Suppose you inherited $200,000 and invested it at 6% per year. How much could you withdraw at the end of each year for the next 15 years?

Question

BlackTom AI · Accepted Answer

Question restatement: You inherit 200,000 dollars and invest it at 6% per year. How large a level amount can be withdrawn at the end of each year for 15 years?

First, recognize this as an ordinary annuity (withdrawals at the end of each year) with a present value of 200,000, an interest rate of 6% per year, and a term of 15 years. The goal is to find the annual payment (PMT) that makes the present value of all withdrawals equal to 200,000.

Key formula: For an ordinary annuity, PV = PMT × [1 – (1 + i)^-n] / i, where i is the interest rate per period and n is the number of periods. We solve for PMT: PMT = PV × i / [1 – (1 + i)^-n].

Plug in the values: PV = 200,000, i = 0.06, n = 15.

Compute the discount factor: (1 + i)^-n = (1.06)^-15. Evaluating (1.06)^15 ≈ 2.396, so (1.06)^-15 ≈ 1 / 2.396 ≈ 0.417.

Next, compute the denominator: 1 – (1.06)^-15 ≈ 1 – 0.417 ≈ 0.583.

Compute the numerator: PV × i = 200,000 × 0.06 = 12,000.

Now divide: PMT ≈ 12,000 / 0.583 ≈ 20,590 to 20,600 range. A more precise calculation yields PMT ≈ 20,592.55.

Putting it together: The annual withdrawal that corresponds to an ordinary annuity with these terms is about $20,592.55 per year.

Why this is consistent: The $200,000 grows at 6% annually, and the withdrawals each year deplete the principal while earning interest. The present value of the 15 equal end-of-year withdrawals at 6% must equal 200,000, which is exactly what the PMT calculation enforces.

Why other approaches would be wrong (conceptual checks): If you used a simple division 200,000 / 15, you’d ignore the interest earned during the years, underestimating the amount; if you tried a perpetuity formula, you’d assume infinite withdrawals; if you treated withdrawals as beginning-of-year (an annuity due), the payment would be larger, because you’d forgo one year of interest on the first withdrawal. The ordinary annuity framework used here correctly accounts for interest accrual between withdrawals over 15 years.

In summary, applying the ordinary annuity PV formula with i = 6%, n = 15, and PV = 200,000 yields a withdrawal of about $20,592.55 per year, which matches the provided value.

Suppose you inherited $200,000 and invested it at 6% per year. How much could you withdraw at the end of each year for the next 15 years?单项选择题

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