Calculating cumulative payments with regular increases

[kwestman]

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.

Thank you for your help!
Kellen

 

[joeu2004]

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?

 

[kwestman]

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]

Correct! You're welcome.

 

[kwestman]

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]

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



<b>Excel 2010</b><table cellpadding="2.5px" rules="all" style=";background-color: rgb(255,255,255);border: 1px solid;border-collapse: collapse; border-color: rgb(187,187,187)"><colgroup><col width="25px" style="background-color: rgb(218,231,245)" /><col /><col /><col /><col /><col /></colgroup><thead><tr style=" background-color: rgb(218,231,245);text-align: center;color: rgb(22,17,32)"><th></th><th>A</th><th>B</th><th>C</th><th>D</th><th>E</th></tr></thead><tbody><tr ><td style="color: rgb(22,17,32);text-align: center;">1</td><td style=";">Rent</td><td style="text-align: right;;">1,000.00</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">2</td><td style=";">Term</td><td style="text-align: right;;">6</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">3</td><td style=";">Escalation</td><td style="text-align: right;;">3%</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">4</td><td style=";">Cumulative</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">5</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">6</td><td style="text-align: right;;">1</td><td style="text-align: right;;">1,000.00</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">7</td><td style="text-align: right;;">2</td><td style="text-align: right;;">1,030.00</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">8</td><td style="text-align: right;;">3</td><td style="text-align: right;;">1,060.90</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">9</td><td style="text-align: right;;">4</td><td style="text-align: right;;">1,092.73</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">10</td><td style="text-align: right;;">5</td><td style="text-align: right;;">1,125.51</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">11</td><td style="text-align: right;;">6</td><td style="text-align: right;;">1,159.27</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">12</td><td style="text-align: right;;"></td><td style="text-align: right;;">6,468.41</td><td style="text-align: right;;">6,468.41</td><td style="text-align: right;;">6,468.41</td><td style="text-align: right;;">6,468.41</td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">13</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td></tr></tbody></table><p style="width:1.2em;font-weight:bold;margin:0;padding:0.2em 0.6em 0.2em 0.5em;border: 1px solid rgb(187,187,187);border-top:none;text-align: center;background-color: rgb(218,231,245);color: rgb(22,17,32)">3a</p><br /><br /><table width="85%" cellpadding="2.5px" rules="all" style=";border: 2px solid black;border-collapse:collapse;padding: 0.4em;background-color: rgb(255,255,255)" ><tr><td style="padding:6px" ><b>Worksheet Formulas</b><table cellpadding="2.5px" width="100%" rules="all" style="border: 1px solid;text-align:center;background-color: rgb(255,255,255);border-collapse: collapse; border-color: rgb(187,187,187)"><thead><tr style=" background-color: rgb(218,231,245);color: rgb(22,17,32)"><th width="10px">Cell</th><th style="text-align:left;padding-left:5px;">Formula</th></tr></thead><tbody><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">B7</th><td style="text-align:left">=B6*(<font color="Blue">1+$B$3</font>)</td></tr><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">B8</th><td style="text-align:left">=B7*(<font color="Blue">1+$B$3</font>)</td></tr><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">B9</th><td style="text-align:left">=B8*(<font color="Blue">1+$B$3</font>)</td></tr><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">B10</th><td style="text-align:left">=B9*(<font color="Blue">1+$B$3</font>)</td></tr><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">B11</th><td style="text-align:left">=B10*(<font color="Blue">1+$B$3</font>)</td></tr><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">B12</th><td style="text-align:left">=SUM(<font color="Blue">B6:B11</font>)</td></tr><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">C12</th><td style="text-align:left">=FV(<font color="Blue">B3,B2,-B1,0,0</font>)</td></tr><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">D12</th><td style="text-align:left">=B1*SUMPRODUCT(<font color="Blue">(<font color="Red">1+B3</font>)^(<font color="Red">ROW(<font color="Green">X1:INDEX(<font color="Purple">X:X,B2,1</font>)</font>)-1</font>)</font>)</td></tr><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">E12</th><td style="text-align:left">= (<font color="Blue">B1* (<font color="Red">1+0.03</font>)^0</font>) + (<font color="Blue">B1 * (<font color="Red">1+0.03</font>)^1</font>) + (<font color="Blue">B1 *(<font color="Red"> 1+0.03</font>)^2</font>)+ (<font color="Blue">B1 *(<font color="Red"> 1+0.03</font>)^3</font>)+ (<font color="Blue">B1 *(<font color="Red"> 1+0.03</font>)^4</font>) + (<font color="Blue">B1 * (<font color="Red">1+0.03</font>)^5</font>)</td></tr></tbody></table></td></tr></table><br />

