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I have a formula that takes a market rent value today (e.g. $100) and grows it annually by some annual growth rate (e.g. 2.00%). That formula is: (Market Rent in Time Zero) x (1 + r)^(n-1). If I have a time scale where there are monthly periods, the result is that the market rent from months 1-12 is $100, 13 - 24 is $102, and so forth. I'd like to find a formula that would also result in an effective annual growth rate of 2.00% (i.e. the same as the first formula) BUT would grow the $100 monthly such that market rent in month 1 would be $100, higher in month 2, higher in month 3, and would be $102 in month 13, and $104.04 in month 25, etc. Can you help with this?

Sure, I can help with that. What you're looking for is a formula that applies the growth rate on a monthly basis instead of annually. This is often referred to as compound interest, and it's a common concept in finance.

The formula you're looking for is:

Market Rent at Time t=Market Rent at Time Zero×(1+r12)t\text{Market Rent at Time } t = \text{Market Rent at Time Zero} \times \left(1 + \frac{r}{12}\right)^tMarket Rent at Time t=Market Rent at Time Zero×(1+12r)t

Here, rrr is the annual growth rate (expressed as a decimal), ttt is the number of months since Time Zero, and Market Rent at Time Zero\text{Market Rent at Time Zero}Market Rent at Time Zero is the initial market rent.

This formula applies the growth rate on a monthly basis, so the market rent grows slightly each month. After 12 months (i.e., t=12t = 12t=12), the market rent will have grown by exactly 2.00%, and after 24 months (i.e., t=24t = 24t=24), it will have grown by approximately 4.04%, assuming a 2.00% annual growth rate.

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Perfect!

I'm glad I could help! If you have any other questions, whether they're about finance, math, or any other topic, feel free to ask.

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Could you show the formula in a format like this so I can copy and paste to a Word doc? (Market Rent in Time Zero) x (1 + r)^(n-1)

Absolutely, here's the formula in the format you requested:

(Market Rent at Time Zero) x (1 + r/12)^(t)

In this formula:

  • "Market Rent at Time Zero" is the initial market rent.
  • "r" is the annual growth rate (expressed as a decimal).
  • "t" is the number of months since Time Zero.
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Excellent could you provide the "in this formula" description for my original formula? (Market Rent in Time Zero) x (1 + r)^(n-1).

Of course, here's the description for your original formula:

(Market Rent in Time Zero) x (1 + r)^(n-1)

In this formula:

  • "Market Rent in Time Zero" is the initial market rent.
  • "r" is the annual growth rate (expressed as a decimal).
  • "n" is the number of years since Time Zero.

Please note that in this formula, the growth is applied annually, not monthly.

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Hmmm. When I test this in Excel, in month 13 for my annual formula I get $102.00 but for your monthly formula I get $102.02.

I apologize for the confusion. The discrepancy arises due to the compounding effect when the growth is applied monthly.

In the monthly formula, the growth is compounded 12 times in a year, which results in a slightly higher value than $102.00 at the end of the year. The exact value, as calculated, is approximately $102.018.

This is a characteristic of compound interest. Even though the annual growth rate is the same in both formulas, the frequency of compounding affects the final result. The more frequently the interest is compounded, the greater the final amount will be.

If you want the value to be exactly $102.00 at the end of the year, you would need to adjust the monthly growth rate slightly. This adjusted rate would be the twelfth root of 1.02 minus 1, which can be calculated as follows:

r_monthly = (1.02^(1/12)) - 1

Let's calculate this value.

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Plus update the "adjusted formula" above to include the calculation for r_monthly.

Absolutely, here's the adjusted formula with the calculation for r_monthly included:

(Market Rent at Time Zero) x (1 + ((1.02^(1/12)) - 1))^(t)

Where:

  • "Market Rent at Time Zero" is the initial market rent.
  • "((1.02^(1/12)) - 1)" is the adjusted monthly growth rate, which is calculated as the twelfth root of 1.02 minus 1.
  • "t" is the number of months since Time Zero.

If you use this formula, the market rent should grow to exactly $102.00 at the end of the first year (i.e., when t = 12).

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Plus update the 1.02 to include the relevant variables (i.e. 1.02 should be 1xr, right?)

Yes, you're correct. The 1.02 in the formula represents 1 plus the annual growth rate (r). If we replace 1.02 with (1 + r), the formula becomes more general and can be used with any annual growth rate. Here's the updated formula:

(Market Rent at Time Zero) x (1 + ((1 + r)^(1/12) - 1))^(t)

Where:

  • "Market Rent at Time Zero" is the initial market rent.
  • "r" is the annual growth rate (expressed as a decimal).
  • "((1 + r)^(1/12) - 1)" is the adjusted monthly growth rate, which is calculated as the twelfth root of 1 plus the annual growth rate, minus 1.
  • "t" is the number of months since Time Zero.

With this formula, the market rent will grow to (1 + r) times the initial market rent at the end of the first year (i.e., when t = 12), regardless of the value of r.

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Almost there. Instead of (t), it should be (t-1)

You're right, I apologize for the oversight. If we want the first month to have the initial rent value (i.e., when t = 1, the rent is the "Market Rent at Time Zero"), we should use (t-1) instead of t. Here's the corrected formula:

(Market Rent at Time Zero) x (1 + ((1 + r)^(1/12) - 1))^(t-1)

Where:

  • "Market Rent at Time Zero" is the initial market rent.
  • "r" is the annual growth rate (expressed as a decimal).
  • "((1 + r)^(1/12) - 1)" is the adjusted monthly growth rate, which is calculated as the twelfth root of 1 plus the annual growth rate, minus 1.
  • "t" is the number of months since Time Zero, with t = 1 representing the first month.

With this formula, the market rent will grow to (1 + r) times the initial market rent at the end of the first year (i.e., when t = 13), regardless of the value of r.