Term — Term Life Insurance

Overview

Term Life Insurance provides death benefit coverage for a fixed period (the “term”). If the insured survives the term, the policy expires with no payout. Level premiums are paid throughout the policy term.

Compared to whole life, term policies have shorter durations and lower premiums because coverage is time-limited. From an ALM perspective, the liability horizon is capped at the policy term, making cashflow matching more tractable.

Who buys it: Working-age individuals seeking affordable coverage during peak earning/family years (e.g., 20-year term to cover a mortgage).

Insurer’s risk: Mortality within the term + interest rate mismatch, but over a bounded horizon.

Key Parameters

Parameter Type Description
face_value float Death benefit amount
annual_premium float Level annual premium
term int Policy term in years
qx list[float] Annual mortality rates (must have at least term values)
frequency int Premium payment frequency per year (default 12 = monthly)
age int | None Issue age (reference only)

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

Formulas

Net Cashflow

Identical to whole life, but limited to the policy term T:

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

Symbol Meaning
F Face value (death benefit)
P Annual premium
f Payment frequency per year
T Policy term in years
{}_tp_x Survival probability to time t

Present Value

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

Duration

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

Convexity

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

Example

Term Life — Example Policy
metric value
Face Value $500,000
Annual Premium $1,200
Term 20 years
Age 35
Frequency 12x/year (monthly)

Cashflow Profile

Expected Cashflows (first 24 periods)
period year survival_prob expected_premium expected_benefit net_cashflow
1 0.083333 0.9999 $100 $33 −$67
2 0.166667 0.9999 $100 $33 −$67
3 0.25 0.9998 $100 $33 −$67
4 0.333333 0.9997 $100 $33 −$67
5 0.416667 0.9997 $100 $33 −$67
6 0.5 0.9996 $100 $33 −$67
7 0.583333 0.9995 $100 $33 −$67
8 0.666667 0.9995 $100 $33 −$67
9 0.75 0.9994 $100 $33 −$67
10 0.833333 0.9993 $100 $33 −$67
11 0.916667 0.9993 $100 $33 −$67
12 1.0 0.9992 $100 $33 −$67
13 1.083333 0.9991 $100 $33 −$67
14 1.166667 0.9991 $100 $33 −$67
15 1.25 0.9990 $100 $33 −$67
16 1.333333 0.9990 $100 $33 −$67
17 1.416667 0.9989 $100 $33 −$67
18 1.5 0.9988 $100 $33 −$67
19 1.583333 0.9988 $100 $33 −$67
20 1.666667 0.9987 $100 $33 −$67
21 1.75 0.9986 $100 $33 −$67
22 1.833333 0.9986 $100 $33 −$67
23 1.916667 0.9985 $100 $33 −$67
24 2.0 0.9984 $100 $33 −$67

Unlike whole life, cashflows stop abruptly at year 20. Expected benefits grow over the term as the insured ages and mortality increases, but the total exposure is bounded.

Term Length Comparison

Impact of Policy Term on Risk Metrics
term_years pv_net_liability duration convexity
10 −$5,958 4.39 27.31
15 −$7,093 5.61 46.31
20 −$6,606 4.64 23.75
25 −$4,749 −2.48 −171.38
30 −$1,279 −84.47 −2,721.04

Longer terms increase both the net liability PV and duration, as more years of mortality exposure are included.

Term vs Whole Life Comparison

The term policy’s cashflows end at year 20, while whole life extends for the insured’s full mortality horizon. This fundamental difference drives the duration and convexity gap between the two products.

Sensitivity Analysis

Duration & Convexity

Risk Metrics @ 4% Discount Rate
metric value
Net Liability PV $-6,606
Macaulay Duration 4.64 years
Convexity 23.75

Term policies have shorter durations than whole life because cashflows are capped at the policy term, making them easier to hedge with medium-term fixed-income assets.