Suppose you are committed to owning a $210,000 Ferrari. If you believe your mutual fund can achieve an annual return of 10.75 percent, and you want to buy the car in 10 years on the day you turn 30, how much must you invest today?




To find out how much you must invest today to have $210,000 in 10 years with an annual return of 10.75%, we'll use the **Present Value (PV) formula**. The Present Value formula calculates the amount you need to invest now to achieve a specific future value given a certain interest rate and time period.

### Step-by-Step Solution

1. **Understand the variables involved:**
   - **Future Value (FV):** $210,000 (the cost of the Ferrari in 10 years)
   - **Annual Interest Rate (r):** 10.75% or 0.1075 as a decimal
   - **Number of Years (n):** 10 years
   - **Present Value (PV):** The amount we need to find (the initial investment required today)

2. **Present Value Formula:**

   The Present Value $ PV $ is given by the formula:

   \[
   PV = \frac{FV}{(1 + r)^n}
   \]

3. **Substitute the values into the formula:**

   - $ FV = 210,000 $
   - $ r = 0.1075 $
   - $ n = 10 $

   \[
   PV = \frac{210,000}{(1 + 0.1075)^{10}}
   \]

4. **Calculate $ (1 + r)^n $:**

   \[
   (1 + 0.1075)^{10} = 1.1075^{10}
   \]

   Let's calculate this:

   \[
   1.1075^{10} \approx 2.788
   \]

5. **Calculate the Present Value (PV):**

   \[
   PV = \frac{210,000}{2.788}
   \]

   \[
   PV \approx 75,345.62
   \]

### Final Answer

You must invest approximately **$75,345.62** today to have $210,000 in 10 years, assuming an annual return of 10.75%.

Would you like more details on any of the steps, or do you have any questions?

### Related Questions

1. How would the amount to invest today change if the interest rate were 9% instead of 10.75%?
2. What would happen to the required initial investment if the car's price increased to $250,000?
3. How much would you have if you invested $75,345.62 at 10.75% for 15 years instead of 10 years?
4. What if the investment was compounded semi-annually instead of annually? How much would you need to invest?
5. How does increasing the investment period affect the present value needed for the same future amount?

### Tip

When calculating investment needs for future goals, always consider the effect of compounding periods and the impact of different interest rates on your investment strategy.

Find out how much you need to invest today to afford a $210,000 Ferrari in 10 years with an annual return of 10.75%. Learn the Present Value formula and step-by-step calculation method.

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