Question

sandra wants to buy a car when she graduates from central university 5 years from now. she believes that she will need \\$29,300 to buy the car.

(a) calculate how much money sandra must put into her savings account today to have \\$29,300 in 5 years, assuming she can earn 10\\% compounded annually. (for calculation purposes, use 4 decimal places as displayed in the factor table provided and round final answer to 0 decimal place, e.g. 58,975.)

amount \\$

(b) calculate how much money sandra must put into her savings account today to have \\$29,300 in 5 years, assuming she can earn 10\\% compounded semiannually. (for calculation purposes, use 4 decimal places as displayed in the factor table provided and round final answer to 0 decimal place, e.g. 58,975.)

amount \\$

Explanation:

🆕 New Concept Discovered: Present Value of a Single Sum
Finding today's value of a future amount.

Step 1: Identify the variables for part (a)

To find how much Sandra must invest today (Present Value, or \( \text{PV} \)) to have a future amount (\( \text{FV} \)) of \( \$29,300 \) in 5 years with annual compounding:

• Future Value (\( \text{FV} \)) = \( \$29,300 \)
• Number of periods (\( n \)) = \( 5 \) years
• Annual interest rate (\( i \)) = \( 10\% \)

The formula for the Present Value is:

$$ \text{PV} = \text{FV} \times \text{PVF}_{i, n} $$

Where \( \text{PVF}_{i, n} \) is the Present Value Factor for a single sum at interest rate \( i \) for \( n \) periods:

$$ \text{PVF}_{10\%, 5} = \frac{1}{(1 + 0.10)^5} = \frac{1}{1.61051} \approx 0.6209 $$

Using the standard 4-decimal place factor table value:

$$ \text{PVF}_{10\%, 5} = 0.6209 $$

Step 2: Calculate the amount for part (a)

Multiply the future value by the present value factor:

$$ \text{PV} = \$29,300 \times 0.6209 $$

$$ \text{PV} = \$18,192.37 $$

Rounding to the nearest whole dollar (0 decimal places):

$$ \text{PV} \approx \$18,192 $$

Step 3: Identify the variables for part (b)

When interest is compounded semiannually (twice a year), we must adjust the interest rate per period (\( i \)) and the total number of compounding periods (\( n \)):

• Number of periods (\( n \)) = \( 5 \text{ years} \times 2 = 10 \text{ periods} \)
• Interest rate per period (\( i \)) = \( \frac{10\%}{2} = 5\% \text{ per period} \)

The Present Value Factor for a single sum at \( 5\% \) for \( 10 \) periods is:

$$ \text{PVF}_{5\%, 10} = \frac{1}{(1 + 0.05)^{10}} = \frac{1}{1.62889} \approx 0.6139 $$

Using the standard 4-decimal place factor table value:

$$ \text{PVF}_{5\%, 10} = 0.6139 $$

Step 4: Calculate the amount for part (b)

Multiply the future value by the adjusted present value factor:

$$ \text{PV} = \$29,300 \times 0.6139 $$

$$ \text{PV} = \$17,987.27 $$

Rounding to the nearest whole dollar (0 decimal places):

$$ \text{PV} \approx \$17,987 $$

Answer:

(a) Amount: $18,192

(b) Amount: $17,987