Part 1 of 2
a. Use the appropriate formula to find the value of the annuity.
b. Find the interest.
| Periodic Deposit | Rate | Time |
| :--------------- | :--------------------------------- | :-------- |
| $100 at the end of every six months | 5.5 % compounded semiannually | 25 years |
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a. The value of the annuity is $ [ ]
(Do not round until the final answer. Then round to the nearest dollar as needed.)
Answer (2)
The value of the annuity after 25 years of semiannual deposits of $100 at a 5.5% interest rate is approximately $10,481, and the interest earned over this period is approximately $5,481.
;
Calculate the periodic interest rate: i = 2 5.5% = 0.0275 .
Calculate the total number of periods: N = 25 × 2 = 50 .
Calculate the future value of the annuity: F V = 100 × 0.0275 ( 1 + 0.0275 ) 50 − 1 = 10481.17 .
Calculate the interest earned: I n t eres t = 10481.17 − ( 100 × 50 ) = 5481.17 . The value of the annuity is $10481 and the interest is $5481 .
Explanation
Understanding the Problem We are given a periodic deposit of $100 at the end of every six months, an interest rate of 5.5% compounded semiannually, and a time period of 25 years. We need to find the value of the annuity and the interest earned.
Calculating Periodic Interest Rate and Total Periods First, we need to determine the periodic interest rate and the total number of periods. The interest rate is 5.5% per year, compounded semiannually, so the periodic interest rate is 2 5.5% = 2.75% = 0.0275 . The time period is 25 years, with deposits made semiannually, so the total number of periods is 25 × 2 = 50 .
Calculating the Future Value of the Annuity Next, we use the future value of an ordinary annuity formula to find the value of the annuity: F V = P × i ( 1 + i ) N − 1 where P = 100 , i = 0.0275 , and N = 50 . Plugging in these values, we get: F V = 100 × 0.0275 ( 1 + 0.0275 ) 50 − 1 F V = 100 × 0.0275 ( 1.0275 ) 50 − 1 F V = 100 × 0.0275 3.98117 − 1 F V = 100 × 0.0275 2.98117 F V = 100 × 108.40618 F V = 10481.17 Therefore, the value of the annuity is $10481.17 .
Calculating Total Deposits Now, we calculate the total amount of deposits made over the 25 years. Since the deposit is $100 every six months, and there are 50 periods, the total amount of deposits is: T o t a l De p os i t s = P × N = 100 × 50 = 5000 So, the total amount of deposits is $5000 .
Calculating the Interest Earned Finally, we calculate the interest earned by subtracting the total deposits from the future value of the annuity: I n t eres t = F V − T o t a l De p os i t s = 10481.17 − 5000 = 5481.17 Therefore, the interest earned is $5481.17 .
Final Answer a. The value of the annuity is $10481 . b. The interest is $5481 .
Examples
Annuities are commonly used in retirement planning. For example, if you deposit a certain amount of money regularly into an annuity account, it grows over time due to the interest earned. This problem demonstrates how to calculate the future value of such an annuity and the total interest earned, which helps in estimating the potential returns from a retirement savings plan. Understanding these calculations allows individuals to make informed decisions about their financial future and plan for a comfortable retirement.
