A lender's 12,000 personal loans sit in four score buckets with 30 defaults of 4,000, 60 of 4,000, 90 of 2,500 and 150 of 1,500. Compute default rates, 95% intervals and expected loss at 60% loss given default, and say whether the buckets are well ordered.

The buckets are well ordered: default rates of 0.75%, 1.50%, 3.60% and 10.0% with 95% intervals that do not overlap. Expected loss is Rs 3.96 crore on Rs 240 crore of loans, 1.65%. Bucket D is one loan in eight but carries 45% of that loss. The smallest buckets have the widest intervals, and A against B is the closest call, still about 3.2 standard errors apart.

Toss a fair coin 20 times and you might see 13 heads; nobody concludes the coin is biased. A default rate from a finite book is an estimate of a proportion, and its error shrinks only with the square root of the number of loans. The standard error is the square root of p times (1 - p) divided by n, and a 95% interval is the rate plus or minus 1.96 of those. Each bucket here has at least 30 defaults, enough for this normal approximation to be reasonable; with only a handful of defaults you would use an exact or Wilson interval instead.

Read the intervals, not the point estimates. No two intervals overlap, so each bucket's default rate is distinguishable from its neighbour's, and the score ranks risk in the right order. The closest pair is A and B: 0.48% to 1.02% against 1.12% to 1.88%. A direct test of the difference gives about 3.2 standard errors, comfortably significant. Bucket D has the widest interval, plus or minus 1.5 points, because it is both the smallest bucket and the one with the highest rate, but its whole interval sits far above C's.

Expected loss is exposure times default rate times loss given default. Loss given defaultThe share of a loan that is lost once a borrower defaults, after collections and recoveries. of 60% turns bucket D's 10% default rate into a 6% expected loss rate, against 0.45% for bucket A. Bucket D is 12.5% of the loans but about 45% of the Rs 3.96 crore of expected loss. That is the number the credit team will care about: pricing or cutting bucket D moves the book's losses far more than any change in A or B.

Finish with what the file cannot tell you. These are one-year defaults from a single two-year-old cohort, so the rates carry that year's economy. A 60% loss given default is an assumption, not an estimate from this file. And an interval measures sampling error only; if applicants have changed since, the true rate can move outside it.