Question

UNIVERSAL SAVINGS & LOAN has \(1000 to lend. Risk-free loans will be paid back in full next year with4% 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 bailout, but one will occur with probability P. For what values of Pwill the lending institution make risky loans?

Short answer

a) The lending institution is expected to earn $40 as profits from each risky and risk-free loan.

b) The institution will choose risky loans. The expected cost to the government would be $200 per loan.

c) The Universal will make risky loans for the value of 40 + 200P.

Step-by-step solution

Explanation of part (a)

The institutions have $1000 to lend on which he earns $40 on risk-free loans since they yield 4% interest, and earns $300 on risky loans that are paid back since they yield 30% interest. Therefore, expected profits on risk-free and risky loans would be;

Risk-free;$E\left(\mathrm{\pi }\right)=\$40$

For risky loans, they yield an interest of $300, which has an 80% chance to be paid in full and a 20% chance of defaulting; therefore, the default will be calculated on the lending amount to ascertain how much it amounted to bad loans and 80% will be calculated on $300.

Risky;$E\left(\mathrm{\pi }\right)=\text{0.80}\left(\text{300}\right)\text{- 0.20}\left(\text{- 1000}\right)\text{= \$ 40}$

Hence, the expected profits on both the loans are the same.

Explanation of part (b)

In this case, if the government bails out, then there will be no default. Therefore, the expected profit on the risky loan would become;

$E\left(\mathrm{\pi }\right)=\text{0.80}\left(\text{300}\right)\text{- 0.20}\left(\text{0}\right)\text{= \$ 240}$

The institution would choose risky loans since the expected profit is $240, which is much more than the expected profits on the risk-free loan which is $40.

The cost to the government would be 0.20 (1000) = $200.

Explanation of part (c)

The expected profits on risky loans would be $40 without bailout and $240 with a bailout. Considering the probability of no bailout is P, the probability of bailout would be 1-P. Therefore, expected profits on risky loans would be given by: $\text{240P + 40}\left(\text{1 - P}\right)=\text{240P + 40P}$