Problem 79 - College Algebra with Modeling and Visualization

Question

Annuity If \(x\) dollars is deposited every 2 weeks \((26\) times per year) into an account paying an annual interest rate \(r,\) expressed in decimal form, then the amount \(A\) in the account after \(t\) years can be approximated by the formula $$ A=x\left(\frac{(1+r / 26)^{26 t}-1}{(r / 26)}\right) $$ If \(\$ 50\) is deposited every 2 weeks into an account paying \(8 \%\) interest, approximate the amount after 10 years.

Step-by-Step Solution

Answer

The approximate amount after 10 years is $19,800.

Step 1: Identify Variables

In the given formula for the annuity, identify the key variables. Here, we have the deposit amount per period \(x = 50\) dollars, the annual interest rate \(r = 0.08\), and the number of years \(t = 10\).

Step 2: Determine Interest Periods and Rate

Since deposits are made every 2 weeks, the number of periods per year is 26. Thus, the interest for each period is \(\frac{r}{26} = \frac{0.08}{26}\approx 0.00308\).

Step 3: Substitute Values into Formula

Substitute the known values into the annuity formula: \[A = 50 \left(\frac{(1+0.00308)^{260}-1}{0.00308}\right)\] where 260 is the total number of periods over 10 years (\(26 \times 10\)).

Step 4: Calculate Compound Interest Factor

Calculate the term \((1+0.00308)^{260}\). With a calculator, this will be approximately \(e^{(260\ln(1.00308))} \approx 2.21964\).

Step 5: Evaluate Final Expression

Compute the final amount by evaluating \[A = 50 \left(\frac{2.21964-1}{0.00308}\right)\].First, find the difference: \(2.21964 - 1 = 1.21964\).Then divide by the interest per period: \(\frac{1.21964}{0.00308} \approx 396\).Finally, multiply by the deposit amount: \(50 \times 396 = 19800\).

Step 6: Conclude Calculation

The approximate amount in the account after 10 years is \(\$19800\).

Key Concepts

Compound Interest

Compound interest is a key concept when it comes to understanding how an investment grows over time. Unlike simple interest, which only earns interest on the initial principal, compound interest earns interest on the initial principal and on any accumulated interest from previous periods.
This means your money grows at an increasing rate over time. For example, when you deposit money into an account earning compound interest, the interest is calculated on the new total after each period rather than the original amount deposited.

• Includes both interest on the initial deposit and on the interest that has been added to the account in previous periods.
• Grows exponentially over time, meaning the longer your money is invested, the more it will grow.

In the context of annuities, compound interest is crucial as it allows the small, regular deposits to accumulate into a sizeable amount over time.

Financial Formulas

Financial formulas are mathematical expressions used to calculate values like the future worth of an investment. In the case of annuities, these formulas allow us to estimate the amount expected in an account after a certain number of years.
The annuity formula applied here involves a set of inputs:

• Periodic deposit amount.
• Interest rate per period.
• Total number of periods.

The formula mentioned combines these factors into a powerful tool to predict the future value by taking into consideration the impact of compound interest over recurrent deposit intervals. Accurately substituting values into these formulas will directly lead to a correct estimation of your financial future.

Interest Rates

Interest rates are pivotal in calculating how much your investment will grow. They represent the cost of borrowing money or the reward for saving money, expressed as a percentage of the principal.
In the annuity example, the annual interest rate of \( 8\% \) is converted into a per-period rate by dividing it by the number of compounding periods per year.

• Reflect the growth potential of your investment.
• Higher rates typically increase the amount of interest earned.

Understanding how to derive per-period interest rates is important when working with annuities. The interest rate impacts how much additional money will be gained through the interest earned with each period, which subsequently affects the total value of the account over time.

Time Value of Money

The time value of money is a foundational principle in finance that describes how the value of money changes over time. A dollar today is worth more than a dollar tomorrow because it can be invested and earn interest.
In the context of our annuity calculation, the principle underscores why regular deposits grow more significatively over the investment period. Time, combined with interest, adds layers of value to the initial deposits.

• Money today has more purchasing power than the same amount in the future.
• Early and regular investments make the most of the compounding effect.

This principle is integrated into annuity formulas, which inherently recognize that deposits made sooner hold more value as they have more time to grow under compound interest. Recognizing this concept helps in making strategic financial decisions for future savings or investments.