Why Fractional Age Assumptions Are Needed

Life tables provide mortality rates at integer ages, but many actuarial calculations require values at non-integer ages or durations. For example, computing a continuous insurance value A-bar_x from a discrete life table, or finding the probability of surviving 2.5 years, requires an assumption about how deaths are distributed within each year of age. The three standard fractional age assumptions each interpolate differently between the integer-age values provided by the life table.

Each assumption produces different survival probabilities, forces of mortality, and insurance values at fractional ages. The choice of assumption affects computed values, especially at older ages where mortality is high. Understanding all three assumptions and their properties is essential for Exam LTAM.

Uniform Distribution of Deaths (UDD)

UDD assumes deaths occur uniformly within each year of age. For 0 <= s < 1, the fractional-year mortality probability is: s_q_x = s * q_x. The survival function is linear between integer ages: s_p_x = 1 - s * q_x. Under UDD, the force of mortality within the year is mu_{x+s} = q_x / (1 - s * q_x), which increases within the year. This increasing force is reasonable because the population becomes riskier as healthier individuals survive through the year.

The most important UDD relationship connects continuous and discrete insurance values: A-bar_x = (i/delta) * A_x, where i is the effective annual interest rate and delta = ln(1+i) is the force of interest. This formula allows you to compute continuous insurance values directly from discrete values in the life table. Similarly, for annuities: a-bar_x = (id/delta^2) * a-ddot_x - (i - delta)/delta^2. A simpler approximation is a-bar_x approximately equals a-ddot_x - 1/2. For mthly insurance under UDD: A_x^(m) = (i/i^(m)) * A_x.

Constant Force of Mortality

The constant force assumption sets mu_{x+s} = mu = -ln(p_x) for 0 <= s < 1. This produces an exponential survival curve within each year: s_p_x = (p_x)^s = e^(-mu*s). The fractional-year mortality probability is s_q_x = 1 - (p_x)^s. Under this assumption, the force of mortality is flat within each year, though it may jump at integer ages.

The constant force assumption is mathematically convenient because the survival function takes an exponential form. For computing insurance values, the continuous whole life insurance under constant force uses A-bar_x = sum over k of v^(k+1) * k_p_x * q_{x+k} * delta/(delta + mu_{x+k}), where each year's contribution adjusts for the continuous timing of death within that year using the ratio delta/(delta + mu). This assumption is less commonly tested than UDD but appears in problems that specifically require constant force calculations.

Balducci (Hyperbolic) Assumption

The Balducci assumption, also called the hyperbolic assumption, sets 1/(1 - s * q_{x+s}) to be linear in s. Equivalently, the assumption can be stated as: (1-s)_q_{x+s} = (1-s) * q_x, which means the probability of dying in the remaining fraction of the year is proportional to the remaining time. This gives s_q_x = s * q_x / (1 - (1-s) * q_x).

Under Balducci, the force of mortality is mu_{x+s} = q_x / (1 - (1-s) * q_x), which decreases within each year of age. This decreasing force is generally considered unrealistic because it implies that the instantaneous risk of death drops as the year progresses. For this reason, the Balducci assumption is rarely used in practice but is included on Exam LTAM for completeness and for problems that require comparison across all three assumptions.

Comparing the Three Assumptions

For a given q_x, the three assumptions produce different fractional-age probabilities. At s = 0 and s = 1, all three agree (they match the integer-age values). For 0 < s < 1, the Balducci assumption gives the highest mortality probability s_q_x, UDD gives an intermediate value, and constant force gives the lowest. Equivalently, for the survival probability: constant force gives the highest s_p_x, UDD is in the middle, and Balducci gives the lowest.

The forces of mortality behave differently within the year. Under UDD, mu increases. Under constant force, mu is flat. Under Balducci, mu decreases. These different behaviors affect the timing of expected death payments within the year, which in turn affects continuous insurance values. The practical differences between the assumptions become more significant at older ages where q_x is large.

Worked Example

Suppose q_70 = 0.04 and you need to find 0.5_p_70 under each assumption. Under UDD: 0.5_p_70 = 1 - 0.5 * 0.04 = 0.98. Under constant force: 0.5_p_70 = (0.96)^0.5 = 0.97980. Under Balducci: 0.5_q_70 = 0.5 * 0.04 / (1 - 0.5 * 0.04) = 0.02 / 0.98 = 0.020408, so 0.5_p_70 = 0.97959. As expected, constant force gives the highest survival probability and Balducci the lowest.

For the force of mortality at age 70.5: Under UDD, mu_{70.5} = 0.04 / (1 - 0.5 * 0.04) = 0.04 / 0.98 = 0.04082. Under constant force, mu_{70.5} = -ln(0.96) = 0.04082 (coincidentally similar at this q_x level). Under Balducci, mu_{70.5} = 0.04 / (1 - 0.5 * 0.04) = 0.04082. At this particular midpoint and q_x level the values are very close, but they diverge more at higher mortality rates and at points other than the midpoint.

Common Pitfalls

A frequent mistake is applying the UDD continuous-to-discrete relationship A-bar_x = (i/delta) * A_x when a different fractional age assumption is specified. This formula is specific to UDD and does not hold under constant force or Balducci. Another common error is computing (1-s)_q_{x+s} when the problem asks for s_q_x, or vice versa. Be careful to identify exactly which probability the problem requests.

Students sometimes confuse the force of mortality behavior (increasing, constant, or decreasing) across the three assumptions. A helpful mnemonic: UDD gives Uniform deaths, so force Increases (to maintain uniform deaths from a shrinking population). Constant force is self-explanatory. Balducci gives decreasing force, which is counterintuitive and unrealistic.

Study Tips

Exam LTAM tests all three assumptions and may ask you to compare results across them. Memorize the key formulas for s_q_x, s_p_x, and mu_{x+s} under each assumption. Know the UDD relationship A-bar_x = (i/delta) * A_x and be prepared to use it in insurance and annuity calculations. Practice converting between the different fractional-age formulas and work problems that require applying each assumption to the same underlying life table data.

Fractional Age Assumptions: UDD, Constant Force, Balducci