Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 16
What is the future value of \(\$ 600\) in 20 years assuming an interest rate of 11 percent compounded semiannually?
Short Answer
Expert verified
The future value of \(600 in 20 years, assuming an interest rate of 11 percent compounded semiannually, is approximately \)5,527.20.
Step by step solution
01
Analyze the given information
We are given:
- Present value (PV) = $600
- Interest rate (annual) = 11% = 0.11
- Compounding frequency = Semiannually
- Time (t) = 20 years
Now, we need to calculate the number of compounding periods (n) and the effective interest rate (r) for semiannual compounding.
02
Calculate the number of compounding periods (n)
Since the interest is compounded semiannually, there will be 2 compounding periods in a year. We need to find the total number of compounding periods for 20 years.
n = Compounding periods in a year * Number of years
n = 2 (compounding periods) * 20 (years)
n = 40 compounding periods
03
Calculate the effective interest rate (r)
Now, we need to find the effective interest rate per compounding period. Since the annual interest rate is 11 percent, we need to divide this by the number of compounding periods in a year to get the interest rate per compounding period.
r = (Annual Interest Rate) / (Number of compounding periods in a year)
r = 0.11 / 2
r = 0.055
04
Use the Future Value formula
The formula for the Future Value (FV) is:
FV = PV * (1 + r)^n
From Steps 2 and 3, n = 40 and r = 0.055.
Plugging these values into the formula, we can calculate the future value of the investment.
FV = 600 * (1 + 0.055)^40
05
Calculate the Future Value (FV)
Now we just have to compute the value for the FV:
FV 鈮� 600 * (1.055)^40
FV 鈮� 600 * 9.212
FV 鈮� $5,527.20
The future value of \(600 in 20 years, assuming an interest rate of 11 percent compounded semiannually, is approximately \)5,527.20.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Compounding Interest
Compounding interest is a powerful concept in finance that refers to earning interest on not just the initial principal but also on accumulated interest from previous periods. Essentially, interest is calculated on top of interest, which can significantly increase the growth of an investment over time.
To understand how compounding works, imagine putting money in a savings account. Initially, you earn interest on your principal. Over time, as interest is added to your principal, future interest calculations include the additional interest already earned. This snowball effect is what makes compounding so attractive.
Key points include:
To understand how compounding works, imagine putting money in a savings account. Initially, you earn interest on your principal. Over time, as interest is added to your principal, future interest calculations include the additional interest already earned. This snowball effect is what makes compounding so attractive.
Key points include:
- Frequency of compounding: Determines how often interest is added to the principal. Common frequencies include annually, semiannually, quarterly, and monthly.
- Higher compounding frequency leads to a higher future value.
- Understanding compounding helps in making informed financial and investment decisions.
Present Value
Present value (PV) is the current worth of a future sum of money, given a specific rate of return. It answers the question: how much is a certain amount of money in the future worth today?
The concept of present value is based on the principle that money today is worth more than the same amount in the future due to its earning potential. This is closely tied to the opportunity cost of having money tied up in investments that could have generated returns.
Here's how it works:
The concept of present value is based on the principle that money today is worth more than the same amount in the future due to its earning potential. This is closely tied to the opportunity cost of having money tied up in investments that could have generated returns.
Here's how it works:
- PV calculations help you evaluate investment opportunities by providing a way to compare future cash flows to today's values.
- It determines what you would need to invest now to reach a certain future amount.
- Utilizing the correct discount rate is crucial, as it reflects the risk and opportunity cost associated with the investment.
Effective Interest Rate
The effective interest rate (EIR) represents the true annual interest rate, taking into account the effects of compounding. It's essential for accurately comparing financial products with different compounding periods.
Unlike the nominal rate, which doesn't consider the compounding effect, the EIR provides a more realistic view of the interest you'll earn or pay. To calculate the effective rate, use the formula:
\[EIR = (1 + \frac{r}{n})^n - 1\]
Where:
Unlike the nominal rate, which doesn't consider the compounding effect, the EIR provides a more realistic view of the interest you'll earn or pay. To calculate the effective rate, use the formula:
\[EIR = (1 + \frac{r}{n})^n - 1\]
Where:
- \(r\) is the nominal annual interest rate.
- \(n\) is the number of compounding periods per year.
Financial Formulas
Financial formulas serve as tools to solve a variety of monetary problems and make informed decisions. They simplify complex calculations, aiding in the planning and evaluation of financial strategies.
Several essential formulas are widely used:
Several essential formulas are widely used:
- Future Value (FV): Calculates how much an investment is worth after a certain period. The formula is:
\[FV = PV \times (1 + r)^n\] - Present Value (PV): Finds the value of a future sum of money in today's terms, aiding in investment choices.
- Effective Interest Rate (EIR): Converts nominal rates to a true rate that considers compounding, crucial for comparison.
