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Chapter 5: Problem 8

Your friend invests \(\$ 2000\) at \(5 \frac{1}{4} \%\) per annum compounded semiannually. You invest an equal amount at the same yearly rate, but compounded daily. How much larger is your account than your friend's after 8 years?

Short Answer

Expert verified
Your account is approximately $29.82 larger than your friend's after 8 years.

Step by step solution

01

Convert the interest rates to decimal form

First, convert the interest rate from a percentage to a decimal. The interest rate given is \(5 \frac{1}{4}\%\) which is equivalent to \(5.25\%\). To convert this to a decimal, divide by 100, yielding \(0.0525\).
02

Calculate final amount for semiannual compounding

For your friend's investment, which is compounded semiannually, use the compound interest formula: \[A = P \left(1 + \frac{r}{n}\right)^{nt}\]where \(P = 2000\), \(r = 0.0525\), \(n = 2\) (since interest is compounded twice a year), and \(t = 8\).Substitute the values into the formula:\[A = 2000 \left(1 + \frac{0.0525}{2}\right)^{2 \cdot 8}\]\[A = 2000 \left(1 + 0.02625\right)^{16}\]\[A = 2000 \times 1.02625^{16}\]Calculate \(A\).
03

Calculate final amount for daily compounding

For your investment, which is compounded daily, use the same compound interest formula:\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]where \(n = 365\) since interest is compounded daily.Substitute the values into the formula:\[A = 2000 \left(1 + \frac{0.0525}{365}\right)^{365 \cdot 8}\]\[A = 2000 \times \left(1 + 0.0001438356\right)^{2920}\]Calculate \(A\).
04

Compare the two investment amounts

Subtract your friend's final amount from your own final amount to find the difference. Let \(A_{friends}\) be the friend's final amount and \(A_{your}\) be your final amount.\(\text{Difference} = A_{your} - A_{friends}\)Calculate this amount to find out how much larger your account is compared to your friend's.

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Key Concepts

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

Semiannual Compounding
Semiannual compounding means that interest is added to the principal twice a year. This approach impacts how quickly your investment grows.
  • Principal amount: The starting sum, here, \$2000\.
  • Interest rate: Given as \5.25\% annually.
  • Frequency: Semiannual means the interest application is twice a year, or every 6 months.
The formula to determine the future value with semiannual compounding is the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where \(n = 2\) because the compounding occurs semiannually. Each half year, the interest previously computed gets added to your balance, which starts earning interest itself, so over long periods, the total amount, \(A\), grows faster than simple interest.
Daily Compounding
Daily compounding takes the concept of compound interest to the next level by adding interest every single day. Each day, a tiny fraction of interest is calculated and added to the balance.
  • Principal: Same initial investment, \$2000\.
  • Annual interest rate: \5.25\%.
  • Compounding frequency: Here, each day, making it \365\ times a year.
The formula remains \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] In this case, \(n = 365\). The frequency of calculation means more frequent additions to the principal, which makes daily compounding typically more beneficial over longer periods. Since the balance is updated daily, it continuously earns slightly more each subsequent day. It demonstrates the power of compound interest over time.
Interest Rate Conversion
Interest rate conversion is crucial in understanding how different compounding periods affect the total interest earned or paid. By transforming the percentage rate into a practical rate for compounding frequency, you accurately reflect how interest accumulates.
The interest listed as \5.25\% per annum doesn't apply directly in all cases since that rate must be adjusted according to the number of compounding periods within a year.
For semiannual compounding, divide \5.25\% by \2\ to get a per-period rate. For daily, divide by \365\. This step ensures that the calculations in the compound interest formula accurately reflect the situation, providing a usable interest amount to be inserted into \[ \frac{r}{n} \].
Investment Comparison
Investment comparison allows you to determine which investment strategy yields a better return. By comparing the end balances of two different compounding methods, you make informed financial decisions.
  • Final Amount (Semiannual): Calculated using \ n = 2\.
  • Final Amount (Daily): Calculated using \ n = 365\.
  • Difference: Subtract the semiannual amount from the daily to find extra earnings.
In this case, the exercise asks to find out which investment grows more over \8\ years. The \[ A_{your} - A_{friends} \] formula demonstrates the additional earnings from daily compounding. Even a slight increase each day can lead to a significant difference, especially over more extended periods.

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