Ben Garrison invested $15,000 at 5% compounded daily in a credit union account that matures in 1 year. He also invested 20000 at 5.25% compounded monthly in a Silver Screen account that matures in 4 years. (a) What is the interest earned at maturity for both accounts? (b) What is the annual percentage yield for each account?
a) Well, it looks like Ben Garrison is quite the savvy investor! Let's calculate the interest earned for each account.
For the first account, the formula for compound interest is A = P(1 + r/n)^(nt), where:
A = the final amount
P = the principal amount
r = the annual interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the number of years
Using the formula, the interest earned on the first account would be:
A = 15000(1 + 0.05/365)^(365*1)
A ≈ $15,769.86
For the second account, we'll use the same formula, but make sure to adjust for the different rate and compounding period:
A = 20000(1 + 0.0525/12)^(12*4)
A ≈ $23,599.08
Therefore, the interest earned at maturity for both accounts would be:
$15,769.86 for the first account, and
$23,599.08 for the second account.
b) To calculate the annual percentage yield (APY), we can use the formula APY = (1 + r/n)^(n*t) - 1, where:
r = the nominal interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the number of years
For the first account:
APY = (1 + 0.05/365)^(365*1) - 1
APY ≈ 0.0509 or 5.09%
And for the second account:
APY = (1 + 0.0525/12)^(12*4) - 1
APY ≈ 0.0540 or 5.40%
So, the annual percentage yield for the first account is approximately 5.09%, and for the second account, it is approximately 5.40%.
To calculate the interest earned at maturity for both accounts, we can use the formula:
A = P(1 + r/n)^(nt)
Where:
A is the final amount accumulated (including principal and interest)
P is the principal amount
r is the annual interest rate (expressed as a decimal)
n is the number of times that interest is compounded per year
t is the time in years
For the first account:
Principal (P) = $15,000
Annual interest rate (r) = 5% or 0.05 (converted to decimal)
Compounding frequency (n) = 365 (daily compounding)
Time (t) = 1 year
Using the formula:
A = 15000(1 + 0.05/365)^(365*1)
= 15000(1 + 0.00013699)^(365)
= $15,767.60
The interest earned for the first account is:
Interest earned = A - P
= $15,767.60 - $15,000
= $767.60
For the second account:
Principal (P) = $20,000
Annual interest rate (r) = 5.25% or 0.0525 (converted to decimal)
Compounding frequency (n) = 12 (monthly compounding)
Time (t) = 4 years
Using the formula:
A = 20000(1 + 0.0525/12)^(12*4)
= 20000(1 + 0.004375)^(48)
= $25,586.65
The interest earned for the second account is:
Interest earned = A - P
= $25,586.65 - $20,000
= $5,586.65
Now, to calculate the annual percentage yield (APY) for each account. The APY is the effective annual interest rate that takes into account compound interest over the course of one year.
For the first account:
APY = (1 + r/n)^(n) - 1
= (1 + 0.05/365)^(365) - 1
= 5.127%
For the second account:
APY = (1 + r/n)^(n) - 1
= (1 + 0.0525/12)^(12) - 1
= 5.413%
To find the interest earned at maturity for each account, we can use the formula for compound interest:
(a) For the first account, we have:
Principal amount (P) = $15,000
Annual interest rate (r) = 5% or 0.05 (expressed as a decimal)
Compounding periods in a year (n) = 365 (since it is compounded daily)
Time period (t) = 1 year
Using the formula for compound interest:
A = P * (1 + r/n)^(n*t)
A = 15,000 * (1 + 0.05/365)^(365*1)
A ≈ 15,000 * (1.0001369863)^365
A ≈ 15,000 * 1.051267
A ≈ $15,790.01
The interest earned at maturity is therefore approximately $15,790.01 - $15,000 = $790.01.
For the second account, we have:
Principal amount (P) = $20,000
Annual interest rate (r) = 5.25% or 0.0525 (expressed as a decimal)
Compounding periods in a year (n) = 12 (since it is compounded monthly)
Time period (t) = 4 years
Using the same formula for compound interest:
A = P * (1 + r/n)^(n*t)
A = 20,000 * (1 + 0.0525/12)^(12*4)
A ≈ 20,000 * (1.004375)^48
A ≈ 20,000 * 1.221623
A ≈ $24,432.46
The interest earned at maturity is therefore approximately $24,432.46 - $20,000 = $4,432.46.
(b) The Annual Percentage Yield (APY) represents the effective annual interest rate when compounding is taken into account. It takes into consideration the compounding frequency, which differs for each account.
For the first account, compounded daily, you can calculate the APY using the following formula:
APY = (1 + r/n)^n - 1
APY = (1 + 0.05/365)^365 - 1
APY ≈ 0.051267 or 5.13%
For the second account, compounded monthly, you can calculate the APY using the same formula:
APY = (1 + r/n)^n - 1
APY = (1 + 0.0525/12)^12 - 1
APY ≈ 0.054673 or 5.47%
So, the Annual Percentage Yield (APY) for the first account is approximately 5.13%, and for the second account, it is approximately 5.47%.