Percentile

[Anonymous]

Hi All!

I am getting confused with this percentile stuff. Which way do I calculate it?

Lets say - the 80th percentile -
a) .2 = tPx
OR
b) .8 = tPx

:???:

I have seen similar questions done both ways - or maybe I am reading this wrong? But how do I know what way to use?

THANKS!

[Daisy]

80th percentile means only 20% of the population is "better" or "survives" therefore 80% must be dead by time t - so .20=tPx - hope that helps

[Anonymous]

Yes, that does help - thanks Daisy!

Can I ask another question??? I am not too sure how to solve the following question.

Anyone wanna help?

A Whole life policy is issued to (30) with Z=10v^T 0<= T. If mu = 0.05 and the 80th percentile of Z is 5.12, find the APV of the benefit. The force of interest is constant. (ANS 2.5)

SO - where do I start? If I say .2 = tPx
and then Constant force gives me
0.2=e^(-mu*t)

NOW what? I don't understand since I am given mu and what the 80th percentile is? So, I want to find delta but how?

Thanks!

[rabbit]

Since Z is biggest if the insured dies sooner (less time to discount the benefit payment), the 80th percentile of Z will actually occur at the 20th percentile of `time until death'. So you want to begin by solving 0.8 = tpx = e^(-mu * t). This gives
t = 4.4629.

Knowing the 80th percentile of Z is 5.12 tells us that 5.12 = 10*e^(-delta * t) which solves to give delta = 0.15.

Now, for constant force, the APV for a whole life policy with benefit of 1 is
(mu)/(mu + delta) = (0.05)/(0.2) = 0.25. So the benefit you are after is 10 * this amount, which is 2.5.

[Anonymous]

Okay, but I don't like the first bit now! I follow through after but I don't see how you know to use .8 = tPx?

That was my initial problem - and found that for the 80th percentile, I want to say .2 = tPx not .8 = tPx.

So, I like everything but the first step. I don't know how I would know to do it that way? and how do you know it will occur at the 20th percentile?

I do appreciate your help.... THANKS and sorry I am a pain!

[rabbit]

The way I remember it is that the sooner someone dies, the worse it is for the insurance company. Remember that the insurance company wants a small Z, since that is the PV of the death benefit. But a Z at the 80th percentile is a `big' Z (bigger than 80%, anyway).

Z is big when T is small. (Z is biggest if the insured dies right away.) So the 80th percentile of Z is at the 20th percentile for T.

It is a little confusing, and to be honest, my first attempt started with 0.2 = tpx also!

Hope this helps ....

[Anonymous]

Thanks - that makes a bit more sense anyways. I will have to go through it a few times to get my head around it.

I think I need to keep my T's and Z's straight.

Thanks for your help

[phdmom]

I think a picture helps. Z is a decreasing function of t (since v<1). So if you draw a horizontal line at approximately the 80th percentile of Z, you see it intersects your graph at the 20th percentile of t.

Can you tell my math background is much stronger than my insurance background?! I can never remember which way is better/worse for the company...I just know it's an exponential function with base <1!