Aaron D. Deo — Actuarial Risk, Reserving & Loss Distribution Model

Operating boundary

This is an educational actuarial decision reference. It connects stochastic compound Poisson-Gamma frequency/severity modeling, balance sheet loss development reserving triangles, and microeconomic rate elasticity waterfall bridges into an inspectable, auditable system.

Feature-to-Proof Map

Each core analytical capability has a stable ID, a user action, named mathematical proof, and an explicit boundary.

F001 · control

Seeded fleet loss scenario

Purpose: Demonstrate controlled stochastic loss, retention, premium, and tail calculations.

User action: Change fleet assumptions or select a scenario and inspect loss financing metrics.

Boundary: Seeded synthetic simulation with deductible, premium, tail, chart, and export controls.

Proof: Known-answer, boundary, direction, repeatability, and current-state equality checks.

F002 · control

Loss-development reconciliation

Purpose: Transparent development-factor controls and ultimate-loss reconciliation.

User action: Change selected development factors and inspect ultimate loss, IBNR, and cumulative factors.

Boundary: Three editable synthetic factors feeding four reserving outputs.

Proof: Known-answer, reconciliation, and conservation identity (Ultimate = Paid + IBNR).

F003 · control

Rate and unit-economics view

Purpose: Explain relationships among rate action, elasticity attrition, inflation, and combined ratio.

User action: Change rate, elasticity, inflation, and expense assumptions and review the waterfall walk.

Boundary: Constant-elasticity quantity response walking Gross Premium to Net Underwriting Margin.

Proof: Known-answer, direction, control-coverage, and scenario-reset checks.

1. Compound Loss Distribution

Fleet exposure (vehicle-years)

Frequency (claims/veh-yr)

Gamma shape

Gamma scale (USD/claim)

Deductible (USD/claim)

Expense ratio (% loaded)

Simulated years

Seed

Mean claims/year—

Mean gross loss—

Mean retained loss—

Mean insurer loss—

Selected loaded premium—

Modeled reduction—

99.5% tail loss—

2. Chain-Ladder Reserving Triangle

Fixed illustrative latest-paid amounts: AY1 $8.5M, AY2 $7.4M, AY3 $5.1M.

12→24 selected factor

24→36 selected factor

36→ultimate tail

Total ultimate—

Total IBNR—

12→ult cumulative—

24→ult cumulative—

3. Underwriting & Rate Elasticity

Base premium ($M/year)

Base losses ($M/year)

Base retention (%)

Rate change (%)

Demand elasticity

Claim inflation (%)

Expense ratio (%)

Post-rate premium—

Post-rate retention—

Post-rate losses—

Combined ratio—

Underwriting profit—

4. Current Loss & Layer Distribution

GrossRetainedInsurerLoaded Premium99.5% Tail

Current decision interpretation

—

Model logic

Fleet: seeded compound Poisson-Gamma simulation with per-claim deductible. Reserving: deterministic selected development factors applied to fixed paid losses. Economics: constant-elasticity rate/quantity scenario walking Gross Premium down to Net Profit.

Spreadsheet Companion

The complete 6-tab formula workbook Urban_Fleet_Risk_Decision_Model.xlsx contains matching live formulas with zero errors.

5. Reserving Triangle: Cumulative Development ($M)

Accident Year (AY) paid-loss development through 36 months, cumulative link ratios (CDF), and estimated unpaid IBNR liabilities.

Accident Year	12 Months ($M)	24 Months ($M)	36 Months ($M)	Selected CDF	Estimated Ultimate ($M)	Latest Paid ($M)	Unpaid IBNR Liability ($M)
PORTFOLIO TOTAL	—	—	—	—	$29.41M	$21.00M	$8.41M

Reserving Conservation Identity: Total Ultimate ($29.41M) = Cumulative Paid ($21.00M) + Unpaid IBNR Liability ($8.41M).

6. Underwriting Margin & Combined Ratio Waterfall Walk ($M)

Executive P&L bridge walking baseline gross premium through rate revision, price elasticity attrition, claim inflation, and operating expenses to net underwriting profit.

Base Premium Rate Lift / Gain Elasticity / Losses / Expense Subtotals / Profit

Waterfall Step	Delta ($M)	Balance ($M)

7. Deductible Sensitivity & Risk-Financing Ladder

Multi-layer schedule evaluating retention financing, transferred risk, loaded premium, and Total Cost of Risk (TCOR) across retention layers.

