Problem 3 - Algebra and Trigonometry

Question

Annuity Find the amount of an annuity that consists of 10 annual payments of \(\$ 1000\) each into an account that pays 6\(\%\) interest per year.

Step-by-Step Solution

Answer

The annuity amounts to approximately $13,147.45 after 10 years.

Step 1: Understand the Formula

To find the future value of an annuity, we use the formula: \( FV = P \times \frac{(1 + r)^n - 1}{r} \) where \( P \) is the annual payment, \( r \) is the interest rate per period, and \( n \) is the number of payments.

Step 2: Identify and Substitute Values

In this exercise, \( P = 1000 \), \( r = 0.06 \) (since 6% in decimal form is 0.06), and \( n = 10 \). Substitute these values into the formula: \( FV = 1000 \times \frac{(1 + 0.06)^{10} - 1}{0.06} \).

Step 3: Calculate the Future Value

First, calculate \((1 + 0.06)^{10}\). This evaluates to approximately \(1.790847\). Now substitute this into the formula: \( FV = 1000 \times \frac{1.790847 - 1}{0.06} \). This becomes \( 1000 \times \frac{0.790847}{0.06} \).

Step 4: Simplify the Expression

Simplify \( \frac{0.790847}{0.06} \) to get approximately \(13.14745\). So, the future value becomes \( 1000 \times 13.14745 \), which equals approximately \( 13147.45 \).

Key Concepts

Future Value

The future value of an annuity is the total amount you will have in the future after making all your regular payments into an interest-earning account over a set period. It represents the sum of all payments made, along with the interest earned on those payments.
For example, if you make regular deposits into a savings account, the future value is the amount you expect to have at the end of your saving period. It's calculated using the interest rate, the number of deposits, and the amount of each deposit.

• Future value increases with more payments: The more payments you make, the higher your future value will be.
• Higher interest rates also increase future value: A higher rate means more interest, building up your future value over time.
• The amount is greater over time: The further ahead the time horizon, the more future value will grow.

Interest Rate

The interest rate is the percentage at which your investments grow per period. It's a key factor in determining how much your money will grow over time. In the context of an annuity, it's the rate that applies to each period’s payment.
Interest rates can vary depending on economic conditions and financial institutions. In simple terms, if you have an interest account, a 6% interest rate means your money grows by 6% each year.

• Expressed as a decimal: When using formulas, convert percent to decimal (e.g., 6% becomes 0.06).
• Compounds over time: Interest doesn't just earn on the original sum; it earns on the accumulated interest as well.
• Varies among accounts: Different savings accounts or investments may offer different rates.

Annual Payments

Annual payments refer to fixed amounts that are paid or deposited into an account each year. This is a crucial part of calculating an annuity's future value since the regular contributions compound over time with interest.
For instance, in our scenario, an annual payment of $1000 is made every year for ten years. Even though the same amount is paid each year, the accumulated value at the end is more than just the sum of payments due to interest.

• Regular and consistent: Payments are made yearly without fail.
• Allow compounding: Over time, these payments build on themselves, increasing the total sum significantly.
• Predictable growth: The annual nature makes it easier to predict final value over other variable systems.

Annuity Formula

The annuity formula is a mathematical expression used to calculate the future value of an annuity. It considers the regular payment amount, interest rate per period, and number of payments.
The formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Here:

• \( P \): The payment made in each period.
• \( r \): The interest rate per period.
• \( n \): The number of payments.

It's crucial to input these correctly to get an accurate future value.
In practice, it shows how the combination of interest and regular deposits grows your investment.