You want to have a $\$ 100,000$ college fund in $20$ years. How much will you have to deposit now in an account with an APR of $5 \%$ and daily compounding?

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Q. You want to have a

\$ 100,000

college fund in

20

years. How much will you have to deposit now in an account with an APR of

5 \%

and daily compounding?

Identify Variables: First, let's identify the variables for the formula for compound interest: $\newline$ Future Value (FV) = $\$100,000$ $\newline$ Annual Interest Rate (r) = $5\%$ or $0.05$ $\newline$ Number of Years (t) = $20$ $\newline$ Number of times the interest is compounded per year (n) = $365$ (daily compounding)
Use Compound Interest Formula: We use the formula for compound interest to find the present value (PV), which is the amount we need to deposit now: $\newline$ $FV = PV \times (1 + \frac{r}{n})^{n\times t}$ $\newline$ We need to rearrange the formula to solve for PV: $\newline$ $PV = \frac{FV}{(1 + \frac{r}{n})^{n\times t}}$
Plug in Values: Now plug in the values: $\newline$ $PV = \frac{100,000}{(1 + \frac{0.05}{365})^{365\times20}}$
Calculate Denominator: Calculate the denominator $(1 + \frac{0.05}{365})^{365\times20}$ : $\newline$ First, calculate $\frac{0.05}{365}$ : $\newline$ $\frac{0.05}{365} = 0.0001369863$
Add $1$ : Now add $1$ to the result from the previous step: $\newline$ $1 + 0.0001369863 = 1.0001369863$
Exponentiation: Next, raise this sum to the power of $365 \times 20$ : $\newline$ $(1.0001369863)^{(365 \times 20)} = (1.0001369863)^{7300}$
Calculate Present Value: Calculate the exponentiation: $\newline$ $(1.0001369863)^{7300} \approx 2.653297705$
Perform Division: Now divide the future value by the result to find the present value: $PV = \frac{100,000}{2.653297705}$
Perform Division: Now divide the future value by the result to find the present value: $\newline$ $PV = \frac{100,000}{2.653297705}$ Perform the division to get the present value: $\newline$ $PV \approx 37,689.76$