Monte-Carlo IBNR Reserving — Canadian P&C Market

Author: Reda Hakkani | PhD Candidate, Applied Mathematics | Montréal, QC
Domain: Actuarial Reserving · Canadian P&C Insurance
Regulatory context: OSFI MCT · OSFI A-4 · CIA Standards · IFRS 17

---

Overview

IBNR reserve estimation using Bootstrap Chain-Ladder with 10,000 Monte-Carlo simulations for Canadian personal auto bodily injury claims.

Line of business: Ontario Personal Auto — Bodily Injury (BI)
Accident years: 2015–2024 (10-year triangle)
Regulatory requirement: OSFI Minimum Capital Test (MCT) at 99.5% VaR

---

Key Results

Metric	Value
Development Triangle	10 accident years × 10 development periods
Chain-Ladder Best Estimate	CAD 772.6M
Bornhuetter-Ferguson	CAD 789.0M
Bootstrap Mean Reserve	CAD 822.7M
Coefficient of Variation	10.69%
VaR 75% (Going-concern CIA)	CAD 880.2M
VaR 90%	CAD 938.8M
VaR 99.5% (OSFI MCT SCR)	CAD 1,068.1M
Risk Margin above CL	CAD 295.5M (+38.3%)
Bootstrap Simulations	10,000

---

Development Triangle — Ontario Personal Auto BI (CAD millions)

	D1	D2	D3	D4	D5	D6	D7	D8	D9	D10
AY2015	95.8	159.1	195.6	224.4	250.4	265.3	265.6	268.2	282.2	285.5
AY2016	99.8	170.0	196.9	233.1	262.4	274.9	277.8	285.3	296.7	—
AY2017	96.5	172.9	225.3	240.4	263.2	292.3	292.6	301.9	—	—
AY2018	93.6	165.9	216.1	247.1	269.7	272.3	288.1	—	—	—
AY2019	102.1	180.2	235.9	264.2	285.4	303.2	—	—	—	—
AY2020	117.9	191.3	245.9	274.8	302.8	—	—	—	—	—
AY2021	109.3	194.7	240.5	266.7	—	—	—	—	—	—
AY2022	114.0	201.8	259.8	—	—	—	—	—	—	—
AY2023	131.0	216.4	—	—	—	—	—	—	—	—
AY2024	129.7	—	—	—	—	—	—	—	—	—

Bold = latest diagonal (observed). Blanks = IBNR to estimate.

---

Link Development Factors (LDFs) — Ontario BI Pattern

D1→D2	D2→D3	D3→D4	D4→D5	D5→D6	D6→D7	D7→D8	D8→D9	D9→D10	Tail
1.721	1.265	1.125	1.101	1.058	1.017	1.023	1.046	1.012	1.000

Fast initial development (litigation + direct compensation) → slow tail (CAT BI, long-term disability)

---

Methodology

Step 1 — Chain-Ladder (Volume-Weighted)

Observed cumulative triangle (10×10)
              │
              ▼
Age-to-Age LDF (volume-weighted average)
              │
              ▼
CDF-to-Ultimate by accident year
              │
              ▼
IBNR = Ultimate − Latest Paid Diagonal

Step 2 — Bornhuetter-Ferguson (CIA Credibility Blend)

CIA a priori ELR (FSRA filing history)
              │
              ├── × % Unreported (1 − 1/CDF)
              │
              ▼
BF IBNR = A Priori × (1 − 1/CDF)

CIA recommendation:
- Recent years (AY2022+): high BF weight (low credibility)
- Older years (AY2015–2018): high CL weight (full credibility)

Step 3 — Bootstrap Chain-Ladder (10,000 simulations)

Observed triangle
              │
              ▼
Pearson residuals (observed vs CL fitted)
              │
              ▼
Repeat 10,000 times:
  ├── Resample residuals (bootstrap with replacement)
  ├── Reconstruct pseudo-triangle
  ├── Re-estimate LDFs on pseudo-triangle
  ├── Project future payments + process variance
  │   (overdispersed Poisson structure)
  └── Store simulated IBNR
              │
              ▼
Reserve distribution
  → VaR 75%  (CIA going-concern)
  → VaR 90%
  → VaR 99.5% (OSFI MCT SCR requirement)

---

IBNR by Accident Year

Accident Year	Latest Paid	% Developed	CL Ultimate	IBNR (CL)	IBNR (BF)
AY2015	285.5M	100.0%	285.5M	0.0M	0.0M
AY2016	296.7M	98.8%	300.2M	3.5M	3.8M
AY2017	301.9M	94.5%	319.5M	17.6M	18.1M
AY2018	288.1M	92.4%	311.9M	23.8M	24.5M
AY2019	303.2M	90.8%	334.0M	30.8M	32.1M
AY2020	302.8M	85.8%	352.8M	50.1M	53.4M
AY2021	266.7M	78.0%	342.1M	75.4M	78.9M
AY2022	259.8M	69.3%	374.9M	115.1M	121.2M
AY2023	216.4M	54.8%	395.0M	178.6M	184.3M
AY2024	129.7M	31.8%	407.4M	277.7M	272.7M
TOTAL			3,423.3M	772.6M	789.0M

---

Regulatory Framework

Standard	Requirement	Application in this model
OSFI MCT	Capital at 99.5% VaR	✓ VaR 99.5% = CAD 1,068.1M
OSFI A-4	P&C reserve standards	✓ Bootstrap uncertainty
CIA P&C	Appointed Actuary Report	✓ CL + BF dual methods
IFRS 17	Risk adjustment	✓ Reserve percentiles
Solvency II	SCR equivalent	✓ Same 99.5% confidence

---

Installation

git clone https://github.com/RedaHakkani/monte-carlo-ibnr-reserving.git
cd monte-carlo-ibnr-reserving
pip install -r requirements.txt
python src/ibnr_reserving.py

Requirements

numpy>=1.24.0
pandas>=2.0.0
scipy>=1.11.0
matplotlib>=3.7.0

Output

• Full console report (triangle, LDFs, IBNR by year, VaR matrix)
• ibnr_reserving_results.png — 6-panel actuarial dashboard

---

References

• England, P.D. & Verrall, R.J. (2002). Stochastic Claims Reserving in General Insurance. IoA.
• Mack, T. (1993). Distribution-free Calculation of the Standard Error of Chain-Ladder Estimates. ASTIN.
• CIA (2020). Practice-Specific Standards for Property and Casualty Insurance.
• OSFI (2022). Guideline A-4 — Property and Casualty Insurance.
• FSRA (2023). Ontario Automobile Insurance Reporting Requirements.

---

Reda Hakkani — PhD Candidate, Applied Mathematics | Montréal, QC
Available for actuarial and quantitative risk roles — hakkanireda@hotmail.com