A company wants to finance a new depot by means of a 30-year mortgage.

Bank 2 offers an **annuity** mortgage of 360.000 Euro at a yearly nominal interest rate of 6%. A payment (redemption and interest) has to be made at the end of each month.
 
Compute the sum of payments for the mortgage. What is the total sum of payments over the 30 years? 

To compute the total sum of payments for the 30-year mortgage, we need to follow the steps outlined below:

### Given Data:
- **Loan amount (Principal)**: \( P = 360,000 \) Euros
- **Annual nominal interest rate**: \( i = 6\% \) (or 0.06 as a decimal)
- **Mortgage term**: 30 years
- **Payment frequency**: Monthly

### Step 1: Convert the interest rate to a monthly rate.
The interest rate given is a yearly nominal interest rate. Since the payments are made monthly, we need to convert this to a monthly interest rate:

\[
r = \frac{i}{12} = \frac{0.06}{12} = 0.005 \quad \text{(monthly interest rate)}
\]

### Step 2: Calculate the number of total payments.
The mortgage lasts for 30 years with monthly payments, so the total number of payments \( n \) is:

\[
n = 30 \times 12 = 360 \quad \text{(monthly payments over 30 years)}
\]

### Step 3: Calculate the monthly payment using the annuity formula.
The formula for calculating the monthly payment \( M \) in an annuity mortgage is:

\[
M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}
\]

Substitute the values we have:

\[
M = 360,000 \times \frac{0.005(1 + 0.005)^{360}}{(1 + 0.005)^{360} - 1}
\]

We can now calculate this.The monthly payment for the mortgage is approximately **2,158.38 Euros**.

To find the total sum of payments over the 30 years:

\[
\text{Total sum of payments} = 2,158.38 \times 360 = 777,017.48 \, \text{Euros}
\]

Thus, the total sum of payments over the 30 years is approximately **777,017.48 Euros**.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How much of the total payment is interest versus principal repayment?
2. What would the total payment be if the interest rate were 5% instead of 6%?
3. How does the monthly payment change if the loan term is 20 years instead of 30 years?
4. What is the impact of making extra monthly payments on the total interest paid?
5. How does the annuity formula differ for other types of loans, such as fixed-interest or variable-rate loans?

**Tip:** The longer the loan term, the more interest you'll pay over time. Reducing the term can lead to significant savings on interest!

A company wants to finance a new depot by means of a 30-year mortgage. Bank 2 offers an annuity mortgage of 360.000 Euro at a yearly nominal interest rate of 6%. A payment (redemption and interest) has to be made at the end of each month. Compute the sum of payments for the mortgage. What is the total sum of payments over the 30 years?

Learn how to compute the total sum of payments for a 30-year annuity mortgage of 360,000 Euros at a 6% yearly interest rate. This finance problem demonstrates key concepts like the time value of money, annuity mortgage formulas, and monthly interest rate calculations. Suitable for finance and business students.

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