Problem 52 - Calculus

Question

You are the loan shark. What is \(\$ 1\) worth after a year of continuous compounding at \(1 \%\) per day?

Step-by-Step Solution

Answer

After a year, $1 grows to approximately $38.78 with continuous compounding at 1% per day.

Step 1: Understanding the Problem

We need to calculate the value of \(1 after a year with continuous compounding at an interest rate of 1% per day. The formula for continuous compounding is: \[ A = Pe^{rt} \] where \( A \) is the amount after time \( t \), \( P \) is the principal amount (\)1), \( r \) is the daily interest rate, and \( t \) is the time in years.

Step 2: Calculating the Daily Interest Rate

Since the interest is compounded at 1% per day, the daily interest rate \( r \) used in the formula is \( 0.01 \) (because 1% is equivalent to 0.01 in decimal form).

Step 3: Determining the Time Period in Years

The problem indicates the time period is one year. However, since the rate is given per day, we consider \( t \) in terms of years, which is still 1 year since we are directly using per day rates.

Step 4: Applying the Continuous Compounding Formula

Insert the values into the formula: \[ A = 1 imes e^{0.01 imes 365} \] since there are approximately 365 days in a year.

Step 5: Calculating the Exponent

Calculate the exponent: \[ 0.01 imes 365 = 3.65 \] This is the power to which \( e \) will be raised.

Step 6: Final Calculation

Now calculate the final amount using the calculation \[ A = e^{3.65} \] Using a calculator, we find \[ e^{3.65} \ \approx 38.78. \]

Key Concepts

Interest Rate

The interest rate is a fundamental concept in financial mathematics. When you see a percentage like 1% per day, it represents the growth of an investment or debt. This rate indicates how much extra money you earn on top of the initial amount (principal) over a specified time. In our problem, this is

• the rate at which the borrowed \(\$1\) will grow day by day,
• signifying a consistent profit.

Most interest rates are given annually, but in some cases, like with continuous compounding,they can be broken down into smaller time frames, such as daily.
Continuous compounding uses the mathematical constant \(e\), which helps calculate the precise growth as if the investment grows instantly at every moment, which is more efficient than yearly or monthly focus.

Exponential Growth

Exponential growth describes a process where the growth rate of a value is proportional to its current size, which results in increasingly rapid growth as time passes. In the case of continuous compounding, the formula involved

• uses an exponential function, expressed as \(e^{rt}\),
• where \(e\) is a constant approximately equal to 2.718.

This is used to model situations where growth happens concurrently, such as in population numbers or financial investments.
Exponential growth functions are powerful, showing how quickly things can grow under constant compounding. For example, in our problem, although the daily interest rate is a modest 1%, when continuously compounded, it results in significant growth over a year, calculating at approximately 38.78 dollars from just one dollar!

Financial Mathematics

Financial mathematics is a critical area of study that involves calculating and understanding growth, risk, and return on investments. It uses formulas and principles to predict how current investments will grow over time.

In our specific problem, the focus is on continuous compounding, a more advanced financial concept.

• It helps investors understand how their money will grow if the interest rate accumulates at every moment.
• Financial mathematics takes human emotion out of financial decision-making and relies purely on formulas and calculations.

By grasping these principles, individuals can make well-informed financial decisions, assessing the potential future value of their financial assets, and optimizing growth under specific conditions, like our example of replacing simple interest with compounding to realize more rewarding returns.