .8 each loan generates a cashflow of 100k and with .2 each loan generates a cash flow of 0. Similarly, if the L-type makes 3 loans then each loan generates a cash flow of 100k with prob .5 and each loan generates a cashflow of 0 with probability .5).

Question:

Suppose that the regulator cannot tell which bank is H-type and which is L-type. How much will the regulator optimally allow the bank’s to borrow? What is the expected social value of this optimal policy?

Suppose that you are a bank regulator who knows that there are two types of banks under your jurisdiction, high quality (H) and low quality (L), who differ in their ability to screen loan applications for credit-worthy borrowers. Using past data you estimate that roughly 70% of banks are high quality and 30% are low quality. High-quality banks generate a cash flow per loan of 100k with probability .8 and 0 with probability .2. A low-quality bank on the other hand generates a cash flow per loan of 100k with probability .5 and 0 with probability .5.

 

These banks primarily make small loans to local businesses which creates a social benefit of 4% per dollar lent (i.e. if the bank lends $10 then the social benefit is 4%*$10). When a bank fails though, it generates a social cost of 15% per dollar of losses to the bank's creditors (i.e. if creditors lose $10 then the social cost is 15%*$10).

As a regulator you must choose a “capital requirement”, dictating how much the banks can borrow to make loans, in order to maximize the expected social value. Each loan costs $50k and generates the returns described above. Additionally, the banks start with $50k of their own money to invest. You, as the regulator, are debating whether to let the banks borrow money to issue 0, 1, or 2 additional loans. We assume the banks find it optimal to issue as many loans as possible but can only raise additional funds by borrowing. This means banks will always make at least 1 loan with their initial $50k and will make an additional 0, 1, or 2 loans with borrowed money depending on how much the regulator allows them to borrow. All loans generate the same cash flow (e.g. if the H-type makes 3 loans then with prob .8 each loan generates a cash flow of 100k and with .2 each loan generates a cash flow of 0. Similarly, if the L-type makes 3 loans then each loan generates a cash flow of 100k with prob .5 and each loan generates a cash flow of 0 with probability .5).

Question:

The regulator is deciding whether to invest in a “stress testing” technology that will allow them to distinguish H-type banks from L-type banks. How much would the regulator be willing to pay for this technology?

Suppose that you are a bank regulator who knows that there are two types of banks under your jurisdiction, high quality (H) and low quality (L), who differ in their ability to screen loan applications for credit-worthy borrowers. Using past data you estimate that roughly 70% of banks are high quality and 30% are low quality. High-quality banks generate a cash flow per loan of 100k with probability .8 and 0 with probability .2. A low-quality bank on the other hand generates a cash flow per loan of 100k with probability .5 and 0 with probability .5.

These banks primarily make small loans to local businesses which creates a social benefit of 4% per dollar lent (i.e. if the bank lends $10 then the social benefit is 4%*$10). When a bank fails though, it generates a social cost of 15% per dollar of losses to the bank's creditors (i.e. if creditors lose $10 then the social cost is 15%*$10).

As a regulator you must choose a “capital requirement”, dictating how much the banks can borrow to make loans, in order to maximize the expected social value. Each loan costs $50k and generates the returns described above. Additionally, the banks start with $50k of their own money to invest. You, as the regulator, are debating whether to let the banks borrow money to issue 0, 1, or 2 additional loans. We assume the banks find it optimal to issue as many loans as possible but can only raise additional funds by borrowing. This means banks will always make at least 1 loan with their initial $50k and will make an additional 0, 1, or 2 loans with borrowed money depending on how much the regulator allows them to borrow. All loans generate the same cash flow (e.g. if the H-type makes 3 loans then with prob .8 each loan generates a cash flow of 100k and with .2 each loan generates a cash flow of 0. Similarly, if the L-type makes 3 loans then each loan generates a cash flow of 100k with prob .5 and each loan generates a cash flow of 0 with probability .5).

 

Question:

Assume the regulator knows which banks are H-type and which are L-type. What is the expected social value (i.e. social benefit minus expected social cost) if the regulator allows the H-type bank to borrow 0, $50k, or $100k to make additional loans? What bout the L-type bank? How much will the regulator optimally allow the H-type and L-type banks to borrow? (remember all bank always makes at least 1 loan, so they will either make 1, 2, or 3 loans depending on whether the regulator allows them to borrow 0, $50k, or $100k. Also note that the social cost only applies to borrowed money lost in default, not the bank’s initial $50k, and the social benefit applies to money lent, not the realized cashflow).

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