How to prove the following result if Balducci assumption holds? $$P(T_x > t) = \frac{ 1 - q_x}{1 - (1 - t)q_x}$$
Hello Deepesh, You can prove the result as follows: $$\begin{align*}L.H.S &= \mathbb{P}[T_x > t] = {}_{t}p_{x} \\&= \frac{p_x}{{}_{1-t}p_{x+t}}\end{align*}$$ The first equality comes from the fact that, \(T_x > t\) essentially means that a person currently aged \(x\) will survive for at least \(t\) more years. Now, assuming that \(0<t<1\) we have, $$p_x = {}_{t}p_x\times{}_{1-t}p_{x+t}$$ thus giving us the second expression. Finally, using the Balducci Assumption, ie, $$(1-t)q_x = {}_{1-t}q_{x+t}$$ we get: $$\begin{align}\frac{p_x}{{}_{1-t}p_{x+t}}&= \frac{1-q_x}{1-{}_{1-t}q_{x+t}}\\\\&=\frac{1-q_x}{1-(1-t)q_{x}}=R.H.S\end{align}$$ Hence, we proved: $$P(T_x > t) = \frac{ 1 - q_x}{1 - (1 - t)q_x}$$
