Class 4 — CARVM Reserve Fundamentals¶

This session introduces the Commissioners Annuity Reserve Valuation Method (CARVM), the statutory reserving framework historically used for fixed and deferred annuity products, including Multi-Year Guaranteed Annuities (MYGA).

The goal of this class is to build a structural and algorithmic understanding of how statutory annuity reserves are determined under conservative assumptions.

Scope of this class

This class focuses on CARVM / AG 33 methodology only. VM-22 will be introduced in Class 5 as an extension and modernization.

1. Regulatory Context: Where CARVM Comes From¶

1.1 What Is CARVM?¶

CARVM stands for:

Commissioners Annuity Reserve Valuation Method

It is a statutory reserving methodology designed to determine the minimum reserve required for certain annuity products under U.S. insurance regulation.

CARVM applies primarily to:

Fixed deferred annuities
MYGA and traditional fixed annuity contracts
Products without equity-linked guarantees

1.2 Regulatory Source¶

CARVM is formally described and governed by:

Actuarial Guideline 33 (AG 33)

AG 33 provides:

the valuation framework
permitted assumptions
benefit definitions
and methodology for determining reserves

2. Statutory Reserve Concept¶

2.1 Traditional Reserve Formula¶

\[ \text{Reserve} = PV(\text{Future Benefits}) - PV(\text{Future Considerations}) \]

This formulation works well for life insurance, but annuities require additional structure due to embedded options.

2.2 Why Annuities Require a Different Method¶

Annuity contracts differ because:

benefits are path-dependent
policyholders have multiple behavioral choices
future cash flows depend on timing and actions

Warning

CARVM is not an expected value calculation. It is designed to capture the worst-case benefit path.

3. CARVM Core Philosophy¶

3.1 Worst-Case Valuation¶

CARVM assumes that:

At each decision point, the policyholder will act in a manner that is most favorable to themselves and most adverse to the insurer.

This results in a deterministic, conservative reserve.

3.2 Time-First Logic¶

CARVM operates using a time-first framework:

Fix a policy year
Compare all allowable benefits at that time
Select the maximum benefit
Discount backward one period

Maximization occurs within each time point, not across time.

4. MYGA as the Base Product Example¶

4.1 MYGA Benefit Structure¶

Benefit Type	Description
Accumulation Value (AV)	Account value with guaranteed interest
Nonforfeiture Value	Statutory minimum value
Guaranteed Fund	Floor under nonforfeiture rules
Surrender Benefit	AV minus charges ± MVA
Death Benefit	Typically AV
Maturity Benefit	Value at end of guarantee period
Annuitization Option	Conversion to income payments

4.2 Annuitization Options Considered¶

Option	Description
15-Year Certain	Fixed payments for 15 years
Life Only	Payments contingent on survival

Annuitization introduces mortality-contingent cash flows and requires separate valuation logic.

5. CARVM Methodology: Integrated Path and Backward Framework¶

5.1 Two-Level Structure¶

CARVM is evaluated in two nested layers:

Across policyholder paths
Within each path, backward induction over time

5.2 Benefit Definition at a Given Time¶

At policy year \(t\), allowable actions may include:

\[ B_t = \max \left( \text{Surrender}_t,\ \text{Death}_t,\ \text{Annuitization}_t,\ \text{Maturity}_t,\ PV_t(\text{Continue}) \right) \]

5.3 Backward Discounting¶

Within each path:

\[ R_{t-1} = \frac{B_t}{1 + i_v} \]

where \(i_v\) is the valuation interest rate.

5.4 CARVM Conceptual Algorithm Flow¶

flowchart TD

    A[Start CARVM Valuation] --> B[Enumerate Policyholder Paths]

    B --> P1[Path 1: No Partial Withdrawal]
    B --> P2[Path 2: Full Free Partial Withdrawal]

    P1 --> C1A[Start at Final Duration]
    C1A --> C1B[Enumerate Allowed Benefits]
    C1B --> C1C[Calculate Value of Each Benefit]
    C1C --> C1D[Take Maximum Benefit]
    C1D --> C1E[Discount Back One Period]
    C1E --> C1F{Reached Time 0?}
    C1F -- No --> C1B
    C1F -- Yes --> R1[Reserve for Path 1]

    P2 --> C2A[Start at Final Duration]
    C2A --> C2B[Enumerate Allowed Benefits]
    C2B --> C2C[Calculate Value of Each Benefit]
    C2C --> C2D[Take Maximum Benefit]
    C2D --> C2E[Discount Back One Period]
    C2E --> C2F{Reached Time 0?}
    C2F -- No --> C2B
    C2F -- Yes --> R2[Reserve for Path 2]

    R1 --> M[Take Maximum Across Paths]
    R2 --> M
    M --> Z[CARVM Reserve at Issue]

6. Path-Dependent Behavior¶

Partial withdrawal behavior affects future contract structure, not just current cash flows.

For MYGA products without GLWB, practice typically evaluates:

Path	Description
No Partial Withdrawal	Maximizes accumulation
Full Free Partial Withdrawal	Maximizes early cash extraction

The CARVM reserve is the maximum across evaluated paths.

7. Assumptions and Terminology¶

7.1 Interest Rates¶

Term	Purpose
Guaranteed Rate	Builds AV
Nonforfeiture Rate	Guaranteed floor
Valuation Interest Rate	Backward discounting
Annuitization Rate	Income conversion

These rates apply at different layers of the calculation.

7.2 Mortality¶

Mortality assumptions are used primarily for annuitization valuation. For deferred MYGA contracts, mortality is secondary and prescribed by statutory annuity tables.

Section A — Looking Ahead: From CARVM to VM-22¶

This class established a deterministic, worst-case framework for annuity reserving under statutory regulation.

In the next class, we will transition to VM-22, which extends these same ideas using a stochastic, distribution-based approach.

Specifically, the next section will cover:

Why CARVM is considered overly conservative for modern products
How VM-22 replaces single-path worst-case logic with scenario distributions
The role of CTE (Conditional Tail Expectation) in reserve determination
How policyholder behavior is modeled probabilistically under VM-22
The conceptual mapping:
CARVM paths ⟶ VM-22 stochastic scenarios
Worst-case path ⟶ Tail of the distribution

This transition will show that VM-22 is not a replacement of CARVM logic, but a generalization built on the same structural foundation.