Question

Compound Interest In Exercises \(45-48\) , find the principal \(P\) that must be invested at rate \(r,\) compounded monthly, so that \(\$ 1,000,000\) will be available for retirement in \(t\) years. $$ r=6 \%, \quad t=40 $$

Short answer

To retire with $1,000,000 at a 6% annual interest rate compounded monthly, an initial investment of approximately $47,392 must be made

Step-by-step solution

Identify the given variables

In this problem, the variables given are the amount available at retirement \(A = 1,000,000\), the interest rate \(r = 6\% = 0.06\) (converted from percentage to a decimal), the time period \(t = 40\) years, as well as the fact that interest is compounded monthly (equivalent to \(n = 12\) times a year).

Rearrange the formula to solve for P

The compound interest formula in terms of interest being compounded monthly will be \(A = P(1 + r/n) ^ {nt}\). To find the principal \(P\), we need to rearrange the formula to \(P = A / {(1 + r/n)}^{nt}\)

Substitute the values into the formula

Substituting the values into the formula gives \(P = 1,000,000 / {(1 + 0.06/12)}^{12 * 40}\)

Calculate the Principal

Upon calculating the value, it comes out to be approximately $47,392. This is the principal that must be invested to get a return of $1,000,000 upon retirement

Key concepts

Principal Amount

The principal amount is the initial sum of money that is invested or loaned. In the context of compound interest, it's crucial as it serves as the base on which interest is calculated. When planning for financial goals like retirement, finding the right principal amount helps meet future targets like a nest egg of $1,000,000.
The principal is not a static value; it interacts with other financial elements like interest rates and time periods. Here, it acts as the starting point in the compound interest formula:

• For the compound interest formula, the principal is denoted as \(P\).
• In the rearranged formula \(P = A / {(1 + r/n)}^{nt}\), solving for \(P\) gives the initial investment required.

This means knowing how much to invest today based on expected growth through compounding. Calculating the right principal provides financial clarity and aligns your investment strategy with your future goals.

Interest Rate

The interest rate is the percentage at which your principal amount grows annually. It plays a pivotal role in determining how fast your investment will reach its future value, such as a retirement fund.
Understanding interest rates means understanding both their potential and their risks.

• In the compound interest formula, the interest rate is often represented by \(r\), and it's typically expressed as a decimal. For instance, 6% converts to 0.06.
• A higher interest rate means more significant growth over time, but it can also mean higher risk depending on the investment.

Interest rates impact not just the growth of your funds, but also the principal required to reach a specific future goal. In cases where interest is compounded, the effective rate might differ due to the frequency of compounding, influencing the end total significantly.

Compounding Frequency

Compounding frequency refers to how often interest is calculated and added back to the principal balance. This frequency impacts the amount of interest accrued over the life of your investment, making it a vital factor in compound interest calculations.
For compound interest, a key consideration is how often compounding occurs:

• Monthly compounding, as used in financial exercises, involves calculating interest 12 times per year.
• The more frequent the compounding, the more interest is accumulated over time. This results in a higher future value of the investment.

In our exercise, compounding monthly means using \(n = 12\) in the formula. This frequent calculation can increase the return significantly compared to annual compounding, affecting the amount initially needed to invest.

Future Value

Future value is the amount your investment is worth after a certain period, once all compounding interest has been applied. It's the target you're aiming for when deciding your principal and understanding your interest rate and compounding frequency.
The future value is what guides your financial planning decisions:

• In the context of retirement, for instance, it's often set as a specific amount, like the $1,000,000 in our exercise.
• Calculating the future value helps determine if your current savings strategy will meet your financial needs at a specific point in the future.

This concept is essential because it is the aim of utilizing compound interest. By applying future value calculations, you can decide the appropriate amount to save now, using current interest rates and compounding methods to align with your long-term goals.