The Equivalence Principle

The net premium is the premium that makes a policy actuarially fair at inception, meaning the insurer expects to break even before expenses and profit. Formally, the equivalence principle sets the expected present value (APV) of future premiums equal to the APV of future benefits: APV(Premiums) = APV(Benefits). This condition ensures that the expected loss on the policy is zero at the time of issue.

The loss-at-issue random variable L_0 represents the present value of the insurer's net obligation: L_0 = PV(benefits) - PV(premiums). For a fully discrete whole life insurance with annual premiums, L_0 = v^{K_x+1} - P * a-ddot_{K_x+1}, where K_x is the curtate future lifetime and P is the net annual premium. Setting E[L_0] = 0 yields the equivalence principle premium: E[v^{K+1}] = P * E[a-ddot_{K+1}], which gives A_x = P * a-ddot_x, and therefore P_x = A_x / a-ddot_x.

Whole Life Insurance Premium

For a fully discrete whole life insurance, the net annual premium is P_x = A_x / a-ddot_x. Using the identity A_x = 1 - d*a-ddot_x, this can be rewritten as P_x = (1 - d*a-ddot_x) / a-ddot_x = 1/a-ddot_x - d. This alternative form is often faster to compute when a-ddot_x is given directly.

For the continuous case, P-bar(A-bar_x) = A-bar_x / a-bar_x = (1 - delta*a-bar_x) / a-bar_x = 1/a-bar_x - delta. The structure is identical, with delta replacing d.

Under constant force of mortality mu with force of interest delta: P-bar(A-bar_x) = mu/(mu+delta) divided by 1/(mu+delta) = mu. This elegant result says the continuous net premium rate equals the force of mortality. Intuitively, the annual premium needed to fund a unit benefit payable upon death equals the instantaneous death rate, because in a continuous framework, premiums are collected continuously and benefits paid continuously.

Premiums for Term and Endowment Insurance

The equivalence principle applies to any benefit structure. For n-year term insurance with premiums payable for n years: P_{x:n}^1 = A_{x:n}^1 / a-ddot_{x:n}. Since the term APV is typically small (especially for young ages and moderate terms), the term premium is correspondingly small.

For n-year endowment insurance: P_{x:n} = A_{x:n} / a-ddot_{x:n} = (1 - d*a-ddot_{x:n}) / a-ddot_{x:n} = 1/a-ddot_{x:n} - d. The endowment premium is always larger than the term premium because the endowment includes the pure endowment benefit that is certain to be paid.

The relationship between these premiums is: P_{x:n} = P_{x:n}^1 + P_{x:n}^{~1}, where P_{x:n}^{~1} = n_E_x / a-ddot_{x:n} is the premium for the pure endowment component. This decomposition shows that the endowment premium funds both the death protection and the savings component.

Limited Payment Premiums

Sometimes premiums are paid over a shorter period than the benefit coverage. For an h-payment whole life insurance (premiums for h years, coverage for life), the premium is h_P_x = A_x / a-ddot_{x:h}. Since a-ddot_{x:h} < a-ddot_x, the limited payment premium h_P_x > P_x. The policyholder pays more per year but for fewer years.

The extreme case is the single premium (h=1), where 1_P_x = A_x / 1 = A_x. The entire cost of the insurance is paid upfront. At the other extreme, as h approaches the maximum possible lifetime, h_P_x approaches P_x (the level premium for life).

For pension products, limited payment structures are natural: premiums are paid during the working years (say from age 25 to 65) to fund a retirement annuity that continues for life. The premium calculation follows the same equivalence principle: G * a-ddot_{25:40} = annuity benefit * deferred annuity APV.

Variance of the Loss Variable

The variance of L_0 measures the insurer's risk on a single policy. For a whole life insurance with annual premiums: Var(L_0) = Var(v^{K+1} - P*a-ddot_{K+1}). Since a-ddot_{K+1} = (1 - v^{K+1})/d, we get L_0 = v^{K+1}*(1 + P/d) - P/d. The variance is Var(L_0) = (1 + P/d)^2 * Var(v^{K+1}) = (1 + P/d)^2 * (^2A_x - A_x^2).

Using P_x = 1/a-ddot_x - d, we get 1 + P/d = 1 + (1/(d*a-ddot_x) - 1) = 1/(d*a-ddot_x). So Var(L_0) = (^2A_x - A_x^2) / (d*a-ddot_x)^2. This formula shows that the variance depends on both the second moment of the insurance (through ^2A_x) and the annuity value.

For a portfolio of n independent identical policies, the variance of the average loss is Var(L_0)/n. The probability of an aggregate loss can be approximated using the normal distribution for large n, which is the basis for determining the insurer's required surplus.

Percentile Premium Principle

An alternative to the equivalence principle is the percentile premium, which sets the premium so that the probability of a loss exceeds zero by at most a specified amount: P(L_0 > 0) <= alpha for some small alpha (say 0.05). This premium exceeds the equivalence principle premium because it includes an implicit risk loading. Solving P(v^{K+1} - P*a-ddot_{K+1} > 0) <= alpha is equivalent to finding the premium that ensures the policy is profitable with probability at least 1-alpha.

Common Pitfalls

Students sometimes confuse the net premium (based on the equivalence principle with no expense or profit provision) with the gross premium (which includes loadings). On Exam LTAM, "net premium" always means the equivalence principle premium unless otherwise stated. Another common error is using the wrong annuity in the premium formula: the premium annuity must match the premium payment pattern (number of payments, timing). If premiums are paid for h years, use a-ddot_{x:h}, not a-ddot_x.

When computing the variance of L_0, a frequent mistake is forgetting the (1 + P/d) factor or incorrectly computing it. Always derive this factor from the relationship between L_0 and v^{K+1} before plugging in numbers.

Study Tips

Exam LTAM tests premium calculation for various benefit and payment patterns. Memorize the key formulas: P_x = A_x/a-ddot_x = 1/a-ddot_x - d for whole life, and the corresponding formulas for term, endowment, and limited-payment products. Practice computing premiums from given insurance and annuity values. Understand the variance formula and be able to compute the probability of loss for a single policy or a portfolio. The simplified forms (using 1/a-ddot - d) are faster and less prone to error than computing A and a-ddot separately.

Net Premium Calculation for Life Insurance Products