4.4 Premiums and reserves for life contingencies

Types of Life Insurance

Life insurance provides financial protection for beneficiaries upon the insured's death. Different policy types exist to meet varying needs around coverage duration, premium flexibility, and cash value accumulation.

Term vs. Permanent Life Insurance

Term life insurance covers a specified period (10, 20, or 30 years) and pays a death benefit only if the insured dies during that term. Premiums start low but typically increase at renewal.

Permanent life insurance provides lifelong coverage and includes a cash value component that grows tax-deferred. Premiums are higher than term but generally remain level throughout the policy's life.

Whole Life Insurance

Whole life is a permanent policy with guaranteed level premiums, a guaranteed death benefit, and guaranteed cash value growth at a fixed rate. A portion of each premium goes into the cash value account. Policyholders can borrow against the cash value or surrender the policy for its accumulated value, though surrender charges often apply in the early years.

Universal Life Insurance

Universal life (UL) offers flexibility within permanent coverage. Policyholders can adjust premiums and death benefits within certain limits. Cash value grows based on a declared interest rate set by the insurer, which may exceed the guaranteed minimum but can change over time. Some UL policies include a no-lapse guarantee rider that keeps coverage in force even if cash value is depleted, provided certain conditions are met.

Variable Life Insurance

Variable life combines permanent insurance with an investment component. Policyholders allocate premiums among sub-accounts (similar to mutual funds), and both the cash value and death benefit fluctuate with investment performance. The policyholder bears the investment risk. Some policies offer a guaranteed minimum death benefit or optional riders to cushion against market downturns.

Endowment Insurance

Endowment insurance pays a lump sum either to the policyholder if they survive the policy term or to beneficiaries if the insured dies during the term. Premiums are higher than other policy types because they're designed to accumulate a target amount by the end of the endowment period (e.g., 10, 20, or 30 years). These policies are sometimes used as savings vehicles for goals like retirement or education funding, with life insurance protection built in.

Assumptions for Pricing Life Insurance

Pricing life insurance requires assumptions about future mortality, interest rates, and expenses. These assumptions draw on historical data, industry trends, and the insurer's own experience, and they're reviewed and updated regularly. Conservative assumptions help ensure the insurer can meet future obligations.

Mortality Assumptions

Mortality assumptions estimate the expected number and timing of deaths among the insured population, based on factors like age, gender, health status, and smoking habits. Insurers rely on mortality tables that show the probability of death at each age, developed from past experience and adjusted for anticipated improvements in life expectancy. More stringent underwriting leads to lower assumed mortality rates for the insured group.

Interest Rate Assumptions

Interest rate assumptions estimate the expected return on the insurer's invested assets over the life of its policies. Insurers typically invest premiums in diversified portfolios of fixed-income securities (bonds, mortgages) to generate income for future claims and expenses. This assumption is a major lever in pricing: higher assumed rates produce lower premiums and reserves, while lower assumed rates produce higher premiums and reserves.

Expense Assumptions

Expense assumptions estimate the costs of acquiring, administering, and servicing policies over their lifetime. These include:

• Commissions and underwriting costs
• Policy issuance and maintenance
• Claims processing
• Overhead expenses

Expense assumptions are based on the insurer's actual experience, industry benchmarks, and anticipated trends. They're allocated to each policy based on factors such as age, premium size, and policy type.

Life Tables and Mortality Laws

Life tables and mortality laws are foundational tools for pricing and valuing life insurance. Life tables summarize the probability of death and survival at each age for a specific population. Mortality laws are mathematical models that describe how mortality rates change with age, enabling construction of life tables and projection of future mortality trends.

Select vs. Ultimate Life Tables

Select life tables reflect the mortality experience of newly underwritten individuals. Because these people recently passed health screening, their mortality rates are initially lower than the general insured population. This is called the selection effect.

Ultimate life tables reflect mortality after the selection effect has worn off, typically 15–25 years after policy issue. In practice, actuaries use select tables for the initial policy years and transition to ultimate tables for later durations.

Makeham's Law

Makeham's law models the force of mortality (the instantaneous rate of death) as a function of age:

$\mu(x) = A + Bc^x$

• $A$ is a constant representing age-independent mortality (accidents, etc.)
• $Bc^x$ is the exponential term representing age-dependent mortality that increases with age

This law is particularly useful for modeling mortality at older ages, where the exponential component dominates.

Gompertz's Law

Gompertz's law is a special case of Makeham's law where $A = 0$:

$\mu(x) = Bc^x$

The key observation behind this law is that the logarithm of mortality rates increases roughly linearly with age for most adults. Empirically, mortality approximately doubles every 8 years at adult ages. Gompertz's law is widely used due to its simplicity and good fit to data, particularly for ages 30–90.

