# Calculating cumulative payments with regular increases

#### kwestman

##### New Member
If I had rental income from a house, payment x, and the rent increased y% annually over z years, how can I calculate the cumulative rental income in one formula?

As an example, if the rent was 1000, and annual increases were 3%, I would use:
Cumulative rental income = (1000 * 1+0.03^0) + (1000 * 1+0.03^1) + (1000 * 1+0.03^2)... + (1000 * 1+0.03^5).

How can I automate that in ONE formula where the inputs are just X, Y, and Z?

I suspect it will use one of the financial formulas.

Kellen

#### joeu2004

##### Well-known Member
Cumulative rental income = (1000 * 1+0.03^0) + (1000 * 1+0.03^1) + (1000 * 1+0.03^2)... + (1000 * 1+0.03^5)
With annual payment in X1 (1000), annual percentage increase in Y1 (3%), and number of years in Z1 (6):

=X1*SUMPRODUCT((1+Y1)^(ROW(X1:INDEX(X:X,Z1,1))-1))
or
=FV(Y1,Z1,-X1)

probably formatted as Currency.

Obviously, the second formula better. The first formula is provided as "proof of concept"; it is the same as your method of calculation.

Question: is the rent truly 1000 per year(!), the same frequency as the annual increases? Or is it 1000 per month, with annual increases?

Last edited:

#### kwestman

##### New Member
This is exactly the answer I needed, thank you!

Yes, rent would be 1000/month, but I just multiplied by 12 for the FV formula.

#### joeu2004

##### Well-known Member
Correct! You're welcome.

#### kwestman

##### New Member
Upon further review, the FV formula appears to be applying the interest rate to the balance, and not to the regular payment to the account (ie. rent).

FV = PV (1 + r)^n

I need:

Total Rent = (pmt * 1+n^0) + (pmt* 1+n^1) + (pmt * 1+n^2) + ... + (pmt * 1+n^z)
where pmt is payment, n is payment inflation rate per year, and z is the number of years

I don't think those are the same things.

Can anybody offer a solution?

#### Dave Patton

##### Well-known Member
Did you try joe2004's suggestions?
Copy the example below and use Excel's Formula Evaluate to review the formulas.

#### joeu2004

##### Well-known Member
the FV formula appears to be applying the interest rate to the balance, and not to the regular payment to the account (ie. rent).
FV = PV (1 + r)^n
Only if there are no periodic payments. With periodic payments, the correct formula is (all values are positive):

FV=PV*(1+r)^n + PMT*((1+r)^n - 1)/r

where PV=0, PMT=12000 (1000*12), r=3% and n=6 in your case. See the Excel PV Function help page; simplified here.

I need:
Total Rent = (pmt * 1+n^0) + (pmt* 1+n^1) + (pmt * 1+n^2) + ... + (pmt * 1+n^z)
where pmt is payment, n is payment inflation rate per year, and z is the number of years
The two are equivalent, when you use the same variable names: r is payment inflation rate, and n is number of years. Also use proper parentheses (also note the last exponent): pmt*(1+r)^0 +...+ pmt*(1+r)^(n-1)

(The formula in your original posting was more correct. I did not realize your mistake in parentheses, unconsciously reading 1+0.03 as 1.03.)

I provided a SUMPRODUCT formula that mimics that calculation for the purpose of demonstrating that my FV formula calculates the same result.

I could provide mathematical proof, but I think that will overwhelm you. Refer to Dave's response for a demonstration.

#### Dave Patton

##### Well-known Member
Kellen may not be following his thread but I will post a minor edit.

You can try the alternative or alternatives that you prefer.

Last edited:

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