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. $$ r=7 \frac{1}{2} \%, \quad t=20 $$

Short answer

Use a calculator to perform the specific operations in step 4 to find the required starting investment amount, P.

Step-by-step solution

Understand the compound interest formula

The standard formula for compound interest is: \[ A=P\left(1+\frac{r}{n}\right)^{nt} \] where A is the end amount, P is the principal amount, r is the rate of interest, n is the number of times interest is compounded per time period, and t is the time in years. However, in this case we are given the end amount A and we need to solve for the principal P, so we will need to rearrange the formula to: \[ P = \frac{A}{(1+\frac{r}{n})^{nt}} \]

Convert percentage rate to a decimal and identify values

We should convert the given percentage rate into a decimal, by dividing by 100. Thus, \(r=7.5\%\), or \(7.5 \:divided \:by\: 100= 0.075\). As the interest is compounded monthly, \(n = 12\) times per year, for \(t = 20\) years and \(A = 1,000,000\).

Substitute values into the formula

Substituting \(A=1,000,000, r = 0.075, n = 12, t = 20\) into the formula, \[P = \frac{1{,}000{,}000}{(1+\frac{0.075}{12})^{12*20}} \]

Calculate the principal

To solve for P, perform the operations according to the standard order of operations: parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right). After calculation, the required principal amount P will be obtained.

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 refers to the initial investment from which interest begins to accumulate. Often, finding the principal amount involves working backward from the desired future value, or end amount, of an investment.
To do this, we rearrange the standard compound interest formula. Knowing the future amount, interest rate, compounding frequency, and time period, allows us to calculate the necessary principal. Apprehending this concept is crucial as it enables you to determine how much you need to invest today to achieve a specific financial goal in the future.

Interest Rate

The interest rate in a compound interest scenario is the percentage that the principal earns in a specific time period. It's often expressed as an annual rate, and it plays a significant role in determining how your money grows over time.
To use it in calculations, the interest rate is usually converted into a decimal by dividing it by 100. In our exercise, the interest rate is given as 7.5%, which converts to 0.075. This interest rate is then divided by the number of times it compounds in a year to find the periodic rate. Understanding this concept helps you to see how much return you're gaining on your investment relative to the principal amount.

Time Period

The time period in the context of compound interest refers to the total duration for which the money is invested or borrowed. It is usually measured in years. This concept is essential for calculating the amount by which your investment will grow, as it includes the effect of compounding over the total time span.
In our example, a time period of 20 years is provided. This tells us the investment is long-term, allowing for the interest to compound multiple times, maximizing the growth of the investment over such a span. Understanding the time period is vital for planning purposes and anticipating the growth of your investment.

Monthly Compounding

Monthly compounding implies that the interest is calculated and added to the principal every month. This frequency of compounding can significantly impact the amount of interest accumulated over time. The higher the frequency of compounding, the greater the total amount of interest will be.
In the formula, the compounding frequency is denoted as "n." For monthly compounding, this means 12 times a year. It effectively divides the annual interest rate by 12 to find the monthly interest rate. The impact of this can be seen in how regularly you're earning interest on interest, which accelerates the growth of your investment.

• Increases your investment's growth rate.
• Optimizes returns over the long term.
• Provides a more significant compounding effect compared to other frequencies like quarterly or annually.

Appreciating this concept allows you to leverage the benefit of frequent compounding to maximize your future financial outcomes.