GF582 Unit 3 Lending Probability Problem Set Example

Probability Through Loss: A Unit 3 Problem Set From Loan Pools to Linked Failures

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Purdue University Global

GF582: Statistical Methods for Decision Making

Unit 3 Probability Problem Set

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[Date]

The loans, the warning flag, the portfolio and every probability are illustrative; the articles and textbook cited are real.

The Layout Used Throughout

Each of the six problems runs in four parts: the setting, the probability rule or distribution and why it fits, the working with the matching spreadsheet function, and a closing line reading the result for a lender or investor. Choosing the distribution from the setting, rather than from the chapter heading, is the skill the unit tests (Anderson et al., 2020).

Problem 1: Defaults in a Loan Pool

Setting. A lender holds 40 similar equipment loans, each with a 2.5 percent chance of defaulting this year, independently.

Distribution. Binomial: forty separate yes-or-no outcomes, all sharing one default chance and none affecting another.

Working. The chance of no defaults is 0.975 raised to the 40th power, about 36.3 percent, from BINOM.DIST(0, 40, 0.025, FALSE). The chance of exactly one is about 37.2 percent and of exactly two about 18.6 percent. The chance of three or more is one less the sum, about 7.8 percent.

Reading. Even a sound pool with a 2.5 percent default rate will see three or more defaults in about one year out of thirteen. A lender who treats such a year as evidence of poor underwriting may be reacting to chance.

Problem 2: A Warning Flag and Bayes' Rule

Setting. A monitoring system flags 70 percent of loans that will default, but also flags 7.5 percent of loans that will not. The base default rate is 2.5 percent.

Rule. Bayes' rule, from an essay published after its author's death (Bayes, 1763): the probability of default given a flag equals the probability of a flag given default times the base rate, divided by the overall probability of a flag.

Working. Flags from defaulting loans: 0.70 times 0.025, or 0.0175. Flags from sound loans: 0.075 times 0.975, about 0.0731. Total flagged: about 0.0906. Probability of default given a flag: 0.0175 divided by 0.0906, about 19.3 percent.

Reading. About four flagged loans in five are false alarms. The flag is useful for deciding which files to review first, not for declining loans automatically.

Problem 3: A Bad Year for a Portfolio

Setting. A balanced portfolio has an expected annual return of 7 percent and a standard deviation of 15 percent. Assume returns are roughly normal.

Working. A loss of more than 10 percent is 17 points below the mean, about 1.13 standard deviations. NORM.DIST(-10, 7, 15, TRUE) gives about 12.9 percent.

Reading. Roughly one year in eight would bring a loss worse than 10 percent. If returns have fatter tails than normal, as many do, the true chance is higher.

Problem 4: Fraud Cases per Month

Setting. A card program averages 1.8 confirmed fraud cases a month, occurring independently.

Distribution. Poisson, because events occur one at a time at a steady average rate.

Working. The chance of four or more cases in a month is one less the chance of zero to three: one less 0.1653 times the sum of 1, 1.8, 1.62 and 0.972, about 10.9 percent, from POISSON.DIST(3, 1.8, TRUE).

Reading. A month with four cases is not unusual enough on its own to suggest a new fraud pattern; about one month in nine will have four or more by chance.

Problem 5: Expected Loss on a Single Loan

Setting. A $50,000 loan has a 4 percent chance of default, and if it defaults the lender recovers 60 percent.

Working. Expected loss equals the probability times the loss if default occurs: 0.04 times $20,000, or $800. The standard deviation of the loss is $20,000 times the square root of 0.04 times 0.96, about $3,919.

Reading. The expected loss is small, but the outcome is all or nothing: the lender loses either nothing or $20,000. Pricing should cover the expected loss; capital should cover the variability.

Problem 6: Two Loans Whose Failures Are Linked

Setting. Two loans to suppliers of the same factory each have a 5 percent chance of default. If they were independent, the chance both default would be 0.25 percent. Because they depend on the same customer, the joint probability is 1.5 percent.

Working. The chance at least one defaults is 0.05 plus 0.05 less 0.015, or 8.5 percent, compared with 9.75 percent under independence. The correlation between the two default events is 0.015 less 0.0025, divided by 0.05 times 0.95, about 0.26.

Reading. Linking lowers the chance that at least one fails but raises the chance that both fail sixfold. For a lender, the second effect matters more, because simultaneous losses are what strain capital.

Every problem asks the same question, how likely a loss is and how bad it could be, but the answer changes with the structure of the risk: independent defaults spread out, false alarms crowd out true ones, and linked defaults arrive together.

Why Linked Defaults Matter

Research on credit portfolios shows that correlation between defaults, often driven by shared exposure to an industry or region, dominates the risk of a loan book far more than the average default rate (Vasicek, 2002). Problem 1's binomial model assumes independence; in a real pool of loans to similar businesses, the chance of a bad year is higher than the binomial suggests.

Conventions

Probabilities are rounded to one decimal place in percentages. Spreadsheet functions are shown in the form used by Excel.

References

Anderson, D. R., Sweeney, D. J., Williams, T. A., Camm, J. D., & Cochran, J. J. (2020). Statistics for business and economics (14th ed.). Cengage.

Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53, 370-418. https://doi.org/10.1098/rstl.1763.0053

Vasicek, O. (2002). The distribution of loan portfolio value. Risk, 15(12), 160-162.