GF582 Unit 4 Card Portfolio Sampling Exercise Example

200 of 6,240 Accounts: A Unit 4 Sampling Exercise for a Card Portfolio Due Diligence

[Student Name]

Purdue University Global

GF582: Statistical Methods for Decision Making

Unit 4 Sampling Exercise

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[Date]

The card program, the buyer and every figure are composite and illustrative; the articles and textbook cited are real.

The Decision Behind the Sample

A regional bank is considering buying a home furnishings retailer's private-label card portfolio. The seller reports an average balance and a delinquency rate, but the buyer's credit team wants its own estimates before bidding. Reviewing every account file is too slow and costly, so the team will sample.

Step 1: Design

Population. The 6,240 open accounts with a balance on the cutoff date, March 31.

Sampling frame. The servicer's account list as of that date, checked against the general ledger total so no accounts are missing.

Parameters. The mean balance per account and the share of accounts 30 or more days past due.

Method. A simple random sample of 200 accounts, chosen by random number from the frame, with each account's file reviewed in full.

Why random selection matters. Choosing files that are easy to pull, or the largest accounts, would bias the estimates. Random selection makes every account equally likely to be chosen, which is what allows the sample to stand in for the whole (Cochran, 1977).

Step 2: The Simple Random Sample

Mean balance. The 200 accounts average $3,087.67 with a standard deviation of $2,220.84.

Finite population correction. Because the sample is a noticeable share of a modest population, the standard error is scaled down to reflect the large share of accounts already drawn (Anderson et al., 2020), multiplying by the square root of 6,040 over 6,239, about 0.9839.

Standard error. The raw standard error is $2,220.84 over 14.142, the root of the sample size, roughly $157.04; after the correction it is about $154.51.

95 percent interval. With a t value of about 1.972 for 199 degrees of freedom, the margin is about $304.70, giving an interval from about $2,783 to $3,392.

Delinquency. Seventeen of the 200 accounts, 8.5 percent, are 30 or more days past due. The standard error, after correction, is about 1.94 points, giving a 95 percent interval of about 4.7 to 12.3 percent.

Reading. The balance interval is reasonably tight, about 10 percent either side of the estimate. The delinquency interval is wide: the true rate could plausibly be half or one and a half times the estimate.

Step 3: A Stratified Redraw

Balances differ sharply by credit score. The team divides the frame into three score bands, below 620, 620 to 699, and 700 and above, holding about 22, 41 and 37 percent of accounts. It draws 200 accounts in the same proportions: 44, 82 and 74.

Within each band, balances vary less than across the whole portfolio, because lower-score accounts tend to carry higher balances relative to their limits. Combining the band estimates, weighted by each band's share of the population, gives a mean balance of about $3,121 with a standard error of about $121, roughly 22 percent smaller than the simple random sample's $154.51.

Stratifying did not change the estimate much, about $3,121 against $3,088, but it narrowed the interval by about a fifth for the same 200 files, because most of the variation in balances lies between score bands rather than within them.

Step 4: How Many Files for a Tighter Estimate

If the buyer wants the mean balance within plus or minus $200 at 95 percent confidence using simple random sampling, the required sample before correction is 1.96 times $2,220.84 divided by $200, squared, about 474. After the finite population correction, about 440 files. Stratifying by score band would bring that down to roughly 270.

For delinquency, a margin of plus or minus 2 points would need about 750 accounts before the correction, about 670 after it, because rates near 8 percent have high relative variability.

Checking the Sample Against Known Totals

One check is available without reviewing more files. The servicer's ledger reports total balances of about $19.6 million across the 6,240 accounts, an average of about $3,141. Both the simple random estimate and the stratified estimate fall within about $55 of that figure, and the ledger average sits comfortably inside both intervals. That agreement supports the sample design and suggests the frame is complete.

Errors Sampling Cannot Fix

Random sampling controls the error that comes from looking at only part of the portfolio. It cannot fix errors in the files themselves, such as a delinquency date recorded wrongly or a payment posted late. The team checked ten of the sampled files against payment histories and found one misdated delinquency, so it plans a slightly larger check before the final bid.

What the Estimates Mean for the Bid

The seller reported an average balance of $3,200 and a delinquency rate of 6.0 percent. The seller's balance figure sits inside the buyer's interval. The seller's delinquency figure sits near the lower end of the interval, so the buyer should either review more files to narrow the delinquency estimate or price the bid on a higher rate. Classic work on markets where sellers know more than buyers explains why: a buyer who relies on the seller's figures will tend to overpay, so independent sampling and conservative pricing protect against that gap (Akerlof, 1970).

Conventions

Intervals use t values. Proportions use the normal approximation, reasonable here because both the number delinquent and the number current exceed ten.

References

Akerlof, G. A. (1970). The market for "lemons": Quality uncertainty and the market mechanism. The Quarterly Journal of Economics, 84(3), 488-500. https://doi.org/10.2307/1879431

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

Cochran, W. G. (1977). Sampling techniques (3rd ed.). Wiley.