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The problem is written in German, and here's the translation to English:

**Problem:**

Due to outstanding performance in her master's thesis, a student receives a scholarship at the end of the year; for five years she will receive an annual advance payment of €10,000 (starting next year).

2a) The student wants to convert the annual advance payments into monthly arrears payments with a term of two years.

What is the amount of the monthly payment at an interest rate of 4%?

**Solution:**

To solve this problem, we need to find the equivalent monthly payment amount given the annual advance payment, interest rate, and time frame. Here's the step-by-step approach:

1. **Identify the given values:**
   - Annual advance payment (A): €10,000
   - Number of years (N): 5
   - Interest rate (annual, compounded monthly): 4% or 0.04

2. **Convert annual interest rate to monthly interest rate:**
   \[
   i = \frac{0.04}{12} = 0.003333 \text{ or } 0.3333\%
   \]

3. **Calculate the present value of the annual payments:**

   The present value of an annuity (advance payments) is calculated using the formula:
   \[
   PV = A \times \frac{1 - (1 + r)^{-N}}{r}
   \]
   where \( r \) is the interest rate per period and \( N \) is the total number of periods.

   For advance payments, the formula is adjusted to:
   \[
   PV = A \times \left( \frac{1 - (1 + r)^{-N}}{r} \right) \times (1 + r)
   \]

   Substituting the given values:
   \[
   PV = 10,000 \times \left( \frac{1 - (1 + 0.04)^{-5}}{0.04} \right) \times (1 + 0.04)
   \]

   Calculate:
   \[
   PV = 10,000 \times \left( \frac{1 - (1.04)^{-5}}{0.04} \right) \times 1.04
   \]

   Let's compute \( (1.04)^{-5} \):
   \[
   (1.04)^{-5} \approx 0.821927
   \]

   Therefore:
   \[
   PV = 10,000 \times \left( \frac{1 - 0.821927}{0.04} \right) \times 1.04
   \]
   \[
   PV = 10,000 \times \left( \frac{0.178073}{0.04} \right) \times 1.04
   \]
   \[
   PV = 10,000 \times 4.451825 \times 1.04
   \]
   \[
   PV \approx 46370.99
   \]

4. **Convert the present value to monthly payments for two years:**

   The formula for the present value of an ordinary annuity (monthly arrears payments) is:
   \[
   PV = PMT \times \frac{1 - (1 + i)^{-n}}{i}
   \]
   where \( PMT \) is the monthly payment, \( i \) is the monthly interest rate, and \( n \) is the total number of monthly payments.

   Here, \( n = 2 \times 12 = 24 \).

   Rearrange to solve for \( PMT \):
   \[
   PMT = PV \times \frac{i}{1 - (1 + i)^{-n}}
   \]
   Substituting the values:
   \[
   PMT = 46370.99 \times \frac{0.003333}{1 - (1.003333)^{-24}}
   \]

   Calculate \( (1.003333)^{-24} \):
   \[
   (1.003333)^{-24} \approx 0.9234
   \]

   Therefore:
   \[
   PMT = 46370.99 \times \frac{0.003333}{1 - 0.9234}
   \]
   \[
   PMT = 46370.99 \times \frac{0.003333}{0.0766}
   \]
   \[
   PMT \approx 2018.26
   \]

Therefore, the monthly payment at an interest rate of 4% is approximately €2018.26.

Would you like further details or have any questions?

Here are 5 more questions you might be interested in:
1. How do you calculate the future value of an annuity?
2. What is the difference between present value and future value?
3. How do you determine the effective annual rate from a nominal interest rate?
4. What are the formulas for compound interest and simple interest?
5. How can you use the amortization formula for loan repayments?

**Tip:** When working with financial calculations, always ensure you correctly convert interest rates and time periods to match the context of the problem (e.g., annual to monthly).

Learn how to convert annual advance payments to monthly payments with a detailed financial calculation. This problem involves determining monthly payments from annual advance payments with an interest rate of 4% over a two-year period.

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