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Outline the data problems commonly seen that make estimates of loss frequency and severity distributions inaccurate or biased.

FRM® Part I · Valuation and Risk Models · Operational Risk · FRM1.VR.7.e

The idea

Frequency and severity estimates are only as good as the loss data behind them, and operational loss data have well-known problems. Internal data contain few or no extreme losses, so the tail is poorly estimated. External data from consortia or public sources help but bring problems of their own. Databases usually record only losses above a collection threshold. Fitting severity to them without adjustment overstates average severity, and counting only recorded events understates frequency. The true frequency is the recorded count divided by the probability of exceeding the threshold. Publicly reported external losses are skewed toward large, newsworthy events, biasing severity upward. They also come from banks of different size and business mix, so they need scaling. Losses are often discovered late, so recent years look artificially quiet. Short histories, changing businesses and pooling of unlike units weaken the estimates further.

Formulas

Loss frequency corrected for a collection threshold

true frequency = recorded count ÷ P(loss > threshold)

recorded count
losses a year above the collection threshold
P(loss > threshold)
the probability that a loss exceeds the threshold, from a severity fit adjusted for truncation

Counting only recorded losses understates frequency, and fitting severity to them without adjustment overstates average severity.

A practice question

From the question bank, with the reason each wrong option is wrong.

Why do banks supplement internal loss data with external data when estimating severity distributions?

  1. Internal data contain few large losses, so the tail of the severity distribution is poorly estimated - correct
  2. External data are free of reporting bias and so are always more accurate than internal dataWhy not: External data have their own biases, such as selective reporting of large losses and differences in bank size.
  3. Internal data contain too many extreme losses relative to the industry, which overstates capitalWhy not: Reverses the problem: internal histories typically have too few extreme events.
  4. External data measure the bank's own loss frequency more reliably than internal dataWhy not: Frequency is specific to the bank's own business and controls; external data mainly inform severity.

Extreme losses are rare, so one bank's history contains few or none. Pooled external data add tail observations, though they must be scaled and checked for bias.

A harder one

A bank estimates severity using a database built from losses reported in the press and regulatory filings by other institutions. Which bias is most likely?

  1. Severity is understated because firms are reluctant to disclose their largest lossesWhy not: Reluctance exists, but public sources skew toward large, visible losses; the dominant direction is the opposite.
  2. No bias arises because many banks contribute, so errors diversify awayWhy not: Selection by size is systematic and does not average out across contributors.
  3. Severity is overstated because large, newsworthy losses are more likely to be reported than small ones - correct
  4. Frequency is overstated but severity is unbiasedWhy not: Selective reporting by loss size distorts the severity distribution, not only the counts.

Publicly reported losses over-represent big events, so a severity distribution fitted to them is shifted upward.

The hardest tier

A database records only losses above USD 50,000 and shows 40 losses per year. A lognormal severity fitted with an adjustment for truncation implies a 25 percent probability that a loss exceeds USD 50,000. The implied annual frequency of all losses is closest to:

  1. 160 per year - correct
  2. 10 per yearWhy not: Multiplies the recorded count by 0.25 instead of dividing.
  3. 53 per yearWhy not: Divides by 0.75, the probability of being below the threshold, instead of the probability of exceeding it.
  4. 40 per yearWhy not: Treats the recorded count as the true frequency, ignoring that losses below the threshold are not recorded.

Recorded losses are the fraction above the threshold: true frequency = 40 / 0.25 = 160 per year.

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