Questions: Question 3 20 Points Time Value of Money. Find P=Present worth when given F=Future value Given: You wish to withdraw 10,000, 8 years in the future. What is your lump sum investment in a savings account today if the account compounds every 6 months at a 9% interest. Find: What is the present value, to achieve 10,000 in 8 years? (A) 54,881 (B) 54,945 (C) 55,019 (D) 55,103

Solution Steps

To find the present value (P) given the future value (F), interest rate, and compounding frequency, we use the formula for the present value of a future sum:

\[ P = \frac{F}{(1 + \frac{r}{n})^{nt}} \]

where:

• \( F \) is the future value (\$10,000),
• \( r \) is the annual interest rate (9% or 0.09),
• \( n \) is the number of compounding periods per year (2, since it compounds every 6 months),
• \( t \) is the number of years (8).

We are given the following values:

• Future value \( F = 10000 \)
• Annual interest rate \( r = 0.09 \)
• Compounding periods per year \( n = 2 \)
• Number of years \( t = 8 \)

To find the present value \( P \), we use the formula:

\[ P = \frac{F}{\left(1 + \frac{r}{n}\right)^{nt}} \]

Substituting the given values into the formula:

\[ P = \frac{10000}{\left(1 + \frac{0.09}{2}\right)^{2 \cdot 8}} \]

First, we calculate the term inside the parentheses:

\[ 1 + \frac{0.09}{2} = 1 + 0.045 = 1.045 \]

Next, we raise this to the power of \( 16 \) (since \( 2 \cdot 8 = 16 \)):

\[ (1.045)^{16} \approx 1.811364 \]

Now, we can calculate \( P \):

\[ P = \frac{10000}{1.811364} \approx 5524.6932 \]

Final Answer