FIA — Fixed Indexed Annuity

Overview

A Fixed Indexed Annuity (FIA) is a single-premium deferred annuity whose account value grows based on an external reference rate (e.g., an Equity index), subject to an annual floor and cap. The policyholder’s principal is protected by the floor (typically 0%), while upside is capped.

FIAs combine elements of fixed annuities (guaranteed minimum) with equity-linked products (market participation). From an ALM perspective, FIAs create path-dependent liabilities that depend on both mortality and future index returns.

Who buys it: Pre-retirees seeking principal protection with some market upside during the accumulation phase.

Insurer’s risk: Market risk (hedging the index-linked crediting), mortality risk, and interest rate risk.

Key Parameters

Parameter Type Description
premium float Single premium paid at issue
term int Accumulation period in years
qx list[float] Annual mortality rates (must have at least term values)
floor float Minimum annual credited rate (default 0%)
cap float Maximum annual credited rate (default 6%)
participation_rate float Fraction of index return credited (default 1.0 = 100%)
age int | None Issue age (reference only)

Code: FIA class in src/alm/liability.py

Formulas

Credited Rate

Each year, the credited rate is derived from the external index rate:

r_{\text{credited}} = \max\!\Big(\text{floor},\; \min\!\big(\text{cap},\; r_{\text{index}} \times \text{participation}\big)\Big)he

Account Value Accumulation

AV_t = AV_{t-1} \times (1 + r_{\text{credited},t})

with AV_0 = \text{premium}.

Expected Cashflows

Each year has two potential outflows:

\text{Death Benefit}_t = AV_t \times ({}_{t-1}p_x - {}_tp_x)

\text{Maturity Benefit} = AV_T \times {}_Tp_x \quad \text{(final year only)}

\text{Net Cashflow}_t = \text{Death Benefit}_t + \text{Maturity Benefit}_t

Present Value

PV = \sum_{t=1}^{T} \frac{\text{Net Cashflow}_t}{(1 + r)^t}

Duration

D = \frac{1}{PV} \sum_{t=1}^{T} \frac{t \cdot \text{Net Cashflow}_t}{(1 + r)^t}

Example

FIA — Example Policy
metric value
Premium $500,000
Term 10 years
Floor 0%
Cap 6%
Participation Rate 100%
Age 55

Credited Rate Mechanics

The credited rate is clamped between the floor and cap. Negative index returns are floored at 0% (principal protection), while strong returns are capped at 6%.

Account Value Growth

Account Value Accumulation
year index_rate credited_rate account_value
0 0.0% 0.0% $500,000
1 3.0% 3.0% $515,000
2 5.0% 5.0% $540,750
3 −2.0% 0.0% $540,750
4 7.0% 6.0% $573,195
5 4.0% 4.0% $596,123
6 6.0% 6.0% $631,890
7 1.0% 1.0% $638,209
8 8.0% 6.0% $676,502
9 −1.0% 0.0% $676,502
10 5.0% 5.0% $710,327

Scenario Analysis

The floor provides downside protection: even in the bear scenario, the account value never declines (worst case is 0% credited rate per year). The cap limits upside in the bull scenario.

Expected Cashflows

Expected Cashflows (Moderate Scenario)
year account_value survival_prob death_prob expected_death_benefit expected_maturity_benefit net_cashflow
1 $515,000 0.9964 0.0036 $1,862 $0 $1,862
2 $540,750 0.9925 0.0039 $2,113 $0 $2,113
3 $540,750 0.9882 0.0042 $2,293 $0 $2,293
4 $573,195 0.9836 0.0046 $2,652 $0 $2,652
5 $596,123 0.9785 0.0051 $3,017 $0 $3,017
6 $631,890 0.9730 0.0055 $3,501 $0 $3,501
7 $638,209 0.9669 0.0061 $3,873 $0 $3,873
8 $676,502 0.9603 0.0066 $4,483 $0 $4,483
9 $676,502 0.9531 0.0072 $4,879 $0 $4,879
10 $710,327 0.9453 0.0078 $5,565 $671,447 $677,013

The maturity benefit dominates in the final year — most policyholders survive the 10-year term. Death benefits are small each year because mortality is low at ages 55–65.

Participation Rate & Cap Sensitivity

FIA Sensitivity to Cap & Participation Rate
cap participation terminal_av pv_liability
4% 80% $648,003 $439,055
4% 100% $658,156 $445,846
6% 80% $675,784 $457,402
6% 100% $710,327 $480,309
8% 80% $678,334 $459,068
8% 100% $730,557 $493,624
10% 80% $678,334 $459,068
10% 100% $730,557 $493,624

Higher caps and participation rates increase the terminal account value and thus the insurer’s liability.

Sensitivity to Discount Rate

Duration & Convexity

Risk Metrics @ 4% Discount Rate
metric value
Liability PV $480,309
Macaulay Duration 9.79 years
Convexity 98.74

FIAs have relatively short durations (bounded by the accumulation term) compared to whole life or SPIA products. The maturity benefit concentrated in the final year pulls duration toward the term endpoint.