Chapter 2: Problem 22
Given that the future value of \(\$ 950\) subject to continuous compounding will be \(\$ 1,000\) after half a year, find the interest rate.
Short Answer
Expert verified
The interest rate is approximately 10.26%.
Step by step solution
01
- Write the formula for continuous compounding
The formula for continuous compounding is given by \[ A = P e^{rt} \]where:- \(A\) is the future value,- \(P\) is the present value,- \(r\) is the interest rate,- \(t\) is the time in years.
02
- Substitute the given values into the formula
Substitute the given values into the formula: \[ 1000 = 950 e^{r \times 0.5} \]where \(A = 1000\), \(P = 950\), and \(t = 0.5\).
03
- Solve for the interest rate \(r\)
First, isolate the exponential expression: \[ \frac{1000}{950} = e^{0.5r} \]Next, take the natural logarithm on both sides to solve for \(r\):\[ \frac{1000}{950} = e^{0.5r} \]\[ \text{ln}\frac{1000}{950} = 0.5r \]Finally, solve for \(r\):\[ r = 2 \times \text{ln}\frac{1000}{950} \]Calculating \(r\) will give the interest rate.
04
- Calculate the numerical value
Compute the natural logarithm and the interest rate:\[ \text{ln}\frac{1000}{950} \text{ is approximately } 0.0513 \]Thus:\[ r = 2 \times 0.0513 = 0.1026 \] which is approximately 10.26%.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Future Value
To understand continuous compounding, you first need to grasp the concept of Future Value (FV). Essentially, FV represents the value of an investment at a specific point in the future. For example, if you invest \(950 today, and that investment grows to \)1000 in six months, then \(1000 is the Future Value. This estimation helps you to understand how much an investment is worth after interest is applied over a set period of time.
Generally, the formula used to calculate the future value under continuous compounding is \[ A = P e^{rt} \], where \(A\) is the Future Value.
Continuous compounding means that interest is calculated and added to the principal at every possible instant, making it the most frequent form of compounding.
In our exercise, the future value was given as \)1000, and we were asked to find the interest rate.
Generally, the formula used to calculate the future value under continuous compounding is \[ A = P e^{rt} \], where \(A\) is the Future Value.
Continuous compounding means that interest is calculated and added to the principal at every possible instant, making it the most frequent form of compounding.
In our exercise, the future value was given as \)1000, and we were asked to find the interest rate.
Interest Rate Calculation
Calculating the interest rate with continuous compounding requires you to manipulate the formula \[ A = P e^{rt} \].
Let's walk through the given exercise to better understand this concept.
Given: \[ A = 1000 \], \[ P = 950 \], and time, \( t = 0.5 \) years.
Step 1: Substitute the known values into the formula to get \[ 1000 = 950 e^{r \times 0.5} \].
Step 2: Isolate the exponential term to get \[ \frac{1000}{950} = e^{0.5r} \].
Step 3: Take the natural logarithm (ln) of both sides. This gives us \[ \text{ln}\frac{1000}{950} = 0.5r \].
Step 4: Solve for \( r \), we get \[ r = 2 \times \text{ln} \frac{1000}{950} \]. This simplified equation shows how you can find the interest rate for continuous compounding given the present value, future value, and time.
Let's walk through the given exercise to better understand this concept.
Given: \[ A = 1000 \], \[ P = 950 \], and time, \( t = 0.5 \) years.
Step 1: Substitute the known values into the formula to get \[ 1000 = 950 e^{r \times 0.5} \].
Step 2: Isolate the exponential term to get \[ \frac{1000}{950} = e^{0.5r} \].
Step 3: Take the natural logarithm (ln) of both sides. This gives us \[ \text{ln}\frac{1000}{950} = 0.5r \].
Step 4: Solve for \( r \), we get \[ r = 2 \times \text{ln} \frac{1000}{950} \]. This simplified equation shows how you can find the interest rate for continuous compounding given the present value, future value, and time.
Natural Logarithm
To solve for the interest rate in continuous compounding, you need to understand the natural logarithm (ln). This mathematical function helps in converting the exponential equations so you can easily isolate the variable you are solving for.
In our exercise, the equation \[ \frac{1000}{950} = e^{0.5r} \] was converted to \[ \text{ln}\frac{1000}{950} = 0.5r \] by taking the natural logarithm on both sides.
The natural logarithm has the base e (approximately 2.718), making it ideal for situations involving continuous growth.
In simpler terms, ln is the opposite of the exponential function, and it helps to bring down the exponent so you can solve for the interest rate, as shown in the exercise.
In our exercise, the equation \[ \frac{1000}{950} = e^{0.5r} \] was converted to \[ \text{ln}\frac{1000}{950} = 0.5r \] by taking the natural logarithm on both sides.
The natural logarithm has the base e (approximately 2.718), making it ideal for situations involving continuous growth.
In simpler terms, ln is the opposite of the exponential function, and it helps to bring down the exponent so you can solve for the interest rate, as shown in the exercise.
Financial Mathematics
Financial mathematics is the application of mathematical methods to financial problems. One such problem is calculating the future value of an investment with continuous compounding.
Financial mathematics provides various tools, like the continuous compounding formula \[ A = P e^{rt} \], to help investors and analysts predict future values, understand compound interest, and calculate growth rates.
By understanding and applying these methods, you can make more informed financial decisions.
For instance, knowing how to isolate the interest rate in the continuous compounding formula helps when you need to compare different investment options. Similarly, understanding how to use the natural logarithm in financial calculations allows for a more profound comprehension of growth patterns and interest dynamics.
In sum, financial mathematics equips you with robust tools to break down complex financial problems into simpler, more manageable calculations.
Financial mathematics provides various tools, like the continuous compounding formula \[ A = P e^{rt} \], to help investors and analysts predict future values, understand compound interest, and calculate growth rates.
By understanding and applying these methods, you can make more informed financial decisions.
For instance, knowing how to isolate the interest rate in the continuous compounding formula helps when you need to compare different investment options. Similarly, understanding how to use the natural logarithm in financial calculations allows for a more profound comprehension of growth patterns and interest dynamics.
In sum, financial mathematics equips you with robust tools to break down complex financial problems into simpler, more manageable calculations.
