Moses' goal, when he retires from work in seven years, is to have $400,000 in his Retirement Fund. Assuming he achieves his goal and the fund earns 7% compounded semi-annually after he retires, Moses will, at the end of every six months, take $20,000 out of his Retirement Fund. For how long will he be able to do that before the money runs out?

Question

Quizplus · Accepted Answer

To solve this, we use the formula for the present value of an annuity because Moses is withdrawing a fixed amount periodically, and the fund earns a fixed interest rate. The formula is $PV = PMT \times \left(\frac{1 - (1 + r)^{-n}}{r}\right)$, where PV is the present value (initial amount in the retirement fund), PMT is the payment amount per period, r is the interest rate per period, and n is the number of periods. Here, $PV = $400,000$, $PMT = $20,000$, $r = 0.07/2 = 0.035$ (because the interest is compounded semi-annually), and we need to solve for n.Rearranging the formula to solve for n gives us $n = \frac{\log(1 - \frac{PV \cdot r}{PMT})}{\log(1 + r)}$.Substituting the given values, we get $n = \frac{\log(1 - \frac{400,000 \cdot 0.035}{20,000})}{\log(1 + 0.035)}$.This calculation will give us the total number of periods (n) Moses can withdraw the money. Since the withdrawals are semi-annual, to find the number of years, we divide n by 2.After performing the calculation, we find that Moses will be able to withdraw the money for approximately 17.5 years before the fund runs out, which corresponds to choice E.

Moses' Goal, When He Retires from Work in Seven Years