Mike wants to donate $5,000,000 to establish a fund to provide an annual scholarship in perpetuity. The fund will earn an interest rate of 4.54% p.a. compounded half-yearly (*j*2=4.54% p.a.) and the first scholarship will be first awarded 3.5 years after the date of the donation.

(a) What is the amount of the annual scholarship (rounded to two decimal places)?


To determine the amount of the annual scholarship that Mike can provide in perpetuity, we need to calculate the value of an annuity based on the given conditions.

### Step 1: Understand the Interest Rate and Scholarship Timing
The interest rate provided is $ j_2 = 4.54\% $ compounded semi-annually. To find the equivalent effective annual interest rate:

\[
i = \left(1 + \frac{j_2}{2}\right)^2 - 1 = \left(1 + \frac{0.0454}{2}\right)^2 - 1 = \left(1 + 0.0227\right)^2 - 1 = 1.04620129 - 1 = 0.04620129
\]

So, the effective annual interest rate $ i $ is approximately $ 4.6201\% $.

### Step 2: Perpetuity Calculation
For a perpetuity, the amount of the scholarship $ S $ is determined by the formula:

\[
S = \text{Fund Value} \times \text{Effective Interest Rate}
\]

### Step 3: Adjust the Fund Value
Since the first scholarship is awarded 3.5 years after the donation, the initial fund of $5,000,000 will accumulate interest for 3.5 years before the first disbursement. Therefore, we need to adjust the initial fund value to reflect this interest accumulation.

#### Step 3.1: Determine the Future Value of the Fund After 3.5 Years
3.5 years include 7 semi-annual periods. The future value $ FV $ of the fund after 3.5 years is calculated using the formula:

\[
FV = \text{Fund Value} \times \left(1 + \frac{j_2}{2}\right)^n
\]
where $ n $ is the number of semi-annual periods.

\[
FV = 5,000,000 \times \left(1 + \frac{0.0454}{2}\right)^7 = 5,000,000 \times \left(1.0227\right)^7 \approx 5,000,000 \times 1.1687 = 5,843,500
\]

### Step 4: Calculate the Annual Scholarship Amount
Using the future value of the fund, the annual scholarship $ S $ can be calculated as:

\[
S = 5,843,500 \times 0.04620129 \approx 269,885.37
\]

### Final Answer
The amount of the annual scholarship is **$269,885.37**, rounded to two decimal places.

Would you like more details or have any other questions? Here are some related problems you might be interested in:

1. How would the scholarship amount change if the interest rate were lower?
2. What if Mike wanted the first scholarship awarded after 5 years?
3. How does the semi-annual compounding affect the calculation compared to annual compounding?
4. What happens to the scholarship amount if the fund earns interest at a varying rate?
5. How much would Mike need to donate if he wanted the scholarship amount to be $300,000 annually?
6. What is the impact of inflation on the real value of the scholarship?
7. How much interest will the fund accumulate in the first 3.5 years?
8. What would be the scholarship amount if it were paid monthly instead of annually?

**Tip**: When calculating perpetuities, always ensure that the interest rate is appropriately matched to the compounding period to avoid errors in the annuity calculation.

Learn how to calculate the annual scholarship amount in perpetuity using compounded interest rates. This problem involves determining the effective annual interest rate, adjusting the initial fund value, and applying the perpetuity formula. Suitable for advanced mathematics students.

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