Question

Person A opens an IRA at age 25 dollar contributes 2000 dollar year for 10 years, but makes no additional contributions thereafter. Person B waits until age 35 to open an IRA and contributes 2000 dollar year for 30 years. There is no initial investment in either case. $$ \begin{array}{l}{\text { (a) Assuming a return rate of } 8 \% \text { , what is the balance in each IRA at age } 65 ?} \\ {\text { (b) For a constant, but unspecified, return rate } r \text { , determine the balance in each IRA at age }} \\ {65 \text { as function of } r .} \\ {\text { (c) Plot the difference in the balances from part (b) for } 0 \leq r \leq 0.10 \text { . }} \\\ {\text { (d) Determine the return rate for which the two IRA's have equal value at age } 65 .}\end{array} $$

Short answer

Question: Calculate the return rate at which both IRAs have equal value at age 65. Answer: To find the return rate where both IRAs have equal value, we need to solve the equation $$ (1+r)^{10(65-25)} = (1+r)^{10(65-35)} $$ using logarithmic functions or numerical methods.

Step-by-step solution

a) Calculate the balance of each IRA at age 65 for a return rate of 8%

For person A, find the future value of the initial 10-year investment at age 65: Use the Future Value of an Ordinary Annuity formula: $$ FV_A = P \frac{(1+r)^{n(65-25)}-1}{r} $$ For person B, find the future value of the 30-year investment at age 65: $$ FV_B = P \frac{(1+r)^{n(65-35)}-1}{r} $$ Use the given values P = 2000 and r = 0.08 to find the future values: $$ FV_A = 2000 \frac{(1+0.08)^{10(65-25)}-1}{0.08} $$ $$ FV_B = 2000 \frac{(1+0.08)^{10(65-35)}-1}{0.08} $$ Calculate the balance of each IRA at age 65: $$ IRA_A = FV_A $$ $$ IRA_B = FV_B $$

b) Calculate the balance of each IRA at age 65 as a function of the return rate r.

Use the same formulas for FV_A and FV_B, replacing 0.08 with r: $$ FV_A = P \frac{(1+r)^{n(65-25)}-1}{r} $$ $$ FV_B = P \frac{(1+r)^{n(65-35)}-1}{r} $$ The balance of each IRA at age 65 as a function of the return rate r: $$ IRA_A(r) = FV_A(r) $$ $$ IRA_B(r) = FV_B(r) $$

c) Plot the difference in balances for a range of return rates.

You will need to use graphing software to plot the difference in the balances from 0 <= r <= 0.10. You simply need to plot the function: $$ Difference(r) = IRA_B(r) - IRA_A(r) $$

d) Determine the return rate for which both IRAs have an equal value at age 65.

To find the return rate for which both IRAs have equal value at age 65, set IRA_A(r) equal to IRA_B(r): $$ IRA_A(r) = IRA_B(r) $$ Solve for r: $$ P \frac{(1+r)^{n(65-25)}-1}{r} = P \frac{(1+r)^{n(65-35)}-1}{r} $$ Simplifying, canceling P and r: $$ (1+r)^{n(65-25)} = (1+r)^{n(65-35)} $$ To find the return rate where both IRAs have equal value, solve for r using logarithmic functions or numerical methods.

Key concepts

Future Value of an Annuity

The concept of the Future Value of an Annuity is crucial when calculating how much money a series of regular payments, called annuities, will grow to over time, considering compound interest. An annuity is a sequence of equal payments made at regular intervals. The future value essentially tells us how much those contributions will amount to after a certain time period.
To find this future value, we use a formula specific to ordinary annuities, which assumes payments happen at the end of each period. This formula is given by:\[FV = P \frac{(1 + r)^n - 1}{r}\]where:- \(FV\) is the future value of the annuity.- \(P\) is the payment amount per period.- \(r\) is the return rate per period.- \(n\) is the total number of payment periods.In the context of an IRA, understanding this formula helps us compute the accumulated value of regular contributions, assuming they earn interest over years until retirement. This knowledge enables us to compare different investment strategies, such as contributing earlier but less frequently versus contributing later for a longer period.

Return Rate Calculation

Return rate calculation involves determining the percentage at which an investment grows annually. It is essential in understanding how different types of investments will perform over time, especially for retirement accounts like IRAs.
The return rate, often denoted as \(r\), is fundamental to calculating the future value of deposits, as it reflects how effectively contributions at each interval grow due to interest earnings.- If you know the desired future value, contribution amount, and time period, you can backtrack to solve for the return rate that will make those numbers work.- For irregular equations or when solving problems like equating two differing investments' values at a future point, we might need to use numerical methods or graphical solutions to find \(r.\)In practical terms, understanding return rates helps evaluate whether investing early with higher interest or contributing steadily with potentially fluctuating rates yields the desired retirement fund balance.

IRA (Individual Retirement Account)

An IRA (Individual Retirement Account) is a financial tool used by individuals to save for retirement. Contributions to an IRA may be tax-deferred or tax-free depending on the type of IRA.
When you open an IRA, you can choose between traditional and Roth IRAs: - **Traditional IRAs**: Contributions are often tax-deductible, and taxes are paid when withdrawals are made in retirement. - **Roth IRAs**: Contributions are made with post-tax income, but withdrawals, including earnings, are tax-free at retirement. IRAs allow individuals to invest in stocks, bonds, mutual funds, and other assets to grow their retirement savings. The growth is primarily driven by compound interest, making the start time and duration of contributions critical.
In the original problem, two individuals start their IRAs at different ages and make varying contributions, allowing us to evaluate how time and consistent investing impact the final retirement amount. Understanding these dynamics encourages informed decisions about when and how much to save for the future.