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Chapter 6: Problem 2

An investment: You open an account by investing \(\$ 250\) with a financial institution that advertises an APR of \(5.25 \%\), with continuous compounding. What account balance would you expect 1 year after making your initial investment?

Short Answer

Expert verified
The expected account balance after 1 year is approximately $263.48.

Step by step solution

01

Understand the Formula for Continuous Compounding

Continuous compounding involves using the formula for calculating the future value of an investment with continuous compounding, which is \( A = Pe^{rt} \) where \(A\) is the amount of money accumulated after n years, including interest, \(P\) is the principal amount (initial investment), \(r\) is the annual interest rate (in decimal), and \(t\) is the time (in years).
02

Define the Variables

In this case, \(P = 250\), \(r = 0.0525\) (since 5.25% is 5.25/100 in decimal), and \(t = 1\) year.
03

Substitute the Variables into the Formula

Plug the values of \(P\), \(r\), and \(t\) into the formula: \[ A = 250 \times e^{0.0525 \times 1} \].
04

Solve the Exponent

Calculate \( e^{0.0525} \) using a calculator. The result is approximately \( 1.0539 \).
05

Calculate the Final Amount

Multiply the principal \(P = 250\) by the result from Step 4: \( A = 250 \times 1.0539 \approx 263.475 \).
06

Round to the Nearest Cent

Since we are dealing with money, round the result to two decimal places: \( A \approx 263.48 \).

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Key Concepts

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

Future Value Calculation
The future value of an investment tells us how much an investment will be worth after a certain period, based on the present investment amount and interest accrued over time.
This metric is essential because it helps investors predict the financial benefits of their investments in the future.
Understanding future value calculations is crucial when comparing different investment opportunities.
  • It allows you to see the possible growth of different investment amounts over time.
  • You can also assess the impact of different interest rates on your investments.
With continuous compounding, the future value is calculated using the formula:
\[ A = Pe^{rt} \] Where,
  • \(A\) is the future value of the investment, including interest.
  • \(P\) is the principal amount, or initial investment.
  • \(r\) is the annual interest rate as a decimal.
  • \(t\) is the time the money is invested for in years.
  • \(e\) is the base of natural logarithms, approximately equal to 2.71828.
Grasping this formula is key to estimating how an investment will perform, allowing you to make informed financial decisions.
Interest Rate
An interest rate is the percentage at which interest is paid by a borrower for the use of money that they borrow from a lender, or earned on an investment.
It's a critical factor in calculating how investments grow over time.
When using continuous compounding, the interest rate determines how fast your investment grows.
In our example, the annual interest rate is 5.25%, which in decimal form becomes 0.0525. This conversion is crucial for calculations.
  • To convert a percentage rate to a decimal, divide by 100.
  • An interest rate in decimal form allows you to use it directly in mathematical formulas.
Understanding how to manipulate and use interest rates in calculations helps you compare and contrast different investment opportunities.
Compounding Formula
The concept of compounding involves earning interest on previously earned interest.
Continuous compounding is a specific and very powerful type of compounding.
Unlike simpler compounding methods, which might calculate interest monthly or annually, continuous compounding assumes the investment compounds an infinite number of times per year.
This can result in slightly higher returns compared to other compounding methods.
The compounding formula used here, \(A = Pe^{rt} \), utilizes the mathematical constant \(e\) to represent continuous growth.
  • The term \(e^{rt}\) represents the continuous growth factor over time \(t\) at a rate \(r\).
  • Continuous compounding allows the investment to grow at an exponential rate.
Understanding the compounding formula is important because it calculates how both the principal and previously accumulated interest grow together over time, thus maximizing potential investment returns.
Investment Growth
Investment growth reflects how much an initial investment increases over time.
This growth depends greatly on the interest rate and the method of compounding.
Using continuous compounding, your investment's growth accelerates as interest is constantly being calculated and added back into the account.
This results in a constant and fluid growth pattern.
  • Investments benefitting from continuous compounding can grow faster than those with simple or periodic compounding.
  • Future value estimations can help you set goals as you understand how different variables affect growth.
By understanding investment growth through continuous compounding, you can make strategic decisions to optimize your investments for better returns.

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Most popular questions from this chapter

Looking up: The constant \(g=32\) feet per second per second is the downward acceleration due to gravity near the surface of the Earth. If we stand on the surface of the Earth and locate objects using their distance up from the ground, then the positive direction is up, so down is the negative direction. With this perspective, the equation of change in velocity for a freely falling object would be expressed as $$ \frac{d V}{d t}=-g $$ (We measure upward velocity \(V\) in feet per second and time \(t\) in seconds.) Consider a rock tossed up- ward from the surface of the Earth with an initial velocity of 40 feet per second upward. a. Use a formula to express the velocity \(V=V(t)\) as a linear function. (Hint: You get the slope of \(V\) from the equation of change. The vertical intercept is the initial value.) b. How many seconds after the toss does the rock reach the peak of its flight? (Hint: What is the velocity of the rock when it reaches its peak?) c. How many seconds after the toss does the rock strike the ground? (Hint: How does the time it takes for the rock to rise to its peak compare with the time it takes for it to fall back to the ground?)

Mileage for an old car: The gas mileage \(M\) that you get on your car depends on its age \(t\) in years. a. Explain the meaning of \(\frac{d M}{d t}\) in practical terms. b. As your car ages and performance degrades, do you expect \(\frac{d M}{d t}\) to be positive or negative?

Investing in the stock market: You are considering buying three stocks whose prices at time \(t\) are given by \(P_{1}(t), P_{2}(t)\), and \(P_{3}(t)\). You know that \(\frac{d P_{1}}{d t}\) is a large positive number, \(\frac{d P_{2}}{d t}\) is near zero, and \(\frac{d P_{3}}{d t}\) is a large negative number. Which stock will you buy? Explain your answer.

The acceleration due to gravity: From the time of Galileo, physicists have known that near the surface of the Earth, gravity imparts a constant acceleration of 32 feet per second per second. Explain how this shows that if air resistance is ignored, velocity for a falling object is a linear function of time.

Magazines: Two magazines, Alpha and \(B e t a\), were introduced at the same time with the same circulation of 100 . The circulation of Alpha is given by the function \(A\), which has the equation of change $$ \frac{d A}{d t}=0.10 A $$ The circulation of Beta is given by the function \(B\), which has the equation of change $$ \frac{d B}{d t}=10 $$ Here \(t\) is the time, in years, since the magazines were introduced. a. One of these functions is growing in a linear way, whereas the other is growing exponentially. Identify which is which, and find formulas for both functions. b. Which magazine is growing more rapidly in circulation? Be sure to explain your reasoning.
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