Mary B deposits $8500 into a savings account, compounded monthly at a nominal interest rate of 9 percent, as part of a savings plan that she would like to undertake in 5 years.
1) What is the effective interest rate per annum?
2) How much money would she have saved after 5 years?
3) Mary has an an emergency at the end of the second year after she had invested the money.She withdraws $5000 How much money would she receive after 5 years?
1) The balance after one year with compounding is increased by a factor
(1 +(.09/12)^12 = (1.0075)^12 = 1.093807
The effective annual interest rate is therefore 9.3807%
2) 8500*(1.093807)^5 = $ 13,308.29
3) After two years and withdrawing $5000, she is left with
8500*(1.093807)^2 - 5000
= 10,169.52 - 5000 = 5169.52
After three more years, that becomes
5159.52*(1.093807)^3 = $ 6751.98
To answer these questions, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = the nominal interest rate in decimal form
n = the number of times interest is compounded per year
t = the number of years
Let's tackle each question one by one.
1) To find the effective interest rate per annum, we need to use the formula:
Effective Annual Interest Rate = (1 + r/n)^n - 1
Given:
Nominal interest rate (r) = 9%
Compounding frequency (n) = 12 (monthly)
Substituting the values into the formula:
Effective Annual Interest Rate = (1 + 0.09/12)^12 - 1
Calculating this yields an effective annual interest rate of approximately 9.3806%.
2) To calculate how much money Mary would have saved after 5 years, we use the compound interest formula again:
A = P(1 + r/n)^(nt)
Given:
P = $8,500 (principal amount)
r = 9% (interest rate in decimal form)
n = 12 (compounding frequency)
t = 5 years
Substituting the values into the formula:
A = $8,500(1 + 0.09/12)^(12 * 5)
Calculating this yields an amount of approximately $12,997.38.
Therefore, Mary would have approximately $12,997.38 saved after 5 years.
3) To calculate how much money Mary would receive after 5 years, taking into account the withdrawal after the second year, we need to adjust our approach.
First, let's calculate how much money Mary would have after 2 years:
A2 = P(1 + r/n)^(nt)
Given:
P = $8,500 (principal amount)
r = 9% (interest rate in decimal form)
n = 12 (compounding frequency)
t = 2 years
Substituting the values into the formula:
A2 = $8,500(1 + 0.09/12)^(12 * 2)
Calculating this yields an amount of approximately $9,936.09.
Now, let's subtract the withdrawal amount from this:
A_final = A2 - $5,000
Calculating this yields an amount of approximately $4,936.09.
Therefore, after 5 years and considering the $5,000 withdrawal, Mary would receive approximately $4,936.09.