Balducci assumption proof

[Deepesh]

How to prove the following result if Balducci assumption holds?
$$P(T_x > t) = \frac{ 1 - q_x}{1 - (1 - t)q_x}$$

 

[KaustavSen]

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:
$$P(T_x > t) = \frac{ 1 - q_x}{1 - (1 - t)q_x}$$