How many months would it take for Harry to save an adequate amount for retirement if he deposits $2,000 per month into an account beginning 1 month from now that pays 9 percent per year if he wishes to have a total of $1,000,000 at retirement? (Round to nearest month)

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BlackTom AI · Accepted Answer

The problem asks how long it takes, in months, for monthly deposits of $2,000 to reach a future value of $1,000,000, given an annual interest rate of 9% with monthly compounding, starting one month from now.

First, identify the monthly interest rate: i = 0.09 / 12 = 0.0075.

This is an ordinary (end-of-period) annuity problem. The future value after n months is:
FV = P * [((1 + i)^n - 1) / i], where P = 2,000.

We set FV = 1,000,000 and solve for n:
1,000,000 = 2,000 * [((1 + 0.0075)^n - 1) / 0.0075].

Divide both sides by 2,000: 500 = ((1.0075)^n - 1) / 0.0075.

Multiply both sides by 0.0075: 3.75 = (1.0075)^n - 1.

Thus (1.0075)^n = 4.75.

Take natural logarithms:
n = ln(4.75) / ln(1.0075).

Numerically, ln(4.75) ≈ 1.5581 and ln(1.0075) ≈ 0.007472, giving n ≈ 208.6 months.

Since you can only have whole months, round to the nearest month: n ≈ 209 months.

Note: This analysis assumes monthly compounding at 9% nominal annual rate and end-of-period deposits starting one month from now. If the compounding or timing conventions differ, the result would change accordingly.

Regarding the provided answer options: no answer choices were supplied in the prompt, so the calculation above stands as the derivation to reach 209 months under the stated assumptions.

FIN-2800-4 Exam 1

How many months would it take for Harry to save an adequate amount for retirement if he deposits $2,000 per month into an account beginning 1 month from now that pays 9 percent per year if he wishes to have a total of $1,000,000 at retirement? (Round to nearest month)

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