Question

Find the principal \(P\) that must be invested at rate \(r\), compounded monthly, so that \(\$ 500,000\) will be available for retirement in \(t\) years. $$ r=7 \frac{1}{2} \%, \quad t=20 $$

Short answer

To have \(\$500,000\) at retirement with an interest rate of \( 7.5\% \) compounded monthly for \(20\) years, the required principal to be invested initially would be about \(\$125,632.35\).

Step-by-step solution

Convert the Interest Rate to Decimal

First, it is necessary to convert the interest rate from percentage to decimal. This is done by dividing the given rate by 100, i.e., \(r = 7.5/100 = 0.075\).

Plug the Known Values Into the Formula

Next, substitute all the provided values in the compound interest formula: \( A = P(1 + r/n)^{nt} \) and reformat it to solve for \(P\). Thus the equation becomes: \( P = A / (1 + r/n)^{nt} \)

Insert Values Into the Equation

Now, substitute the known values into the obtained equation: \( P = \$500,000 / (1 + 0.075/12)^(12*20) \)

Evaluating the Equation

Evaluate the right side of the equation to find the value of \( P \) which will be the principal that needs to be invested to have the desired amount at retirement. It can be calculated using a calculator.

Key concepts

Principal Investment Calculation

When preparing for events like retirement, understanding how to calculate the principal investment is crucial. It is the initial amount required to reach a future financial goal based on a certain interest rate and time period. To determine this, we use a compound interest formula which can be rearranged to solve for the principal amount, represented as P.

Given the problem, we're planning for a future value (also known as the maturity value) of \$500,000 in 20 years with an interest rate of 7.5%, compounded monthly. The rearranged compound interest formula is:
\[ P = \frac{A}{(1 + \frac{r}{n})^{nt}} \]
Here, \(A\) is the amount of money we want after \(t\) years, \(r\) is the annual interest rate in decimal, \(n\) is the number of times the interest is compounded per year, and \(t\) is the time in years. For our example, we need to find \(P\) such that \(A = \$500,000\), \(r = 0.075\), \(n = 12\) (monthly compounding), and \(t = 20\) years. Calculating \(P\) requires rearranging and then substituting these values into the compound interest formula.

Interest Rate Conversion

Interest rates are usually given in percentage form, but to use them in financial formulas, they need to be converted into decimal form. This is because percentage form is not a direct mathematical representation of the rate but rather a ratio per hundred for ease of understanding.

To convert an interest rate from a percentage to a decimal, simply divide by 100. For example, with the rate given as 7.5%, the conversion is as follows:
\[ r = \frac{7.5}{100} = 0.075 \]
This conversion is critical when working with financial equations, such as the compound interest formula, which require the rate to be in decimal form to accurately calculate the growth of the investment over time.

Time Value of Money

The time value of money is a financial concept that illustrates the idea that a sum of money is worth more now than an identical sum in the future due to its potential earning capacity. This core principle underlines the importance of compound interest when evaluating investments.

Compound interest takes into account not only the initial principal and the interest accrued over time but also the interest earned on the interest, which can significantly increase the future value of the investment. This is why the present value of a sum to be received in the future is less than its future value. The compound interest formula, essential for time value of money calculations, reflects this:
\[ A = P(1 + \frac{r}{n})^{nt} \]
In the formula, as \(t\) increases, the value of \(A\) grows exponentially due to compounding. Thus, the earlier an investment is made (a higher value of \(t\)), the greater the future value due to the time value of money.