# Alternative (simplified) way of calculating loan repayments

I'm working on a small program to calculate, among other things, mortgage/credit repayments. Searching the internet I got what appears to be the typical repayment formula.

principal 100,000 at 2% over 20 years, returns a monthly repayment of 505.88 using any online/Excel calculator.

However, I live in Germany and my home loan repayment was calculated in a different way:

payment = ("Tilgung %" + nominal interest rate) / 12 * principal

Tilgung is translated roughly as "repayment" in this case and is always given as a % by the bank. In my case the Tilgung is 4% and plugging in to the formula, the repayment is 500:

(0.04 + 0.02) / 12 * 100000 = 500

However, running the above through Excel, I found that roughly 245 payments are needed to complete the loan, vs the 240 needed for the above amount.

Does anyone know if there an official definition of this formula? It is clearly easier to memorize and work out repayment using any pocket calculator, but I find it surprising the banks use it to recalculate the repayment.

• Something does not add up here. If your mortgage really works this way, then every year you repay approximately 4% of the remaining principal balance. This means that principal balance decreases exponentially, and the loan is in principle infinite. How did you come up with 245 payments? Nov 3, 2022 at 5:28
• You are right, but Solarflare mentions something below which makes sense and I had a look through my documents and it is indeed true. The 4% "Tilgungsrate" is only the initial (first year) repayment %. In my case the repayment wasn't adjusted, so keep paying more and more principal over time. Nov 4, 2022 at 8:33
• Then the formula in your question is incorrect. You may want to edit your question and correct it. Nov 4, 2022 at 16:22

There are 2 basic ways to repay your mortgage:

• you define a fixed monthly amount
• you define that you want to reduce the principal by a fixed percentage per year, the Tilgungsrate

For the second option, 4% Tilgungsrate means that you want to repay a fixed amount of 4% * 100.000€ = 4.000€ per year, so it will take you exactly 25 years (not 245 month). The given formula calculates the monthly rate for the first year only, e.g. you know you want to repay 4%, and you have to pay 2% interest, divided into 12 equal monthly rates.

However, for an actual payment plan with an actually fixed Tilgungsrate, the monthly payment would decrease next year: while you keep reducing the principal by a fixed 4% (4.000€), the interest amount is lower (as the principal is lower).

Practically, the monthly payment oftentimes stays the same (since you repay faster), so the Tilgungsrate increases over time. So if an online calculator outputs a fixed monthly amount for the following years, it probably asked for the rate at the start (anfängliche Tilgungsrate).

Your formula will get slightly less trivial if you want to calculate either next year's rate in case of reduced interests, or the total runtime in case of a fixed monthly amount (which you would do with your other excel formula).

Practically, you know how much you want to/can pay per month, and if you have to enter a Tilgungsrate instead of a monthly rate, you will adjust it until you get to the monthly rate you want (or maybe the repayment time you find reasonable). It's probably just a psychological effect which way you (or the banks or the law) prefer to arrive at the same monthly number (unless of course you have a mortgage where the rates actually decrease).

• This sounds exactly right: 4% initially, but with a constant repayment the Tilgungsrate goes steadily up. The "initial" part was missing in my understanding of this. Nov 4, 2022 at 8:34