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. $$ r=8 \%, \quad t=35 $$

Short answer

After calculation, the principal amount (P) that needs to be invested is approximately \$89,073.87.

Step-by-step solution

Understanding given values

In order to solve the problem, we must first identify the given values and the required formula. The given values are \(A=\$1,000,000\), \(r=8\%\), \(t=35\years\), and \(n=12\) times per year (once monthly).

Converting the rate to decimal

The annual rate r given as a percent needs to be converted to its decimal form by dividing it with 100, i.e. \(r=\frac{8}{100}=0.08\).

Substitute the values into the formula

Now we substitute the values into the compound interest formula, which gives us: \(P= \frac{\$1,000,000}{(1+\frac{0.08}{12})^{12*35}}\).

Solve for P

Calculating the above expression will give us the value of P. This is the principal amount that needs to be invested.

Key concepts

Present Value Calculation

Understanding how to calculate the present value of money is essential when dealing with investments and savings. It allows you to determine how much you would need to invest now in order to reach a specific financial goal in the future.

Using the compound interest formula is a common method to perform this calculation. The formula is \[ P = \frac{A}{(1 + \frac{r}{n})^{n*t}} \] where

• \(P\) is the present value or principal,
• \(A\) is the future value,
• \(r\) is the annual interest rate in decimal form,
• \(n\) is the number of times interest is compounded per year,
• and \(t\) is the time in years.

For instance, to find out how much you would need to invest today to have \(1,000,000 in 35 years at an annual interest rate of 8%, compounded monthly, you would set \(A\) as \)1,000,000, \(r\) as 0.08, \(n\) as 12, and \(t\) as 35. Then you solve for \(P\), which gives you the present value required.

To improve student understanding, it's helpful to provide examples with different interest rates and compounding frequencies, showing the impact these variables have on the present value.

Time Value of Money

The concept of the time value of money is a foundational principle in financial mathematics. It recognizes that a dollar today is worth more than a dollar in the future because of its potential earning capacity.

The ability to earn interest makes today's money potentially worth more, and the time value of money equation helps highlight this potential. We see this principle in action when we calculate compound interest. Over time, money can grow significantly due to reinvestment of earned interest, which effectively earns more interest, known as compound interest.

To apply this concept practically and improve learning outcomes, consider comparing scenarios where money is invested over different periods of time, illustrating how more time allows compound interest to work more effectively, increasing the future value of an investment.

Exponential Growth and Decay

In financial mathematics, exponential growth and decay models describe how investments grow over time with compound interest or decrease in value, as with depreciation of assets.

When dealing with compound interest, the formula we use shows exponential growth—as the amount of interest earned increases over time, it grows at a rate that becomes progressively faster. This is because the interest earned in the previous periods also earns interest.

Understanding the principles of exponential functions is key to grasping compound interest. By analyzing different investment timeframes and rates, students can observe the 'snowball' effect, leading to a deeper understanding of exponential growth in financial contexts.

Financial Mathematics

Financial mathematics encompasses a wide range of topics used in the financial industry, including the analysis of investments, the management of financial risks, and the time value of money.

Key formulas and functions are at the heart of financial mathematics, like the compound interest formula used in our exercise. This specific area of study is practical because it can be applied to everyday decision-making, from saving for retirement to understanding loan repayment plans.

To support better student comprehension, practical application exercises that include real-world scenarios—such as comparing different retirement plans or exploring different mortgage terms—can be particularly illuminating.