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Chapter 3: Problem 9

Find the future value of \(\$ 20000\) in 2 years' time if compounded quarterly at \(8 \%\) interest.

Short Answer

Expert verified
The future value is approximately \(\$ 23433.18\).

Step by step solution

01

- Understand the formula

The formula to find the future value with compound interest is given by: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:- \(A\) is the future value.- \(P\) is the principal amount (initial investment).- \(r\) is the annual interest rate (decimal).- \(n\) is the number of times interest is compounded per year.- \(t\) is the number of years.
02

- Assign the given values

From the problem statement, we have:- Principal amount, \(P = 20000\)- Annual interest rate, \(r = 8\text{\text{ or }}0.08\) (as a decimal)- Compounding frequency, \(n = 4\) (quarterly)- Time period, \(t = 2\)
03

- Substitute the values into the formula

Now, substitute the values into the formula:\[ A = 20000 \left(1 + \frac{0.08}{4}\right)^{4 \times 2} \]Simplify inside the parentheses first:\[ A = 20000 \left(1 + 0.02\right)^{8} \]
04

- Solve the exponent

Next, solve the exponent:\[ 1.02^{8} \approx 1.171659 \]
05

- Multiply by the principal amount

Finally, multiply by the principal amount:\[ A = 20000 \times 1.171659 \]\[ A \approx 23433.18 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compounding Interest Formula
Compound interest grows your investment faster over time by applying interest on both the initial principal and the previously earned interest. The formula for it is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] This formula helps find out the future value (\(A\)) of an investment. The variables involved are:
  • \(P\) - Principal Amount
  • \(r\) - Annual Interest Rate in decimal form
  • \(n\) - Compounding frequency
  • \(t\) - Number of years
By substituting these values into the formula, we can calculate how much the investment will be worth after a certain time.
Annual Interest Rate
The annual interest rate (\r) is the percentage rate at which your money grows in one year. To ensure accuracy in the compound interest formula, you must convert it from a percentage to a decimal. For example, an 8% interest rate becomes \(0.08\) in decimal form. This rate determines how much interest is added to the original principal and previously earned interest each compounding period.
Compounding Frequency
Compounding frequency () refers to how often the interest is applied to your investment throughout the year. Popular frequencies include:
  • Annually (once a year)
  • Semi-annually (twice a year)
  • Quarterly (four times a year)
  • Monthly (twelve times a year)
In our example, the interest is compounded quarterly, so \(n = 4\) . The higher the compounding frequency, the more interest is added to the investment, causing it to grow faster.
Principal Amount
The principal amount (\(P\)) is the initial amount of money you invest or borrow. It is the starting point for growth through compound interest. In our example, the principal amount is \(\text{\textdollar}20000\). This value is critical because the compound interest will be calculated on it and the interest accrued in each period.
Interest Calculation
Interest calculation involves more than just applying the annual interest rate; it compounds over the periods within the term. Here's a brief run-through using our example:
  1. Identify the principal amount: \(P = \text{\textdollar}20000\)
  2. Convert the annual interest rate to decimal: \(r = 0.08\)
  3. Determine compounding frequency: \(n = 4\)
  4. Calculate number of periods: \(nt = 4 \times 2 = 8\)
  5. Substitute into the formula: \( A = 20000 \left(1 + \frac{0.08}{4}\right)^{8} \)
  6. Solve inside the parentheses: \( 1 + 0.02 = 1.02 \)
  7. Calculate the exponent: \( 1.02^{8} \approx 1.171659 \)
  8. Multiply by the principal: \(\text{\textdollar}20000 \times 1.171659 \approx \text{\textdollar}23433.18\)
This gives the future value of the investment after 2 years, compounded quarterly.

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