How can I find out a annual increasing PMT for a value

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msvariar

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May 10, 2019
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I have a future value of 2307936 (after 5 years) and need to make an investment every month, with an annual increase of 10%. The rate of return on investment is 9% per annum.

If there is no annual increase of 10% I am getting 30882 as PMT. How do I calculate with 10% annual increase.
 

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2307936 is pretty specific. How did you arrive at that requirement?

I presume that you calculated 30882 as follows: =PMT((1+9%)^(1/12)-1, 5*12, 0, -2307936, 0).

That is good to know, because it tells us how you want to convert the annual rate (yield) to a monthly rate. The result is 30882.4752290111.

However, I believe you should use: =PMT((1+9%)^(1/12)-1, 5*12, 0, -2307936, 1). That results in 30661.4879718442.

Presumably, you want to earn a return on each payment. Therefore, payments should occur at the beginning of each period.

-----

Off-hand, I'm not sure that we can use a formula to calculate the payment exactly.

We might be able to derive a formula that provides a good-enough estimate. But I don't have time to look at the math right now.

In the meantime, I think the easiest and possibly best approach is to use Goal Seek or Solver. Then the initial payment is 25526.5305148194.

The set-up:

[TABLE="class: grid, width: 200"]
<tbody>[TR]
[TH][/TH]
[TH]A[/TH]
[TH]B[/TH]
[TH]C[/TH]
[/TR]
[TR]
[TD="align: center"]1[/TD]
[TD="align: right"][/TD]
[TD="align: right"]Annual[/TD]
[TD="align: right"]Monthly[/TD]
[/TR]
[TR]
[TD="align: center"]2[/TD]
[TD="align: right"]Yield[/TD]
[TD="align: right"]9.00%[/TD]
[TD="align: right"]0.7207%[/TD]
[/TR]
[TR]
[TD="align: center"]3[/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[TD="align: right"][/TD]
[/TR]
[TR]
[TD="align: center"]4[/TD]
[TD="align: right"]Pmt#[/TD]
[TD="align: right"]Pmt[/TD]
[TD="align: right"]End Bal[/TD]
[/TR]
[TR]
[TD="align: center"]5[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]25,526.53[/TD]
[TD="align: right"]25,710.51[/TD]
[/TR]
[TR]
[TD="align: center"]6[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]25,526.53[/TD]
[TD="align: right"]51,606.32[/TD]
[/TR]
[TR]
[TD="align: center"]
[/TD]
[TD="align: right"]
[/TD]
[TD="align: right"]
[/TD]
[TD="align: right"]
[/TD]
[/TR]
[TR]
[TD="align: center"]15[/TD]
[TD="align: right"]11[/TD]
[TD="align: right"]25,526.53[/TD]
[TD="align: right"]293,230.88[/TD]
[/TR]
[TR]
[TD="align: center"]16[/TD]
[TD="align: right"]12[/TD]
[TD="align: right"]25,526.53[/TD]
[TD="align: right"]321,054.80[/TD]
[/TR]
[TR]
[TD="align: center"]17[/TD]
[TD="align: right"]13[/TD]
[TD="align: right"]28,079.18[/TD]
[TD="align: right"]351,650.31[/TD]
[/TR]
[TR]
[TD="align: center"]18[/TD]
[TD="align: right"]14[/TD]
[TD="align: right"]28,079.18[/TD]
[TD="align: right"]382,466.32[/TD]
[/TR]
[TR]
[TD="align: center"]
[/TD]
[TD="align: right"]
[/TD]
[TD="align: right"]
[/TD]
[TD="align: right"]
[/TD]
[/TR]
[TR]
[TD="align: center"]63[/TD]
[TD="align: right"]59[/TD]
[TD="align: right"]37,373.39[/TD]
[TD="align: right"]2,254,047.59[/TD]
[/TR]
[TR]
[TD="align: center"]64[/TD]
[TD="align: right"]60[/TD]
[TD="align: right"]37,373.39[/TD]
[TD="align: right"]2,307,936.00[/TD]
[/TR]
</tbody>[/TABLE]

Formulas
C2: =(1+B2)^(1/12)-1
A5: =N(A4)+1
B5: empty (initially)
C5: =(N(C4)+B5)*(1+$C$2)
B6: =IF(MOD(A6-1,12)=0,B5*(1+10%),B5)
Copy A5 into A6
Copy C5 into C6
Copy A6:C6 into A7:C64

Solver:
Objective: C64
To Value of: 2307936
By Changing: B5

(In hindsight, I should have put 10% into a cell, and referenced the cell in the formula in B6.
 
