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?
To find the amount Erica's father should deposit every year on August 1st, we need to calculate the future value of the $1,000 withdrawals over the academic year, taking into account the interest earned from the savings account.
a) Calculation for deposit on August 1st:
Since the bank pays 5% per annum interest compounded monthly, the monthly interest rate is 5% / 12 = 0.4167%.
The time period for the withdrawals is 9 months (September 1st to June 1st).
We will use the future value of an ordinary annuity formula:
FV = PMT * ((1 + r)^n - 1) / r
Where:
FV = future value
PMT = withdrawal amount ($1,000)
r = interest rate per period (0.4167%)
n = number of periods (9)
Substituting the values into the formula:
FV = 1000 * ((1 + 0.4167%)^9 - 1) / 0.4167%
Using a financial calculator or spreadsheet, the future value comes out to be approximately $9,255.56.
So, Erica's father should deposit $9,255.56 every year on August 1st to cover the $1,000 monthly withdrawals.
b) If the father wishes to deposit every year on August 31st, the deposit amount will not be the same as in part (a). This is because the time period for the withdrawals will change.
If the deposit is made on August 31st, the time period will be from September 1st of the current year to June 1st of the next year, which is 10 months.
We need to recalculate the future value using the new time period:
FV = 1000 * ((1 + 0.4167%)^10 - 1) / 0.4167%
Using the financial calculator or spreadsheet, the future value comes out to be approximately $9,619.80.
Thus, if the deposit is made on August 31st, Erica's father should deposit $9,619.80 every year to cover the $1,000 monthly withdrawals.
In summary, the deposit amount will be different if the deposit date changes because the time period for the withdrawals will change, resulting in a different future value.
a) To calculate the amount Erica's father should deposit every year on August 1st, we need to consider the interest earned on the savings account. The interest is calculated using compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial deposit)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case, Erica's father wants to withdraw $1,000 every month over the academic year from September 1st to June 1st, which is a total of 10 months.
The annual interest rate is 5%, and it is compounded monthly, so n = 12.
The number of years is 10 months / 12 months (since the interest is calculated annually).
Now let's calculate the deposit amount:
A = $1,000 (withdrawal amount)
P = ? (deposit amount)
r = 5% = 0.05
n = 12
t = 10/12
$1,000 = P(1 + 0.05/12)^(12*(10/12))
Simplifying the equation:
1 = (1 + 0.05/12)^(10)
1 = (1.0041667)^(10)
1 = 1.04166
To solve for P, we divide both sides by (1 + 0.05/12)^10:
P = $1,000 / 1.04166
P ≈ $959.03
So, Erica's father should deposit approximately $959.03 every year on August 1st.
b) If Erica's father wishes to deposit every year on August 31st instead of August 1st, the deposit amount will be different. This is because the interest is calculated based on the number of days in a month, and the interest earned for the month of August will be different depending on the deposit date.
Since the interest is compounded monthly, the deposit date will affect the interest earned for that specific month. If Erica's father deposits on August 31st, the interest earned for August will be minimal, as the days between August 31st and September 1st are very few. Therefore, the deposit amount will need to be higher to reach the $1,000 withdrawal amount for the remaining months.
In conclusion, the deposit amount will not be the same as in part (a) if Erica's father wants to deposit on August 31st. The deposit amount will be higher to compensate for the lower interest earned in August.