Problem 31 - Calculus and its applications

Question

Bob and Ann Mackenzie have a new grandchild, Brenda, and want to create a trust fund for her that will yield \(\$ 250,000\) on her 24 th birthday. a) What lump sum should they deposit now at \(5.8 \%,\) compounded continuously, to achieve \(\$ 250,000 ?\) b) The amount in part (a) is more than they can afford, so they decide to invest a constant amount, \(R(t)\) dollars per year. Find \(R(t)\) such that the accumulated future value of the continuous money stream is \(\$ 250,000,\) assuming an interest rate of \(5.8 \%\) compounded continuously.

Step-by-Step Solution

Answer

1. Deposit: \( \$62,130.47 \). 2. Annual contribution: \( \$19,272.99 \).

Step 1: Understanding Continuous Compounding Formula

Continuous compounding is described by the formula \( A = Pe^{rt} \), where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal (initial) amount, \( r \) is the annual interest rate, and \( t \) is the time in years. We need to find \( P \) such that \( A = 250,000 \) after 24 years with \( r = 0.058 \).

Step 2: Solving for Principal Amount (P)

Substitute \( A = 250,000 \), \( r = 0.058 \), and \( t = 24 \) into the formula: \( 250,000 = Pe^{0.058 imes 24} \). Then solve for \( P \):\[ P = \frac{250,000}{e^{0.058 \times 24}} \]

Step 3: Calculating Remaining Terms and Result

Calculate \( e^{0.058 \times 24} \), approximately \( e^{1.392} = 4.024 \). Thus, \( P \approx \frac{250,000}{4.024} \approx 62,130.47 \). Bob and Ann need to deposit approximately \( \$62,130.47 \) now.

Step 4: Understanding Continuous Money Stream

Continuous money stream with continuous compounding is given by the formula \( A = \int_0^t R(t) e^{r(t-x)} \, dx \), where \( R(t) \) is the constant investment rate. We need to find \( R(t) \) such that \( A = 250,000 \) after 24 years.

Step 5: Solving for Constant Payment Rate R

Using the integration formula, set up the equation \( 250,000 = R(t) \int_0^{24} e^{0.058(24-x)} \, dx \). Calculate the integral, \( \int_0^{24} e^{0.058(24-x)} \, dx = \frac{1 - e^{-0.058 \times 24}}{0.058} \approx \frac{1 - 0.2488}{0.058} \approx 12.9724 \). Thus, solved for \( R(t) \), \[ R(t) = \frac{250,000}{12.9724} \approx 19,272.99 \]

Step 6: Result of R Calculation

Bob and Ann need to contribute approximately \( \\(19,272.99 \) annually to reach a future value of \( \\)250,000 \) on Brenda’s 24th birthday.

Key Concepts

Future Value Calculation

The concept of future value is crucial when planning investments or saving for big events, like creating a trust fund for a loved one's future. Future value calculation is about determining how much money a current investment will grow over a specified period, given a certain interest rate. In our exercise, Bob and Ann Mackenzie want to ensure they have \( \\(250,000 \) ready for their granddaughter Brenda by her 24th birthday. To achieve this, they utilize a method called continuous compounding, which provides a way to earn interest on an investment that is constantly reinvested. For a lump sum investment, you would typically use the formula:

• \( A = Pe^{rt} \)

Here, the goal is to find the present investment \( P \) that will grow to \( \\)250,000 \) in 24 years. This involves rearranging the formula to solve for \( P \), which represents the starting amount you need to invest today. By plugging in the given values into the equation (\( r = 0.058 \) for the interest rate and \( t = 24 \) years), you can determine that Bob and Ann need to invest around \( \$62,130.47 \) right now.

Trust Fund Investment

Trust funds are a wise investment choice for those looking to secure a financial future for someone special, like a grandchild. For Bob and Ann Mackenzie, their trust fund objective is clear: expand a savings plan to ensure Brenda receives \( \$250,000 \) when she turns 24. A trust fund provides a structured way to meet such financial goals, whether through a one-time lump sum investment or regular contributions over time.
Flexibility is key in financial planning, and trust funds offer this by allowing contributors like Bob and Ann to invest in ways that suit their financial situation. A one-time investment might not always be feasible, so spreading out contributions with a continuous money stream might be more practical. This approach still aims for the trust fund's future goal, ensuring it accrues to the desired amount thanks to consistent additions over time and the power of continuous compounding interest.

Continuous Money Stream

In scenarios where a large initial investment isn't possible, a continuous money stream can be a useful alternative. This involves making consistent, smaller contributions over time rather than putting down a single, substantial amount upfront. The money stream concept leverages continuous compounding, meaning that each contribution is immediately earning interest.
To calculate the needed contribution rate for a continuous money stream, Bob and Ann can use the formula:

• \( A = \int_0^t R(t) e^{r(t-x)} \, dx \)

This integral ensures that every year's contribution \( R(t) \), grows at a rate compounded continuously until year 24 when the target amount needs to be reached. By integrating over the 24 years, they can discover how much needs to be contributed each year. In this instance, Bob and Ann find they should deposit approximately \( \\(19,272.99 \) annually to reach \( \\)250,000 \) by Brenda's 24th birthday. This strategy takes advantage of the exponential power of compounding, allowing regular, smaller investments to achieve significant financial goals over time.