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?
credit union investment
= 15000(1 + .05/365)^365 = $15,769.01
silver screen account
= 20000(1 + .0525/12)^12
= 21,075.64
total after 1 year = 36,844.65 on an investment of 35,000
To make any comparison, we have to evaluate at the same "time spot".
thus a) is not a valid question, we cannot compare the interest at the end of year 1 with that at end of year 4
b)
so 35000(1+i) = 36844.65
1+i = 1.0527
the equivalent annual rate is 5.27%
To calculate the interest earned at maturity for each account, we can use the formula:
Interest = Principal * (1 + Rate/100)^(Time)
For the first account, Ben Garrison invested $15,000 at 5% compounded daily for 1 year.
(a) Calculating the interest earned for the first account:
Principal = $15,000
Rate = 5% (converted to decimal form, 5/100 = 0.05)
Time = 1 year
Interest = 15000 * (1 + 0.05/365)^(365*1)
Interest ≈ $15,760.24
Therefore, the interest earned at maturity for the first account is approximately $15,760.24.
For the second account, Ben Garrison invested $20,000 at 5.25% compounded monthly for 4 years.
(a) Calculating the interest earned for the second account:
Principal = $20,000
Rate = 5.25% (converted to decimal form, 5.25/100 = 0.0525)
Time = 4 years
Interest = 20000 * (1 + 0.0525/12)^(12*4)
Interest ≈ $25,344.53
Therefore, the interest earned at maturity for the second account is approximately $25,344.53.
To calculate the Annual Percentage Yield (APY) for each account, we can use the following formula:
APY = (1 + Rate/100/N)^N - 1
Where:
Rate is the annual interest rate
N is the number of compounding periods in one year (365 for daily compounding and 12 for monthly compounding)
(b) Calculating the APY for the first account:
Rate = 5% (converted to decimal form, 5/100 = 0.05)
N = 365 (daily compounding)
APY = (1 + 0.05/365)^365 - 1
APY ≈ 5.13%
Therefore, the annual percentage yield for the first account is approximately 5.13%.
(b) Calculating the APY for the second account:
Rate = 5.25% (converted to decimal form, 5.25/100 = 0.0525)
N = 12 (monthly compounding)
APY = (1 + 0.0525/12)^12 - 1
APY ≈ 5.39%
Therefore, the annual percentage yield for the second account is approximately 5.39%.
To calculate the interest earned at maturity for each account, you can use the formula for compound interest:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
where:
A = the final amount (including principal and interest)
P = the principal amount (initial investment)
r = annual interest rate (expressed as a decimal)
n = number of times interest is compounded per year
t = number of years
For the first account:
P = $15,000
r = 5% = 0.05 (as a decimal)
n = compounded daily, so 365 times in a year
t = 1 year
Plugging in the values, we get:
\[A = 15,000 \left(1 + \frac{0.05}{365}\right)^{(365)(1)}\]
To calculate the interest earned, we subtract the initial principal from the final amount, so:
Interest earned at maturity = A - P
For the second account:
P = $20,000
r = 5.25% = 0.0525 (as a decimal)
n = compounded monthly, so 12 times in a year
t = 4 years
\[A = 20,000 \left(1 + \frac{0.0525}{12}\right)^{(12)(4)}\]
Again, we subtract the initial principal from the final amount to get the interest earned.
Now, let's calculate the annual percentage yield (APY) for each account. The APY is a measure of the annual rate of return, taking into account the effect of compounding.
The formula for APY is:
\[APY = (1 + \frac{r}{n})^n - 1\]
For the first account:
r = 5% = 0.05 (as a decimal)
n = compounded daily, so 365 times in a year
For the second account:
r = 5.25% = 0.0525 (as a decimal)
n = compounded monthly, so 12 times in a year
Now you have all the information to calculate the interest earned at maturity and the annual percentage yield for each account.