Check my work.
Sam Long anticipates he will need approximately $225,000 in 15 years to cover his 3-year-old daughter's college bills for a 4-year degree.
How much would he have to invest today at an interest rate of 8% compounded semiannually? (Use the Table provided.) Note: Do not round intermediate calculations. Round your answer to the nearest cent.
Answer (2)
Calculate the semiannual interest rate: i = 2 0.08 = 0.04 .
Calculate the number of compounding periods: n = 15 × 2 = 30 .
Apply the present value formula: P V = ( 1 + 0.04 ) 30 225000 .
Calculate the present value and round to the nearest cent: P V ≈ $69 , 371.70 .
Explanation
Understanding the Problem We need to determine the present value (PV) of an investment that will grow to $225,000 in 15 years, given an annual interest rate of 8% compounded semiannually.
Calculating Semiannual Interest Rate First, we need to find the semiannual interest rate. Since the interest is compounded semiannually, we divide the annual interest rate by 2: i = 2 r = 2 0.08 = 0.04
Calculating Number of Compounding Periods Next, we need to find the total number of compounding periods. Since the investment is for 15 years and the interest is compounded semiannually, we multiply the number of years by 2: n = t × 2 = 15 × 2 = 30
Applying the Present Value Formula Now, we use the present value formula for compound interest: P V = ( 1 + i ) n F V where FV is the future value, i is the interest rate per period, and n is the number of periods.
Substituting Values Substitute the given values into the formula: P V = ( 1 + 0.04 ) 30 225000
Calculating the Denominator Calculate ( 1 + 0.04 ) 30 : ( 1 + 0.04 ) 30 = 1.0 4 30 ≈ 3.24339751
Calculating Present Value Now, calculate the present value: P V = 3.24339751 225000 ≈ 69371.70029
Rounding the Result Round the present value to the nearest cent: P V ≈ $69 , 371.70
Final Answer Therefore, Sam Long would have to invest approximately $69,371.70 today to have $225,000 in 15 years.
Examples
Understanding present value is crucial for financial planning. For instance, if you want to buy a house in 10 years and estimate needing $100,000 for a down payment, you can calculate how much to invest today. Assuming a 5% annual return, compounded annually, you'd determine the present value needed. This calculation helps in setting realistic savings goals and making informed investment decisions, ensuring you meet your future financial needs.
Sam Long needs to invest approximately $69,371.70 today to have $225,000 in 15 years at an 8% interest rate compounded semiannually. This was calculated using the present value formula, considering the semiannual compounding of interest. The final value was obtained after calculating the necessary factors and rounding to the nearest cent.
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