Problem 47 - Calculus Early Transcendentals

Question

(a) If \(\$ 1000\) is invested at \(8 \%\) per year compounded continuously (Exercise \(46),\) what will the investment be worth after 5 years? (b) If it is desired that an investment at \(8 \%\) per year compounded continuously should have a value of \(\$ 10,000\) after 10 years, how much should be invested now? (c) How long does it take for an investment at \(8 \%\) per year compounded continuously to double in value?

Step-by-Step Solution

Answer

(a) Approximately $1491.82, (b) About $4493.29, and (c) Around 8.66 years.

Step 1: Understand the formula for continuous compound interest

The formula for continuous compounding is given by \\[ A = Pe^{rt} \] \where \( A \) is the amount of money accumulated after \( n \) years, including interest. \( P \) is the principal amount (the initial amount), \( r \) is the annual interest rate (as a decimal), \( t \) is the time in years, and \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Step 2: Solve part (a) - Future value of the investment

We need to find the future value after 5 years of a $1000 investment at an 8% interest rate. \( P = 1000, r = 0.08, t = 5 \). Substitute these values into the formula: \\[ A = 1000e^{0.08 \times 5} \] \Calculate \( e^{0.4} \), and then multiply by 1000 to find \( A \).

Step 3: Calculate (a)

Calculate \( e^{0.4} \approx 1.49182 \). \Thus, \\[ A = 1000 \times 1.49182 \approx 1491.82 \] \The investment will be worth approximately $1491.82 after 5 years.

Step 4: Solve part (b) - Present value of the investment

To find how much should be invested now to have $10,000 after 10 years, given an 8% rate, rearrange the formula to solve for \( P \), using \( A = 10000, r = 0.08, t = 10 \): \\[ P = \frac{10000}{e^{0.08 \times 10}} \]

Step 5: Calculate (b)

Calculate \( e^{0.8} \approx 2.22554 \). \Then, \\[ P = \frac{10000}{2.22554} \approx 4493.29 \] \So, approximately $4493.29 should be invested now.

Step 6: Solve part (c) - Time to double the investment

We need to find \( t \) when \( A = 2P \), \( r = 0.08 \). Rearrange the continuous compounding formula: \\[ 2P = Pe^{0.08t} \Rightarrow 2 = e^{0.08t} \] \Take the natural logarithm of both sides: \\[ \ln(2) = 0.08t \] \Solve for \( t \).

Step 7: Calculate (c)

\( \ln(2) \approx 0.69315 \). \Thus, \\[ 0.08t = 0.69315 \Rightarrow t = \frac{0.69315}{0.08} \approx 8.6644 \] \It takes about 8.66 years for the investment to double.

Key Concepts

Investment Growth

Investment growth refers to how the value of an investment increases over time. It is a critical concept to understand when dealing with compounded interests, especially in continuous compounding, where interest is calculated continuously over infinite time intervals. Growth depends on factors like the initial amount invested, the interest rate, and the duration for which the investment is held.
To grasp investment growth using continuous compounding, consider the formula: \[ A = Pe^{rt} \] where:

• \( A \) is the future value of the investment after time \( t \)
• \( P \) is the principal amount, or initial investment
• \( r \) is the annual interest rate expressed as a decimal
• \( t \) is the time in years for which the money is invested

The factor \( e^{rt} \) represents the exponential growth of the investment, which is more pronounced the longer the money is invested and the higher the interest rate.

Interest Rate

The interest rate is the percentage at which your investment grows annually. In the context of continuous compounding, it's key to represent this rate as a decimal when using it in formulas. For example, an \( 8\% \) interest rate is written as \( 0.08 \).
The interest rate significantly impacts how fast an investment grows. In continuous compounding, the rate implies that interest is added to the principal non-stop, theoretically at every possible infinitesimal point in time.
When planning an investment, understanding how different interest rates can affect the future value is crucial. Here's why:

• Higher interest rates accelerate the rate of growth, yielding greater returns over the same period.
• Even minor changes in the interest rate can lead to substantial differences in the final amount.
• Constant compounding effects can result in a higher accumulated amount compared to other compounding methods like annual or quarterly.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases over time, creating a curve that represents ever-accelerating growth. In financial terms, continuous compounding embodies exponential growth. This happens because the accumulated interest compounds at a constant rate, constantly being added to the principal, which then earns more interest.
To see exponential growth in action, consider an investment with continuous compounding:

• Initial investment grows at a steady rate,
• Interest is computed as interest-on-interest as time progresses.
• The longer the time, the more pronounced the growth; thus, the graph of the amount over time becomes steeper.

Continuous compounding increases the effectiveness of interest rates, as the compound becomes more frequent. Hence, continuous compounding is especially beneficial for long-term investments. This exponential characteristic indicates that even a small increase in the rate or time can lead to high growth in the investment amount.