Actuarium

Adaptive AI research engine

Journal Scout

Search journals in any field, blitz any article into key findings, real-life applications, practice translations and protocols, then turn the ideas into 5K–400K-word books, handouts and presentations β€” exported as print-ready PDF (Amazon KDP interior rules), Word and PowerPoint. Upload your own PDF or Word document to start from there. Every blitz you open is kept in your Blitz library to read, listen to, quiz on and reuse.

Also: Actuarial bibliography (every curated textbook, paper and journal) Β· Webinar library Β· Practice dashboard Β· Publish to Amazon KDP

Real actuarial data
Genuine NAIC Schedule P triangles from the CAS Loss Reserve Database (Meyers & Shi) β€” 1988–1997 accident years, paid and incurred, with earned premium and the realised ultimates for hindsight testing. Pick a company and line; the analysis below runs live (chain ladder, Bornhuetter–Ferguson, Mack, ODP bootstrap, lognormal severity, VaR/TVaR). Turn on grounding and every new blitz applies its article to these numbers.
Earned premium
2,238,741
Ultimate loss ratio
78.6%
CL IBNR
193,320
Mack CV
30.3%
Bootstrap 95th pct
270,136
CL hindsight error
9.1%
Workers' Compensation
$000
1988–1997
CAS source
Open in Decision Studio

Real data: Allstate Ins Co Grp β€” Workers' Compensation (paid basis, amounts in thousands of USD)

Source: Meyers, G.G. and Shi, P. (2011). The Retrospective Testing of Stochastic Loss Reserve Models. CAS E-Forum, using data pulled from NAIC Schedule P Parts 1-4, 1988-1997. https://www.casact.org/publications-research/research/research-resources/loss-reserving-data-pulled-naic-schedule-p

Development, reserves and loss ratios by accident year

AYEarned premiumLatest paidAgeCDFCL ultimateBF ultimateCL IBNRUlt. loss ratioActual ultimateCL hindsight error
1988394,742325,322101.000325,322325,322082.4%325,3220.0%
1989374,252273,87391.011276,864277,1492,99174.0%277,574-0.3%
1990280,320256,78881.047268,961267,07012,17395.9%263,0002.3%
1991313,982239,19571.080258,402258,11019,20782.3%248,3194.1%
1992252,698159,49661.130180,151182,97720,65571.3%168,8446.7%
1993201,05587,21551.196104,286113,88917,07151.9%90,68615.0%
1994174,38191,07741.307119,003124,24327,92668.2%94,73025.6%
1995146,36687,31131.514132,157127,56544,84690.3%91,16145.0%
199693,29444,91622.02590,94883,18546,03297.5%49,25584.6%
19977,65169114.5013,1105,5142,41940.7%2,9096.9%
Total2,238,7411,565,8841,759,2041,765,024193,32078.6%1,611,8009.1%

Selected volume-weighted age-to-age factors: 2.2230, 1.3377, 1.1584, 1.0927, 1.0586, 1.0455, 1.0314, 1.0361, 1.0109 (tail 1.000).

Reserve risk

MeasureValue
Mack total standard error58,633
Mack CV of total reserve30.3%
ODP bootstrap mean reserve195,949
ODP bootstrap s.d.42,833
75th / 95th / 99th percentile221,978 / 270,136 / 302,665
VaR 99.5% / TVaR 99%310,739 / 323,642

Incremental-cell severity (lognormal MLE on 55 positive cells): ΞΌ=9.819\mu = 9.819, Οƒ=1.091\sigma = 1.091, mean 33,316, CV 1.51.

Key formulas: f^k=βˆ‘iCi,k+1/βˆ‘iCi,k\hat f_k = \sum_i C_{i,k+1} / \sum_i C_{i,k}; U^iCL=Ci,nβˆ’i∏kβ‰₯nβˆ’if^k\hat U_i^{CL} = C_{i,n-i}\prod_{k\ge n-i}\hat f_k; U^iBF=Ci,nβˆ’i+(1βˆ’1/CDFi) ELRβ‹…Pi\hat U_i^{BF} = C_{i,n-i} + (1 - 1/\mathrm{CDF}_i)\,\mathrm{ELR}\cdot P_i; TVaR99%=E[R∣R>VaR99%]\mathrm{TVaR}_{99\%} = \mathbb E[R \mid R > \mathrm{VaR}_{99\%}].

Observations

  • Volume-weighted age-to-age factors: 2.223, 1.338, 1.158, 1.093… β€” slow early development typical of a long-tailed line.
  • Chain-ladder IBNR on the paid basis is 193,320 (thousands of USD), 12.3% of the latest diagonal.
  • Premium-weighted ultimate loss ratio 78.6%; by year it ranges 40.7%–97.5%.
  • Mack standard error of total reserve 58,633 (CV 30.3%); ODP bootstrap 95th percentile 270,136 vs mean 195,949.
  • Hindsight test: chain ladder over-estimated the realized ultimate by 9.1% (actual data through age 10 is known for every year).
  • Bornhuetter–Ferguson (ELR 81.0%) is 0.3% above chain ladder in total β€” the gap sits in the immature years.
  • Incremental paid amounts fit a lognormal with Οƒ = 1.09 (CV 1.51), i.e. heavy-tailed cell values.
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