My Math Forum present value question

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 April 4th, 2015, 09:46 AM #1 Newbie   Joined: Apr 2015 From: miami Posts: 18 Thanks: 0 present value question Is there a formula for the present value of a stream of payments that increases by a specific amount at the beginning of each new year? For example, Mr. Smith gets \$100 every 2 weeks but at the beginning of each new year the bi-weekly payment goes up \$15 so for the new year he gets \$115 every 2 weeks, and the next year his bi-weekly payments goes up to \$130 and so on..... So far, the formulas I found only calculate present value of an increasing annuity using a compounding factor. Last edited by skipjack; April 5th, 2015 at 01:40 AM.
April 4th, 2015, 11:38 AM   #2
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Quote:
 Originally Posted by cb123 Is there a formula for the present value of a stream of payments that increases by a specific amount at the beginning of each new year? For example, Mr. Smith gets \$100 every 2 weeks but at the beginning of each new year the bi-weekly payment goes up \$15 so for the new year he gets \$115 every 2 weeks, and the next year his bi-weekly payments goes up to \$130 and so on..... So far, the formulas I found only calculate present value of an increasing annuity using a compounding factor.
Yes there is.
Too hammered right now.
Tell you later.
Unless Sir Denis or Sir Dexter/Abraham beat me to it.

Last edited by skipjack; April 5th, 2015 at 01:41 AM.

 April 4th, 2015, 07:50 PM #3 Member     Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5 Did this in a hurry so any errors or omissions are at my fault Code: A = 100 G = 15 i = annual nominal rate, i.e. 12% annual compounded bi-weekly n = number of years p = period (2/52 = 1/26) c = interest compounding could be anything, bi-weekly assumed = 1/26 aey(i,c) = (1 + i*c)^(1/c) - 1 x = (1 + aey(i,c))^p y = x^-(1/p) PV = [1 - x^-(1/p)]/[x - 1] ( A*y [y^n - 1]/[y - 1] + G*y/[y - 1] { [y^n - 1]/[y - 1] - n } ) PV = [1 - x^-(26)]/[x - 1] ( 100*y [y^n - 1]/[y - 1] + 15*y/[y - 1] { [y^n - 1]/[y - 1] - n } ) Last edited by skipjack; April 9th, 2015 at 03:17 PM.
 April 4th, 2015, 10:17 PM #4 Newbie   Joined: Apr 2015 From: miami Posts: 18 Thanks: 0 Thank you, I will try it out tomorrow. This is a big help!!
April 5th, 2015, 01:07 AM   #5
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Quote:
 Originally Posted by cb123 Thank you, I will try it out tomorrow. This is a big help!!
You welcome

However after posting, I noticed a slight glitch in my formula

I will wait for either Sir Denis or Sir Jonah to compare my formula results with their own results, I am sure there will be a slight difference

I will amend my formula once I am notified that my reply has been taken to task

 April 5th, 2015, 11:32 AM #6 Newbie   Joined: Apr 2015 From: miami Posts: 18 Thanks: 0 In your example, G is the annual increase, i is the discount factor?, c is the discount factor divided by the number of periods(p)? Is that correct? Last edited by skipjack; April 9th, 2015 at 03:18 PM.
April 5th, 2015, 06:40 PM   #7
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Quote:
 Originally Posted by cb123 In your example, G is the annual increase, i is the discount factor?, c is the discount factor divided by the number of periods(p)? Is that correct?
Code:
A=100 : initial payment at the end of each two weeks
G=15   : annual increase
i         : annual nominal interest rate such as 12%
c        : interest compounding frequency such as 1/26 biweekly then periodic rate is i*c = 0.12/26 = 0.46% is the biweekly rate
p        : payment period, in this case bi-weekly = 2/52 = 1/26
n        : n is the number of years
As for the formula, there are couple of issues I will address a bit later

Firstly formula is used when n is complete number of years, I will later fix this to include years with fractional part

Second, I made an error in deriving the formula which I will fix.

Last edited by skipjack; April 9th, 2015 at 03:19 PM.

 April 5th, 2015, 08:55 PM #8 Newbie   Joined: Apr 2015 From: miami Posts: 18 Thanks: 0 got it, thanks I will wait for the final fix before running.
April 6th, 2015, 01:28 AM   #9
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Quote:
 Originally Posted by cb123 got it, thanks I will wait for the final fix before running.
For complete number of periods in a year, the following is a general case formula to find present value of an ordinary annuity (end of period payments) that have payments which may increase or decrease by a money amount per period

Code:

A=100
g=15
i=13% annual cpd biweekly
n=3 years
c=1/26 biweekly
p=1/26 period is a biweek

Find PV

PV = [1 - x^(-1/p)]/(x-1) * [ (A+g*n-g) * [y^n - 1]/(y-1) ] - [ g/(y-1) * { [y^n - 1]/(y-1) - n } ]

aey(i,c) = (1 + i*c)^(1/c) - 1

i=0.13
c=1/26

aey(13%,1/26) = (1 + 0.13/26)^(26) - 1
aey(13%,1/26) = 0.138459553

aey(13%,1/26) = 13.85%

x=[1 + aey(i,c)]^p

p=1/26

x=[1 + 13.85%]^(1/26)
x=[1.1385]^(0.038462)

x=1.005

y=x^(-1/p)

y=(1.005)^(-26)

y=0.87837991

PV = [1 - x^(-1/p)]/(x-1) * [ (A+gn-g) * [y^n - 1]/(y-1) ] - [ g/(y-1) * { [y^n - 1]/(y-1) - n } ]

PV = [1 - 1.005^(-26)]/(1.005-1) * [ (100+15*3-15) * [0.87837991^3 - 1]/(0.87837991-1) ] - [ 15/(0.87837991-1) * { [0.87837991^3 - 1]/(0.87837991-1) - 3 } ]

PV = \$7,329.20

Last edited by skipjack; April 9th, 2015 at 03:58 PM. Reason: added missing -1 to aey(i,c) formula

 April 6th, 2015, 06:29 AM #10 Newbie   Joined: Apr 2015 From: miami Posts: 18 Thanks: 0 Much thanks, this will help big time

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