Actuarial Notes

Claim Count Triangle

Study guide: Exam 5

Claim Count Triangle is a development triangle of claim counts rather than dollars. Counts are what make the dollar triangles interpretable: they separate "more claims" from "costlier claims", and they are the denominator of every severity diagnostic.

\text{Ultimate Counts} = N_n \times \text{CDF}^{\text{count}}_{n \to \text{ult}}

\text{Severity} = \frac{\text{Ultimate Losses}}{\text{Ultimate Counts}}

Four count triangles are built, and each answers a different question:

  • Reported counts — how fast claims come in. Develops to ultimate; the basis for pure IBNR counts.
  • Closed counts — how fast claims settle. The disposal rate is closed counts over ultimate (or reported) counts.
  • Open counts — the inventory. Divided into case reserves it gives average case outstanding, the sharpest case adequacy diagnostic.
  • Counts closed with payment (vs. closed without) — a shift in the CWP proportion changes average paid severity without any change in claim cost.

Further points:

  • Count factors are smaller than dollar factors at the same age, because a claim is counted as soon as it is reported while its dollars accrue for years afterward.
  • Counts must be defined consistently: per claim or per claimant, counting claims closed without payment or not, treating reopened claims as new or as continuing. A definition change mid-triangle looks exactly like a frequency trend.
  • Counts split broad IBNR into its two parts: pure IBNR (unreported claims, valued at an expected severity) and IBNER (development on known claims). Dollar triangles alone cannot separate them.
  • Counts are the input to the Frequency-Severity Method and the disposal-rate variant of Berquist-Sherman.

Developing Counts and Deriving Severity Cumulative reported claim counts:

AY 12 mo 24 mo 36 mo
2021 450 520 545
2022 475 550
2023 500

The tail beyond 36 months is 1.010. Ultimate losses for AY 2023 are projected at \$612{,}000.

Project ultimate counts and severity.

Answer

\begin{align*}
f_{12\text{–}24} &= \frac{520 + 550}{450 + 475} = \frac{1{,}070}{925} = 1.157 \\[4pt]
f_{24\text{–}36} &= \frac{545}{520} = 1.048
\end{align*}

\text{CDF}^{\text{count}}_{12} = 1.157 \times 1.048 \times 1.010 = 1.225

\begin{align*}
\text{Ultimate counts (AY 2023)} &= 500 \times 1.225 = 612 \\[4pt]
\text{Ultimate severity} &= \frac{\$612{,}000}{612} = \$1{,}000
\end{align*}

Note the count CDF of 1.225 against a dollar CDF that would typically be well above 1.5 at 12 months for the same line: counts are 82\% complete while dollars are much less so, because the claims already reported have years of payment ahead of them.

Counts Reveal What Dollars Hide Two accident years at 24 months:

AY 2022 AY 2023
Reported losses \$5{,}000{,}000 \$6{,}000{,}000
Reported counts 1{,}000 1{,}500
Closed counts 600 900
Open counts 400 600
Case reserves \$2{,}000{,}000 \$2{,}100{,}000

Reported losses rose 20\%. What actually happened?

Answer

\begin{align*}
\text{Severity: } \frac{5{,}000{,}000}{1{,}000} = \$5{,}000 &\to \frac{6{,}000{,}000}{1{,}500} = \$4{,}000 \\[6pt]
\text{Disposal rate: } \frac{600}{1{,}000} = 60\% &\to \frac{900}{1{,}500} = 60\% \\[6pt]
\text{Avg case O/S: } \frac{2{,}000{,}000}{400} = \$5{,}000 &\to \frac{2{,}100{,}000}{600} = \$3{,}500
\end{align*}

Losses rose 20\% but counts rose 50\% while average severity fell 20\%. The book has grown, and it has grown into smaller claims — a mix shift toward a lower-severity segment, not an increase in claim cost.

The average case outstanding falling from \$5{,}000 to \$3{,}500 is consistent with that story (smaller claims), but it is also what a case reserve weakening would look like. The disposal rate holding at 60\% rules out a settlement-speed change. Distinguishing mix from adequacy requires looking at the claims themselves — which is exactly the conversation with claims management that Friedland's Chapter 4 prescribes.

Without the count triangles, the dollar triangle shows only "+20\%" and none of this is visible.