Deductible Option (d)	Retained Loss / Claim	Insurer Loss / Claim	Annual Retained Loss	Annual Insurer Loss	% Transferred	Commercial Premium	Total Cost of Risk (TCOR)	Savings vs $0 Ded

Executable acceptance report

Rendered from window.__HTML_ACCEPTANCE_REPORT__. Any failure blocks state export.

Group	Status	Observed vs expected evidence

Model Assumptions & Baseline Register

Controlled baseline parameters, exposure units, statistical distributions, and actuarial rationale calibrated for the commercial fleet loss model.

Assumption	Baseline Value	Unit	Class	Rationale & Actuarial Intuition
Fleet Exposure (E)	2,500	Vehicle-years	Exposure Base	Active fleet size; volume base for per-unit frequency scaling and cross-period comparisons.
Claim Frequency (λ)	0.1200	Claims / veh-yr	Poisson Rate	Annual claim reporting rate producing expected claim count E[N] = 300 claims/year.
Severity Gamma Shape (α)	2.20	Dimensionless	Gamma Parameter	Controls dispersion around the mean; calibrated from CV = 0.6742 via α = 1 / CV².
Severity Gamma Scale (θ)	$4,500	USD / claim	Gamma Parameter	Scale parameter producing mean severity E[X] = αθ = $9,900 per reported claim.
Per-Claim Deductible (d)	$25,000	USD / claim	Policy Retention	Retention threshold splitting claims into retained min(X, d) vs insurer max(0, X - d).
Underwriting Expense Load (e)	20.0%	Percentage	Mechanical Load	Gross-up ratio converting modeled pure premium into loaded commercial premium indication.
12-to-24 Link Ratio (f12)	1.6100	Factor	Reserving CDF	Historical Chain-Ladder paid loss development factor from 12 to 24 months.
24-to-36 Link Ratio (f24)	1.2500	Factor	Reserving CDF	Historical Chain-Ladder paid loss development factor from 24 to 36 months.
36-to-Ultimate Tail Factor	1.0500	Factor	Reserving Tail	Expected development beyond 36 months to final closure and settlement.
Demand Price Elasticity (ε)	-1.20	Ratio	Microeconomic	Policyholder demand response (% ΔQuantity / % ΔRate) governing renewal retention attrition.

Mathematical Specifications & Derivations

Analytical formulations, closed-form probability identities, and balance-sheet conservation proofs governing the fleet model.

Model Component	Governing Formulation & Analytical Specification	Reconciliation & Actuarial Proof
1. Compound Poisson-Gamma Aggregate Loss	Claim count follows N ∼ Poisson(λE) with E[N] = 2,500 × 0.12 = 300 claims. Severity follows X_i ∼ Gamma(α = 2.20, θ = $4,500) with E[X] = αθ = $9,900. Total annual gross loss is S = ∑_i=1^N X_i.	Theoretical analytic mean: E[S] = E[N] × E[X] = 300 × $9,900 = $2,970,000.
2. Per-Claim Deductible Layer Partitioning	For each claim X_i under deductible d ($25,000 baseline): Retained Loss: Y_i^L = min(X_i, d) Insurer Loss: Y_i^C = max(0, X_i − d)	Conservation identity guaranteed: X_i = Y_i^L + Y_i^C and S = S_retained + S_insurer identically.
3. Loaded Commercial Premium Indication	Modeled Pure Premium: P₀ = E[S_insurer]. Loaded Indication: P = P₀ / (1 − e) where e is the expense load ratio (20.0%).	Indicated Unit Rate: Rate = P / E per vehicle-year.
4. Loss Development Chain-Ladder Reserving	Cumulative Development Factors: CDF_t = ∏_{k ≥ t} f_k. Estimated Ultimate: U_t = Paid_t × CDF_t. Unpaid IBNR Reserve: IBNR_t = U_t − Paid_t.	Balance sheet identity reconciles exactly: Ultimate = Cumulative Paid + Total IBNR.
5. Price Elasticity Waterfall Walk	Demand Response: %ΔQ = ε × %ΔRate (ε = −1.20). Earned Premium: P^* = P₀ × (1 + %ΔRate) × (1 + %ΔQ). Loss Cost: L^* = L₀ × (1 + Inflation) × (1 + %ΔQ).	Net Underwriting Margin: Margin = P^* − L^* − Expenses.