marie is 65 years old and ready to retire. she has a $1 million nest egg and wishes to spend $90,000 per…

marie is 65 years old and ready to retire. she has a $1 million nest egg and wishes to spend $90,000 per year as an ordinary annuity (i.e., she will withdraw $90,000 per year at the end of each year). if her investment portfolio earns 6% annually, at what age will she run out of money? (longevity risk). a around 76 years old b around 84 years old c around 98 years old d never e none of these are correct

marie is 65 years old and ready to retire. she has a $1 million nest egg and wishes to spend $90,000 per year as an ordinary annuity (i.e., she will withdraw $90,000 per year at the end of each year). if her investment portfolio earns 6% annually, at what age will she run out of money? (longevity risk). a around 76 years old b around 84 years old c around 98 years old d never e none of these are correct

Answer

Explanation:

Step1: Use the present - value of an ordinary annuity formula

$PV = A\times\frac{1-(1 + r)^{-n}}{r}$, where $PV$ is the present value ($PV=$1000000$), $A$ is the annual payment ($A = $90000$), and $r$ is the interest rate per period ($r=0.06$). We need to solve for $n$. First, rewrite the formula for $n$: $\frac{PV\times r}{A}=1-(1 + r)^{-n}$ $(1 + r)^{-n}=1-\frac{PV\times r}{A}$ $-n=\ln(1 - \frac{PV\times r}{A})\div\ln(1 + r)$

Step2: Substitute the values

$PV = 1000000$, $A = 90000$, $r=0.06$ $1-\frac{PV\times r}{A}=1-\frac{1000000\times0.06}{90000}=1-\frac{60000}{90000}=1 - \frac{2}{3}=\frac{1}{3}$ $n=-\frac{\ln(\frac{1}{3})}{\ln(1 + 0.06)}=\frac{\ln(3)}{\ln(1.06)}\approx18.85$

Step3: Calculate the age

Marie is 65 years old. Add the number of years $n$ to her current age. $65 + 18.85\approx84$ years old

Answer:

B. Around 84 years old