Actuarial Notes

Actual vs Expected Analysis

Study guides: Exam 5, Exam CP-321, Exam GH-101

Actual vs. Expected Analysis (A/E) compares the emergence that actually occurred in a period against what the previous valuation's assumptions implied should occur. It is the primary monitoring tool between full reserve reviews, and the fastest way to detect that an estimate is drifting.

\text{Expected emergence} = U_{\text{prior}} \times \left(\frac{1}{\text{CDF}_{n+1}} - \frac{1}{\text{CDF}_{n}}\right)

\text{A/E} = \frac{\text{Actual emergence}}{\text{Expected emergence}}

  • The expected figure comes from the prior ultimate and the emergence pattern: the share of ultimate the pattern says should report (or be paid) between the two valuations. It is not last year's reported losses grown by a factor — it is a forecast made in advance and then scored.
  • A/E is run by accident year and by maturity, on paid and reported separately. Where the deviation sits is as informative as its size: concentrated in recent years points at the current diagonal; spread across all years points at a calendar-year effect; concentrated at old maturities points at the tail.
  • Paid and reported A/E together diagnose the cause. Reported running high with paid on track suggests case strengthening; paid running high with reported on track suggests faster settlement.
  • One period's deviation is noise; a run of them is a signal. Emergence is volatile, so a single A/E of 1.15 means little. Four consecutive quarters above 1.0, in the same direction, is evidence the assumptions are wrong.
  • A/E is the mechanism that would have caught the drifting ultimates on Ultimate Loss in year one instead of year four, and it feeds directly into whether factors, the tail or the a priori ELR need re-selection.

Scoring a Quarter's Emergence At 12/31/2023, AY 2022 was estimated at ultimate \$10{,}000{,}000 with 60\% reported at 24 months and 75\% at 36 months. Reported losses at 12/31/2023 were \$6{,}000{,}000; at 12/31/2024 they are \$8{,}100{,}000.

Compute the A/E ratio and interpret.

Answer

\begin{align*}
\text{Expected emergence} &= \$10{,}000{,}000 \times (0.75 - 0.60) \\
&= \$1{,}500{,}000 \\[4pt]
\text{Actual emergence} &= \$8{,}100{,}000 - \$6{,}000{,}000 \\
&= \$2{,}100{,}000 \\[6pt]
\text{A/E} &= \frac{\$2{,}100{,}000}{\$1{,}500{,}000} = 1.40
\end{align*}

Emergence ran 40\% above expectation. Re-estimating from the new diagonal at the same pattern:

\frac{\$8{,}100{,}000}{0.75} = \$10{,}800{,}000

an 8\% increase in the indicated ultimate.

Before booking that, two questions: is the 40\% overage confined to AY 2022 or visible across all accident years on this diagonal (calendar-year effect), and is it in paid, in case reserves, or both? The answers determine whether this is one year's adverse experience or a systematic issue affecting the whole reserve.

Reading a Full A/E Table A/E ratios on reported losses for the year ending 12/31/2024:

AY Maturity A/E (reported) A/E (paid)
2020 48–60 1.32 1.04
2021 36–48 1.28 0.99
2022 24–36 1.35 1.02
2023 12–24 1.30 1.01

Diagnose.

Answer Reported emergence exceeded expectation by about 30\% in every accident year at every maturity, while paid emergence was on expectation throughout.

That pattern rules out most explanations:

  • Not a single bad accident year — it is uniform across years.
  • Not increased loss payments — paid A/E is \approx 1.0.
  • Not a reporting-speed change — that would show in paid as well, and would not be uniform across mature years where reporting is complete.

What is left is case reserve strengthening during calendar 2024: reserves on already-known claims were raised across the board, with no corresponding change in payments.

The reserving consequence is that reported development factors are now biased high, and applying them unadjusted would develop the strengthening a second time — the Berquist-Sherman situation. The strengthened reserves themselves are not the error; the factors are.

The wider consequence: paid A/E at 1.0 means the ultimate cost of these claims may not have changed at all. Whether the strengthening was warranted is a question for the claims department, and the answer determines whether the ultimate estimate should move.