Last edited:
Upvote 0
a)Try building a table with your information and then use Goal Seek.
b)Try the UDF shown below


Excel 2010
ABCDE
1Payment25,301.99
2Rate9%
3Term5
4Escalating10%
5Future$2,307,936.00
6
7
8125,301.99189.7625,491.75
3aaa
Cell Formulas
RangeFormula
D5=nSave(B1,B2,B3,1+B4)
D8=B8*B2/12
B8=B1
E8=B8+D8

Code: 
Function nSave(InSave As Double, PC As Double, Yrs As Double, Incpc As Double) As Variant

Dim n As Double, Tot As Double, st As Double
st = InSave
For n = 1 To Yrs * 12
   Tot = InSave * (1 + PC / 12)
   If n Mod 12 = 0 Then st = st * Incpc
   InSave = Tot + st
Next n
nSave = Tot
End Function
 
Last edited:
Upvote 0
As I suspected, we can indeed derive formulas to calculate the initial payment directly and exactly, not an estimate.

Granted, the formulas are complicated. So as I said before, my previous solution using Goal Seek or Solver might be easier.

For the direct calculation....

First some important assumptions, which I noted in my previous posting.

1. The monthly rate should be derived from the annual rate as follows: (1+9%)^(1/12)-1. This is based on msvariar's assertion that without the 10% annual increase, the monthly payment would be 30882 [sic]. That is derived from the formula =PMT((1+9%)^(1/12)-1, 5*12, 0, -2307936, 0), which results in 30882.4752290111.

2. IMHO, payments should be at the beginning of periods, not at the end, so that we earn a return on all payments. This is contrary to msvariar's implicit assumption that payments are at the end of periods. So the correct formula should be =PMT((1+9%)^(1/12)-1, 5*12, 0, -2307936, 1). That results in 30661.4879718442.

Based on those assumptions, the initial payment can be calculated as follows.

[TABLE="class: grid, width: 250"]
<tbody>[TR]
[TD][/TD]
[TD="align: center"]A[/TD]
[TD="align: center"]B[/TD]
[TD="align: center"]C[/TD]
[/TR]
[TR]
[TD="align: center"]1[/TD]
[TD="align: right"][/TD]
[TD="align: right"]Annual[/TD]
[TD="align: right"]Monthly[/TD]
[/TR]
[TR]
[TD="align: center"]2[/TD]
[TD="align: right"]Yield[/TD]
[TD="align: right"]9.00%[/TD]
[TD="align: right"]0.7207%[/TD]
[/TR]
[TR]
[TD="align: center"]3[/TD]
[TD="align: right"]Nper[/TD]
[TD="align: right"]5[/TD]
[TD="align: right"]60[/TD]
[/TR]
[TR]
[TD="align: center"]4[/TD]
[TD="align: right"]%Pmt inc[/TD]
[TD="align: right"]10.00%[/TD]
[TD="align: right"][/TD]
[/TR]
[TR]
[TD="align: center"]5[/TD]
[TD="align: right"]FV goal[/TD]
[TD="align: right"]2,307,936.00[/TD]
[TD="align: right"][/TD]
[/TR]
[TR]
[TD="align: center"]6[/TD]
[TD="align: right"]Init pmt[/TD]
[TD="align: right"]294,545.69[/TD]
[TD="align: right"]25,526.53[/TD]
[/TR]
</tbody>[/TABLE]

Formulas:
C2: =(1+B2)^(1/12) - 1
C3: =B3*12
B6: =B5 / SUMPRODUCT((1+B2)^(B3+1-ROW(A1:INDEX(A:A,B3,1))) * (1+B4)^(ROW(A1:INDEX(A:A,B3,1))-1))
C6: =PMT(C2,12,0,-B6*(1+B2),1)

The correctness of the initial annual payment in B6 can be demonstrated as follows:

[TABLE="class: grid, width: 400"]
<tbody>[TR]
[TD][/TD]
[TD="align: center"]E[/TD]
[TD="align: center"]F[/TD]
[TD="align: center"]G[/TD]
[TD="align: center"]H[/TD]
[TD="align: center"]I[/TD]
[TD="align: center"]J[/TD]
[/TR]
[TR]
[TD="align: center"]8[/TD]
[TD="align: right"]Pmt#[/TD]
[TD="align: right"]Pmt/yr[/TD]
[TD="align: right"]End Bal[/TD]
[TD="align: right"][/TD]
[TD="align: right"]Pmt/mo[/TD]
[TD="align: right"]%Pmt inc[/TD]
[/TR]
[TR]
[TD="align: center"]9[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]294,545.69[/TD]
[TD="align: right"]321,054.80[/TD]
[TD="align: right"][/TD]
[TD="align: right"]25,526.53[/TD]
[TD="align: right"][/TD]
[/TR]
[TR]
[TD="align: center"]10[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]324,000.26[/TD]
[TD="align: right"]703,110.02[/TD]
[TD="align: right"][/TD]
[TD="align: right"]28,079.18[/TD]
[TD="align: right"]10.00%[/TD]
[/TR]
[TR]
[TD="align: center"]11[/TD]
[TD="align: right"]3[/TD]
[TD="align: right"]356,400.29[/TD]
[TD="align: right"]1,154,866.23[/TD]
[TD="align: right"][/TD]
[TD="align: right"]30,887.10[/TD]
[TD="align: right"]10.00%[/TD]
[/TR]
[TR]
[TD="align: center"]12[/TD]
[TD="align: right"]4[/TD]
[TD="align: right"]392,040.31[/TD]
[TD="align: right"]1,686,128.13[/TD]
[TD="align: right"][/TD]
[TD="align: right"]33,975.81[/TD]
[TD="align: right"]10.00%[/TD]
[/TR]
[TR]
[TD="align: center"]13[/TD]
[TD="align: right"]5[/TD]
[TD="align: right"]431,244.35[/TD]
[TD="align: right"]2,307,936.00[/TD]
[TD="align: right"][/TD]
[TD="align: right"]37,373.39[/TD]
[TD="align: right"]10.00%[/TD]
[/TR]
</tbody>[/TABLE]

