Actuarial Outpost Fractional Age Assumptions
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#1
10-09-2007, 09:01 AM
 ndaka26 Member Join Date: May 2007 Posts: 302
Fractional Age Assumptions

How do we deal with a fractional age assumption question (udd, constant, hyperbolic) when s+t > 1? Plugging the values into given formulas does not work.
#2
10-09-2007, 09:05 AM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 7,250

Take care of the integral part, then use the fractional age assumption for the fractional part.

For example, express $_{2.2}q_x$ as ${}_2p_x\,{}_{0.2}q_{x+2}$

or express $_{0.7}q_{x+0.5}$ as $_{0.5}p_{x+0.5}\,{}_{0.2}q_{x+1}$
#3
10-09-2007, 09:27 AM
 ndaka26 Member Join Date: May 2007 Posts: 302

Quote:
 Originally Posted by Abraham Weishaus Take care of the integral part, then use the fractional age assumption for the fractional part. For example, express $_{2.2}q_x$ as ${}_2p_x\,{}_{0.2}q_{x+2}$ or express $_{0.7}q_{x+0.5}$ as $_{0.5}p_{x+0.5}\,{}_{0.2}q_{x+1}$
Thanks Professor for the response. But I still don't get why we use survival propabilities when we are trying to find the probability of death/failure the expression 2.2qx = 2Px * 0.2qx+2 implies that 2.2qx = 2/2.2qx?
#4
10-09-2007, 09:50 AM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,348

He was just giving you the formula for the final fractional piece. E.g., the complete expression for the second would be $_{0.7}q_{x+0.5}$ = $_{0.5}q_{x+0.5}$ + $_{0.5}p_{x+0.5}\,{}_{0.2}q_{x+1}$
#5
10-09-2007, 01:14 PM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,724

Once a probability takes you outside a single integer year of age, it's often simplest to find the equivalent set of lx values from your year-long probabilities, express your complicated probability in terms of lx values, and then interpolate as needed (linear on lx, on 1/lx, or on ln(lx)) to get those lx values.

Jim Daniel
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#6
10-09-2007, 01:54 PM
 ndaka26 Member Join Date: May 2007 Posts: 302

Quote:
 Originally Posted by Gandalf He was just giving you the formula for the final fractional piece. E.g., the complete expression for the second would be $_{0.7}q_{x+0.5}$ = $_{0.5}q_{x+0.5}$ + $_{0.5}p_{x+0.5}\,{}_{0.2}q_{x+1}$
Gandalf thanks, I get it
#7
10-09-2007, 02:04 PM
 ndaka26 Member Join Date: May 2007 Posts: 302

Quote:
 Originally Posted by Gandalf He was just giving you the formula for the final fractional piece. E.g., the complete expression for the second would be $_{0.7}q_{x+0.5}$ = $_{0.5}q_{x+0.5}$ + $_{0.5}p_{x+0.5}\,{}_{0.2}q_{x+1}$
would that be the same as qx + px * 0.2qx+1, though?
#8
10-09-2007, 02:31 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,348

No, because $_{0.5}q_{x+0.5}$ is not the same as qx (and similarly for the term you changed to px).
#9
01-15-2019, 10:03 PM
 urbjhawk SOA Join Date: Oct 2017 Studying for LTAM Posts: 14

Can anyone help me with what u_(x+t) would be if t>1? We have that u(x+t) = q_x/(1-t*q_x) for when t<1 under the UDD assumption, but I can't seem to find a solution for when t>1.
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#10
01-15-2019, 10:19 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,348

If you stop and think about it, you're asking a rather silly question. Fractional age assumptions are for the pattern of mortality between integral ages. So they tell you how to get (for example) mu_50.2 from q50. They wouldn't tell you how to get mu_51.2 (that's 50+1.2) from q50, because that's not in the year starting with age 50.

You could say mu_(50+1.2)=mu_(51.2)=mu_(51+.2) and then evaluate it under udd by the formula you already gave as q_51/(1-.2*q_51).