How can it be shown mathematically the statement: (p/a,i,2
n)=i(p/a,i,n)^2+2(p/a,i,n)/(1+i)^n.   I would like to see this as a
mathematical demonstration and not as an example.

---

Request for Question Clarification by livioflores-ga on 05 Mar 2005 14:24 PST

What is the meaning of these expressions: 
(p/a,i,2n)
i(p/a,i,n)^2
2(p/a,i,n)/(1+i)^n
i

Thank you.

---

Clarification of Question by drf-ga on 05 Mar 2005 14:51 PST

This statement is from an economics class.  p/a signifies "present
value/a single payment".  "i" signifies "annual interest rate", "n"
signifies the "number of annual interest periods".  Some clarifying
info further explaining this topic can be found on:
http://www.paranzasoft.com/help/pages/caInterestFactors.html
or  http://mathforum.org/dr.math/faq/faq.interest.html
I'm just not understanding how this works.  Thanks!

---

Clarification of Question by drf-ga on 06 Mar 2005 11:53 PST

the original statement can be re-written as: [((1+i)^2n) -1 /
i((1+i)^2n)] = i [((1+i)^n) -1 / i((1+i)^n)]^2 + 2 [((1+i)^n) -1 /
i(1+i)^n] /(1+i)^n

The portion of the statement "(p/a,i,n)" is the same as 
"[((1+i)^n) -1 / i(1+i)^n]"

Hi drf!!


The expression "(p/a,i,n)" is one type of notation for the "present
value of an annuity factor" or PVAF_n, it is equal to:

                      (1+i)^n - 1 
(p/a,i,n) = PVAF_n = --------------       Eq.1
                       i*(1+i)^n
 

See the following documents for reference:

"Derivation of Time Value of Money Formulas" by Peter F. Colwell:
Look at the center of the second page.
http://www.business.uiuc.edu/orer/V13-2-3.pdf

"The Time Value of Money"
Look at page 7 the table of 'Time Value Equivalence Factors' for the
"Present worth of an annuity factor":
http://ocw.mit.edu/NR/rdonlyres/Nuclear-Engineering/22-812JSpring2004/BF4BDE00-2E68-4CA7-B2C4-57E609C22089/0/lec02slides.pdf


Now the problem. What we want to demonstrate is the following proposition:

PVAF_2n = i*(PVAF_n)^2 + 2*PVAF_n/(1+i)^n


From Eq.1 we have that:

(PVAF_n)^2 = [(1+i)^n - 1]^2/[i^2*(1+i)^2n] =
           
           = [(1+i)^2n - 2*(1+i)^n + 1]/[i^2*(1+i)^2n] =

           = [(1+i)^2n]/[i^2*(1+i)^2n] - 2*[(1+i)^n - 1]/[i^2*(1+i)^2n] =
  
Then:

i*(PVAF_n)^2 = [(1+i)^2n]/[i*(1+i)^2n] - 2*[(1+i)^n - 1]/[i*(1+i)^2n] =

             = PVAF_2n - 2*[(1+i)^n - 1]/[i*(1+i)^2n] =

             = PVAF_2n - 2*[(1+i)^n - 1]/[i*(1+i)^n]*[1/(1+i)^n] =

             = PVAF_2n - 2*PVAF_n/(1+i)^n

From the last equation we can isolate PVAF_2n to obtain:

PVAF_2n = i*(PVAF_n)^2 + 2*PVAF_n/(1+i)^n

This is what we want to demonstrate.


I hope that this helps you. feel free to request for a clarification
if you need it. I will gladly respond your requests for further
assistance on this topic before you rate this answer.

Best regards.
livioflores-ga

---

Request for Answer Clarification by drf-ga on 30 Mar 2005 12:08 PST

I have another question out there: 502721 
If you could help with that, that would be great!

---

Clarification of Answer by livioflores-ga on 30 Mar 2005 20:28 PST

Hi drf!!

Thank you for the explicit confidence but I am not skilled in the
topic of your new question so I am unable to give you an answer and/or
some clues about how to solve the problem.
I suggest you to add to the question as a clarification which is the
related topic (course + chapter) to the problem, may be one of us can
research inside text lectures and find some examples or solve the
problem.

Regards.
livioflores-ga

Great work and very prompt.  Did a great job of providing me
references so I can further understand how this works.  I'm a
believer!