Annuities immediate and annuities due

The payments are constant and are made at the end of each year starting from the first, so the annuity is an annuity immediate, and the period of the payments coincides with that of the rate. The present value is the payment multiplied by $a_{\overline{8}|0.03}$, the accumulated value is the payment multiplied by $s_{\overline{8}|0.03}$.

As a check, the accumulated value must be the present value carried to the end of the eighth year, and indeed $10529.54\cdot 1.03^8\approx 13338.50$. The present value is less than the sum of the payments, $12000$ euros, because each discounted payment is worth less than its amount; the accumulated value is greater, because each payment earns interest until the end.

The payments are monthly and the rate is annual, so first one computes the equivalent monthly rate, because the annuity formulas require the rate to refer to the same period as the payments. In the computation of $a_{\overline{12}|i_{12}}$ one uses the fact that $(1+i_{12}{)}^{12}=1.06$, which avoids raising an approximate number to the twelfth power.

The rent is paid at the beginning of each month, so the annuity is an annuity due, and its present value is that of the annuity immediate multiplied by $1+i_{12}$. The present value of about $7011.24$ euros is the sum that, paid in a single amount at the beginning of the year, is equivalent to the twelve payments.

The deposits are made at the beginning of each year, so they form an annuity due, and the balance required at the end of the tenth year is its accumulated value, $R(1+i)s_{\overline{10}|i}$. As the diagram shows, the last deposit is made at time $9$ and earns interest for one year, while the first earns interest for $10$ years.

The equation is linear in $R$, and dividing gives a payment of about $1790.72$ euros. As a check, the ten deposits add up to about $17907$ euros, and the difference up to $20000$ euros is due to the interest.

The debt is the present value of the payments, so one imposes $15000=2500a_{\overline{n}|0.05}$ and solves for $1.05^{-n}$. The number $\frac{15000\cdot 0.05}{2500}=0.3$ is less than $1$, because the payment exceeds the interest of the first year, $750$ euros; therefore the problem has a solution, and taking the logarithm gives $n\approx 7.31$.

The number of payments is not an integer, so $7$ full payments are made and an eighth, reduced payment $X$ at time $8$. The seven full payments cover a present value of about $14465.93$ euros; the remaining part of the debt, about $534.07$ euros in present value, must be paid at time $8$, therefore it is multiplied by $1.05^8$. The last payment is therefore about $789.06$ euros.