Question

Suppose that April has access to an investment that will pay 10 % interest compounded continuously. Which is better. to be given 1000 now so that she can take advantage of this investment opportunity or to be given $$ 1325$ after 3 years?

Short answer

April should take the \(1000 now and invest it, as it will grow to \)1349.86, which is more than $1325.

Step-by-step solution

Understand Continuous Compounding Formula

The formula for continuously compounded interest is given by: \( A = P e^{rt} \), where: - \( A \) is the amount of money accumulated after time \( t \), including interest.- \( P \) is the principal amount (the initial amount of money).- \( r \) is the annual interest rate (decimal).- \( t \) is the time the money is invested for in years.- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Calculate Future Value of \(1000 Invested Now

Given \( P = 1000 \), \( r = 0.10 \) (which is 10% as a decimal), and \( t = 3 \) years, substitute these values into the formula: \[ A = 1000 \times e^{0.10 \times 3} \]Calculate the exponent: \( 0.10 \times 3 = 0.3 \)Then, calculate: \[ A = 1000 \times e^{0.3} \]Using a calculator to find \( e^{0.3} \approx 1.3499 \):\[ A = 1000 \times 1.3499 \approx 1349.86 \]

Compare Future Value to Immediate \)1325 Payment

The future value of the \(1000 investment after 3 years at a 10% continuously compounded interest rate is \)1349.86. Compare this to the \(1325 she would receive after 3 years. Since \)1349.86 > \(1325, it is better for April to take the \)1000 now and invest it.

Key concepts

Investments

When it comes to investing, understanding how your money grows over time is crucial. Investments can take various forms, such as stocks, bonds, real estate, or savings accounts. Each type of investment has its own risk and return profile. The key goal of any investment is to make your money work for you. In this example, April has the option to either receive \(1000 now or \)1325 after three years. To make an informed decision, she needs to consider how her money will grow if invested now versus waiting for the fixed amount.

Compound Interest Formula

One essential concept in finance is the compound interest formula, especially continuous compounding. The formula for continuously compounded interest is expressed as: The formula for continuously compounded interest is given by: The variables in this formula are:
- is the principal amount or the initial investment.
- is the annual interest rate in decimal form. For a 10% interest rate, would be 0.10.
- represents the time the money is invested for, in years. In this scenario, .
- is the base of the natural logarithm, with a value of approximately 2.71828.
- is the accumulated amount after time .

Using the values from April's scenario, we need to calculate the future value of \(1000 invested at 10% interest for 3 years: To do this, we plug in the values: Calculate the exponent:
Multiply 0.10 by 3 to get:
Then, calculate
Using a calculator, we find :
Therefore, the accumulated amount would be approximately \)1349.86.
This demonstrates the powerful effect of continuous compounding.

Comparison of Investment Returns

The final step in determining the better option for April is to compare the returns of the two scenarios. If she takes the \(1000 now and invests it with continuous compounding at a 10% interest rate, she will end up with approximately \)1349.86 after three years. On the other hand, if she waits for the fixed payment of \(1325 after three years, she misses out on the additional \)24.86 she could have earned.
By comparing the two amounts:

• \(1000 now invested becomes ~\)1349.86 after 3 years
• \(1325 received after 3 years without investment

Clearly, investing the \)1000 now yields a higher return due to the magic of continuously compounding interest. This simple example illustrates the advantage of understanding investment returns and how effective investing can maximize your wealth over time.