During the 1st year at university, Erica¡¦s father had been sending her $1,000 per month for incidental expenses.Starting from the 2nd academic year, her father decided instead to make a
deposit into a savings account on August 1st every year so that Erica could withdraw $1,000 on the
first day of each month over the academic year from September 1st
to June 1st.
a)If the bank pays 5% p.a. interest compounded monthly, how much should Erica¡¦s father deposit every year on August 1st?
b)If the banking condition remains the same but the father wishes to deposit every year on August 31st
, will the deposit amount be the same as in part (a)? Why?
a) To calculate how much Erica's father should deposit every year on August 1st, we'll use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the desired withdrawn amount ($1,000 per month) x 10 months = $10,000
P = the principal amount (deposit)
r = annual interest rate (5%)
n = number of times the interest is compounded per year (monthly, so 12)
t = number of years
We want to solve for P, so we rearrange the formula:
P = A / (1 + r/n)^(nt)
Plugging in the values:
P = $10,000 / (1 + 0.05/12)^(12*1)
P ≈ $9,396.55
Erica's father should deposit approximately $9,396.55 every year on August 1st.
b) If Erica's father wishes to deposit every year on August 31st instead, the deposit amount will not be the same as in part (a). This is because the interest will accumulate for an additional month before Erica starts withdrawing on September 1st. Therefore, the deposit amount will need to be larger to account for the additional interest earned.
To answer these questions, we need to calculate the amount Erica's father should deposit every year on August 1st to be able to withdraw $1,000 on the first day of each month over the academic year from September 1st to June 1st.
a) To calculate the deposit amount, we can use the formula for the future value of an ordinary annuity:
Future Value = A * [(1 + r)^n - 1] / r
Where:
A = Amount of periodic payment (in this case, $1,000)
r = Interest rate per period (5% p.a. compounded monthly = 5% / 12)
n = Number of periods (9 months from September to June)
Let's substitute the values into the formula and solve for Erica's father's deposit:
Future Value = $1,000 * [(1 + 5%/12)^9 - 1] / (5%/12)
Calculating this, we find that the future value is approximately $9,034.19
So, Erica's father should deposit $9,034.19 on August 1st every year to ensure Erica can withdraw $1,000 each month from September 1st to June 1st.
b) If the father wishes to deposit every year on August 31st instead of August 1st, the deposit amount will not be the same as in part (a). This is because by delaying the deposit by one month, there will be an additional month of interest earned on the previous year's deposit.
To calculate the deposit amount for August 31st, we need to calculate the future value of the annuity over 10 months instead of 9 months (from September to June).
Future Value (August 31st) = $1,000 * [(1 + 5%/12)^10 - 1] / (5%/12)
Calculating this, we find that the future value is approximately $9,456.05.
Therefore, if the father wishes to deposit every year on August 31st, the deposit amount should be approximately $9,456.05. This amount is higher than in part (a) because of the additional month of interest earned on the previous year's deposit.