What lump sum was deposited first in a bank that offers a 5% interest compounded monthly to be able to withdraw 4605 birr per month at the end of each month for 13 years?
The lump sum deposited first would be 4605 birr x (1 + 0.05/12)^156 = 845,945.45 birr.
To determine the lump sum that was initially deposited, we can use the formula for the present value of an annuity:
PV = PMT * [(1 - (1+r)^(-n)) / r]
Where:
PV = Present value (lump sum initially deposited)
PMT = Amount withdrawn each month (4605 birr)
r = Monthly interest rate (5% divided by 12)
n = Total number of months (13 years multiplied by 12)
Let's plug in the values and calculate:
PMT = 4605
r = 0.05/12
n = 13 * 12
PV = 4605 * [(1 - (1+0.05/12)^(-(13*12))) / (0.05/12)]
PV = 4605 * [(1 - (1+0.00417)^(-156)) / 0.00417]
PV = 4605 * [1 - 0.1176649 / 0.00417]
PV = 4605 * (0.8823351 / 0.00417)
PV = 4605 * 211.3334728
PV ≈ 972,927.19 birr
Therefore, the lump sum initially deposited in the bank was approximately 972,927.19 birr.
To calculate the lump sum that was deposited initially, we need to use the formula for calculating the future value of an annuity.
The future value of an annuity formula is given by:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future value of the annuity
P = Payment amount per period
r = Interest rate per period
n = Number of periods
In this case, the payment amount per period is 4605 birr, the interest rate is 5% compounded monthly, and the number of periods is 13 years (which is equivalent to 12*13 = 156 months).
Let's calculate the future value of the annuity first:
FV = 4605 * ((1 + (0.05/12))^156 - 1) / (0.05/12)
FV ≈ 4605 * (1.004167^156 - 1) / (0.004167)
Now, let's solve for the initial lump sum (P):
FV = P * ((1 + r)^n - 1) / r
P = FV * r / ((1 + r)^n - 1)
Plugging in the values, we get:
P ≈ ((4605 * (1.004167^156 - 1) / (0.004167)) * (0.05/12) / ((1 + (0.05/12))^156 - 1)
Calculating this expression will give you the lump sum that was deposited initially to be able to withdraw 4605 birr per month for 13 years.