You invest $1,000 in a certificate of deposit that matures after 10 years and pays 5 percent interest, which is compounded annually until the certificate matures.
a) How much interest will the saver earn if the interest is left to accumulate?
b) How much interest will the saver earn if the interest is withdrawn each year?
c) Why are the answers to a and b different
To calculate the answers to these questions, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
Now, let's solve the questions step by step:
a) How much interest will the saver earn if the interest is left to accumulate?
In this case, since the interest is left to accumulate, we are considering the total amount including both the initial principal and the interest earned.
Using the formula mentioned above, we substitute the given values:
P = $1,000, r = 5% (0.05 as a decimal), n = 1 (since interest is compounded annually), t = 10 years.
A = $1,000(1 + 0.05/1)^(1*10)
A = $1,000(1 + 0.05)^10
A = $1,000(1.05)^10
A ≈ $1,628.89
To find the interest earned, we subtract the initial principal from the total amount:
Interest earned = A - P
Interest earned = $1,628.89 - $1,000
Interest earned ≈ $628.89
Therefore, the saver will earn approximately $628.89 in interest when it is left to accumulate.
b) How much interest will the saver earn if the interest is withdrawn each year?
In this case, we assume that the interest is withdrawn at the end of each year. This means the interest earned in a year is not added back into the investment.
We can use the same formula, but for each year, we calculate the interest earned, and then subtract it from the initial principal for the next year's calculation.
Again, using the initial values:
P = $1,000, r = 5% (0.05 as a decimal), n = 1 (since interest is compounded annually), t = 10 years.
To find the interest earned each year, we calculate:
Year 1: Interest earned = P * r = $1,000 * 0.05 = $50
Year 2: Interest earned = ($1,000 - $50) * r = $950 * 0.05 = $47.50
Year 3: Interest earned = ($950 - $47.50) * r = $902.50 * 0.05 = $45.13
We repeat this calculation for all 10 years, and then sum up the interest earned each year:
Total interest earned = $50 + $47.50 + $45.13 + ... + interest earned in year 10
The final total will be the interest earned if withdrawn each year.
c) Why are the answers to a and b different?
The answers to a and b are different because in scenario a, the interest is left to accumulate, meaning it is reinvested each year and added to the initial principal, resulting in compounding interest. This leads to higher total earnings over time.
In scenario b, the interest is withdrawn each year and not reinvested. As a result, the interest earned each year is not added back into the principal amount, causing the total interest earned to be lower compared to scenario a.
Essentially, scenario a takes advantage of the compounding effect, while scenario b does not.