WL — Whole Life Insurance

Overview

Whole Life Insurance provides lifetime death benefit coverage in exchange for level premium payments. The policy remains in force for the insured’s entire life — whenever death occurs, the face value is paid to the beneficiary.

From an ALM perspective, whole life creates a very long-duration liability. The insurer collects premiums for decades while the death benefit obligation stretches to the end of the mortality table. This makes WL policies highly sensitive to interest rate changes.

Who buys it: Individuals seeking permanent life insurance and estate planning.

Insurer’s risk: Mortality + interest rate mismatch over a very long horizon.

Key Parameters

Parameter Type Description
face_value float Death benefit amount
annual_premium float Level annual premium
qx list[float] Annual mortality rates from insured’s current age
frequency int Premium payment frequency per year (default 12 = monthly)
age int | None Issue age (reference only)

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

Formulas

Equivalence Principle (Net Level Premium)

The net level premium P is set so that the present value of premiums equals the present value of benefits:

\underbrace{\sum_{t=1}^{n} \frac{P/f \cdot {}_{{(t-1)/f}}p_x}{(1 + r/f)^t}}_{\text{PV of premiums}} = \underbrace{\sum_{t=1}^{n} \frac{F \cdot {}_{(t-1)/f|1/f}q_x}{(1 + r/f)^t}}_{\text{PV of benefits}}

Symbol Meaning
P Annual premium
F Face value (death benefit)
f Payment frequency per year
r Discount rate
{}_tp_x Probability of survival from issue to time t
{}_{t\|s}q_x Probability of death between times t and t+s

Net Cashflow (Insurer’s Perspective)

Each period’s net cashflow is:

\text{Net}_t = F \cdot ({}_{(t-1)/f}p_x - {}_{t/f}p_x) - \frac{P}{f} \cdot {}_{(t-1)/f}p_x

Positive = net outflow for the insurer (early on, premium income exceeds expected claims; later, claims dominate).

Duration

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

Convexity

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

Example

Whole Life — Example Policy
metric value
Face Value $1,000,000
Annual Premium $9,730
Age 40
Frequency 12x/year (monthly)
Horizon 81 years

Cashflow Profile

Expected Cashflows (first 24 periods)
period year survival_prob expected_premium expected_benefit net_cashflow
1 0.083333 0.9999 $811 $80 −$731
2 0.166667 0.9998 $811 $80 −$731
3 0.25 0.9998 $811 $80 −$731
4 0.333333 0.9997 $811 $80 −$731
5 0.416667 0.9996 $811 $80 −$731
6 0.5 0.9995 $811 $80 −$731
7 0.583333 0.9994 $810 $80 −$731
8 0.666667 0.9994 $810 $80 −$731
9 0.75 0.9993 $810 $80 −$731
10 0.833333 0.9992 $810 $80 −$731
11 0.916667 0.9991 $810 $80 −$731
12 1.0 0.9990 $810 $80 −$731
13 1.083333 0.9990 $810 $86 −$724
14 1.166667 0.9989 $810 $86 −$724
15 1.25 0.9988 $810 $86 −$724
16 1.333333 0.9987 $810 $86 −$724
17 1.416667 0.9986 $810 $86 −$724
18 1.5 0.9985 $810 $86 −$724
19 1.583333 0.9984 $810 $86 −$724
20 1.666667 0.9984 $810 $86 −$724
21 1.75 0.9983 $810 $86 −$724
22 1.833333 0.9982 $809 $86 −$724
23 1.916667 0.9981 $809 $86 −$724
24 2.0 0.9980 $809 $86 −$724

Early in the policy, expected premiums exceed expected benefits (net cashflow is negative for the insurer = net inflow). As the insured ages, mortality increases and expected benefits dominate.

Premium vs Benefit — Cumulative View

The crossover point — where cumulative expected benefits exceed cumulative expected premiums — is the breakeven year from the insurer’s perspective.

Premium Sensitivity to Issue Age

WL Premium & Duration by Issue Age
issue_age annual_premium duration
25.0 $5,203 −751,367,731,085,050,880.0
30.0 $6,345 −66,592,358,865,272,352.0
35.0 $7,806 22,691,580,207,564,164.0
40.0 $9,730 −36,285,299,520,206,368.0
45.0 $12,227 7,730,845,651,242,143.0
50.0 $15,487 −26,431,088,638,698,028.0
55.0 $19,705 −5,769,792,563,612,866.0
60.0 $25,333 −7,903,717,415,596,538.0
65.0 $32,923 6,489,510,753,310,977.0
70.0 $43,748 −18,607,520,464,368,164.0

Older issue ages require higher premiums because there are fewer years to collect premiums before the expected claim.

Sensitivity Analysis

Duration & Convexity

Risk Metrics @ 4% Discount Rate
metric value
Net Liability PV $-0
Macaulay Duration -36285299520206368.00 years
Convexity -2010016520538589440.00

Whole life policies typically have very long durations due to the lifetime coverage horizon, making them among the most interest-rate-sensitive liabilities an insurer holds.