Problem 32 - Calculus and its applications

Question

Ted and Edith Markey have a new grandchild, Kurt, and want to create a trust fund for him that will yield \(\$ 1,000,000\) on his 22 nd birthday. a) What lump sum should they deposit now at \(4.2 \%\), compounded continuously, to achieve \(\$ 1,000,000 ?\) 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 \(\$ 1,000,000,\) assuming an interest rate of \(4.2 \%\) compounded continuously.

Step-by-Step Solution

Answer

a) Deposit approximately $396,252.27 now. b) Invest approximately $27,652.79 per year.

Step 1: Understanding the Formula for Continuous Compounding

The formula for continuous compounding is given by the equation \( A = P e^{rt} \), where \( A \) is the amount of money accumulated after time \( t \), \( P \) is the principal amount (initial deposit), \( r \) is the annual interest rate, and \( t \) is the time in years. We need to solve for \( P \) that gives \( A = 1,000,000 \) after 22 years at an interest rate of 4.2%. Convert the interest rate to decimal by dividing by 100: \( r = 0.042 \).

Step 2: Calculating the Lump Sum Deposit

Substitute the known values into the continuous compounding formula: \( 1,000,000 = P e^{0.042 \times 22} \). First, calculate the exponent: \( 0.042 \times 22 = 0.924 \). Therefore, the equation becomes \( 1,000,000 = P e^{0.924} \). Now solve for \( P \) by dividing both sides by \( e^{0.924} \): \( P = \frac{1,000,000}{e^{0.924}} \). Calculate \( e^{0.924} \), which is approximately 2.519. Thus, \( P \approx \frac{1,000,000}{2.519} \approx 396,252.27 \).

Step 3: Setting up the Formula for a Continuous Money Stream

The formula to calculate the amount accumulated from a continuous money stream with continuous compounding is \( A = R \frac{e^{rt} - 1}{r} \), where \( R \) is the amount deposited per year. We set \( A = 1,000,000 \), \( r = 0.042 \), and \( t = 22 \).

Step 4: Solving for the Annual Continuous Payment R(t)

Substitute the known values into the continuous money stream formula: \( 1,000,000 = R \frac{e^{0.042 \times 22} - 1}{0.042} \). We already calculated \( e^{0.924} \approx 2.519 \). Hence, \( e^{0.924} - 1 \approx 1.519 \). The equation becomes \( 1,000,000 = R \frac{1.519}{0.042} \). Simplify to find \( 1,000,000 = R \times 36.1667 \). Divide both sides by 36.1667 to solve for \( R \): \( R \approx \frac{1,000,000}{36.1667} \approx 27,652.79 \).

Key Concepts

Trust Fund

A trust fund is a special financial arrangement where assets, such as money or property, are held by a trustee for the benefit of a beneficiary. In this case, Ted and Edith want to create a trust fund for their grandchild, Kurt, to ensure he has financial security when he turns 22 years old. The goal of the trust fund is to reach a future value of $1,000,000, which can serve as a significant financial foundation for Kurt. Creating a trust fund involves careful planning, not only to decide the amount to deposit but also to choose the right investment strategy to achieve the desired future value. Using continuous compounding as a strategy helps maximize the growth of the initial deposit or regular contributions, ensuring the final fund meets the goal. Understanding how to apply financial formulas when setting up a trust fund is critical in making informed investment decisions.

Future Value

Future value (FV) refers to the amount of money an investment will grow to over a period of time when interest is applied. In this situation, Ted and Edith want their initial investment, or a series of regular investments, to grow to \(1,000,000 by Kurt's 22nd birthday. The formula for future value using continuous compounding is \[ A = P e^{rt} \] where:

• \(A\) is the future amount (\)1,000,000).
• \(P\) is the initial principal amount.
• \(r\) is the annual interest rate (converted to a decimal).
• \(t\) is the time in years.

For part (a), we calculate the lump sum needed now to meet the future value. However, if this amount is too large, they may opt for constant contributions over time as addressed in part (b), still using continuous compounding to reach the same future value.

Exponential Growth

Exponential growth describes a situation where the rate of growth is proportional to the current amount. In financial terms, it means the investment grows faster as the principal increases. Continuous compounding is a prime example of exponential growth, as it consistently reinvests earned interest, leading to greater returns over time.The formula for a continuous money stream, applicable when making regular deposits, is: \[ A = R \frac{e^{rt} - 1}{r} \] where:

• \(A\) is the desired future value ($1,000,000).
• \(R\) is the constant amount contributed per year.
• \(r\) is the annual interest rate, as a decimal.
• \(t\) denotes the duration in years.

This formula helps in finding the necessary yearly deposit to achieve the same future value as if a lump sum were deposited initially. Understanding this concept is crucial when individual investments transform into substantial wealth due to the compounding effect. Leveraging exponential growth via continuous compounding can significantly aid in reaching financial goals efficiently.