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Chapter 5: Problem 22

If the present value of \(\$ 1000\) to be received in 5 years is \(\$ 559.90,\) what rate of interest, compounded continuously, was used to compute this present value?

Short Answer

Expert verified
The interest rate is approximately 11.61\%.

Step by step solution

01

Identify the Formula

The formula to find the present value when interest is compounded continuously is given by: \[ PV = FV \times e^{-rt} \] where: PV = Present Value FV = Future Value r = interest rate (in decimal form) t = time (in years) e = Euler's number (approximately equal to 2.71828)
02

Plug in the Given Values

Here, PV = 559.90, FV = 1000, and t = 5. Plug these values into the formula: \[ 559.90 = 1000 \times e^{-r \times 5} \]
03

Isolate the Exponential Term

Divide both sides by 1000 to isolate the exponential term: \[ \frac{559.90}{1000} = e^{-5r} \] Simplify the fraction: \[ 0.5599 = e^{-5r} \]
04

Apply the Natural Logarithm

Take the natural logarithm (ln) of both sides to remove the exponential: \[ \ln(0.5599) = \ln(e^{-5r}) \] Using the property of logarithms, this simplifies to: \[ \ln(0.5599) = -5r \]
05

Solve for the Interest Rate

Divide both sides by -5 to solve for r: \[ r = \frac{\ln(0.5599)}{-5} \] Calculate the value: \[ r \approx \frac{-0.5804}{-5} \approx 0.11608 \] Convert to a percentage: \[ r \approx 11.61\% \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Present Value
The concept of present value (PV) allows you to determine how much a future sum of money is worth right now. This is significant because money available today has more value than the same amount in the future due to its potential earning capacity.

In our example, the present value of \(1000 to be received in 5 years is \)559.90. This tells us that \(559.90 today is equivalent to \)1000 in 5 years considering the rate of continuous compounding.

The general formula to calculate PV when interest is compounded continuously is:

\[ PV = FV \times e^{-rt} \]

Where:
  • PV = Present Value
  • FV = Future Value
  • r = interest rate (annual)
  • t = time (years)
  • e = Euler's number (about 2.71828)
Future Value
The future value (FV) is the sum of money you will receive or have after a certain period, given a particular interest rate. In our problem, the future value is $1000, meaning you expect to get this amount in 5 years.

The relationship between present and future value is crucial for financial planning and investment decisions. Using continuous compounding helps in these calculations because it compounds interest at every moment.

Using the formula:

\[ FV = PV \times e^{rt} \]

You can see how today's amount grows into the future amount. In reverse, the present value gives you an idea of what future funds are worth today.
Natural Logarithm
Natural logarithms (ln) are the logarithms to the base of Euler's number (e). They are used widely in mathematics, especially in problems involving continuous growth or decay.

In the given problem, isolating the exponential term and taking the natural logarithm simplifies the equation and helps to solve for the interest rate. This is done by the logarithmic property:

\[ \text{If} \text{ } a = e^b, \text{ } \text{then} \text{ } \text{ln}(a) = b \]

Using this property allowed us to transform:

\[ 0.5599 = e^{-5r} \]

to:

\[ \text{ln}(0.5599) = -5r \]

This is a powerful tool for solving exponential equations.
Euler's Number
Euler's number, denoted as e, is an irrational number approximately equal to 2.71828. It plays an essential role in calculus and complex analysis due to its unique properties.

One of the key areas where Euler's number is used is in continuous compounding of interest. This means that interest is compounded an infinite number of times per period.

In our formula:

\[ PV = FV \times e^{-rt} \]

Euler's number helps translate continuous growth into exponential terms that are manageable and calculable. The presence of e ensures a more accurate representation of growth compared to simple or periodic compounding.

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Most popular questions from this chapter

Ice Age Many scientists believe there have been four ice ages in the past 1 million years. Before the technique of carbon dating was known, geologists erroneously believed that the retreat of the Fourth Ice Age began about 25,000 years ago. In \(1950,\) logs from ancient spruce trees were found under glacial debris near Two Creeks, Wisconsin. Geologists determined that these trees had been crushed by the advance of ice during the Fourth Ice Age. Wood from the spruce trees contained \(27 \%\) of the level of \(^{14} \mathrm{C}\) found in living trees. Approximately how long ago did the Fourth Ice Age actually occur?

Solve the given differential equation with initial condition. $$5 y=3 y^{\prime}, y(0)=7$$

If an investment triples in 15 years, what yearly interest rate (compounded continuously) does the investment earn?

An investment of \(\$ 2000\) yields payments of \(\$ 1200\) in 3 years, \(\$ 800\) in 4 years, and \(\$ 500\) in 5 years. Thereafter, the investment is worthless. What constant rate of return \(r\) would the investment need to produce to yield the payments specified? The number \(r\) is called the internal rate of return on the investment. We can consider the investment as consisting of three parts, each part yielding one payment. The sum of the present values of the three parts must total \(\$ 2000 .\) This yields the equation $$ 2000=1200 e^{-3 r}+800 e^{-4 r}+500 e^{-5 r} $$ Solve this equation to find the value of \(r\)

Solve the given differential equation with initial condition. $$y^{\prime}-\frac{y}{7}=0, y(0)=6$$
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