My Math Forum  

Go Back   My Math Forum > Science Forums > Economics

Economics Economics Forum - Financial Mathematics, Econometrics, Operations Research, Mathematical Finance, Computational Finance


Thanks Tree1Thanks
Reply
 
LinkBack Thread Tools Display Modes
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 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 12:40 AM.
cb123 is offline  
 
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 knight-errant 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:
Originally Posted by cb123 View Post
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 12:41 AM.
jonah is offline  
April 4th, 2015, 06:50 PM   #3
Member
 
AbrahamA's Avatar
 
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 02:17 PM.
AbrahamA is offline  
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!!
cb123 is offline  
April 5th, 2015, 12:07 AM   #5
Member
 
AbrahamA's Avatar
 
Joined: May 2014
From: Rawalpindi, Punjab

Posts: 69
Thanks: 5

Quote:
Originally Posted by cb123 View Post
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
AbrahamA is offline  
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.
cb123 is offline  
April 5th, 2015, 05:40 PM   #7
Member
 
AbrahamA's Avatar
 
Joined: May 2014
From: Rawalpindi, Punjab

Posts: 69
Thanks: 5

Quote:
Originally Posted by cb123 View Post
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 02:19 PM.
AbrahamA is offline  
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.
cb123 is offline  
April 6th, 2015, 12:28 AM   #9
Member
 
AbrahamA's Avatar
 
Joined: May 2014
From: Rawalpindi, Punjab

Posts: 69
Thanks: 5

Quote:
Originally Posted by cb123 View Post
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 02:58 PM. Reason: added missing -1 to aey(i,c) formula
AbrahamA is offline  
April 6th, 2015, 05:29 AM   #10
Newbie
 
Joined: Apr 2015
From: miami

Posts: 18
Thanks: 0

Much thanks, this will help big time
cb123 is offline  
Reply

  My Math Forum > Science Forums > Economics

Tags
present, question



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Present value of money Jaider Applied Math 1 February 24th, 2015 02:29 PM
How should I present this to my teacher? n777l Complex Analysis 5 September 15th, 2013 01:11 AM
present value etc HELP :( marmozetka Algebra 3 May 21st, 2013 04:21 PM
Present value gouhayashi Economics 5 May 19th, 2011 09:39 PM
Present value... Sandy123 Economics 1 October 6th, 2009 07:16 PM





Copyright © 2019 My Math Forum. All rights reserved.