Problem 22 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Luis has $$\$ 150,000$$ in his retirement account at his present company. Because he is assuming a position with another company, Luis is planning to "roll over" his assets to a new account. Luis also plans to put $$\$3000$$/quarter into the new account until his retirement 20 yr from now. If the account earns interest at the rate of \(8 \% /\) year compounded quarterly, how much will Luis have in his account at the time of his retirement? Hint: Use the compound interest formula and the annuity formula.

Step-by-Step Solution

Answer

Luis will have a total of \(\$150,000(1 + \frac{0.08}{4})^{(4)(20)} + \$3,000(\frac{(1 + 0.02)^{(4)(20)} - 1}{0.02})\) in his account at the time of his retirement after 20 years, considering the compounded interest and his quarterly contributions.

Step 1: Calculation for Initial Amount

The initial amount is $$\$150,000$$. We will use the compound interest formula to find its value after 20 years. Compound Interest Formula: \(A = P(1 + \frac{r}{n})^{nt}\) Where: A = Amount after time t P = Principal (Initial amount) = \( \$150,000\) r = Annual interest rate = \( 8\% = 0.08\) n = Compounding frequency per year (quarterly compounding) = 4 t = Time in years = 20 We substitute the values to get: \(A = \$150,000(1 + \frac{0.08}{4})^{(4)(20)}\)

Step 2: Calculation for Quarterly Contributions

We will now calculate the total interest generated for Luis' $$\$3,000$$ quarterly investments using the annuity formula: Annuity Formula: \(S = P(\frac{(1 + r)^{nt} - 1}{r})\) Where: S = Future value of annuity P = Quarterly payment = \( \$3,000\) r = Annual interest rate per quarter = \( 0.08/4 = 0.02\) n = Number of times the interest is compounded per year = 4 t = Time in years = 20 We substitute the values to get: \(S = \$3,000(\frac{(1 + 0.02)^{(4)(20)} - 1}{0.02})\)

Step 3: Find the total amount in the account

To find the total amount in Luis' account at the end of his retirement, we will sum up the amount we calculated in Steps 1 and 2. Total Amount = \( A + S \) Using the values from Steps 1 and 2, we can complete the calculations to find the amount Luis will have in his account at the time of his retirement.

Key Concepts

Annuity Formula

When planning for retirement, especially with periodic contributions like quarterly payments, the annuity formula is a valuable tool. An annuity is basically a series of equal payments made at regular intervals. These payments can grow substantially over time if compounded at a consistent interest rate. The annuity formula to find the future value, when contributions are made regularly, is expressed as:\[S = P \left(\frac{(1 + r)^{nt} - 1}{r} \right)\]- **\(S\)** is the future value of the annuity after the investment period.- **\(P\)** represents each regular payment made, like Luis' quarterly payments of $3,000.- **\(r\)** is the interest rate per period. For Luis, who has quarterly compounding, this is the annual rate divided by the number of quarters.- **\(nt\)** indicates the total number of compounding periods over the investment duration.This formula helps calculate how those regular payments accumulate and grow, factoring in the effect of compound interest. It considers how each payment earns interest over time, which can significantly enhance the future value of the savings.

Compound Interest Formula

Compound interest is a key concept in finance, particularly important for understanding how investments grow over time. It describes the process of earning interest on both the initial principal and any accumulated interest from previous periods.The formula for compound interest is:\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]Where:- **\(A\)** represents the future value of the investment, accounting for interest earned.- **\(P\)** is the principal amount, or the initial sum of money invested, such as Luis' $150,000.- **\(r\)** stands for the annual interest rate.- **\(n\)** indicates the number of times interest is compounded per year.- **\(t\)** is the time in years the money is invested or borrowed.For example, if interest is compounded quarterly, the frequency \(n\) would be 4, and the interest rate per period \(\frac{r}{n}\) is accounted for accordingly.Compound interest is powerful because it grows not only on your principal but also on the interest earned, making savings grow faster than simple interest, which only charges interest on the initial amount. This is why it's an important part of building your savings for retirement.

Retirement Savings

Planning for retirement requires a solid understanding of how much you will need and how to grow your savings effectively. Setting up a plan involves assessing current assets, determining future financial needs, and taking advantage of compounding returns over time. 1. **Starting with Initial Savings:** - As Luis did, starting with a lump sum provides a strong foundation. For him, it was $150,000 already saved, ready to be rolled over. 2. **Contributions Over Time:** - Regular contributions, such as quarterly deposits, significantly build up the retirement fund. These steady inputs ensure that the fund grows consistently over time. 3. **Leveraging Compound Interest:** - Using the compound interest formula helps you assess how initial savings grow over the years due to reinvesting earned interest. - Regular contributions over the long term are magnified by the power of compound interest, often leading to substantial increases in total retirement funds. 4. **Setting Clear Goals:** - Goal setting helps determine how much needs to be saved and what interest rates to aim for in order to ensure there are enough funds available upon retirement. By carefully planning and applying both the annuity and compound interest formulas, like in Luis’ plan, individuals can effectively prepare for a comfortable retirement, ensuring financial stability in their later years.