Premiums for Life Insurance

Premiums are the payments policyholders make in exchange for insurance coverage. Actuaries determine premiums using the equivalence principle: at policy inception, the expected present value of premiums paid must equal the expected present value of benefits provided (plus expenses, in the case of gross premiums).

Net Single Premium

The net single premium (NSP) is the present value of future benefits, calculated using assumed mortality and interest rates only (no expenses or profit). For a whole life insurance policy on a life aged $x$ with benefit amount 1:

$A_x = \sum_{k=0}^{\infty} v^{k+1} \cdot {}_kp_x \cdot q_{x+k}$

where $v = \frac{1}{1+i}$ is the discount factor, ${}_kp_x$ is the probability of surviving $k$ years, and $q_{x+k}$ is the probability of dying in year $k+1$.

The NSP is a theoretical building block used to derive other premium types and reserves.

Net Level Annual Premium

The net level annual premium is a constant annual payment whose present value equals the net single premium. You calculate it by dividing the NSP by the appropriate life annuity-due factor:

$P_x = \frac{A_x}{\ddot{a}_x}$

where $\ddot{a}_x$ is the present value of a life annuity-due of 1 per year payable while $(x)$ survives. Like the NSP, this is a theoretical quantity that serves as the foundation for gross premium and reserve calculations.

Gross Premium

The gross premium is what the policyholder actually pays. It equals the net premium plus loadings for:

• Expenses: acquisition costs, maintenance, and claims settlement
• Profit margin: a buffer for adverse deviations and a return on capital

Gross premiums may be expressed as a percentage of the net premium or as a flat amount per policy, and they vary by age, gender, policy type, and other rating factors.

Premium Payment Frequency

Premiums can be paid annually, semi-annually, quarterly, or monthly. More frequent payments result in slightly higher total annual premiums because of the time value of money (the insurer receives funds later on average) and additional administrative costs. Payment frequency also affects cash value growth and the policyholder's ability to maintain coverage if payments are missed.

Reserves for Life Insurance

Reserves are the funds an insurer sets aside to meet future policy obligations. The basic formula is:

$\text{Reserve} = \text{PV(future benefits)} - \text{PV(future net premiums)}$

Adequate reserves are essential for solvency and are closely regulated by insurance authorities.

Net Premium Reserves

Net premium reserves use the net level annual premium (no expense or profit loadings) in their calculation. For a whole life policy at duration $t$:

${}_tV_x = A_{x+t} - P_x \cdot \ddot{a}_{x+t}$

This represents the theoretical amount needed to cover future benefits, assuming net premiums continue to be received and invested at the valuation interest rate. Net premium reserves are a conservative liability measure and form the basis for regulatory reporting.

Gross Premium Reserves

Gross premium reserves use the actual gross premiums charged to policyholders. They provide a more realistic picture of the insurer's liabilities because they account for expected future expenses and profit margins. These reserves are primarily used for internal purposes: pricing analysis, profitability assessment, and cash flow projections.

Prospective vs. Retrospective Reserves

These are two different ways to compute the same quantity:

• Prospective method: Looks forward from the valuation date. Reserve = PV(future benefits) − PV(future net premiums).
• Retrospective method: Looks backward from the valuation date. Reserve = Accumulated value of past net premiums − Accumulated value of past benefits paid.

Under consistent assumptions (level net premium, no changes in mortality or interest), the two methods produce identical results. In practice, they may differ if actual experience deviates from assumptions.

Modified Reserves

Modified reserves allow a more gradual reserve build-up in early policy years. The idea is straightforward: use a modified net premium that is lower than the true net level premium in the first year (or first few years) and correspondingly higher in later years.

This reduces the initial reserve strain caused by high first-year acquisition costs. The Full Preliminary Term (FPT) method is a common approach, where the first-year net premium is set equal to the cost of one-year term insurance, with higher premiums in subsequent years to compensate.

Reserves for Varying Insurance Amounts

Some policies have benefits that change over time (increasing term, decreasing term, paid-up additions, dividend accumulations). Reserves for these policies must account for the changing benefit amounts and corresponding premium adjustments. Actuaries use specialized techniques such as the Z-factor or K-factor methods to handle these calculations.

Valuation of Life Insurance Liabilities

Valuation determines the present value of an insurer's policy liabilities at a specific point in time. It requires assumptions about future mortality, interest rates, expenses, lapses, and other factors affecting cash flows. Valuation is central to financial reporting, solvency assessment, and risk management.

Regulation of Life Insurance Reserves

Insurance regulators set minimum reserve standards to protect policyholders and maintain industry stability. In the U.S., the National Association of Insurance Commissioners (NAIC) develops model laws and regulations, including the Standard Valuation Law and the Valuation Manual. These prescribe valuation methods, assumptions, and documentation requirements. Insurers must hold reserves at least as high as those calculated under the prescribed standards.

