SPIA — Single Premium Immediate Annuity

Overview

A Single Premium Immediate Annuity (SPIA) is a contract where the policyholder pays a single lump-sum premium and in return receives periodic income payments for life, starting immediately. An optional certain period guarantees payments for a fixed number of years regardless of survival.

SPIAs are the bread-and-butter liability for retirement income. They expose the insurer to longevity risk (the annuitant lives longer than expected) and interest rate risk (the discount rate used to price the annuity changes after issue).

Who buys it: Retirees converting savings into guaranteed income.

Insurer’s risk: Longevity + interest rate mismatch.

Key Parameters

Parameter Type Description
premium float Single lump-sum payment at issue
annual_payout float Total annual payout to the annuitant
qx list[float] Annual mortality rates from current age
frequency int Payouts per year (default 12 = monthly)
certain_period int Guaranteed payment period in years (0 = life only)
age int | None Issue age (reference only)

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

Formulas

Actuarially Fair Premium

The fair single premium equals the present value of all expected future payouts:

\text{Premium} = \sum_{t=1}^{n} \frac{E[\text{Payout}_t]}{(1 + r/f)^t}

where:

Symbol Meaning
E[\text{Payout}_t] Expected payout at period t (full payout during certain period, payout \times survival probability after)
r Annual discount rate
f Payment frequency per year
n Total number of periods (= \text{years} \times f)

Expected Payout

E[\text{Payout}_t] = \begin{cases} \text{pmt} & \text{if } t/f \leq \text{certain period} \\ \text{pmt} \times {}_tp_x & \text{otherwise} \end{cases}

where \text{pmt} = \text{annual\_payout} / f and {}_tp_x is the probability of surviving to time t/f.

Duration (Macaulay)

D = \frac{1}{PV} \sum_{t=1}^{n} \frac{t}{f} \cdot \frac{E[\text{Payout}_t]}{(1 + r/f)^t}

Convexity

C = \frac{1}{PV \cdot f^2} \sum_{t=1}^{n} \frac{t(t+1) \cdot E[\text{Payout}_t]}{(1 + r/f)^{t+2}}

Example

SPIA — Example Policy
metric value
Annual Payout $120,000
Fair Premium $1,772,870
Certain Period 10 years
Age 65
Frequency 12x/year (monthly)

Cashflow Profile

Expected Cashflows (first 24 periods)
period year payout survival_prob expected_payout
1 0.083333 $10,000 0.9994 $10,000
2 0.166667 $10,000 0.9989 $10,000
3 0.25 $10,000 0.9983 $10,000
4 0.333333 $10,000 0.9977 $10,000
5 0.416667 $10,000 0.9972 $10,000
6 0.5 $10,000 0.9966 $10,000
7 0.583333 $10,000 0.9960 $10,000
8 0.666667 $10,000 0.9954 $10,000
9 0.75 $10,000 0.9949 $10,000
10 0.833333 $10,000 0.9943 $10,000
11 0.916667 $10,000 0.9937 $10,000
12 1.0 $10,000 0.9932 $10,000
13 1.083333 $10,000 0.9926 $10,000
14 1.166667 $10,000 0.9920 $10,000
15 1.25 $10,000 0.9914 $10,000
16 1.333333 $10,000 0.9908 $10,000
17 1.416667 $10,000 0.9902 $10,000
18 1.5 $10,000 0.9896 $10,000
19 1.583333 $10,000 0.9890 $10,000
20 1.666667 $10,000 0.9884 $10,000
21 1.75 $10,000 0.9877 $10,000
22 1.833333 $10,000 0.9871 $10,000
23 1.916667 $10,000 0.9865 $10,000
24 2.0 $10,000 0.9859 $10,000

The gap between nominal and expected payouts widens over time as mortality reduces the expected payment. During the 10-year certain period, both lines overlap because payments are guaranteed.

Certain Period Comparison

Impact of Certain Period on SPIA Pricing
certain_period premium pv duration convexity
0 years $1,734,014 $1,734,014 10.60 175.39
5 years $1,743,854 $1,743,854 10.56 174.47
10 years $1,772,870 $1,772,870 10.51 172.66
15 years $1,822,129 $1,822,129 10.57 172.42
20 years $1,895,298 $1,895,298 10.85 177.91

Longer certain periods increase the premium because more payments are guaranteed regardless of survival.

Sensitivity Analysis

Duration & Convexity

Risk Metrics @ 4% Discount Rate
metric value
Present Value $1,772,870
Macaulay Duration 10.51 years
Convexity 172.66

As discount rates increase, duration decreases because distant cashflows contribute less to the present value.