The actuarial premium calculator is a specialized tool used in the insurance industry to determine the appropriate premium rates for insurance policies. This tool calculates the cost of insuring someone based on the probability of certain events, such as death, and combines this with other financial factors, including operational costs and expected investment returns. By using such calculators, insurers can set premiums that are both competitive and sufficient to cover the anticipated claims and expenses.
Pure Premium (or Net Premium) Calculation
A fundamental concept in actuarial science is the calculation of the pure premium. The pure premium reflects the cost of claims per policy, not accounting for the operational expenses or profits. Here is how it's calculated:
Pure Premium = Σ (q_x * B_x) from x = 0 to n
Where:
- q_x is the probability of death at age x
- B_x is the benefit amount at age x
- n is the number of years
Gross Premium Calculation
To arrive at the gross premium, which is the actual amount charged to the policyholder, the pure premium is adjusted for operational expenses and expected income from investments:
Gross Premium = Pure Premium + Expense Loadings - Expected Investment Income
Expense loadings might include acquisition costs, maintenance costs, and other expenses related to the issuance and maintenance of the policy.
Present Value of Future Benefits (PVFB)
An important aspect of actuarial calculations is determining the present value of future benefits, calculated as follows:
PVFB = Σ (v^t * q_x * B_x) from t = 1 to n
Where:
- v is the discount factor (1 / (1 + i))
- t is the year of the policy
Present Value of Future Premiums (PVFP)
Similarly, the present value of future premiums is calculated to ensure the policy remains profitable under expected future conditions:
PVFP = Σ (v^t * p_x * P) from t = 1 to n
Where:
- p_x is the probability of survival at age x
- P is the premium amount
Actuarial Premium Calculation
The actuarial premium ensures that the present value of future premiums equals the present value of future benefits:
P = PVFB / Σ (v^t * p_x) from t = 1 to n
Quick Reference Table for Actuarial Premium Calculator
The following table provides typical values and results for actuarial premium calculations, helping users understand the dependencies without performing detailed calculations:
| Age | Probability of Death (q_x) | Survival Probability (p_x) | Benefit Amount (B_x) | Pure Premium | Gross Premium |
|---|---|---|---|---|---|
| 30 | 0.001 | 0.999 | $50,000 | $50 | $120 |
| 40 | 0.002 | 0.998 | $75,000 | $150 | $240 |
| 50 | 0.005 | 0.995 | $100,000 | $500 | $550 |
| 60 | 0.010 | 0.990 | $125,000 | $1250 | $1350 |
This table can be use as a handy guide for typical scenarios encountered in actuarial calculations.
Example of Actuarial Premium Calculator
Let's consider a simplified example to illustrate how an actuarial premium is calculated:
- Assumptions:
- Probability of death (q_x) at age 50: 0.005
- Benefit amount (B_x) at age 50: $100,000
- Expense loading: $200
- Expected investment income: $150
- Discount rate (i): 5%
- Calculations:
- Pure Premium: 0.005 times 100,000 = $500
- Gross Premium: 500 + 200 - 150 = $550
- Present Value of Future Benefits (PVFB) for one year: (0.005 times 100,000) / (1 + 0.05) approximately $476.19
- Present Value of Future Premiums (PVFP) for one year, assuming survival probability (p_x) of 0.995: (0.995 times 550) / (1 + 0.05) approximately $519.05
- Actuarial Premium (P) required to balance PVFP and PVFB: P = PVFB / ((1 / (1 + 0.05)) times 0.995) approximately $505
This example shows how each element contributes to the final premium calculation, using straightforward assumptions for clarity.
Most Common FAQs
The pure premium is the cost that reflects the expected claims cost per policy, exclusive of operational expenses and profit margins.
Actuaries calculate the present values of expected future benefits and premiums to ensure the insurance policy is priced to be financially viable over its term.
Besides the pure premium, factors like administrative expenses and expected returns from invested premiums significantly impact the gross premium.