GF582 Unit 5 Confidence Interval Workup in Two Parts Example

An Interval That Crosses Zero, and One That Does Not: A Unit 5 Confidence Interval Workup

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Purdue University Global

GF582: Statistical Methods for Decision Making

Unit 5 Confidence Interval Workup

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[Date]

The pension plan, the manager, the lender's workout files and every figure are composite and illustrative; the articles and textbook cited are real.

Part 1: Has the Manager Added Value?

The data. A pension plan has held a composite international equity manager for five years. The 60 monthly differences between the manager's return and its benchmark's average 0.2375 percent, with a sample standard deviation of 1.0639 percent.

Hand calculation. Dividing 1.0639 by 7.746, the root of 60, gives a standard error of about 0.1374. With 59 degrees of freedom, a 95 percent interval uses a t multiplier of about 2.0010. The margin of error is 2.0010 times 0.1374, about 0.2748. The interval runs from 0.2375 less 0.2748, about negative 0.0373 percent, to 0.2375 plus 0.2748, about 0.5123 percent a month.

Software check. A spreadsheet's CONFIDENCE.T(0.05, 1.0639, 60) returns 0.2748, matching the hand calculation.

In annual terms. Multiplying by twelve, the interval runs roughly from negative 0.45 to positive 6.15 percent a year around a point estimate of about 2.85 percent.

Reading. The best estimate is that the manager added about 2.85 points a year, a meaningful amount. But the interval includes zero, so five years of data cannot rule out that the manager added nothing and the excess came from luck. Research on fund performance has long found that distinguishing skill from chance takes many years of returns, because monthly excess returns are noisy relative to their average (Fama & French, 2010).

What the plan committee should conclude. Keep the manager, since the evidence leans positive, but do not treat the record as proof of skill. Another two to three years of similar results would narrow the interval enough to exclude zero if the true added return is near the estimate.

Part 2: How Much Does the Lender Lose When a Loan Defaults?

The data. An equipment lender's 34 completed workout files from defaulted loans show an average loss, after recoveries and costs, of 39.4 percent of the balance at default, with a standard deviation of 18.2 percentage points.

Hand calculation. Here 18.2 over 5.831, the root of 34, gives about 3.121, and the multiplier for 33 degrees of freedom is about 2.0345. The margin is about 6.35 points. The interval runs from about 33.0 to 45.8 percent.

Software check. CONFIDENCE.T(0.05, 18.2, 34) returns about 6.35.

Reading. The lender can say with reasonable confidence that its average loss severity lies between about a third and just under half of the balance. Severity varies this widely in most lending because it depends on collateral, the time a workout takes and the state of the economy (Schuermann, 2004). That range matters: at a 4 percent default rate on $100 million of loans, it means expected losses of about $1.3 million to $1.8 million a year.

The two intervals answer different kinds of question: the manager's interval tests whether an effect exists at all, so crossing zero is the key fact, while the lender's interval sizes an effect everyone agrees exists, so its width is what matters for pricing.

A One-Sided Reading for the Manager

Some committee members asked whether a one-sided view, testing only whether the manager added value, would change the conclusion. A one-sided 95 percent lower bound is 0.2375 less 1.6711 times 0.1374, about 0.008 percent a month, just above zero. The verdict therefore depends on how the question is framed, which is itself a sign that the evidence is borderline and that the committee should avoid strong conclusions in either direction.

Comparing the Two Precisions

The manager's interval is about 0.55 points wide around an estimate of 0.24, more than twice the estimate itself. The lender's interval is about 12.7 points wide around 39.4, about a third of the estimate. Relative precision, not just width, tells the reader how much the estimate can be trusted for a decision.

Assumptions Behind Both Intervals

Both use the t distribution, which assumes the underlying values are roughly normal or the sample is large enough for the average to behave normally. Sixty monthly differences meet that standard comfortably. Thirty-four loss severities are adequate, though loss data are often skewed, with a cluster of small losses and a tail of near-total losses; a skewed sample of this size can make the interval slightly too narrow on the high side (Anderson et al., 2020).

Both also assume independent observations. Monthly excess returns are close to independent. Workout files may not be: defaults concentrated in one downturn tend to have higher losses together, which would make the true uncertainty larger than the interval shows.

What Would Narrow Each Interval

For the manager, only more time. Precision improves only with the root of the sample size, so cutting the margin in half would take four times as many months, about twenty years.

For the lender, more files or a better model. Adding loans from a partner lender's records, or separating severity by equipment type, would reduce the uncertainty faster than waiting for new defaults.

Conventions

Means and standard deviations are sample statistics. Intervals are two-sided at 95 percent confidence.

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.

Fama, E. F., & French, K. R. (2010). Luck versus skill in the cross-section of mutual fund returns. The Journal of Finance, 65(5), 1915-1947. https://doi.org/10.1111/j.1540-6261.2010.01598.x

Schuermann, T. (2004). What do we know about loss given default? In D. Shimko (Ed.), Credit risk models and management (2nd ed., pp. 249-274). Risk Books.