You want to endow a scholarship now that will pay 

​$6 comma 0006,000 

per year​ forever, starting the first award 

1010 

years from now. If the​ school's endowment discount rate is 

6 %6%​, 

what amount must you donate today to endow the​ scholarship?

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Part 1

The amount you must donate today to endow the scholarship is 

​$enter your response here. 

 ​(Round to the nearest​ cent.)

To calculate how much you must donate today to endow the scholarship, we will use the concept of **present value**. Since this scholarship payment continues indefinitely (forever), we are essentially dealing with a perpetuity, but with a **deferred start** (10 years from now).

Here's how to approach it step-by-step:

1. **Step 1: Present Value of a Perpetuity Formula**
   The formula for the present value of a perpetuity starting **today** is:

   \[
   PV = \frac{C}{r}
   \]

   where:
   - $ C $ is the annual payment ($6,000),
   - $ r $ is the discount rate (6% = 0.06).

   Substituting these values:

   \[
   PV_{\text{today}} = \frac{6000}{0.06} = 100,000
   \]

   However, this is the present value **if** the scholarship started **today**. But since it starts **10 years from now**, we must adjust this by discounting it back 10 years.

2. **Step 2: Adjust for the 10-Year Deferral**
   The present value of the perpetuity must be discounted to the present, taking into account that the payments start 10 years from now. The formula for discounting a future sum is:

   \[
   PV_{\text{today}} = \frac{PV_{\text{future}}}{(1 + r)^t}
   \]

   where:
   - $ PV_{\text{future}} $ is the present value of the perpetuity calculated in step 1 ($100,000),
   - $ r $ is the discount rate (0.06),
   - $ t $ is the number of years (10 years).

   Substituting these values:

   \[
   PV_{\text{today}} = \frac{100,000}{(1 + 0.06)^{10}}
   \]

   Now, let's calculate this.

\[
PV_{\text{today}} = \frac{100,000}{(1.06)^{10}} \approx \frac{100,000}{1.790847} \approx 55,871.97
\]

Therefore, you must donate **$55,871.97** today to endow the scholarship.

---

Would you like further details or clarification on any of these steps? 

Here are 5 questions to deepen your understanding:
1. How would the amount change if the scholarship started after 5 years instead of 10?
2. What would happen if the discount rate was 5% instead of 6%?
3. How does the perpetuity formula apply to other real-world scenarios?
4. How would you calculate the donation if the scholarship amount increased each year?
5. Can you adjust the formula for finite periods instead of perpetuity?

**Tip:** When dealing with perpetuities, the key is to remember that the payments are constant and continue indefinitely, so you can use the simple $ \frac{C}{r} $ formula to find their present value.

You want to endow a scholarship now that will pay $6,000 per year forever, starting the first award 10 years from now. If the school's endowment discount rate is 6%, what amount must you donate today to endow the scholarship?

Find out how much you need to donate today to endow a $6,000-per-year scholarship that starts in 10 years, using a 6% discount rate. This problem covers key financial concepts such as present value, perpetuities, and discounting future value. Suitable for college-level finance or economics students.

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