How much should a father invest at the time his son is born in order to provide him with ​$90,000 at age 21​? Assume that the interest is 3​% compounded monthly.

Part 1
The father should invest ​?

To determine how much the father should invest, we need to calculate the present value (PV) of the future amount of $90,000, given that the interest rate is 3% compounded monthly for 21 years.

The formula for the present value when interest is compounded monthly is:

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

Where:
- $ PV $ = Present Value (the amount to be invested now)
- $ FV $ = Future Value ($90,000)
- $ r $ = Annual interest rate (3% or 0.03)
- $ n $ = Number of times interest is compounded per year (monthly, so 12)
- $ t $ = Number of years the money is invested (21 years)

Plugging in the values:

\[ PV = \frac{90000}{(1 + \frac{0.03}{12})^{12 \cdot 21}} \]

First, calculate the monthly interest rate:

\[ \frac{0.03}{12} = 0.0025 \]

Next, calculate the total number of compounding periods:

\[ 12 \cdot 21 = 252 \]

Then, calculate the compound factor:

\[ (1 + 0.0025)^{252} \]

Now, use this to find the present value:

\[ PV = \frac{90000}{(1.0025)^{252}} \]

We can calculate this step-by-step:

1. Calculate the compound factor:
\[ (1.0025)^{252} \approx 1.9307 \]

2. Divide the future value by the compound factor:
\[ PV = \frac{90000}{1.9307} \approx 46601.19 \]

So, the father should invest approximately $46,601.19.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. What is the future value formula for continuous compounding?
2. How would the investment amount change if the interest rate were compounded quarterly instead of monthly?
3. What if the interest rate were 4% instead of 3%? How would that affect the investment amount?
4. How does the number of compounding periods per year affect the present value?
5. What is the effect of a different time period, say 18 years instead of 21 years, on the present value?
6. How can one calculate the interest earned over the investment period?
7. What are some real-world applications of present value calculations?
8. How does inflation affect the future value and present value calculations?

**Tip:** Always verify the compounding frequency used in interest calculations, as it significantly impacts the results.

Learn how to calculate the present value required to invest at birth to provide $90,000 at age 21, considering 3% interest compounded monthly. Step-by-step solution with key formulas and calculations.

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