Question

You are saving money for retirement. You plan to withdraw \(30,000 at the beginning of each year for 20 years after you retire. Based on the type of investment you are making, you can expect to earn an annual return of 8% on your savings after you retire. a. Let \)a_n\( be your balance \)n\( years after retiring. Write a recursive equation that shows how \)a_n\( is related to \)a_{n-1}\(. b. Solve the equation from part (a) for \)a_{n-1}\(. Find \)a_0\(, the minimum amount of money you should have in your account when you retire. (Hint: Let \)a_{20}=0$.)

Short answer

The minimum amount of money that should be present in your account when you retire is approximately $349,712.95.

Step-by-step solution

Derive Recursive Relation

To depict the balance one year after retiring, \(a_n\), in terms of the balance from the previous year, \(a_{n-1}\), consider the fact that money is withdrawn at the start of each year and an 8% return is earned on the remaining balance for the rest of the year. This gives the relation: \(a_n = 1.08 \cdot (a_{n-1} - 30000)\). The 1.08 factor comes from the 8% return and the 30000 represents the withdrawn amount at the start of the year.

Adapt Equation to Find \(a_{n-1}\)

To find \(a_{n-1}\), we need to solve our derived equation. Rearranging terms to solve for \(a_{n-1}\), we get : \(a_{n-1} = \frac{a_n}{1.08} + 30000\). This equation allows us to relate any year's balance with the balance from the previous year.

Find Initial Balance at Retirement (\(a_0\))

Given \(a_{20} = 0\), we can use our equation from Step 2 to calculate backwards from \(a_{20}\) (at the end of year 20) to \(a_0\) (at the start of retirement). Using \(a_{20} = 0\), it will allow us to compute the values one after the other until \(a_0\). Doing this yields \(a_0 \approx 349,712.95\).

Key concepts

Recursive Equation

Recursive equations are a fantastic tool when it comes to modeling situations where each term in a sequence depends on the previous ones. In the realm of retirement savings, they help us model how our account balance evolves over time. Here, we used a recursive formula to find your balance after n years, considering factors like initial withdrawal and annual return.
A recursive equation can be depicted as:

• \( a_n = 1.08 imes (a_{n-1} - 30000) \)

This means that each year, the balance, \( a_n \), is calculated based on \( a_{n-1} \), the previous year’s balance. We subtract the $30,000 withdrawal and apply the 8% annual return.
To use a recursive equation, start by defining your initial terms, in this case, until you reach the desired endpoint. Recursive equations simplify complex arithmetic sequences by breaking them down into a series of linked operations. This makes it straightforward to compute each year’s balance by repeating the pattern.

Annual Return

An annual return represents the percentage gain or profit on an investment over one year. Here, we're discussing an 8% annual return. This rate is crucial for planning retirement savings, as it dictates how much your money grows year over year.
Annual returns are expressed in terms of a percentage of the initial investment:

• A return of 8% means that for every $100 invested, you earn $8 each year.

In financial planning, such returns are vital in calculating future account balances. They help us understand the power of compounding, where even small percentage gains grow significantly over time. The return rate helps predict how much you'll have or need at future points in life.

Investment Planning

Investment planning involves making decisions about saving and investing to achieve specific financial goals. When planning for retirement, it's essential to consider factors like the initial amount saved, expected withdrawals, risk tolerance, and anticipated annual returns.
Key components of effective investment planning include:

• Assessing your financial goals (e.g., withdrawing $30,000 per year for 20 years).
• Calculating the necessary starting balance (here, around $349,712.95), and
• Evaluating the impact of different returns and withdrawal rates.

Planning helps you set realistic milestones and make informed decisions. It uses models like the recursive equation to track progress over your timeline, ensuring you have enough savings for your retirement needs.

Compound Interest

Compound interest is often described as the interest on the interest. It’s what makes your savings grow faster over time. When you reinvest your earned interest, you allow those earnings to generate their own interest, leading to exponential growth.
The magic formula for compound interest is:

• \( A = P (1 + r/n)^{nt} \)

where:

• \( A \) is the amount of money accumulated after n years, including interest.
• \( P \) is the principal amount (initial balance).
• \( r \) is the annual interest rate (in decimal).
• \( n \) is the number of times interest applied per year.
• \( t \) is the time in years.

In this retirement scenario, the 8% annual return is compounded yearly, increasing the total account balance. Compound interest is a cornerstone concept in financial mathematics, driving the growth of investments and savings over prolonged periods.