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Data Aggregation

Note Section 1.2 • Reading time: ~5 mins

Aggregation Methods: CY, AY, and PY

Actuaries aggregate premium and loss data in three primary ways: Calendar Year (CY), Accident Year (AY), and Policy Year (PY). Understanding these methods is critical for correct ratemaking and reserving.

Definitions & Principles

  • Calendar Year (CY): Aggregates all transactions occurring within a specific calendar year, regardless of when the policy was written or when the accident occurred.
    • Focus: The transaction date defines inclusion.
  • Policy Year (PY): Aggregates all transactions arising from policies that became effective during a specific calendar year.
    • Focus: The policy effective date defines inclusion. If a policy is audited or reopened years later, the transaction is mapped back to its original PY.
  • Accident Year (AY): Aggregates all loss transactions arising from accidents occurring within a specific calendar year, regardless of when the policy was written or when the claim is reported.
    • Focus: The accident date defines inclusion.
  • Written Premium (WP): The total premium billed to policyholders for the coverage term. WP transactions are recorded on the transaction effective date.
  • Earned Premium (EP): The portion of written premium that corresponds to the expired portion of the policy’s exposure period.

Example Case Study

Data Table

Assume the following policies are written by an insurer:

PolicyEffective DateExpiration DateTermInitial Premium
Policy 1June 1, 2012May 31, 201312 months$480
Policy 2July 1, 2012December 31, 20126 months$125
Policy 3March 1, 2013February 28, 201412 months$225
Policy 4August 1, 2013March 31, 20148 months$300
  • Final Audit Rule: Six months after a policy expires, the initial premium increases by 8% due to a final audit.

Audit Transaction Details

  • Policy 1: Expires 5/31/2013 →\to Audit transaction on 11/30/2013 for \480 \times 8% = $38.40$.
  • Policy 2: Expires 12/31/2012 →\to Audit transaction on 6/30/2013 for \125 \times 8% = $10.00$.
  • Policy 3: Expires 2/28/2014 →\to Audit transaction on 8/31/2014 for \225 \times 8% = $18.00$.
  • Policy 4: Expires 3/31/2014 →\to Audit transaction on 9/30/2014 for \300 \times 8% = $24.00$.

Calculations & Solutions

1. CY 2013 Earned Premium (as of Dec 31, 2013)

We calculate the premium earned during the calendar year 2013. This includes the portion of initial premium earned in 2013 plus any audit transactions that occurred in 2013 (which are fully earned immediately since they apply to expired terms).

  • Policy 1: Active in 2013 from Jan 1 to May 31 (5 months).
    • \text{Earned Initial} = \480 \times \frac{5}{12} = $200.00$.
    • Audit occurred on 11/30/2013 (in CY 2013) = $38.40.
    • \text{Total} = \200.00 + $38.40 = $238.40$.
  • Policy 2: Expired in 2012 (no initial premium earned in 2013).
    • Audit occurred on 6/30/2013 (in CY 2013) = $10.00.
    • \text{Total} = \10.00$.
  • Policy 3: Active in 2013 from March 1 to Dec 31 (10 months).
    • \text{Earned Initial} = \225 \times \frac{10}{12} = $187.50$.
    • Audit occurred in 2014 (not in CY 2013).
    • \text{Total} = \187.50$.
  • Policy 4: Active in 2013 from Aug 1 to Dec 31 (5 months out of 8-month term).
    • \text{Earned Initial} = \300 \times \frac{5}{8} = $187.50$.
    • Audit occurred in 2014 (not in CY 2013).
    • \text{Total} = \187.50$.
CY 2013 EP=$238.40+$10.00+$187.50+$187.50=$623.40\text{CY 2013 EP} = \$238.40 + \$10.00 + \$187.50 + \$187.50 = \mathbf{\$623.40}

2. CY 2013 Written Premium (as of Dec 31, 2013)

We sum all premium transactions written/recorded in CY 2013:

  • Policy 1 (Audit): Written 11/30/2013 = $38.40.
  • Policy 2 (Audit): Written 6/30/2013 = $10.00.
  • Policy 3 (Initial): Written 3/1/2013 = $225.00.
  • Policy 4 (Initial): Written 8/1/2013 = $300.00.
CY 2013 WP=$38.40+$10.00+$225.00+$300.00=$573.40\text{CY 2013 WP} = \$38.40 + \$10.00 + \$225.00 + \$300.00 = \mathbf{\$573.40}

3. PY 2013 Earned Premium (as of Dec 31, 2013)

We only include policies effective in 2013 (Policy 3 and Policy 4) and calculate what they earned by Dec 31, 2013:

  • Policy 3 (Eff. 3/1/2013): Earns 10 months of initial premium in 2013.
    • \text{Earned} = \225 \times \frac{10}{12} = $187.50$.
  • Policy 4 (Eff. 8/1/2013): Earns 5 months of initial premium in 2013.
    • \text{Earned} = \300 \times \frac{5}{8} = $187.50$.

(Audits for both policies occur in 2014, so they are not included as of Dec 31, 2013).

PY 2013 EP=$187.50+$187.50=$375.00\text{PY 2013 EP} = \$187.50 + \$187.50 = \mathbf{\$375.00}

4. PY 2013 Written Premium (as of Dec 31, 2014)

We include policies effective in 2013 (Policy 3 and Policy 4) and sum all transactions written on them on or before Dec 31, 2014:

  • Policy 3 (Eff. 3/1/2013):
    • Initial premium written 3/1/2013 = $225.00.
    • Audit premium written 8/31/2014 = $18.00.
  • Policy 4 (Eff. 8/1/2013):
    • Initial premium written 8/1/2013 = $300.00.
    • Audit premium written 9/30/2014 = $24.00.
PY 2013 WP=$225.00+$18.00+$300.00+$24.00=$567.00\text{PY 2013 WP} = \$225.00 + \$18.00 + \$300.00 + \$24.00 = \mathbf{\$567.00}

[!NOTE] Note: There is a common transposition typo in some solutions listing this answer as $576. The correct mathematical calculation is $567 (525×1.08525 \times 1.08).

Key Takeaway Summary

  • CY: Think of a horizontal time slice (transactions occurring in that year).
  • PY: Think of a policy group (policies effective in that year, tracked over their lifetime).
  • Audits: Fully earned when written (since the risk period has already passed).