Following the birth of a child, a parent wants to make an initial investment $P_{0}$ that will grow to $\$ 40,000$ for the child's education at age 18 . Interest is compounded continuously at $5 \%$. What should the initial investment be? Such an amount is called the present value of $\$ 40,000$ due 18 years from now.

Step 1 ：We are given that the future value of the investment (A) is $40,000, the annual interest rate (r) is 5% or 0.05 in decimal form, and the time period (t) is 18 years. We need to find the present value of the investment (P_{0}).

Step 2 ：The formula for continuous compounding is given by: \(A = P_{0}e^{rt}\)

Step 3 ：We can rearrange the formula to solve for \(P_{0}\): \(P_{0} = \frac{A}{e^{rt}}\)

Step 4 ：Substituting the given values into the formula, we get: \(P_{0} = \frac{40000}{e^{0.05*18}}\)

Step 5 ：Calculating the above expression, we find that \(P_{0} \approx 16262.79\)

Step 6 ：Final Answer: The initial investment should be approximately \(\boxed{16262.79}\)