Present value of annuity with payments increasing by 100

Stripey

New member
Joined
Nov 2, 2018
Messages
15
Present value of annuity due with payments increasing by 100

Hi,
We have to solve this problem for class:
What is the present value of annuity due when each payment is $100 higher than the previous one?
The first payment is $10.000, annual interest rate: 5 %, the payment period as well as the interest period is 4 months (EDIT: it's 3 months) and the payments are going to be coming in for 15 years.
(EDIT: this is an annuity due)

So I figured out I need to find the quotient and then use it for calculating the sum of the payment's present value as geometric sequence.
The value of P1 is obviously 10.000
Is P2 = 10.000 * 1/(1+0,0125) + 100 or P2 = (10.000 + 100) * 1/1+(1+0,0125) ? Furthermore, what would P3 be then? And is this approach even correct?


(I hope the terms and symbols are correct (EDIT: they were not :D ) - I am not a native English speaker, but I tried my best when translating them)
 
Last edited:
Hi,
What is the present value of annuity when each payment is 100 higher than the previous one?
Is P2 = 10.000 * 1/(1+0,0125) + 100 or P2 = (10.000 + 100) * 1/1+(1+0,0125) ? Furthermore, what would P3 be then? And is this approach even correct?
[/SUB]
If you start with payment of 10, and payments increase by 100,
then payments will be: 10, 110, 210, 310 .....

Can you clarify further...
 
Hi,
We have to solve this problem for class:
What is the present value of annuity when each payment is $100 higher than the previous one?
The first payment is $10.000, annual interest rate: 5 %, the payment period as well as the interest period is 4 months and the payments are going to be coming in for 15 years.


So I figured out I need to find the quotient and then use it for calculating the sum of the payment's present value as geometric sequence.
The value of P1 is obviously 10.000
Is P2 = 10.000 * 1/(1+0,0125) + 100 or P2 = (10.000 + 100) * 1/1+(1+0,0125) ? Furthermore, what would P3 be then? And is this approach even correct?


(I hope the terms and symbols are correct - I am not a native English speaker, but I tried my best when translating them)

It appears that period means thousands and comma means decimal, so the first payment is $10 000, increasing by $100 each time.
 
In the future, when denoting money, please use "." only this way: 1234.56;
that's one thousand two hundred and thirty-four dollars and fifty-six cents.

To get used to your problem, make up a SIMPLER/SIMILAR case; as example:
an annuity of $100 annually as 1st payment, increasing by $100 each year;
what is the present value at end of 4th year if rate is 12% annually?

Code:
YEAR  PAYMENT INTEREST  BALANCE
  0                         .00
  1   100.00     .00     100.00
  2   200.00   12.00     312.00
  3   300.00   37.44     649.44
  4   400.00   77.93    1127.37
Now "experiment" with that!
 
In the future, when denoting money, please use "." only this way: 1234.56;
that's one thousand two hundred and thirty-four dollars and fifty-six cents.

To get used to your problem, make up a SIMPLER/SIMILAR case; as example:
an annuity of $100 annually as 1st payment, increasing by $100 each year;
what is the present value at end of 4th year if rate is 12% annually?

Code:
YEAR  PAYMENT INTEREST  BALANCE
  0                         .00
  1   100.00     .00     100.00
  2   200.00   12.00     312.00
  3   300.00   37.44     649.44
  4   400.00   77.93    1127.37
Now "experiment" with that!


I actually have the same problem with this as I had with the original task. So the PV of the first payment is 100. With the second payment of 200 - is it 178.57 or 189.29? Now I'm slightly more convinced it is 178.57, but still not 100% sure.

And sorry about the period/comma thing. That might be even more confusing than the maths itself :D:D
 
Last edited:
Hi,
We have to solve this problem for class:
What is the present value of annuity when each payment is $100 higher than the previous one?
The first payment is $10.000, annual interest rate: 5 %, the payment period as well as the interest period is 4 months and the payments are going to be coming in for 15 years.


So I figured out I need to find the quotient and then use it for calculating the sum of the payment's present value as geometric sequence.
The value of P1 is obviously 10.000
Is P2 = 10.000 * 1/(1+0,0125) + 100 or P2 = (10.000 + 100) * 1/1+(1+0,0125) ? Furthermore, what would P3 be then? And is this approach even correct?


(I hope the terms and symbols are correct - I am not a native English speaker, but I tried my best when translating them)
\(\displaystyle P_1 \) occurs one time period (ie 4 months) after the "present".
 
Code:
year  payment interest  balance
  0                         .00
  1   100.00     .00     100.00
  2   200.00   12.00     312.00
  3   300.00   37.44     649.44
  4   400.00   77.93    1127.37
You're using annuity due; my example is annuity deferred,
meaning 1st pay't is one period later (year 1 means end of 1st year).
PV = 1127.37 / 1.12^4 = 716.46
which can also be obtained this way:
100 / 1.12^1 + 200 / 1.12^2 + 300 / 1.12^3 + 400 / 1.12^4 = 716.46

As I told you, I was using this example simply to see where you're at.
Do you understand it?
 
Code:
year  payment interest  balance
  0                         .00
  1   100.00     .00     100.00
  2   200.00   12.00     312.00
  3   300.00   37.44     649.44
  4   400.00   77.93    1127.37
You're using annuity due; my example is annuity deferred,
meaning 1st pay't is one period later (year 1 means end of 1st year).
PV = 1127.37 / 1.12^4 = 716.46
which can also be obtained this way:
100 / 1.12^1 + 200 / 1.12^2 + 300 / 1.12^3 + 400 / 1.12^4 = 716.46

As I told you, I was using this example simply to see where you're at.
Do you understand it?

716.46 is the result for annuity deferred, right? If so, I might actually get it now. The original problem should be annuity due (I corrected it after it was pointed out in the previous comment) so if your simplified verison was due, the PV would equal 802.44.
Thank you for the example. I may have solved the whole thing already using arithmetico-geometric sequence formula, hope it's correct ;)
 
I noticed above that you used i = 0.0125 which is 0.05/4.

Shouldn't it be 0.05/3 as there are 3 lots of "4 months" in a year??

EDIT: Ok forget that - just saw your edit to original post.
 
Well, here's a formula that'll give present value of an annuity
where the payment is increased by a constant amount:

PV = {i*[f*(r^n - 1) - c*(n - 1)] + c*r*[r^(n - 1) - 1]} / (i^2 * r^n)
where r = 1 + i

Using your problem:
i = interest (.05 / 4 = .0125)
n = number of periods (15 * 4 = 60)
f = first payment (100)
c = constant payment increase (100)

That'll result in PV of 112687.74
Add to that the instant payment of 10.000, whatever that means!

Agree Jeff????
 
Top