Task: You got an offer to buy a boat with a present value of 10800€.
The deposit is 6000€ you have to pay 1200€ a year (as instalment) for 3 years.
The rest of your debt is paid in 4 equal instalments (which are also paid at the end of the year once a year, so the whole process takes 7 years).
Calculate the amount of the new instalment if the annual interest rate is 8%.
My Question: Is there any formular that could help me with this task, I tried everything I could for basically the whole day compound interest, normal interest, tables and nothing seems to work. How do I solve this?
Firstly we choose a reference date. All payments are discounted/compounded to this date. I choose $t=0$ (present).
The present value of $n$ equal yearly payments $r$, which a made at the end of the years is
$$PV=r\cdot \frac{(1+i)^n-1}{i\cdot (1+i)^n}, \ \text{where i is the interest rate}$$
Therefore the equation for your exercise is
$$10800=6000+1200\cdot \frac{(1+0.08)^3-1}{0.08\cdot (1+0.08)^3}+\underbrace{\underbrace{X\cdot \frac{(1+0.08)^4-1}{0.08\cdot (1+0.08)^4}}_{\text{value at t=3}}\cdot \frac1{(1+0.08)^3}}_{\text{present value at t=0}}$$
It remains to solve the equation for $X$. Feel free to ask if something is unclear.