 

[joeu2004]

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]

Kellen may not be following his thread but I will post a minor edit.

You can try the alternative or alternatives that you prefer.

<b>Excel 2010</b><table cellpadding="2.5px" rules="all" style=";background-color: rgb(255,255,255);border: 1px solid;border-collapse: collapse; border-color: rgb(187,187,187)"><colgroup><col width="25px" style="background-color: rgb(218,231,245)" /><col /><col /><col /><col /></colgroup><thead><tr style=" background-color: rgb(218,231,245);text-align: center;color: rgb(22,17,32)"><th></th><th>A</th><th>B</th><th>C</th><th>D</th></tr></thead><tbody><tr ><td style="color: rgb(22,17,32);text-align: center;">1</td><td style=";">Rent</td><td style="text-align: right;;">1,000.00</td><td style="text-align: right;;"></td><td style="text-align: right;background-color: #92D050;;">6,468.41</td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">2</td><td style=";">Term</td><td style="text-align: right;;">6</td><td style="text-align: right;;"></td><td style="text-align: right;;">6,468.41</td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">3</td><td style=";">Escalation</td><td style="text-align: right;;">3%</td><td style="text-align: right;;"></td><td style="text-align: right;;">6,468.41</td></tr><tr ><td style="color: rgb(22,17,32);text-align: center;">4</td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;"></td><td style="text-align: right;;">6,468.41</td></tr></tbody></table><p style="width:1.2em;font-weight:bold;margin:0;padding:0.2em 0.6em 0.2em 0.5em;border: 1px solid rgb(187,187,187);border-top:none;text-align: center;background-color: rgb(218,231,245);color: rgb(22,17,32)">3a</p><br /><br /><table width="85%" cellpadding="2.5px" rules="all" style=";border: 2px solid black;border-collapse:collapse;padding: 0.4em;background-color: rgb(255,255,255)" ><tr><td style="padding:6px" ><b>Worksheet Formulas</b><table cellpadding="2.5px" width="100%" rules="all" style="border: 1px solid;text-align:center;background-color: rgb(255,255,255);border-collapse: collapse; border-color: rgb(187,187,187)"><thead><tr style=" background-color: rgb(218,231,245);color: rgb(22,17,32)"><th width="10px">Cell</th><th style="text-align:left;padding-left:5px;">Formula</th></tr></thead><tbody><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">D1</th><td style="text-align:left">=FV(<font color="Blue">B3,B2,-B1,0,0</font>)</td></tr><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">D2</th><td style="text-align:left">=B1* (<font color="Blue">1+0.03</font>)^0+B1 * (<font color="Blue">1+0.03</font>)^1 + B1 *(<font color="Blue"> 1+0.03</font>)^2+ B1 *(<font color="Blue"> 1+0.03</font>)^3+ B1 *(<font color="Blue"> 1+0.03</font>)^4 + B1 * (<font color="Blue">1+0.03</font>)^5</td></tr><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">D3</th><td style="text-align:left">=B1*SUM(<font color="Blue">(<font color="Red">1+B3</font>)^{0;1;2;3;4;5}</font>)</td></tr><tr><th width="10px" style=" background-color: rgb(218,231,245);color: rgb(22,17,32)">D4</th><td style="text-align:left">=B1*SUMPRODUCT(<font color="Blue">(<font color="Red">1+B3</font>)^(<font color="Red">ROW(<font color="Green">X1:INDEX(<font color="Purple">X:X,B2,1</font>)</font>)-1</font>)</font>)</td></tr></tbody></table></td></tr></table><br />