Problem 8 - Calculus Concepts

Question

Doubling Time How long would it take an investment to double under each of the following conditions? a. Interest is \(4.3 \%\) compounded semi-annually. b. Interest is \(5 \%\) compounded daily (use 365 days).

Step-by-Step Solution

Answer

a. 16.31 years; b. 13.87 years.

Step 1: Understand the Doubling Formula

To find out how long it will take for an investment to double, we use the compound interest formula:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Here, - \(A\) is the amount of money accumulated after \(n\) years, including interest.- \(P\) is the principal amount (initial investment).- \(r\) is the annual interest rate (decimal).- \(n\) is the number of times that interest is compounded per year.- \(t\) is the time in years.To find the doubling time, set \(A = 2P\), which simplifies to:\[ 2 = \left(1 + \frac{r}{n}\right)^{nt} \]

Step 2: Solve for Part (a) - Semi-Annually Compounded Interest

For part (a), the interest rate \(r = 0.043\) and it is compounded semi-annually, so \(n = 2\).Substitute these values into the formula:\[ 2 = \left(1 + \frac{0.043}{2}\right)^{2t} \] Now, solve for \(t\) by taking the natural logarithm of both sides:\[ \ln(2) = 2t \cdot \ln \left(1 + \frac{0.043}{2}\right) \]Solving for \(t\):\[ t = \frac{\ln(2)}{2 \cdot \ln \left(1 + \frac{0.043}{2}\right)} \]Calculate this to find the doubling time.

Step 3: Calculate Part (a)

Compute the result for part (a).\[ t = \frac{\ln(2)}{2 \cdot \ln \left(1 + \frac{0.0215\right)} \]Approximately, \( t \approx 16.31 \) years.

Step 4: Solve for Part (b) - Daily Compounded Interest

For part (b), the interest rate \(r = 0.05\) and it is compounded daily, so \(n = 365\).Substitute these values into the formula:\[ 2 = \left(1 + \frac{0.05}{365}\right)^{365t} \]Take the natural logarithm of both sides:\[ \ln(2) = 365t \cdot \ln \left(1 + \frac{0.05}{365}\right) \]Solving for \(t\):\[ t = \frac{\ln(2)}{365 \cdot \ln \left(1 + \frac{0.05}{365}\right)} \]Calculate this to find the doubling time.

Step 5: Calculate Part (b)

Compute the result for part (b).\[ t = \frac{\ln(2)}{365 \cdot \ln \left(1 + \frac{0.00013699}{365}\right)} \]Approximately, \( t \approx 13.87 \) years.

Key Concepts

Doubling Time

Doubling time is a crucial concept in finance, describing how long it takes for an investment to reach twice its original value under a specific interest rate. This measure helps investors understand the growth potential of their investments.

The general formula to discover doubling time uses the properties of compound interest. By setting the future value to double the principal:

• Step 1: Understand the equation - We begin with the compound interest formula \( A = P\left(1 + \frac{r}{n}\right)^{nt} \) and set \( A = 2P \), turning it into \( 2 = \left(1 + \frac{r}{n}\right)^{nt} \)
• Step 2: Solve for \( t \) - This requires taking the natural logarithm of both sides to isolate \( t \) for the equation:
• \[ t = \frac{\ln(2)}{n \cdot \ln\left(1 + \frac{r}{n}\right)} \]

In practice, the doubling time varies based on the interest rate and the compounding frequency, seen in problems setting rates at semi-annual or daily intervals. Each scenario yields different times, reflecting the impact of compounding frequency on growth.

Interest Rate

An interest rate determines how much an investment grows over time. Higher rates imply faster growth, as more interest accumulates on the principal amount. Interest rates are always expressed as a percentage and can be compounded in various ways, such as annually, semi-annually, or daily.

Key Elements of Interest Rates:

• Annual Percentage: This is the rate expressed as a year-long percentage, such as 4.3% or 5%.
• Compounding Frequency: The frequency with which the interest is applied to the principal, impacting overall returns.
• Conversion to Decimal: When used in formulas, the percentage is often converted to decimal form (e.g., 4.3% becomes 0.043).

Understanding how interest rates work is essential when predicting investment doubling time. It's important to consider both the rate and the compounding frequency, as these factors directly influence the overall growth of the investment. By accurately utilizing these concepts, one can effectively forecast the investment potential over time.

Exponential Growth

Exponential growth describes a process where the quantity increases at a rate proportional to its current value. In finance, this refers to a principal investment growing at a steadily compounding rate over time.

This type of growth is described by the compound interest formula, highlighting benefits such as:

• Acceleration: Investment increases rapidly once it reaches a critical mass.
• Continuous Kicks: The effect of accrued interest piling onto prior gains, leading to faster growth.
• Multi-dimensional Impact: Growth is influenced both by the rate of interest and the frequency of compounding.

In practical terms, exponential growth explains why an investment looks small initially but becomes substantial over time. This is common when small amounts invested continually over many years can result in significant wealth due to the power of compounding returns. Understanding exponential growth allows investors to value long-term investing, visualizing future potential benefits rather than short-term gains.