What is the principal invested at 4.75% compounded semi-annually from which monthly withdrawals of $240 can be made at the end of each month for 20 years but deferred for 10 years?
Answers
Given :- What is the principal invested at 4.75% compounded semi-annually from which monthly withdrawals of $240 can be made at the end of each month for 20 years but deferred for 10 years ?
Solution :-
we know that, when rate is compounded semi-annually ,
- Time = 2 times => 10 * 2 = 20 years .
- Rate = Half => 4.75 ÷ 2 = 2.375% annually .
Now, we also know that,
Annuity Formula :-
- A = P{(1 + r/k)^(N•k) - 1} / (r/k)
- A is the balance in the account after N years.
- P is the regular deposit (the amount you deposit each year, each month, etc.)
- r is the annual interest rate in decimal form.
- k is the number of compounding periods in one year.
we have :-
- P = $240
- r = (2.375/100) = (0.02375)
- N = 20 years.
- k = 12 .
- A = Total amount deposited .
Putting all values we get,
→ A = 240•[{1 + (0.02375/12)}^(12*20) - 1] / (0.02375/12)
→ A = 240•[(1 + 0.001)²⁴⁰ - 1] / (0.001)
→ A = 240,000[(1.001)²⁴⁰ - 1]
→ A = 240,000[1.271 - 1]
→ A = 240,000 * 0.271
→ A ≈ $305,040 (Ans.)
Hence, the principal invested is $305,040 .
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Answer:
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