Thiele's Differential Equation for Continuous Reserves
Derive and apply Thiele's differential equation for continuous reserve calculations on Exam LTAM.
Deriving Thiele's Equation
Thiele's differential equation describes how the continuous reserve changes over time. For a fully continuous whole life insurance with premium rate P-bar and benefit b, the equation is d/dt (t_V) = delta * t_V + P-bar minus (b minus t_V) * mu_{x+t}. This states that the rate of reserve change equals interest earned on the reserve plus premium income minus the net amount at risk times the force of mortality. The net amount at risk (b minus t_V) represents the insurer's exposure at time t.
Numerical Solution
Thiele's equation is solved numerically using Euler's method. Working backward from the terminal condition (typically omega_V = 0 for whole life or n_V = b for endowment), each time step gives t_V from t+h_V. The approximation is t_V approximately equals t+h_V minus h*(delta * t+h_V + P-bar minus (b minus t+h_V) * mu_{x+t}). Exam LTAM tests your ability to set up Thiele's equation for various products and to execute a few steps of the numerical solution to find reserves at specific durations.