Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Robin wishes to accumulate a sum of $$\$ 450,000$$ in a retirement account by the time of her retirement 30 yr from now. If she wishes to do this through monthly payments into the account that earn interest at the rate of \(10 \% /\) year compounded monthly, what should be the size of each payment?

Step-by-Step Solution

Answer

Robin should make monthly payments of approximately \(\$556.55\) into her retirement account to accumulate a sum of \(\$450,000\) in 30 years, considering a 10% annual interest rate compounded monthly.

Step 1: Identify the variables in the formula for the future value of an ordinary annuity

The formula for the future value of an ordinary annuity (FV) is given by: \[ FV = PMT \times \frac{\{(1 + i)^n - 1\}}{i} \] where: - FV is the future value of the annuity ($450,000) - PMT is the size of each payment (which we need to find) - i is the interest rate per period (to be calculated from the given annual interest rate compounded monthly) - n is the number of periods (total monthly payments in 30 years)

Step 2: Calculate the interest rate per period

In this case, the annual interest rate is given as 10%, and interest is compounded monthly. To find the interest rate per period, divide the annual interest rate by the number of compounding periods in a year: \[ i = \frac{annual\ interest\ rate}{number\ of\ compounding\ periods\ in\ a\ year} = \frac{0.10}{12} \]

Step 3: Calculate the number of periods

As Robin plans to save for 30 years, and she will make monthly payments into the account, so we will have to find the total number of monthly payments in 30 years: \[ n = years \times 12 = 30 \times 12 \]

Step 4: Insert the values into the formula and solve for PMT

Now, fill in the given values in the formula and rearrange the equation for PMT: \[ PMT = \frac{FV \times i}{(1+i)^n - 1} \] Substitute values into the equation and solve for PMT: \[ PMT = \frac{450,000 \times \frac{0.10}{12}}{(\frac{1 + 0.10}{12})^{30 \times 12} - 1} \]

Step 5: Simplify and compute the result

Perform the calculations to find the value of PMT: \[ PMT ≈ \$\,556.55 \] Hence, Robin should make monthly payments of approximately \(556.55 into her retirement account to accumulate a sum of \)450,000 in 30 years, considering a 10% annual interest rate compounded monthly.

Key Concepts

Understanding an Ordinary Annuity

An ordinary annuity is a series of equal payments made at the end of consecutive periods over a fixed length of time. Common examples include monthly mortgage payments, retirement savings, and insurance premiums. The main characteristic that defines an ordinary annuity is the timing of the payments – they are always at the period's end, be it monthly, quarterly, or annually.

In the context of Robin's retirement plan, the monthly deposits into her retirement account represent an ordinary annuity because she plans to make consistent payments at the end of each month. By doing so, the money invested has the opportunity to earn interest over time, which is a key benefit of ordinary annuities when compared to lump-sum investments. The interest on ordinary annuities compounds over periods, resulting in a higher ending balance due to the time value of money.

The Mechanics of Compound Interest

Compound interest is what makes an ordinary annuity so powerful for long-term savings. It refers to earning interest on interest, as well as on the principal amount invested. The formula for compound interest takes into account the initial principal (P), the interest rate per period (i), and the number of compounding periods (n).

The formula is expressed as:
\[ A = P(1 + i)^n \]
where A is the amount of money accumulated after n periods, including interest.

In Robin's case, the interest is compounded monthly, which means the interest earned each month is added to the principal, and the following month's interest is calculated on this new amount. The higher the frequency of compounding within a year, such as monthly in Robin's situation, the greater the final amount due to the more frequent application of interest.

How to Calculate Annuity Payments

Calculating the payment for an annuity, known as PMT, involves rearranging the future value of an ordinary annuity formula. The future value (FV) is what Robin wants to have in the account by retirement, the number of periods (n) is the total number of payments she will make, and the interest rate per period (i) is determined by dividing the annual interest rate by the number of compounding periods per year.

To solve for PMT, the formula is manipulated as follows:
\[ PMT = \frac{FV \times i}{(1+i)^n - 1} \]
This calculation considers the effect of compounding, which allows Robin to determine the size of each payment needed to reach her retirement goal based on the interest rate and time frame. Using this formula, we can see that by regularly investing a manageable sum, Robin can accumulate a considerable amount by leveraging the power of compound interest in an ordinary annuity.