Question

How much do you need to invest now at \(5 \%\) per annum compounded monthly so that in 1 year you will have $$\$ 10,000 ?

Short answer

\$9513.25

Step-by-step solution

Understand the Compound Interest Formula

The formula for compound interest is given by \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]where \(A\) is the amount of money accumulated after \(n\) years, including interest. \(P\) is the principal amount (the initial amount of money), \(r\) is the annual interest rate (decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time in years.

Identify Given Values

We need to find the initial investment \(P\) given the following values: \(A = 10,000\), \(r = 0.05\), \(n = 12\), and \(t = 1\).

Rearrange the Formula to Solve for \(P\)

We need to isolate \(P\): \[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]

Substitute the Values into the Formula

Plug the values into the formula: \[ P = \frac{10,000}{\left(1 + \frac{0.05}{12}\right)^{12 \cdot 1}} \]Calculate the values inside the parentheses first.

Simplify the Expression Inside the Parentheses

Calculate the expression inside the parentheses: \[ 1 + \frac{0.05}{12} = 1 + 0.0041667 = 1.0041667 \]

Calculate the Exponent

Raise the result to the power of 12: \[ (1.0041667)^{12} = 1.0511618978 \]

Divide to Find the Principal \(P\)

Finally, divide \$10,000 by the result from Step 6: \[ P = \frac{10,000}{1.0511618978} \approx 9513.25 \]

Key concepts

Compound Interest Formula

Compound interest allows your money to grow over time by earning interest not only on the principal amount but also on the interest already earned. To calculate compound interest, we use the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Here, - \(A\) represents the total amount of money after interest - \(P\) is the principal amount or initial investment- \(r\) is the annual interest rate expressed as a decimal (so 5% becomes 0.05)- \(n\) is the number of times the interest is compounded per year- \(t\) is the time the money is invested for, in yearsThis formula helps us see how different variables impact the growth of an investment. For example, more frequent compounding periods (a higher n) will lead to more interest being earned.

Principal Amount

The principal amount is the initial sum of money that you invest or save. In our problem, we need to determine this initial amount that will grow to $10,000 in a year with a 5% annual interest rate, compounded monthly. Using the rearranged compound interest formula, we solve for the principal amount \(P\): \[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]Given that \(A = 10,000\), \(r = 0.05\), \(n = 12\), and \(t = 1\). Substituting these values into the formula gives us the principal amount that you need to invest today to reach your goal in 1 year.

Annual Interest Rate

Interest rates can be tricky, especially when they're given annually but compounding more frequently. The annual interest rate is how much interest you earn over a year. In decimal form, a 5% annual interest rate becomes 0.05. The compound interest formula divides this annual rate by the number of compounding periods per year (n). For our problem, the formula modifies the annual rate to fit the monthly compounding periods: \[ \frac{0.05}{12} \approx 0.004167 \]This tells us the interest rate for each month, showing how even small monthly rates can add up significantly over a year.

Compounded Monthly

When interest is compounded monthly, it means the interest is calculated and added to the principal 12 times per year, once every month. Each month's interest then helps calculate the next month's interest, giving us compound growth. In our formula, \(n = 12\) reflects this monthly compounding frequency. Using monthly compounding with a 5% annual rate and a 1-year period (\(t = 1\)), the interest adds up slightly more than it would with yearly compounding, demonstrating the power of compound interest. Here's a closer look at the transformation: 1. First, convert the annual rate to a monthly rate: \(1 + \frac{0.05}{12} = 1.004167\).2. Next, calculate the compounded amount over 12 months: \((1.004167)^{12} \approx 1.051162\).3. This compounded growth factor tells us how much a principal amount needs to multiply to reach the final amount.By understanding monthly compounding, we can appreciate how even small, frequent gains accumulate over time, leading to significant growth.