  Actuarial Outpost Annuity Questions [pt 12]
 User Name Remember Me? Password
 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

 Financial Mathematics Old FM Forum

 Thread Tools Search this Thread Display Modes
#1
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128 Annuity Questions [pt 12] I get the geometric series approach to this question but I'm wondering if there is a way to solve this problem using the geometric annuity formula. When i=k, the denominator is 0 so it renders the equation unsolvable. This question as well. The way other people solve this problem is also by manipulating the geometric series. But I want to use the formula.

But I got the answer wrong so I assume this isn't the geometric perpetuity PV formula. Again. People used the geometric series method. This chapter in the ASM manual doesn't give any explanation, examples, or formulas for cases when i=k, geometric perpetuity, and this time, accumulated value. What is the FV formula for geometric annuities?
#2 Breadmaker Member SOA  Join Date: May 2009 Studying for CPD - and nuttin' else! College: Swigmore U Favorite beer: Guinness Posts: 4,743 #1. (1 + k)^t / (1+i)^t simplifies to what if i=k?
__________________
"I'm tryin' to think, but nuthin' happens!"
#3
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128 Quote:
 Originally Posted by Breadmaker #1. (1 + k)^t / (1+i)^t simplifies to what if i=k?
1,
That makes the numerator 1-1 = 0.
The denominator is still i-k = 0.
Not sure what you're getting at.
#4 NattyMo Member Non-Actuary Join Date: Nov 2016 Posts: 606 Quote:
 Originally Posted by Futon 1, That makes the numerator 1-1 = 0. The denominator is still i-k = 0. Not sure what you're getting at.
Forget the formula when i=k. First principles...

Simplify (1+k)^t / (1+i)^t
__________________
Favorite Quote(s):

Spoiler:

Progress isn't made by early risers. It's made by lazy men trying to find easier ways to do something.

Don't handicap your children by making their lives easy.

There are no dangerous weapons; there are only dangerous men.

A desire not to butt into other people's business is at least eighty percent of all human 'wisdom'...and the other twenty percent isn't very important.

Belief gets in the way of learning.

The first principle of freedom is the right to go to hell in your own handbasket.

Does history record any case in which the majority was right?

#5
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128 Quote:
 Originally Posted by NattyMo Forget the formula when i=k. First principles... Simplify (1+k)^t / (1+i)^t
Simplify assuming k=i? It's just one.
#6
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,880 When i= k, you can simply sum them to determine present value,.if the first payment is at t=0. If the first payment is at t= 1, you can sum them from t = 1, and then discount the sum from time 1.
#7
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128 Quote:
 Originally Posted by Academic Actuary When i= k, you can simply sum them to determine present value,.if the first payment is at t=0. If the first payment is at t= 1, you can sum them from t = 1, and then discount the sum from time 1.
Oh I see. Thanks. What about cases where i does not equal k and the question asks for accumulated value? How do I approach this?
#8
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,880 Quote:
 Originally Posted by Futon Oh I see. Thanks. What about cases where i does not equal k and the question asks for accumulated value? How do I approach this?
To me the easiest way is to determine the present value and then accumulate at the interest rate. For present value you can either sum the geometric series, or treat the problem as interest only with a rate i' = (i-k)/(1+k) and use the calculator function.

If the first payment is at time 1, the easiest for me is to multiply the series by (1+k)/(1+k). The PV will then be the calculator PV/(1+k).
#9
 Futon Member SOA Join Date: Jul 2016 Studying for FM Posts: 128 Quote:
 Originally Posted by Academic Actuary To me the easiest way is to determine the present value and then accumulate at the interest rate. For present value you can either sum the geometric series, or treat the problem as interest only with a rate i' = (i-k)/(1+k) and use the calculator function. If the first payment is at time 1, the easiest for me is to multiply the series by (1+k)/(1+k). The PV will then be the calculator PV/(1+k).
Oh I see. I guess from what everyone is saying, the geometric annuity immediate formula is not viable for most of the questions.

"If the first payment is at time 1, the easiest for me is to multiply the series by (1+k)/(1+k). "

You mean (1+i)/(1+k)?

Edit: When you say accumulate at the interest rate, does that mean multiplying PV by (1+i)^n?

Last edited by Futon; 07-27-2017 at 03:14 PM..
#10
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,880 Quote:
 Originally Posted by Futon Oh I see. I guess from what everyone is saying, the geometric annuity immediate formula is not viable for most of the questions. "If the first payment is at time 1, the easiest for me is to multiply the series by (1+k)/(1+k). " You mean (1+i)/(1+k)? Edit: When you say accumulate at the interest rate, does that mean multiplying PV by (1+i)^n?
No the present value of the first payment is 1/(1+i)^-1, and the second (1+k)/(1+i)^-2. You want to match up exponents to use an adjusted interest rate.

Basically a angle n @ i and k has the same value as a angle n @ i' /(1+k)

 Thread Tools Search this Thread Show Printable Version Email this Page Search this Thread: Advanced Search Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded 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 06:46 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

Powered by vBulletin®
Copyright ©2000 - 2019, Jelsoft Enterprises Ltd.
*PLEASE NOTE: Posts are not checked for accuracy, and do not
represent the views of the Actuarial Outpost or its sponsors.
Page generated in 0.20146 seconds with 9 queries