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Chapter 5: Problem 71

Investments Compare investing \(\$ 2000\) at \(10 \%\) compounded monthly for 20 years with investing \(\$ 2000\) at \(13 \%\) compounded monthly for 20 years.

Short Answer

Expert verified
Investing at 13% results in approximately $20721.52, which is greater than the 10% investment's $14589.25.

Step by step solution

01

Understand the formula for compound interest

The formula used to calculate the future value of an investment compounded periodically is \( A = P \left( 1 + \frac{r}{n} \right)^{nt} \), 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 (as a decimal), \( n \) is the number of times that interest is compounded per unit year, and \( t \) is the time the money is invested for in years.
02

Calculate the future value for the 10% investment

Use the formula with \( P = 2000 \), \( r = 0.10 \), \( n = 12 \), and \( t = 20 \) years:\[ A = 2000 \left(1 + \frac{0.10}{12}\right)^{12 \times 20} \]First, compute \( 1 + \frac{0.10}{12} = 1.0083333 \). Then calculate \( (1.0083333)^{240} \). Finally, multiply by \( 2000 \). This results in approximately \( A = 14589.25 \).
03

Calculate the future value for the 13% investment

Use the same formula with \( P = 2000 \), \( r = 0.13 \), \( n = 12 \), and \( t = 20 \) years:\[ A = 2000 \left(1 + \frac{0.13}{12}\right)^{12 \times 20} \]Similar to the previous step, compute \( 1 + \frac{0.13}{12} = 1.0108333 \). Then calculate \( (1.0108333)^{240} \). Finally, multiply by \( 2000 \). This results in approximately \( A = 20721.52 \).
04

Compare the results

After calculating both future values, you can compare them: - The 10% interest investment results in approximately \( 14589.25 \).- The 13% interest investment results in approximately \( 20721.52 \).

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Key Concepts

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

Future Value Calculation
The concept of future value calculation is essential when you're looking at investments that grow over time. Future value refers to the amount of money an investment will grow to, after a specified period, at a given interest rate. To determine this, we use the compound interest formula. This formula considers how often the interest is compounded, which makes a significant difference in how much money you will potentially accumulate.
\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]
This formula ensures you account for both the principal and the exponential growth through compounding over multiple periods. The future value gives you a clear idea of your investment's potential by including interest accrued over time.
Investment Comparison
Investment comparison is a strategic approach to evaluate different investment options side-by-side. In our exercise, we are examining two scenarios: investing at 10% versus 13% interest, both compounded monthly over 20 years. The main focus here is to see which option yields a higher return.

To make a fair comparison, the initial amount and duration should be identical, as shown in the problem. With these variables held constant, the difference in future value is solely due to the interest rate differences. This demonstrates the power of higher compounding rates over the same period:
  • 10% compounding results in approximately $14,589.25.
  • 13% compounding leads to around $20,721.52.
Examining these results helps understand how even a small change in interest rate can significantly impact the future value of an investment.
Compound Interest Formula
The compound interest formula is central to calculating future value, as it incorporates the effect of interest on interest. This formula is powerful because, as interest is added to the principal, it also begins to accrue interest, leading to exponential growth:
\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]
Here, the terms make clear how compounding works:
  • \(P\) is the initial investment (principal).
  • \(r\) is the annual interest rate (converted to a decimal for calculation).
  • \(n\) is the number of compounding periods per year.
  • \(t\) represents the number of years the money is invested.
The formula captures how frequently interest is applied (monthly in our exercise), which is crucial as more frequent compounding generally results in higher accumulated interest.
Using this formula correctly can help optimize investment strategies by clearly showing how changes in interest and compounding frequency affect growth.

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Most popular questions from this chapter

Heavier birds tend to have a longer wing span than smaller birds. For one species of bird, the table lists the length \(L\) of the bird's wing span in feet if the bird weighs w pounds. $$\begin{array}{rrrrr}w(\mathrm{b}) & 0.22 & 0.88 & 1.76 & 2.42 \\\L(w)(\mathrm{ft}) & 1.38 & 2.19 & 2.76 & 3.07\end{array}$$ (a) Find a function that models the data. (b) Graph \(L\) and the data. (c) What weight corresponds to a wing span of 2 feet?

Solve each equation. Approximate answers to four decimal places when appropriate. $$16-4 \ln 3 x=2$$

Annuity If \(x\) dollars is deposited every 2 weeks \((26\) times per year) into an account paying an annual interest rate \(r,\) expressed in decimal form, then the amount \(A\) in the account after \(t\) years can be approximated by the formula $$ A=x\left(\frac{(1+r / 26)^{26 t}-1}{(r / 26)}\right) $$ If \(\$ 50\) is deposited every 2 weeks into an account paying \(8 \%\) interest, approximate the amount after 10 years.

Runvay Length There is a relation between an airplane's weight \(x\) and the runway length \(L\) required for takeoff. For some airplanes the minimum runway length \(L\) in thousands of feet may be modeled by \(L(x)=3 \log x,\) where \(x\) is measured in thousands of pounds. (Sourcet. I. Haefner, Introduction to Thangortation Systems.) (a) Graph \(L\) for \(0

Filters Impurities in water are frequently removed using filters. Suppose that a 1 -inch filter allows \(10 \%\) of the impurities to pass through it. The other \(90 \%\) is trapped in the filter. (a) Find a formula in the form \(f(x)=100 a^{x}\) that calculates the percentage of impurities passing through \(x\) inches of this type of filter. (b) Use \(f(x)\) to estimate the percentage of impurities passing through 2.3 inches of the filter.
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