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:$$ PV = P \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] $$Where:- $PV$ is the present value or the initial amount in the retirement fund ($400,000),- $P$ is the payment amount per period ($20,000),- $r$ is the interest rate per period (7% per year compounded semi-annually gives 3.5% per period, or 0.035),- $n$ is the total number of payments.Rearranging the formula to solve for $n$:$$ n = \frac{\log(1 - \frac{PV \times r}{P})}{\log(1 + r)} $$Plugging in the values:$$ n = \frac{\log(1 - \frac{400,000 \times 0.035}{20,000})}{\log(1 + 0.035)} $$$$ n = \frac{\log(1 - 0.7)}{\log(1.035)} $$$$ n = \frac{\log(0.3)}{\log(1.035)} $$$$ n \approx 35 $$Since $n$ represents the total number of semi-annual payments, to find the number of years, we divide by 2:$$ \text{Years} = \frac{35}{2} = 17.5 $$Therefore, Moses will be able to withdraw $20,000 every six months for approximately 17.5 years before the money runs out.

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