Find the future value for the ordinary annuity with the given payment and interest rate. [tex]PMT = $850; 1.45\% compounded semiannually for 2 years.[/tex] The future value of the ordinary annuity is $ $\square$ . (Do not round until the final answer. Then round to the nearest cent as needed.)
Answer (2)
The future value of the ordinary annuity is calculated using the formula for the future value of an annuity with the given payment of $850, a semiannual interest rate of 1.45%, compounded over 4 periods. After working through the calculations, the final amount is approximately $3,457.03. This value represents the total accumulated after 2 years of making regular payments into the annuity at the specified interest rate.
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Calculate the periodic interest rate: i = 2 0.0145 = 0.00725 .
Calculate the number of periods: n = 2 × 2 = 4 .
Apply the future value of an ordinary annuity formula: F V = PMT × i ( 1 + i ) n − 1 .
Substitute the values and calculate: F V = 850 × 0.00725 ( 1 + 0.00725 ) 4 − 1 = 3437.15 . The future value of the ordinary annuity is 3437.15 .
Explanation
Understanding the Problem We are asked to find the future value of an ordinary annuity. We are given the payment amount (PMT), the annual interest rate, the compounding period, and the number of years.
Calculating the Periodic Interest Rate First, we need to determine the interest rate per compounding period and the total number of compounding periods. The annual interest rate is 1.45%, and it is compounded semiannually, which means twice a year. So, the interest rate per period is: i = 2 1.45% = 2 0.0145 = 0.00725
Calculating the Number of Periods The annuity is for 2 years, and the interest is compounded semiannually, so the total number of compounding periods is: n = 2 years × 2 periods/year = 4 periods
Stating the Future Value Formula The formula for the future value (FV) of an ordinary annuity is: F V = PMT × i ( 1 + i ) n − 1
Substituting the Values Now, we substitute the given values into the formula: PMT = $850 i = 0.00725 n = 4 F V = 850 × 0.00725 ( 1 + 0.00725 ) 4 − 1
Calculating the Future Value Now, we calculate the future value: F V = 850 × 0.00725 ( 1.00725 ) 4 − 1 F V = 850 × 0.00725 1.02932 − 1 F V = 850 × 0.00725 0.02932 F V = 850 × 4.04393
F V = 3437.3405
Final Answer Rounding to the nearest cent, the future value of the ordinary annuity is $3437.15.
Examples
Understanding the future value of an annuity is crucial in financial planning. For instance, if you decide to save a fixed amount regularly for your retirement, calculating the future value of this annuity helps you estimate the total savings you'll have at the end of the period. Similarly, businesses use annuity calculations to determine the returns on investment plans or to manage sinking funds for future expenses. This concept is also applicable in understanding loan repayments, where regular payments contribute to reducing the loan amount over time.
