You want to receive 5,000 per month in retirement.

If you can earn 0.75% per month and you expect to need the income for 25 years, how much do you need to have in your account at the beginning of retirement?

Question

You want to receive 5,000 per month in retirement.

If you can earn 0.75% per month and you expect to need the income for 25 years, how much do you need to have in your account at the beginning of retirement?

BlackTom AI · Accepted Answer

Present value of an ordinary monthly annuity is used because the question asks how much you must have at the beginning of retirement to fund monthly withdrawals of 5,000 (i.e., the required sum today before the first month’s interest accrues and assuming payments occur at the end of each month). The key formula: PV = PMT * [1 − (1 + i)^−n] / i.

Relevant concepts
- PMT = periodic payment = 5,000.
- i = monthly interest rate = 0.75% = 0.0075.
- n = total months = 25 years * 12 = 300.
- (1 + i)^−n is the present value discount factor for n periods.

Step-by-step calculation
1) Compute (1 + i)^−n:
   (1.0075)^−300 = 0.1062878339 (this is the discount factor v^n).

2) Compute 1 − (1 + i)^−n:
   1 − 0.1062878339 = 0.8937121661.

3) Divide by i:
   0.8937121661 / 0.0075 = 119.16162148.

4) Multiply by PMT to get PV:
   PV = 5,000 * 119.16162148 = 595,808.1074 ≈ 595,808.11.

Why this is correct
- The formula used is the standard present-value formula for an ordinary annuity (payments at the end of each period). The numeric steps above compute the annuity factor precisely and multiply by the monthly payment to yield the lump sum needed at the beginning of retirement to fund 300 end-of-month withdrawals of 5,000 while earning 0.75% monthly.

Why other interpretations would be wrong
- If you treated the stream as an annuity-due (payments at the beginning of each month, first payment immediately), you would multiply the ordinary-annuity PV by (1 + i), giving a larger required balance (~600,000). That is not the intended interpretation if payments are end-of-month and you want the amount "at the beginning of retirement" to fund them.

Conclusion
The required amount at the beginning of retirement is 595,808.11, matching the provided answer.

You want to receive 5,000 per month in retirement. If you can earn 0.75% per month and you expect to need the income for 25 years, how much do you need to have in your account at the beginning of retirement? 数值题

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