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 riskfree 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

Expected profits are the same, \$40, for both risky and risk-free loans. If the government should bail out Universal, it will choose to offer risky loans since in the worst case, it will get the original \$1000 back. The expected cost to the government will be \$200. The lending institution will choose to make risky loans if the chances of a government bailout occurring are higher than or equal to 20\%.

Step-by-step solution

Calculate expected gain from risk-free loan

For risk-free loan, the expected value is \(1000 \times 0.04 = \$ 40\), which is the profit from lending \(\$ 1000\). The amount returned to the bank will be \( \$ 1000 + \$ 40 = \$ 1040\) as it is certain that the loan will be repaid back.

Calculate expected gain from risky loan

For risky loan, the expected value can be calculated by adding the products of the chance of each event and the amount of money it returns. There is a \(80 \%\) chance to get \(\$ 1000 + \$ 1000 \times 0.30 = \$ 1300\) and a \(20 \%\) chance to get nothing. So, the expected value is \( 0.80 \times \$ 1300 + 0.20 \times \$ 0 =\$ 1040\).

Compare the expected gains

From step 1 and 2, the expected gains from lending risk-free and risky are the same, i.e., \$1040. So, the expected profit is the same whether the lending institution makes risky or risk-free loans.

Decide the type of loan to make if there is a government bailout

Now, once the institution knows about government bail-out in case of default, then taking risky loans become more attractive. This is because in the best case, they make a profit of \$300 and in the worst case, they will get the original \$1000 back.

Calculate expected cost to the government

Now, even if risky loans are taken, the lending institution is benefitted due to government bailout. The government may face a bail cost of \$1000 with a chance of 20\%, hence, its average cost will be \$200.

Find out the values of P for which lending institution will make risky loans

The lending institution will choose risky loans if a bailout will occur with probability P, where P is greater than or equal to 20\%, in which case, the expected loss in the event of default is less than or equal to the profit in case of no default.

Key concepts

Risk-free vs Risky Loans

Understanding the difference between risk-free and risky loans is fundamental for both lenders and borrowers. Risk-free loans are typically government bonds or high-quality corporate bonds that offer a sure, albeit lower, return rate. Lenders who opt for risk-free loans can expect a guaranteed payout without the anxiety of losing their initial investment.

On the other hand, risky loans are akin to venturing into the open sea without a clear idea of weather conditions. These loans could be made to businesses with volatile earnings or individuals with uncertain financial stability. The risk comes with a higher interest rate—a reward for the lender taking a chance on the potential default of the loan. In our exercise, the risk-free loan returns 4%, guaranteeing the lender a profit of \(40, while the risky loan offers a tantalizing 30% interest rate with a chance of a \)300 profit or a \(1000 loss.

The probability of a successful risky loan in this exercise is 80%, which might make it seem smart to take the chance, as the expected returns from both loan types are calculated to be identical (\)1040). However, the certainty associated with risk-free loans is often preferable because real-world scenarios involve multiple loans and default in risky loans could incur additional costs or harm the lender's capital ratio, affecting their ability to make further loans.

Government Bailout Impact

In the financial world, a government bailout is a safety net that acts as a buffer against financial loss during economic downturns or when individual institutions face insolvency risks. When lenders know there is a bailout guarantee, as in our exercise, the calculus of risk assessment changes significantly.

With a government bailout, the risk involved in issuing risky loans diminishes, since the downside—the possibility of a default—is covered by the bailout. The lenders may shift their preference to risky loans, motivated by the higher profit potential without bearing the full brunt of the associated risk. In our scenario, the lender knows that in the event of a default, the government will reimburse the original \(1000. Therefore, the lender is incentivized to issue risky loans, since the maximum profit is higher (\)300 versus \(40) and the risk of losing the principal is overcome by the bailout.

For the government, the impact of bailouts can be profound. It introduces moral hazard, prompting lenders to engage in riskier lending behavior, knowing that losses will be covered. In our example, the expected cost to the government is \)200, calculated as the 20% chance of a $1000 bailout. Comprehensive government bailouts may strain public finances, potentially leading to increased borrowing or taxation to cover these costs.

Probability and Risk in Lending

Probability plays a central role in assessing risk in the lending industry. Lenders use it to estimate the likelihood of a borrower defaulting on a loan. It's a statistical tool that helps in forecasting potential outcomes and making informed decisions on whether to extend credit and at what interest rate.

In lending, a higher probability of default translates to a higher risk, which typically demands a higher reward—or interest rate—to justify the risk. Conversely, a lower probability of default means lower risk and hence, a lower interest rate. In the exercise we're discussing, the risky loan has a 20% chance of default and an 80% chance of being repaid with high interest. Lenders analyze these probabilities to calculate the expected value or profit from the loans they offer.

The probability of a bailout occurring, denoted as 'P' in our exercise, adds another layer to risk assessment. It modifies risk calculations and can influence the behavior of the lender. If there's a high likelihood of a bailout (with P ≥ 20%), then risky loans become more appealing, as they offer the possibility of high returns with minimal actual risk, assuming the bailouts cover potential losses. This interplay of probability and risk is crucial for financial institutions to manage their loan portfolios effectively and avoid insolvency.