Question

Which will deliver a higher future value after one year, a deposit of \(\$ 1,000\) attracting interest at \(15 \%\) compounded daily, or at \(15.5 \%\) compounded semi-annually?

Short answer

The deposit with 15% interest compounded daily delivers a higher future value.

Step-by-step solution

- Identify the Problem

Determine the future value of two different deposits with differing interest compounding methods. Compare the results to see which delivers a higher future value.

- Understanding the Formulas

The formula for compound interest is given by \[ A = P \times \bigg(1 + \frac{r}{n} \bigg)^{nt} \]where: - \(A\) is the amount of money accumulated after n years, including interest.- \(P\) is the principal amount (initial deposit).- \(r\) is the annual interest rate (decimal).- \(n\) is the number of times that interest is compounded per year.- \(t\) is the time the money is invested for in years.

- Calculate Daily Compounded Interest

Substitute the values for the daily compounded interest: - \(P = 1,000\)- \(r = 0.15\)- \(n = 365\)- \(t = 1\)Using the formula, we get: \[ A = 1000 \times \bigg(1 + \frac{0.15}{365} \bigg)^{365 \times 1} \] Calculate: \[ A = 1000 \times (1 + 0.0004109589)^{365} \] \[ A \approx 1000 \times 1.16183424 \] \[ A \approx 1161.83 \]

- Calculate Semi-Annually Compounded Interest

Substitute the values for the semi-annually compounded interest: - \(P = 1,000\)- \(r = 0.155\)- \(n = 2\)- \(t = 1\)Using the formula, we get: \[ A = 1000 \times \bigg(1 + \frac{0.155}{2} \bigg)^{2 \times 1} \] Calculate: \[ A = 1000 \times (1 + 0.0775)^{2} \] \[ A \approx 1000 \times 1.16100625 \] \[ A \approx 1161.01 \]

- Compare the Results

Compare the future values obtained from both compounding methods. \[1161.83 > 1161.01\] The daily compounded interest yields a higher future value.

Key concepts

daily compounding

Daily compounding involves calculating and adding interest to the principal balance every single day.
This method considers how frequently the interest is applied to the principal.
In the case of daily compounding, it is done 365 times a year.
For example, in the exercise provided, a deposit of \(\$1,000\) with a 15% interest rate compounded daily over one year was calculated as follows:

• Principal (P) = 1000
• Annual Interest Rate (r) = 0.15
• Number of times interest is compounded per year (n) = 365
• Time period in years (t) = 1

The formula used was: \[ A = 1000 \times \left(1 + \frac{0.15}{365}\right)^{365} \]
This calculation resulted in an amount of approximately \(\$1,161.83\).
This showcases how daily compounding can significantly increase the final amount due to continuous application of interest.

semi-annual compounding

Semi-annual compounding means the interest is added to the principal twice a year.
Here, the frequency of compounding is reduced compared to daily compounding.
For the given exercise, a deposit of \(\$1,000\) with an interest rate of 15.5%, compounded semi-annually over one year, was resolved as follows:

• Principal (P) = 1000
• Annual Interest Rate (r) = 0.155
• Number of times interest is compounded per year (n) = 2
• Time period in years (t) = 1

Using the formula: \[ A = 1000 \times \left(1 + \frac{0.155}{2}\right)^{2} \]
This yielded an amount of approximately \(\$1,161.01\).
Although less frequent than daily compounding, semi-annual compounding still allows for substantial growth of the initial investment over time.

future value comparison

When comparing future values obtained through different compounding methods, it's evident that the frequency of compounding makes a difference.
Let's break down the comparison:

• For daily compounding at 15%, you get a future value of about \(\$1,161.83\).
• For semi-annual compounding at 15.5%, the future value is approximately \(\$1,161.01\).

Even though the interest rate is slightly higher in semi-annual compounding, the more frequent application of interest in daily compounding delivers a higher future value.
This shows that the compounding frequency can significantly affect the final amount, illustrating the power of

interest rate formulas

Understanding interest rate formulas is crucial for calculating compound interest:
The general formula for compound interest is: \[ A = P \times \left(1 + \frac{r}{n}\right)^{nt} \]

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