To calculate how much more money Seth will have in the second bank after one year due to daily compounding, we need to use the compound interest formula.
First, let's calculate the amount that Seth will have in the first bank after one year, using semiannual compounding.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A is the future value of the investment/loan, including interest
P is the principal investment amount (in this case, $100,000)
r is the annual interest rate (2% or 0.02)
n is the number of times that interest is compounded per year (semiannually, so 2)
t is the number of years the money is invested or borrowed for (1 year)
Using this formula, the amount in the first bank is calculated as:
A1 = 100,000(1 + 0.02/2)^(2*1)
= 100,000(1.01)^2
= 100,000 * 1.0201
= 102,010
So, Seth will have $102,010 in the first bank after one year with semiannual compounding.
Now let's calculate the amount that Seth will have in the second bank after one year, using daily compounding.
The formula for compound interest with continuous compounding is:
A = P(1 + r/n)^(nt)
We need to convert the annual interest rate and the number of times compounded per year to match the daily compounding.
The daily interest rate (rd) can be calculated from the annual interest rate (r) using the formula:
rd = (1 + r/n)^(n/d) - 1
Where:
rd is the daily interest rate
r is the annual interest rate (2% or 0.02)
n is the number of times that interest is compounded per year (365, since it's daily)
d is the number of days in a year (365)
Using this formula, the daily interest rate is calculated as:
rd = (1 + 0.02/365)^(365/365) - 1
= (1.000054795)^1 - 1
= 0.000054795
Now, let's calculate the amount in the second bank:
A2 = 100,000(1 + rd)^(365*1)
= 100,000(1.000054795)^365
= 100,000 * 1.02020134068841
= 102,020.13
So, Seth will have $102,020.13 in the second bank after one year with daily compounding.
To find the difference, we subtract the amount in the first bank from the amount in the second bank:
Difference = A2 - A1
= 102,020.13 - 102,010
≈ $10.13
Therefore, Seth will have approximately $10.13 more in the second bank after one year due to the daily compounding.