Calculating monthly and annual payments with corresponding rate: different results?

Good evening everyone,

I got a small financial problem that I tried to understand and, even though I come to the right answer, I do not understand why my alternative approach does not work.

Problem: you want to buy a flat for €150,000 so you ask for a mortgage to your bank which offers you an effective annual interest rate of 15%. The payments you have to make are monthly and the time span you need for reimbursing is 10 years. How much do you have to pay at the end of each month?

Well, to find the monthly payment (= % for interests + % for principal), we first need to convert that effective annual rate to its monthly counterpart: i = (1+0.15)^(1/12) - 1 = 0.0017149 = 1.1749%

Then, by using the annuity formula, we have a factor of i*(1+i)^n / i so 64.261 where i = 1.1749% from above and n = 120 as 120 months in 10 years.

Then, to get the monthly payment, we simply divide the principal by the factor: €150,000/64.261 = €2334.22 per month to reimburse.

What I do no get is the mistake I make in the alternative approach without using a monthly rate: we simply calculate the annuity factor with r = 15%, compute our yearly payment and we get €150,000/5.284 = €29,887.81. This value, when divided by 12 (to get the monthly payment), however does not give the €2334.22 from above.

When writing those lines, I reckon it is because the payment has to be done monthly and not yearly. Therefore, how could we convert the 15% to get a yearly rate which takes into account the fact that our payments are done monthly ? I guess it has something to do with the compounding interest formula: (1+r) = (1+(i/n))^(n*t) but I am a bit lost...

Any help on how to solve the problem with this alternative view?

Thanks!

You'd have to calculate the monthly annuity factor, multiply it by 12, and solve the annuity factor formula for `i`. Since there's no algebraic way to reverse the annuity factor formula to solve for `i`, there's not a formula to calculate the equivalent "annual" rate based on a periodic payment without going through the monthly conversion first.