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Chapter 3: Problem 8

How much more does \(\$ 1,000\) earn in eight years, compounded daily at \(5 \%,\) than \(\$ 1,000\) over eight years at 5\(\%\) , compounded semiannually?

Short Answer

Expert verified
The return from a daily-compounded investment will be \( \Delta A \) dollars more than a semi-annually compounded one.

Step by step solution

01

Understand and Apply the Compound Interest Formula

The formula for compound interest is \( A = P(1 + r/n)^(nt) \) where: \n\n- A is the amount of money accumulated after n years, including interest. \n- P is the principal amount (the initial amount of money). \n- r is the annual interest rate (in decimal form). \n- n is the number of times that interest is compounded per unit t. \n- t is the time the money is invested for in years.
02

Calculate the Return with Daily Compounding

The frequency of compounding for the daily compounding strategy is \( n = 365 \) times per year, and the interest rate is \( r = 0.05 \). So if you plug these values into the compound interest formula, you get: \n\n- \( A_{daily} = 1000 * (1 + 0.05/365)^(365 * 8) \)
03

Calculate the Return with Semiannual Compounding

The frequency of compounding for the semiannual compounding strategy is \( n = 2 \) times per year. Plug these values into the compound interest formula to get: \n\n- \( A_{semi} = 1000 * (1 + 0.05/2)^(2 * 8) \)
04

Calculate the Difference

Now, all you need to do is subtracting the amount earned with semiannual compounding from the amount earned with daily compounding to get the amount earned over eight years. The difference, \( \Delta A \), in returns can be calculated as: \n\n- \( \Delta A = A_{daily} - A_{semi} \)

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Key Concepts

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

Daily Compounding
When discussing compound interest, daily compounding refers to the interest being added to the principal balance every single day. This means the amount of money you earn in interest is recalculated daily, and each day's interest is calculated on that day's total balance, which includes the principal amount plus any interest that has been added.
  • Daily compounding typically results in earning more interest compared to less frequent compounding periods because you are accruing interest on the interest already earned.
  • Even small amounts of daily compounding can significantly increase how much an investment grows over time.

Being compounded 365 times a year might not seem significant initially, but with each compounding, the investment grows slightly larger. This means that the next day's interest calculation includes this slightly larger amount, leading to exponential growth over time.
Semiannual Compounding
Semiannual compounding means that the interest is compounded twice a year.
  • This type of compounding divides the annual interest rate into two equal parts, applying each half at the end of every six months.
  • It can be beneficial for investments with longer terms, offering more growth than simple annual compounding but less than daily compounding.

The semiannual compounding frequency might result in slightly less interest earned compared to daily compounding since the interest is calculated and added to the principal less frequently.
However, it provides a great balance for investors who prefer moderate risk and return over time.
Interest Rate
The interest rate is a percentage that defines how much interest will be added to an investment over a period of time.
  • With compound interest, the interest rate is usually expressed annually, even if compounding occurs more frequently.
  • The interest rate impacts how quickly investments grow; higher interest rates generally lead to faster growth of your principal amount.

In the context of the original exercise, the rate was 5%, or 0.05 when expressed as a decimal, which needed to be adjusted based on the frequency of compounding.
Understanding how the interest rate affects your investment can guide decisions on choosing the best compounding frequency for your financial goals.
Financial Mathematics
Financial mathematics involves the use of mathematical formulas and models to analyze financial problems. It is essential in understanding processes such as interest calculation and the impact of different compounding periods on investments.
  • Key to financial mathematics is the compound interest formula, which helps in calculating future value by considering variables like principal, interest rate, compounding frequency, and time.
  • Financial mathematics aids in making informed investment decisions by evaluating potential returns based on different scenarios.

In financial mathematics, understanding and applying these tools help in visualizing how money grows and the advantages of certain investment strategies, like choosing daily compounding over semiannual compounding for a potentially higher return.

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Most popular questions from this chapter

Ridgewood Savings Bank charges a \(\$ 27\) per check overdraft protection fee. On July \(8,\) Nancy had \(\$ 1,400\) in her account. Over the next four days, the following checks arrived for payment at her bank: July \(9, \$ 1,380.15,\) July \(10, \$ 670\) and \(\$ 95.67 ;\) July \(11, \$ 130 ;\) and July \(12, \$ 87.60 .\) How much will she pay in overdraft protection fees? How much will she owe the bank after July 12\(?\)

Albert Einstein said that compound interest was 鈥�. . .the most powerful thing I have ever witnessed.鈥� Work through the following exercises to discover a pattern Einstein discovered which is now known as the Rule of 72.. a. Suppose that you invest \(\$ 2,000\) at a 1\(\%\) annual interest rate. Use your calculator to input different values for \(t\) in the compound interest formula. What whole number value of \(t\) will yield an amount closest to twice the initial deposit? b. Suppose that you invest \(\$ 4,000\) at a 2\(\%\) annual interest rate. Use your calculator to input different values for \(t\) in the compound interest formula. What whole number value of \(t\) will yield an amount closest to twice the initial deposit? c. Suppose that you invest \(\$ 20,000\) at a 6\(\%\) annual interest rate. Use your calculator to input different values for \(t\) in the compound interest formula. What whole number value of \(t\) will yield an amount closest to twice the initial deposit? d. Albert Einstein noticed a very interesting pattern when an initial deposit doubles. In each of the three examples above, multiply the value of t that you determined times the percentage amount. For example, in a. multiply t by 1. What do you notice? e. Einstein called this the Rule of 72 because for any initial deposit and for any interest percentage, \(72 \div\) (percentage) will give you the approximate number of years it will take for the initial deposit to double in value. Einstein also said that 鈥淚f people really understood the Rule of 72 they would never put their money in banks.鈥� Suppose that a 10-year-old has $500 to invest. She puts it in her savings account that has a 1.75% annual interest rate. How old will she be when the money doubles?

Ed computes the ending balance for an account he is considering. The principal is \(\$ 20,000,\) and the interest rate is 5.39\(\%\) , compounded continuously for four years. He uses the formula \(B=p e^{t}\) and substitutes directly on his calculator. Look at the keystrokes he entered. $$20,000 \mathrm{e}^{\wedge}(.0539)(4)$$ He presses ENTER and sees this display. $$20000 \mathrm{e}^{\wedge}(.0539)(4)=84430.32472$$ Ed鈥檚 knowledge of compound interest leads him to believe that this answer is extremely unreasonable. To turn \(\$20,000\) into over \(\$84,000\) in just four years at 5% interest seems incorrect to him. a. Find the correct ending balance. b. Explain what part of Ed鈥檚 keystroke sequence is incorrect.

On May 29, Rocky had an opening balance of x dollars in an account that pays 3% interest, compounded daily. He deposits y dollars. Express his ending balance on May 30 algebraically.

Ron estimates that it will cost \(400,000 to send his daughter to a private college in 18 years. He currently has \)90,000 to deposit in an account. What simple interest rate must his account have to reach a balance of $400,000 in 18 years? Round to the nearest percent.
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