Question

Suppose you make monthly deposits of \(P\) dollars into an account that earns interest at a monthly rate of \(p \% .\) The balance in the account after \(t\) years is \(B(P, r, t)=P\left(\frac{(1+r)^{12 t}-1}{r}\right),\) where \(r=\frac{p}{100}\) (for example, if the annual interest rate is \(9 \%,\) then \(p=\frac{9}{12}=0.75\) and \(r=0.0075) .\) Let the time of investment be fixed at \(t=20\) years. a. With a target balance of \(\$ 20,000,\) find the set of all points \((P, r)\) that satisfy \(B=20,000 .\) This curve gives all deposits \(P\) and monthly interest rates \(r\) that result in a balance of \(\$ 20,000\) after 20 years. b. Repeat part (a) with \(B=\$ 5000, \$ 10,000, \$ 15,000,\) and \(\$ 25,000,\) and draw the resulting level curves of the balance function.

Short answer

Short Answer: The set of all points (P, r) that satisfy a given balance B can be found by plugging the target balance value and a fixed time (t) into the formula B(P, r, t) = P( (1+r)^{12t} - 1 ) / r. Solve the equation for P to express P in terms of r. The level curves for different target balances can be plotted by repeating these steps for each balance and plotting the curves on a graph with P on the horizontal axis and r on the vertical axis.

Step-by-step solution

Identify the known variables

We are given the target balance B = $20,000, and the fixed time t = 20 years. The formula for the balance is: B(P, r, t) = P( (1+r)^{12t} - 1 ) / r

Use the given B value to find the equation that relates P and r

We want to find the set of points (P, r) that satisfy B = \(20,000. Plug B = \)20,000 and t = 20 into the balance formula: 20000 = P( (1+r)^{240} - 1 ) / r

Solve this equation for P

To express P in terms of r, we can solve the equation above for P: P(r) = 20000 * r / ( (1+r)^{240} - 1 ) This equation gives us the set of points (P, r) that satisfy a balance of $20,000 after 20 years. #b. Repeat part (a) for target balances of \(5,000, \)10,000, \(15,000, and \)25,000 and draw the resulting level curves. #

Repeat step 2 for the given target balances

For each target balance, plug in the B value and t=20 and solve for P: For B = \$5,000: 5000 = P( (1+r)^{240} - 1) / r. Solve for P: P(r) = 5000 * r / ( (1+r)^{240} - 1 ) For B = \$10,000: 10000 = P( (1+r)^{240} - 1) / r. Solve for P: P(r) = 10000 * r / ( (1+r)^{240} - 1 ) For B = \$15,000: 15000 = P( (1+r)^{240} - 1) / r. Solve for P: P(r) = 15000 * r / ( (1+r)^{240} - 1 ) For B = \$25,000: 25000 = P( (1+r)^{240} - 1) / r. Solve for P: P(r) = 25000 * r / ( (1+r)^{240} - 1 )

Plot the level curves

Plot the level curves for each target balance B(P, r) on a graph with P on the horizontal axis and r on the vertical axis. You can use graphing software or a graphing calculator to help with this step.

Key concepts

Interest Rate

Understanding the interest rate is crucial when working with investments. The interest rate determines how much your deposit will grow over time. Here, we focus on the monthly interest rate, denoted by \( r \), which is derived from the annual interest rate \( p \% \). To convert an annual interest rate to a monthly rate, divide \( p \) by 12 and convert it to a decimal by dividing by 100. For example, an annual rate of 9% becomes a monthly rate of 0.75%, or \( r = \frac{9}{12 \times 100} = 0.0075 \).
It's important to grasp the relationship between interest rates and overall investment growth. The interest rate influences how fast your money accumulates in the account. A higher rate accelerates growth, impacting the balance attained over specific periods, such as the 20-year span in our problem.

Investment

Investment involves committing a resource, such as money, with the expectation of generating a return. In this problem, the investment is the regular monthly deposit \( P \). Understanding how deposits affect growth can help you align your financial goals with reality.
A fundamental consideration in investments is the time period. Here, a fixed 20-year term is used. Longer durations typically result in higher balances, given the power of compound interest. Starting early allows the interest to compound over many periods, building a substantial sum.
Investments make the most of time and interest rates to generate returns. Consider these principles when planning an investment strategy to achieve a specific future balance.

Balance Function

The balance function \( B(P, r, t) = P \left( \frac{(1+r)^{12t} - 1}{r} \right) \) is a key concept in understanding how deposits and interest rates interact over time. This function tells us how much money will accumulate in an account after a set duration, taking into account the regular deposits \( P \) and interest rate \( r \).
The balance is calculated after a certain number of years, \( t \). For this problem, \( t \) is fixed at 20 years, meaning we’re examining the account's growth over this extended period. The function highlights the impact of compounding interest - not only are you earning interest on your initial deposits, but also on the interest already accrued.
This formula is vital for calculating potential future balances, informing decisions on deposit amounts and interest rates needed to achieve desired financial goals.

Level Curves

Level curves in this context are graphical representations of points \((P, r)\) that yield the same balance \( B \) after a specific time. Drawing level curves helps visualize how different combinations of deposits and interest rates meet a target balance.
For example, in the provided problem, we plot level curves for various target balances like \\(5000, \\)10000, \\(15000, \\)20000, and \$25000, by changing the pair \((P, r)\) while maintaining those balances after 20 years.
This visualization aids in understanding the trade-off between the deposit amount \( P \) and the interest rate \( r \). By increasing the deposit \( P \) for a lower rate \( r \) or seeking higher rates with a smaller deposit, these curves help determine feasible financial plans. Thus, level curves are an essential tool in investment planning.