Question

UNIVERSAL SAVINGS \& LOAN has \$1000 to lend. Risk-free loans will be paid back in full next year with \(4 \%\) interest. Risky loans have a \(20 \%\) chance of defaulting (paying back nothing) and an \(80 \%\) chance of paying back in full with \(30 \%\) interest a. How much profit can the lending institution expect to earn? Show that the expected profits are the same whether the lending institution makes risky or risk-free loans. b. Now suppose that the lending institution knows that the government will "bail out" UNIVERSAL if there is a default (paying back the original \(\$ 1000\) ). What type of loans will the lending institution choose to make? What is the expected cost to the government? c. Suppose that the lending institution doesn't know for sure that there will be a bail out, but one will occur with probability \(P\). For what values of \(P\) will the lending institution make risky loans?

Short answer

The lending institution can expect to earn \$240 from risky loans and \$40 from risk-free loans, under normal circumstances. If the government bails out the bank in case of default, the expected profit from risky loans increases, incentivizing the bank to issue risky loans. The lending institution will lend risky loans if the probability of getting bailed out is \(P >= 0.167\).

Step-by-step solution

Calculate Expected Profit from Risky and Risk-Free Loans

To calculate the expected profit from both types of loans, multiply the profit by the likelihood it will be realized. For risk-free loans, the profit is always \$40 (\(1000 * 4\% = 40\)). For risky loans, the expected profit is \(0.8 * 300 + 0.2 * 0 = 240\), where 300 is the profit in the case of payback in full and 0 in case of default.

Compare Expected Profits from Risky and Risk-Free Loans

As calculated in step 1, the lending institution can expect to earn a profit of \$40 from a risk-free loan and \$240 from a risky loan under normal circumstances. These conditions change if the government decides to intervene and bail out the bank in case of default.

Evaluate the Impact of a Government Bail-Out

If the government bails out the bank in case of default, the bank will get back the original \$1000. In this case, the expected profit from risky loans will be \$300 (\(0.8 * 300 + 0.2 * 1000 = 340\)). This raises profitability, incentivizing the bank to issue risky loans.

Find the Probability Threshold for Risky Loans

To find the critical probability (P) threshold at which the bank would choose risky loans, calculate the expected profit for varying probabilities and compare them. This would imply solving the equation \(0.04 * 1000 = (1 - P) * 0.3 * 1000 + P * 1000\) for \(P\).

Understand the Impact of the Threshold Probability

The value calculated in Step 4 indicates the specific probability value which when reached or superseded would make risky loans more attractive for the lending institution.

Key concepts

Expected Profit Calculation

Expected profit calculation is crucial in determining the earnings a bank can anticipate from lending money. When dealing with different types of loans, it's important to account for the possibility and impact of risks.

For risk-free loans, the process is straightforward. The bank loans out \\(1000\ and receives it back with \4\% interest next year. This gives a clear profit of \(1000 \times 0.04 = 40\), making it a safe but not very lucrative option.

On the other hand, risky loans introduce more variability, as there’s a chance of not getting any repayment at all. In our scenario, there’s a \20\% chance the borrower defaults, and the remaining \80\% chance of full repayment with \30\% interest. The expected profit is calculated by considering these probabilities: \(0.8 \times 300 + 0.2 \times 0 = 240\). Although this carries higher profit potential, it also bears the risk of earning nothing.

• Risk-free loan profit: \\)40\
• Risky loan (expected) profit: \$240\

Impact of Government Bailout

A government bailout can significantly alter the dynamics of loan risk assessments. In the event of a loan default, if the government agrees to return the original \\(1000\ to the bank, this dramatically changes the expected outcomes.

Under these circumstances, the risky loan scenario becomes more appealing. Calculating new expected profit includes the bailout option where the bank doesn't lose the principal. The modified expected profit can be calculated as follows: \(0.8 \times 300 + 0.2 \times 1000 = 340\). With a bailout, risky loans yield a higher expected profit and consequently increase the bank's inclination towards risk-taking.

• Old risky loan (expected) profit: \\)240\
• New risky loan (expected) profit with bailout: \$340\

This changes the decision-making landscape and might lead banks to pursue riskier lending strategies if they anticipate governmental support.

Probability and Decision Making

Probability plays a vital role in decision making when outcomes are uncertain. By comparing expected profits of various loan types against potential consequences, banks can make informed choices.

Using the probability (P) of a government bailout, banks seek to determine whether risky loans are advantageous. This involves solving the equation for P where expected profits equalize between loan types:

\0.04 \times 1000 = (1 - P) \times 0.3 \times 1000 + P \times 1000\.

Solving gives us the critical P value. When \(P > 0.84\), risky loans become more profitable even without complete assurance of repayment. This strategic calculation helps the bank determine its lending risk relative to bailout likelihood.

• Critical P value: \0.84\
• P > \0.84\ makes risky loans more profitable

By strategically assessing these factors, banks can better navigate uncertainties in the lending landscape.