Selection Effects

Select mortality tables recognize that recently underwritten lives have lower mortality than the general population at the same age. When an applicant passes medical underwriting, the insurer knows this person is currently in good health. This "selection" effect means newly underwritten lives have lower mortality than the general population of the same age, which includes both healthy and unhealthy individuals. The select period (typically 2 to 15 years) is the duration after underwriting during which mortality rates remain noticeably lower than ultimate rates.

During the select period, mortality depends on both current age and time since selection: q_[x]+t for t = 0, 1, ..., s-1, where s is the select period length. As time passes, the selection effect "wears off" because the underwriting information becomes stale. People who were healthy at underwriting may develop health conditions over time. After the select period, mortality rates merge with the ultimate table: q_[x]+s = q_{x+s}, meaning mortality depends only on attained age.

Structure of Select and Ultimate Tables

A select and ultimate table is organized with rows representing issue age x and columns representing duration since selection t. For a select period of length 2, the table might show q_[x], q_[x]+1, and then q_{x+2} (the ultimate rate). The key feature is that two people of the same attained age but different selection ages have different mortality rates during the select period. For instance, q_[30]+1 (selected at 30, now age 31) differs from q_[31] (just selected at 31), even though both apply to someone currently age 31.

The table typically shows that q_[x] < q_[x-1]+1 < q_{x}, reflecting decreasing selection effect. A person selected at age 30 has the lowest mortality at age 30, a person selected at age 29 and now age 30 has somewhat higher mortality (one year of wear-off), and a person at ultimate rates for age 30 has the highest mortality among these three groups.

Computing Survival Probabilities

To compute survival probabilities from a select table, start with the radix l_[x] and apply mortality rates sequentially. The number alive at duration t is: l_[x]+1 = l_[x] * (1 - q_[x]), then l_[x]+2 = l_[x]+1 * (1 - q_[x]+1), and so on until the select period ends, after which ultimate rates are used. The probability of surviving n years from selection at age x is: n_p_[x] = l_[x]+n / l_[x].

For example, with a 2-year select period, the probability that a life selected at age 40 survives to age 45 is: 5_p_[40] = p_[40] * p_[40]+1 * p_42 * p_43 * p_44, where the first two factors use select rates and the last three use ultimate rates. This calculation requires carefully tracking when the select period ends and switching to ultimate mortality.

Insurance and Annuity Values with Selection

Insurance and annuity values for newly issued policies use select mortality for the appropriate duration. The whole life insurance APV for a life selected at age x is: A_[x] = sum over k from 0 to infinity of v^(k+1) * k_p_[x] * q_[x]+k, where the first s mortality rates are select and the rest are ultimate. Similarly, the whole life annuity-due is: a-ddot_[x] = sum over k from 0 to infinity of v^k * k_p_[x].

The practical impact is that A_[x] < A_x because select lives have lower mortality and therefore lower insurance costs. Conversely, a-ddot_[x] > a-ddot_x because select lives are expected to survive longer, increasing the present value of annuity payments. The net premium P_[x] = A_[x] / a-ddot_[x] is lower than P_x, reflecting the lower mortality risk for recently selected lives.

Reserves with Select Mortality

Reserves for policies issued under selection use select mortality for the remaining select period. The net premium reserve at duration t for a policy issued to [x] is: tV_[x] = A_[x]+t - P_[x] * a-ddot_[x]+t. When t < s (still in the select period), A_[x]+t and a-ddot_[x]+t use remaining select rates followed by ultimate rates. When t >= s, the values use only ultimate rates and the reserve simplifies to tV_[x] = A_{x+t} - P_[x] * a-ddot_{x+t}.

An important consequence is that reserves increase faster once the select period ends because ultimate mortality rates are higher than select rates. The reserve must grow to fund the higher expected claims that emerge when the selection effect wears off.

Common Pitfalls

A frequent mistake is confusing the notation. The bracket notation [x] indicates the age at selection, not the current age. The expression q_[30]+5 means the mortality rate for someone selected at age 30 who is now age 35, with 5 years elapsed since selection. Another common error is failing to switch from select to ultimate rates at the correct duration. If the select period is 2 years, rates for duration 2 and beyond use the ultimate table.

Students also sometimes overlook that different selection ages produce different mortality at the same attained age. When a problem involves multiple policies with different issue dates, each policy has its own position in the select period, and you must track these separately.

Study Tips

Exam LTAM frequently presents select and ultimate tables and asks you to compute survival probabilities, insurance values, or reserves. Practice reading and extracting values from these tables quickly. Make sure you can identify when the select period ends for a given policy and switch to ultimate rates at the right point. Work through problems that compare insurance values computed with select versus ultimate mortality to build intuition for the magnitude of the selection effect. Be comfortable with the bracket notation and its meaning in every context.

Select and Ultimate Mortality Tables for Actuaries