Formulas:
F9: =B6
G9: =(N(G8)+F9)*(1+$B$2)
I9: =PMT($C$2, 12, N(G8), -G9, 1)
F10: =F9*(1+$B$4)
J10: =I10/I9 - 1

Copy G9 into G10. Copy I9 into I10. Copy F10:J10 into F11:J13.

The correctness of the initial monthly payment in C6 can be demonstrated with a table similar to the one in my previous posting, to wit:

[TABLE="class: grid, width: 250"]
<tbody>[TR]
[TH][/TH]
[TH]A[/TH]
[TH]B[/TH]
[TH]C[/TH]
[/TR]
[TR]
[TD="align: center"]8[/TD]
[TD="align: right"]Pmt#[/TD]
[TD="align: right"]Pmt[/TD]
[TD="align: right"]End Bal[/TD]
[/TR]
[TR]
[TD="align: center"]9[/TD]
[TD="align: right"]1[/TD]
[TD="align: right"]25,526.53[/TD]
[TD="align: right"]25,710.51[/TD]
[/TR]
[TR]
[TD="align: center"]10[/TD]
[TD="align: right"]2[/TD]
[TD="align: right"]25,526.53[/TD]
[TD="align: right"]51,606.32[/TD]
[/TR]
[TR]
[TD="align: center"]
[/TD]
[TD="align: right"]
[/TD]
[TD="align: right"]
[/TD]
[TD="align: right"]
[/TD]
[/TR]
[TR]
[TD="align: center"]19[/TD]
[TD="align: right"]11[/TD]
[TD="align: right"]25,526.53[/TD]
[TD="align: right"]293,230.88[/TD]
[/TR]
[TR]
[TD="align: center"]20[/TD]
[TD="align: right"]12[/TD]
[TD="align: right"]25,526.53[/TD]
[TD="align: right"]321,054.80[/TD]
[/TR]
[TR]
[TD="align: center"]21[/TD]
[TD="align: right"]13[/TD]
[TD="align: right"]28,079.18[/TD]
[TD="align: right"]351,650.31[/TD]
[/TR]
[TR]
[TD="align: center"]22[/TD]
[TD="align: right"]14[/TD]
[TD="align: right"]28,079.18[/TD]
[TD="align: right"]382,466.32[/TD]
[/TR]
[TR]
[TD="align: center"]
[/TD]
[TD="align: right"]
[/TD]
[TD="align: right"]
[/TD]
[TD="align: right"]
[/TD]
[/TR]
[TR]
[TD="align: center"]67[/TD]
[TD="align: right"]59[/TD]
[TD="align: right"]37,373.39[/TD]
[TD="align: right"]2,254,047.59[/TD]
[/TR]
[TR]
[TD="align: center"]68[/TD]
[TD="align: right"]60[/TD]
[TD="align: right"]37,373.39[/TD]
[TD="align: right"]2,307,936.00[/TD]
[/TR]
</tbody>[/TABLE]

Formulas:
A9: =N(A8)+1
B9: =C6
C9: =(N(C8)+B9)*(1+$C$2)
B10: =IF(MOD(A10-1,12)=0, B9*(1+$B$4), B9)

Copy A9 into A10. Copy C9 into C10. Copy A10:C10 into A11:C68.


-----
For the curious, the mathematical derivation of the formula in B6 (initial annual payment) is as follows.

PMT*(1+10%)^0*(1+9%)^5 + PMT*(1+10%)^1*(1+9%)^4 +...+ PMT*(1+10%)^4*(1+9%)^1 = FV (2,307,936)

PMT = FV / ( (1+10%)^0*(1+9%)^5 + (1+10%)^1*(1+9%)^4 +...+ (1+10%)^4*(1+9%)^1 )

Note that (1+10%)^0 = 1. I include it in order to make the exponential series clearer.
 
Last edited:
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