Let's substitute the given values into the formula and simplify it step by step.

Formula:

$P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}$

Where:

• $A = 11,000$ (Future Value)
• $r = 0.04$ (Annual Interest Rate)
• $n = 2$ (Compounded Semiannually)
• $t = 3$ (Time in Years)

Step 1: Substitute the values into the formula:

$P = \frac{11,000}{\left(1 + \frac{0.04}{2}\right)^{2 \times 3}}$

Step 2: Simplify the term inside the parentheses:

$1 + \frac{0.04}{2} = 1 + 0.02 = 1.02$

Step 3: Raise this value to the power of $nt$:

$1.02^{2 \times 3} = 1.02^6$

Now, calculate $1.02^6$:

$1.02^6 \approx 1.126162$

Step 4: Divide the future value by this result to find the present value $P$:

$P = \frac{11,000}{1.126162} \approx 9,765.2992$

Step 5: Round to the nearest cent:

$P \approx 9,765.30$

Thus, the amount that should be deposited today is $9,765.30.

Would you like any further clarification or details on this calculation?

Related Questions:

1. How much interest will be earned by the end of three years?
2. What is the formula for future value if the present value and interest rate are known?
3. How does the value change if compounded quarterly instead of semiannually?
4. What is the present value for a future value of $11,000 at 3% compounded semiannually?
5. How do you calculate the effective annual rate (EAR) for 4% compounded semiannually?
6. What is the present value if the future value is $12,000 under the same conditions?
7. How does an increase in the time period to 5 years affect the present value?
8. What would be the required present value if the goal was $20,000 in three years?

Tip: Always double-check calculations when working with compound interest, as small errors in the exponentiation step can lead to incorrect results.