Austin invested
$4600
in an account that pays an annual interest rate of
3.8%
, compounded daily. Assume there are
365
days in each year. Answer each part.
(a) Find the amount in the account after one year, assuming no withdrawals are made.
Do not round any intermediate computations, and round your answer to the nearest cent.
$
(b) Find the effective annual interest rate, expressed as a percentage.
Do not round any intermediate computations, and round your answer to the nearest hundredth of a percent.
(a) We can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the amount after one year, P is the principal (initial investment), r is the annual interest rate (as a decimal), n is the number of times compounded per year, and t is the time in years.
In this case, we have:
A = 4600(1 + 0.038/365)^(365*1)
A = $4,781.53
So the amount in the account after one year is $4,781.53.
(b) The effective annual interest rate takes into account the compounding effect. We can use the formula:
EAR = (1 + r/n)^n - 1
where EAR is the effective annual interest rate and all other variables are as defined above.
In this case, we have:
EAR = (1 + 0.038/365)^365 - 1
EAR = 0.0393 or 3.93%
So the effective annual interest rate is 3.93%.
To find the amount in the account after one year, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount in the account after one year
P = the initial investment amount ($4600)
r = the annual interest rate (3.8% or 0.038)
n = the number of times interest is compounded per year (365)
t = the number of years (1)
(a) Plugging in these values into the formula, we get:
A = 4600(1 + 0.038/365)^(365*1)
Calculating the parentheses first:
(1 + 0.038/365) = 1.00010410959
Plugging this value back into the formula:
A = 4600(1.00010410959)^(365*1)
Calculating the exponent:
(1.00010410959)^(365*1) ≈ 1.04099687516
Finally, multiplying the initial investment by the result:
A ≈ 4600 * 1.04099687516 ≈ $4786.89
Therefore, the amount in the account after one year, assuming no withdrawals are made, is $4786.89.
(b) The effective annual interest rate is the rate of return that would be required for a simple interest calculation to result in the same amount after one year. We can calculate it using the formula:
Effective Annual Interest Rate = (1 + r/n)^n - 1
Plugging in the values:
Effective Annual Interest Rate = (1 + 0.038/365)^365 - 1
Calculating the parentheses:
(1 + 0.038/365) = 1.00010410959
Plugging this value back into the formula:
Effective Annual Interest Rate = (1.00010410959)^365 - 1
Calculating:
(1.00010410959)^365 ≈ 1.04099687516
Finally, subtracting 1:
Effective Annual Interest Rate ≈ 1.04099687516 - 1 ≈ 0.04099687516
Converting this into a percentage:
Effective Annual Interest Rate ≈ 0.04099687516 * 100 ≈ 4.10%
Therefore, the effective annual interest rate is approximately 4.10%.