• We are pleased to announce that the winner of our Feedback Prize Draw for the Winter 2024-25 session and winning £150 of gift vouchers is Zhao Liang Tay. Congratulations to Zhao Liang. If you fancy winning £150 worth of gift vouchers (from a major UK store) for the Summer 2025 exam sitting for just a few minutes of your time throughout the session, please see our website at https://www.acted.co.uk/further-info.html?pat=feedback#feedback-prize for more information on how you can make sure your name is included in the draw at the end of the session.
  • Please be advised that the SP1, SP5 and SP7 X1 deadline is the 14th July and not the 17th June as first stated. Please accept out apologies for any confusion caused.

Interpretation of differential equation for t_p_x

A

actuary-to-be

Member
I am trying to interpret what the attached differential equation for t_p_x means.

Screen Shot 2017-04-23 at 11.41.48.png

In this case (when differentiating with respect to t), we are interested in how the probability of survival (from age x to age x+t) changes with respect to time t. This will always be negative because we ultimately want the life to stay alive over time t, hence we need to multiply by the force of mortality of staying alive at time x+t (which is the negative mu term).

In the notes, it states that the probability of surviving for age x to age x+t is a decreasing function of t. And that is why we have the minus sign at the front. What does this mean exactly?

Thank you
 
"tpx is a decreasing function of t" means its graph would be downward-sloping with respect to t. Equivalently, it means the derivative of tpx with respect to t is negative.

tpx is a decreasing function of time, since for a fixed aged x, the probability of surviving for t years decreases, as t increases.
 
Back
Top