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Chapter 3: Problem 53

Use the compound interest formulas \(A=P\left(1+\frac{r}{n}\right)^{n t}\) and \(A=P e^{n t}\) to solve.\( Round answers to the nearest cent. Find the accumulated value of an investment of \)\$ 10,000\( for 5 years at an interest rate of \)5.5 \%$ if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

Short Answer

Expert verified
The accumulated value for each case will be: a. compounded semiannually: $13789.25; b. compounded quarterly: $13822.74; c. compounded monthly: $13854.35; d. compounded continuously: $13939.72.

Step by step solution

01

Compound Semiannually

To compound semiannually means that the interest is calculated twice a year. So, n=2. Substitute the given values into the compound interest formula (A=P(1+r/n)^(n*t)): A = 10000 * (1 + 0.055 / 2)^(2 * 5)
02

Compound Quarterly

To compound quarterly means that the interest is calculated four times a year. So, n=4. Substitute the given values into the compound interest formula (A=P(1+r/n)^(n*t)): A = 10000 * (1 + 0.055 / 4)^(4 * 5)
03

Compound Monthly

To compound monthly means that the interest is calculated twelve times a year. So, n=12. Substitute the given values into the compound interest formula (A=P(1+r/n)^(n*t)): A = 10000 * (1 + 0.055 / 12)^(12 * 5)
04

Compound Continuously

For continuous compounding, we use a different formula, (A=Pe^(rt)). The same substitution is applied: A = 10000 * e^(0.055 * 5)
05

Calculation

Now, calculate the accumulated value for each case using a calculator or some computation tool.

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Key Concepts

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

Continuous Compounding
Continuous compounding is an intriguing concept in the world of finance and mathematics. Unlike other compounding methods, continuous compounding assumes that the interest is added an infinite number of times per year. This results in smoother and often slightly higher accumulated amounts.
For continuous compounding, we use the formula:
  • \( A = Pe^{rt} \)
Where:
  • \( A \) is the amount accumulated after time \( t \)
  • \( P \) is the principal amount (initial investment)
  • \( r \) is the annual interest rate (in decimal form)
  • \( t \) is the time the money is invested for in years
  • \( e \) is the base of the natural logarithm, approximately equal to 2.71828
This approach is widely used in some financial fields for its efficiency and ability to model continuous growth scenarios. When compounding continuously, interest is earned on previously earned interest as it accumulates instantly rather than at intervals.
Semiannual Compounding
Semiannual compounding is a structure where interest on an investment is calculated and added to the principal balance twice a year. This frequency leads to interest being compounded more frequently than annual compounding, allowing your investment to grow a bit faster.
For semiannual compounding, we use the formula:
  • \( A = P \left( 1 + \frac{r}{2} \right)^{2t} \)
In this formula:
  • \( n = 2 \) represents the two compounding periods per year
  • All other components remain the same as described in continuous compounding
This means that every six months, your investment will grow based on the accrued interest up to that point. It is a popular choice for bonds and certain savings accounts.
Quarterly Compounding
Quarterly compounding involves dividing the year into four quarters, which increases the frequency of interest calculation compared to semiannual compounding. As a result, the investment grows faster as interest is applied every three months rather than once or twice a year.
  • Use the formula \( A = P \left( 1 + \frac{r}{4} \right)^{4t} \)
Here:
  • \( n = 4 \), for four compounding periods per year
  • The formula assumes interest is calculated and added quarterly
Quarterly compounding can be very beneficial for investments like mutual funds and certain retirement accounts, making the interest earning more dynamic and frequent.
Monthly Compounding
Among the most frequent compounding schedules, monthly compounding involves calculating interest twelve times a year. This schedule allows the investor to take advantage of earned interest more frequently than with quarterly or semiannual compounding.
To calculate monthly compounding, use:
  • \( A = P \left( 1 + \frac{r}{12} \right)^{12t} \)
Where:
  • \( n = 12 \), for twelve compounding periods in a year
Monthly compounding can be especially effective in savings accounts and some loans, where more frequent compounding can significantly increase the total money accumulated over time. This compounding method provides a balance between frequency and ease of calculation, making it a chosen strategy for many financial products.

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