In practice, there are two main amortization methods used by and large for retail mortgages:
- The classical amortization formula where all installments have the same value, however the interest % and principal % vary, with the former decreasing over time.
$$A = P \frac{r(1+r)^n}{(1+r)^n - 1}$$ - Decreasing installments with principal equal for all months and a decreasing interest component
Take for example the case of the following mortgage loan:
P = 250,000
r = 8.00%
n = 360 months
A = 1834.41
o.w interest = 1.667.67 (91% of first installment)
o.w principal 167.7 (9% of first installment)
Is it possible to devise a method to fix the interest part (50%) = the principal part (50%) for each installment, when reimbursing a bank loan - but to still reach the interest rate applied (for ex 8%)?
Given principal would be paid in an accelerated fashion in this scenario - what would be the impact on:
- The monthly installment factor A? (would it increase?)
- The duration of time for the loan would increase? (i.e a 30 year old mortgage would reach 60 years in such a system?)
Thank you
Here you are repaying the loan at the rate $r$, where $r$ is the interest rate, so that the repayment and interest rate payment are the same. However, the maturity of the loan will be infinite assuming $r<1$, as you never pay back the loan exactly. The instalment is $2rP_t$, where $P_t$ is the principal that is left at time $t$. This is decreasing over time. The interest payment and debt repayment are both $rP_t$
To add some details, the time $t$ repayment is
$$P_0\times(1-r)^tr$$
Total repayments up to $t$ are
$$P_0\left[r+(1-r)r+(1-r)^2r+(1-r)^3r+\right]=P_0r \frac{1-(1-r)^t}{r}=P_0(1-(1-r)^t)$$
So the principal left at $t$ is
$$P_0(1-r)^t$$
This approaches 0 as $t$ tends to infinity but never reaches it exactly.