To find the lump sum that needs to be deposited in a bank to withdraw a specific amount monthly for a set number of years with compounded interest, we can use the formula for the present value of an annuity.
The formula to calculate the present value (PV) of an ordinary annuity (which is your monthly withdrawals) is:
\[ PV = PMT \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]
Where:
- \( PV \) is the present value (the lump sum to be deposited),
- \( PMT \) is the monthly payment (withdrawals),
- \( r \) is the monthly interest rate (annual interest rate divided by 12),
- \( n \) is the total number of payments (withdrawals).
Given:
- \( PMT = 4605 \) birr per month
- Annual interest rate = 5%, so monthly interest rate \( r = \frac{5%}{12} = \frac{0.05}{12} \approx 0.00416667 \)
- The duration is 13 years, with a total of \( n = 13 \times 12 = 156 \) months
Now, we can substitute the values into the formula:
- Calculate \( r \):
\[ r = \frac{0.05}{12} \approx 0.00416667 \]
- Calculate \( n \):
\[ n = 13 \times 12 = 156 \]
- Now substitute into the PV formula:
\[ PV = 4605 \times \left( \frac{1 - (1 + 0.00416667)^{-156}}{0.00416667} \right) \]
- Calculate \( (1 + r)^{-n} \):
\[ (1 + 0.00416667)^{-156} \approx (1.00416667)^{-156} \approx 0.3983268 \]
- Calculate \( 1 - (1 + r)^{-n} \):
\[ 1 - 0.3983268 \approx 0.6016732 \]
- Now calculate the fraction:
\[ \frac{0.6016732}{0.00416667} \approx 144.400 \]
- Finally, multiply by \( PMT \) to find the present value:
\[ PV \approx 4605 \times 144.400 \approx 665,622 \]
So the lump sum that needs to be deposited first in the bank is approximately 665,622 birr.
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