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Suppose you accumulated $500,000, perhaps from many years of saving.  You put the money in a savings plan earning 6% compounded monthly.  If you want to withdraw $4,000 at the beginning of each month, how long before the savings plan is exhausted?Group of answer choices195 months197 months185 months205 months

Question

Suppose you accumulated 500,000,perhapsfrommanyyearsofsaving. Youputthemoneyinasavingsplanearning6500,000, perhaps from many years of saving.  You put the money in a savings plan earning 6% compounded monthly.  If you want to withdraw 4,000 at the beginning of each month, how long before the savings plan is exhausted?Group of answer choices195 months197 months185 months205 months

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Solution

To solve this problem, we need to use the formula for the present value of an annuity due, which is used to calculate the present value of a series of future payments. The formula is:

PV = PMT * [(1 - (1 + r/n) ^ -nt) / (r/n)] * (1+r/n)

where:

  • PV is the present value, or the total amount that a series of future payments is worth now;
  • PMT is the payment amount per period;
  • r is the annual interest rate (decimal);
  • n is the number of compounding periods per year;
  • t is the time in years.

We know that PV = 500,000,PMT=500,000, PMT = 4,000, r = 6% or 0.06, and n = 12 (since the interest is compounded monthly). We want to find t, the time before the savings plan is exhausted.

First, we need to rearrange the formula to solve for t:

t = -log(1 - (r/n) * (PV / PMT) * (1/(1+r/n))) / (n * log(1 + r/n))

Substituting the given values into the formula:

t = -log(1 - (0.06/12) * (500,000 / 4,000) * (1/(1+0.06/12))) / (12 * log(1 + 0.06/12))

After calculating the above expression, we find that t is approximately 16.25 years. Since we want the answer in months, we multiply this by 12 to get approximately 195 months. Therefore, the savings plan will be exhausted in about 195 months.

This problem has been solved

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