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Chapter 14: Problem 50

Vaving for College When your first child is born, you begin to save for college by depositing $$\$ 400$$ per month in an \(\mathrm{ac}\) count paying \(12 \%\) interest per year. With a continuous stream of investment and continuous compounding, how much will you have accumulated in the account by the time your child enters college 18 years later?

Short Answer

Expert verified
After 18 years, the account will have accumulated approximately \(\$190,171.59\).

Step by step solution

01

Convert annual interest rate to monthly interest rate

Divide the annual interest rate by 12 to get the monthly interest rate: \(r = 0.12 / 12\).
02

Convert time from years to months

Multiply the number of years by 12 to get the number of months: \(t = 18 * 12\).
03

Calculate the future value

Use the formula for the future value of a continuous annuity with continuous compounding: \(FV = P \cdot \frac{e^{rt} - 1}{r}\) Substitute the values into the formula and calculate the future value: \(FV = 400 \cdot \frac{e^{(0.12 / 12) (18 * 12)} - 1}{(0.12 / 12)}\) \(FV = 400 \cdot \frac{e^{(0.01) (216)} - 1}{(0.01)}\) Use a calculator or software to determine the value for the expression: \(FV \approx \$ 190,171.59\) So, after 18 years, the account will have accumulated approximately \( \$190,171.59\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Future Value Calculation
Understanding the future value of money is essential for planning long-term goals like saving for college. In finance, the future value (FV) calculation helps you determine how much your money today will be worth in the future after gaining interest.

To calculate the future value of a lump sum, you can use the formula: \[FV = PV \times (1 + r)^t\] where:
  • \(PV\) is the present value or initial amount,
  • \(r\) is the interest rate per period,
  • \(t\) is the number of periods.
When contributions are made regularly, like a monthly deposit for college savings, the formula adjusts to account for these ongoing investments. This approach reflects the reality that most people save incrementally over time, rather than in a single lump sum.

In our exercise, the regular deposits contribute to an annuity, which is a series of equal payments made at regular intervals. The future value of these payments needs to be calculated accurately to ensure sufficient funds will be available when the child enters college. It's crucial to factor in the interest rate and the compounding frequency to achieve an accurate prediction of future savings.
Continuous Compounding Interest
When saving for college, how your interest is compounded can greatly impact your total savings. Unlike simple or periodic compounding, continuous compounding calculates interest at an infinitesimally small interval, essentially compounding all the time.

The formula to calculate the future value with continuous compounding is given by: \[FV = P \times e^{rt}\] where:
  • \(P\) represents the principal amount,
  • \(e\) is Euler's number (approximately 2.71828), indicative of continuous growth,
  • \(r\) is the annual interest rate, and
  • \(t\) is the time in years.
For our exercise, since we're depositing monthly and compounding continuously, we applied a modified version of this formula to take into account the annuity aspect of the problem. The process showcases the power of exponential growth over time, which significantly benefits long-term investments like college savings plans. By leveraging the concept of continuous compounding, you allow your investment to grow at the fastest rate theoretically possible.
Annuity Savings
An annuity is a financial product that allows individuals to save money over time and then receive payments from that investment in the future. This concept is often applied in the context of retirement plans, but it's equally valuable when saving for college. Annuity savings involve making consistent investments, which accumulate and earn interest over the period until withdrawal.

In the case of saving for college, you would be contributing a fixed amount at regular intervals, such as monthly, into an account that yields interest. The annuity formula for calculating the future value of these series of payments is crucial for determining how much money will be available when the child is ready for college.

The ability to predict how your current savings behavior will impact your future financial position is the primary benefit of understanding annuity savings. A well-managed annuity can ensure that educational expenses are covered, providing peace of mind that funding will be available when the time comes to pay for college.

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