Actuarial Outpost Continuous annuity question
 Register Blogs Wiki FAQ Calendar Search Today's Posts Mark Forums Read
 FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions

 United KingdomActuarial Jobs General Insurance, Life & Other Areas Canada Actuarial JobsCasualty, Life,Pension,Health & Investment Entry LevelActuarial JobsAll Disciplines Health, Life, Pension, Casualty, Investment D.W. Simpson & Company ActuarialRecruitersWorldwide www.dwsimpson.com

#1
10-16-2006, 02:35 PM
 bip30 Join Date: Jul 2005 Posts: 14
Continuous annuity question

A 10-year continuous annuity has a payment rate at time t of:
Rt = 100 - .09t^2, 0<=t<=10.

If deltat = 3/(100+3t), t>= 0, find the present value of the annuity.

Thanks!
#2
10-16-2006, 03:11 PM
 no driver Note Contributor SOA Join Date: Jan 2006 Studying for ERM Posts: 2,073

The PV of an annuity with a continuously variable rate of payment and a continuously variable rate of interest is
$PV=\int_0^n f(t)e^{-\int_0^t \delta_r dr} dt$

So, we have
$PV=\int_0^{10} (100-.09t^2)e^{-\int_0^t \frac{3}{100+3r} dr} dt$

Solving the inner integral:

$u=100+3r$
$du=3dr$

$e^{-\int_{100}^{100+3t} \frac{1}{u} du}=e^{-ln \frac{100+3t}{100}=\frac{100}{100+3t}$

So now we have:

$\int_0^{10} (100-.09t^2)(\frac{100}{100+3t})dt=\int_0^{10} \frac{100000-9t^2}{100+3t}dt=\int_0^{10} \frac{(100-3t)(100+3t)}{100+3t}dt=\int_0^{10} 100-3t dt=850$
#3
10-16-2006, 03:19 PM
 MyKenk Note Contributor Join Date: Nov 2005 Location: http://mhenk.blogspot.com Studying for CAS 9/8/whatever? College: Drake '06 Posts: 8,519

Curse you and you're Tex'ing abilities No Driver!

One of these days... post Nov. 8th, I'm gonna take a look at that stuff...

alternately, you know delta(t) = 3/(100+t) = (3/100)/(1+t/100) = 3 * (.01/(1+.01t)). From that last form, you know that the accumulation function is a(t) = (1+.01t)^3, and that the discount function is (1+.01t)^1/3.

to discount to time zero, you need to divide all payments by the discount function at the time the payment is made, and sum them up (with an integral, over 0 to 10.)

Or, all i've managed to do is make a more complicated integral... well, i have it typed out, so pc=pc+1.

No Driver did it very well, ignore me, been a long day.
__________________

#4
10-16-2006, 03:30 PM
 no driver Note Contributor SOA Join Date: Jan 2006 Studying for ERM Posts: 2,073

Quote:
 Originally Posted by mykenk Curse you and you're Tex'ing abilities No Driver!
Once you get it typed up it's a lot of cut and paste, change a few things, cut and paste again. I started by copying the formula out of the list I posted in the sticky and it went pretty quickly after that. It took me two tries to work the problem because I didn't get the inner integral right the first time (didn't take the reciprocal of the fraction ) so I ended up spending more time on the problem than on answering the post.

When you decide to look into it more, http://www.forkosh.com/mimetex.html is your friend.
#5
10-17-2006, 09:01 AM
 bip30 Join Date: Jul 2005 Posts: 14
.

thank you, got it now.

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off

All times are GMT -4. The time now is 01:26 PM.

 -- Default Style - Fluid Width ---- Default Style - Fixed Width ---- Old Default Style ---- Easy on the eyes ---- Smooth Darkness ---- Chestnut ---- Apple-ish Style ---- If Apples were blue ---- If Apples were green ---- If Apples were purple ---- Halloween 2007 ---- B&W ---- Halloween ---- AO Christmas Theme ---- Turkey Day Theme ---- AO 2007 beta ---- 4th Of July Contact Us - Actuarial Outpost - Archive - Privacy Statement - Top