Valuation Methods

The most common approaches include:

• Net level premium method: PV(future benefits + expenses) − PV(future net premiums). This is the standard approach.
• Gross premium method: Uses actual gross premiums instead of net premiums.
• Full preliminary term (FPT) method: Uses a reduced first-year net premium to lower initial reserve strain.

Product-specific methods also exist, such as the Commissioners Reserve Valuation Method (CRVM) for universal life and the Commissioners Annuity Reserve Valuation Method (CARVM) for deferred annuities.

Valuation Assumptions

The key inputs for reserve calculations are:

• Mortality rates: Based on prescribed tables (e.g., the 2017 CSO table), possibly adjusted for company experience or anticipated improvements
• Interest rates: Based on prescribed formulas reflecting the insurer's portfolio yield, subject to regulatory maximums and minimums
• Lapse rates: Based on company experience studies or industry benchmarks
• Expense assumptions: Based on actual company costs with margins for adverse deviations

Valuation Adjustments

Standard methods don't capture every risk. Additional reserve amounts may be required for:

• Asset adequacy reserves: Cover the risk of asset defaults or market value fluctuations
• Deficiency reserves: Cover the risk of adverse mortality or lapse experience (required when the valuation net premium exceeds the gross premium)
• Deferred acquisition cost (DAC) reserves: Account for the amortization of acquisition expenses

These adjustments are typically determined through asset adequacy testing or cash flow testing, which project future cash flows under multiple scenarios to assess reserve adequacy.

Valuation Reports

The insurer's appointed actuary prepares a formal valuation report summarizing the results. This report typically includes:

• Description of valuation methods, assumptions, and adjustments
• Reconciliation of reserves from the prior valuation date
• Analysis of changes in reserves
• The actuary's opinion on reserve adequacy

Valuation reports are submitted to regulators as part of annual financial statements and are also used internally for management reporting.

Profit Testing for Life Insurance

Profit testing analyzes the expected profitability of a life insurance product over its lifetime. It projects revenues, benefits, expenses, and investment returns to assess whether a product meets the insurer's financial targets. This process involves building a detailed financial model and running projections under various scenarios.

Profit Measures

Several metrics are used to evaluate product profitability:

• Present Value of Future Profits (PVFP): The discounted value of projected after-tax profits over the product's lifetime
• Internal Rate of Return (IRR): The discount rate that makes PVFP equal to zero. A higher IRR indicates better profitability relative to capital invested.
• Return on Equity (ROE): The ratio of after-tax profits to allocated capital
• Break-even year: The first year in which cumulative profits turn positive
• Profit margin: Profit as a percentage of premiums

Each measure offers a different lens on the timing and magnitude of profitability.

Pricing for Profitability

Profit testing is iterative. The actuary sets assumptions for mortality, interest, expenses, and lapses based on company experience and market conditions, then calculates the premiums that achieve target profit measures. This process also considers:

• Competitive pricing in the market
• Regulatory constraints
• Consumer preferences and price sensitivity
• The balance between attractive pricing and adequate risk margins

Sensitivity Analysis

Sensitivity analysis tests how changes in key assumptions affect profitability. The standard approach is to vary one assumption at a time (e.g., increase mortality by 10%, decrease interest rates by 50 basis points) and measure the impact on profit measures.

This identifies which assumptions have the greatest influence on results and informs decisions about where to build in margins. For example, if a product's profitability is highly sensitive to lapse rates, the insurer might add a larger lapse margin or design policy features that encourage persistency.

Profit Margin Targets

Profit margin targets are the minimum profitability levels an insurer requires for its products, typically expressed as a percentage of premiums or reserves. These targets reflect the insurer's risk appetite, capital requirements, and strategic goals, and they may vary by product type, distribution channel, and market segment. Ongoing monitoring of actual experience against assumptions is necessary to ensure targets are met over the long term.

Reinsurance of Life Insurance

Reinsurance transfers a portion of an insurer's risk to another insurer (the reinsurer) in exchange for a share of premiums. It helps insurers manage risk exposure, stabilize financial results, and increase capacity to write new business.

Types of Reinsurance

The two main categories are:

• Proportional (quota share) reinsurance: The insurer and reinsurer share premiums and claims in a fixed proportion (e.g., 60/40). This is straightforward and provides broad risk sharing.
• Non-proportional (excess of loss) reinsurance: The reinsurer pays only for claims above a specified retention limit. The insurer retains all risk below that threshold.

Other arrangements include:

• Surplus reinsurance: A form of proportional reinsurance where the reinsurer's share varies based on the size of each individual policy relative to the insurer's retention
• Stop-loss reinsurance: The reinsurer covers aggregate losses that exceed a specified threshold over a defined period, protecting against unusually adverse overall experience