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Old 07-24-2017, 02:30 PM
Futon Futon is offline
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Default 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?
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Old 07-24-2017, 06:33 PM
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#1. (1 + k)^t / (1+i)^t simplifies to what if i=k?
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Old 07-24-2017, 06:43 PM
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Originally Posted by Breadmaker View Post
#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.
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Old 07-24-2017, 07:45 PM
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Originally Posted by Futon View Post
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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Old 07-24-2017, 08:13 PM
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Originally Posted by NattyMo View Post
Forget the formula when i=k. First principles...

Simplify (1+k)^t / (1+i)^t
Simplify assuming k=i? It's just one.
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Old 07-25-2017, 05:48 PM
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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.
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Old 07-26-2017, 05:17 PM
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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?
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Old 07-26-2017, 06:37 PM
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Originally Posted by Futon View Post
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).
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Old 07-27-2017, 04:08 PM
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Originally Posted by Academic Actuary View Post
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..
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Old 07-27-2017, 04:19 PM
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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)
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