Actuarial Outpost Annuity Questions [pt 12]
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#1
07-24-2017, 02:30 PM
 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.
\begin{align}
PV &= \frac{1-(\frac{1+k}{1+i})^{n}}{i-k}

i&>k

\textrm {My theory:}
\textrm {if} n &\rightarrow \infty

\textrm {then } \frac{1+k}{1+i} &\rightarrow 0

\textrm {since } 1+i &> 1+k

\textrm {thus, geometric perpetuity PV } &= \frac{1}{i-k}
\end{align}

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
07-24-2017, 06:33 PM
 Breadmaker Member SOA Join Date: May 2009 Studying for CPD - and nuttin' else! College: Swigmore U Favorite beer: Guinness Posts: 3,366

#1. (1 + k)^t / (1+i)^t simplifies to what if i=k?
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#3
07-24-2017, 06:43 PM
 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
07-24-2017, 07:45 PM
 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
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#5
07-24-2017, 08:13 PM
 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
07-25-2017, 05:48 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,226

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
07-26-2017, 05:17 PM
 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
07-26-2017, 06:37 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,226

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
07-27-2017, 04:08 PM
 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 04:14 PM..
#10
07-27-2017, 04:19 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,226

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)