Fractional Age Assumptions: UDD, Constant Force, Balducci

Uniform Distribution of Deaths (UDD)

UDD assumes deaths occur uniformly within each year of age. For 0 <= s < 1: s_q_x = s * q_x, and the survival function is linear within each year. Under UDD, the force of mortality within the year is mu_{x+s} = q_x / (1 minus s*q_x), which increases within the year. The key relationship connecting continuous and discrete insurance values under UDD is A-bar_x = (i/delta) * A_x. Similarly, a-bar_x = (i/delta) * (a-ddot_x minus 1/2), approximately. UDD is the most commonly used and most commonly tested fractional age assumption.

Constant Force and Balducci

The constant force assumption sets mu_{x+s} = mu constant for 0 <= s < 1, giving s_p_x = (p_x)^s. This produces a geometric survival curve within each year. The Balducci (hyperbolic) assumption sets 1/(1-s_q_{x+s}) linear in s, giving s_q_x = s*q_x / (1 minus (1-s)*q_x). Under Balducci, the force of mortality decreases within the year, which is generally unrealistic. Exam LTAM tests all three assumptions, requiring you to compute fractional-age survival probabilities, forces of mortality, and their impact on insurance and annuity calculations.