April 4th, 2015, 08:46 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 biweekly payment goes up \$15 so for the new year he gets \$115 every 2 weeks, and the next year his biweekly 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 12:40 AM. 
April 4th, 2015, 10:38 AM  #2  
Senior Member Joined: May 2008 Posts: 301 Thanks: 81  DISCLAIMER: Beer soaked rambling/opinion/observation/reckoning ahead. Read at your own risk. Not to be taken seriously. In no event shall the wandering math knighterrant Sir jonah in his inebriated state be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of his beer (and tequila) powered views. Quote:
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 12:41 AM.  
April 4th, 2015, 06:50 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 biweekly n = number of years p = period (2/52 = 1/26) c = interest compounding could be anything, biweekly 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 02:17 PM. 
April 4th, 2015, 09:17 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, 12:07 AM  #5 
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5  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, 10:32 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 02:18 PM. 
April 5th, 2015, 05:40 PM  #7  
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5  Quote:
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 biweekly = 2/52 = 1/26 n : n is the number of years 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 02:19 PM.  
April 5th, 2015, 07:55 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, 12:28 AM  #9 
Member Joined: May 2014 From: Rawalpindi, Punjab Posts: 69 Thanks: 5  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)]/(x1) * [ (A+g*ng) * [y^n  1]/(y1) ]  [ g/(y1) * { [y^n  1]/(y1)  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)]/(x1) * [ (A+gng) * [y^n  1]/(y1) ]  [ g/(y1) * { [y^n  1]/(y1)  n } ] PV = [1  1.005^(26)]/(1.0051) * [ (100+15*315) * [0.87837991^3  1]/(0.878379911) ]  [ 15/(0.878379911) * { [0.87837991^3  1]/(0.878379911)  3 } ] PV = $7,329.20 Last edited by skipjack; April 9th, 2015 at 02:58 PM. Reason: added missing 1 to aey(i,c) formula 
April 6th, 2015, 05:29 AM  #10 
Newbie Joined: Apr 2015 From: miami Posts: 18 Thanks: 0 
Much thanks, this will help big time 

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