Portfolio Overview
10,000 synthetic policyholders, calibrated to Sub-Saharan African mortality
Policies
10,000
Avg Age
41.8
years
Smoker Rate
25.7%
Avg BMI
26.1
Avg Sum Assured
$49,370
Total Exposure
$493,700,000
Age Distribution
Health Score Distribution
Mortality Model
Gompertz-Makeham hazard model calibrated to Sub-Saharan African mortality patterns
Mortality Rate (qx) by Age
Life Expectancy (ex) by Age
Survivors (lx) out of 100,000
Select and Ultimate Mortality
q[x]+t, the rate for a life underwritten at age x and now t years into the policy. Reading a row left to right shows the effect of underwriting wearing off over the 5-year select period. The ultimate column is the rate the row converges to, which depends on attained age alone.
| Entry age | q[x]+0 | q[x]+1 | q[x]+2 | q[x]+3 | q[x]+4 | Ultimate |
|---|---|---|---|---|---|---|
| 20 | 0.000313 | 0.000400 | 0.000492 | 0.000591 | 0.000697 | 0.000810 @25 |
| 30 | 0.000448 | 0.000585 | 0.000735 | 0.000901 | 0.001086 | 0.001290 @35 |
| 40 | 0.000792 | 0.001055 | 0.001353 | 0.001692 | 0.002075 | 0.002512 @45 |
| 50 | 0.001669 | 0.002252 | 0.002926 | 0.003701 | 0.004592 | 0.005616 @55 |
| 60 | 0.003893 | 0.005290 | 0.006914 | 0.008796 | 0.010971 | 0.013478 @65 |
Survival Analysis
Kaplan-Meier curves across 688 observed deaths (6.9% mortality rate)
Total Deaths
688
Mortality Rate
6.9%
Avg Duration at Death
11.7 yrs
Lapsed
58.4%
censored, not claims
Male / Female Deaths
384 / 304
Overall Survival Curve
Survival by Gender
Survival by Smoker Status
Survival by Health Score
Cox Proportional Hazards
Concordance 0.765 held out, 0.7665 in sample | Proportional hazards: 1 of 5 covariates breach at p < 0.05 | Log-likelihood ratio: 651.44
| Covariate | Coefficient | Hazard Ratio | 95% CI | p-value | Significance |
|---|---|---|---|---|---|
| age | 0.0801 | 1.0833 | [1.075, 1.091] | <0.001 | *** |
| smoker | 0.5232 | 1.6874 | [1.444, 1.971] | <0.001 | *** |
| bmi | -0.0035 | 0.9965 | [0.982, 1.012] | 0.6484 | ns |
| health_score | 0.1605 | 1.1741 | [1.105, 1.247] | <0.001 | *** |
| gender_male | 0.1337 | 1.1431 | [0.983, 1.329] | 0.0818 | ns |
Hazard Ratios
Persistency and Lapses
14.0% lapse in year one, falling to 4.0% ultimate | 34.8% of policies expected to reach the end of their term
Year 1 Lapse
14.0%
Ultimate Lapse
4.0%
from duration 5
In Force at 10 yrs
51.8%
lapses only
In Force at 20 yrs
34.5%
lapses only
Persistency Curve
Lapse Rate by Duration
Select Factors
| Policy year | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Mortality vs ultimate | 45% | 56% | 67% | 78% | 89% | 100% |
| Lapse rate | 14% | 10% | 8% | 6% | 5% | 4% |
Assumptions, not estimates. Nothing here is fitted to experience data. A real basis would come from the insurer's own investigation. Lapses are treated as censoring in the survival analysis, never as claims: a policyholder who stops paying walks away alive, and counting them as deaths would bias the mortality estimate badly.
Actuarial Pricing
Net single premium with 6% discount rate and 15% expense loading
Avg Net Single Premium
$1,766
Avg Annual Premium
$306
gross
Total Annual Income
$3,057,121
Total NSP
$17,661,439
Priced Without Lapses
$370
15.7% lapse credit
Reach End of Term
34.8%
expected
Taking credit for lapses makes the book 15.7% cheaper, because policyholders who leave a term assurance forfeit everything and were never going to claim. That is a real effect and a real exposure: if persistency comes in better than assumed, the book is underpriced. A reserving basis would not take the same credit a pricing basis does.
Average Annual Premium by Age
Average Annual Premium by Risk Factor
Premium by Gender
Premium by Health Score
Monte Carlo Simulation
5,000 simulations over a 5-year horizon
Mean Aggregate Claims
$6,043,996
VaR 99.5%
$8,185,025
Value at Risk
TVaR 99.5%
$8,594,600
Tail Value at Risk
Required Reserve
$8,185,025
Mean + risk margin
Std Deviation
$795,349
Mean Deaths
123
of 10,000
Loss Ratio
1.22%
claims / exposure
Risk Margin
$2,141,029
Aggregate Claims Distribution
Tail beyond VaR 99.5%
Death Count Distribution
Claim Percentiles
p5
$4,785,000
p25
$5,490,000
p50
$6,010,000
p75
$6,585,000
p95
$7,385,250
p99
$7,930,000
p995
$8,185,025
Stress Testing
Monte Carlo under mortality shock scenarios
| Scenario | Mortality Shock | Mean Claims | VaR 99.5% | TVaR 99.5% | Mean Deaths | Required Reserve |
|---|---|---|---|---|---|---|
| Baseline | 1x | $6,043,996 | $8,185,025 | $8,594,600 | 123 | $8,185,025 |
| Mild pandemic | 1.25x | $7,500,003 | $9,950,000 | $10,172,115 | 153 | $9,950,000 |
| Severe pandemic | 1.6x | $9,589,260 | $12,340,000 | $12,678,269 | 196 | $12,340,000 |
| Catastrophic event | 2.5x | $14,714,297 | $17,955,100 | $18,356,800 | 301 | $17,955,100 |