Question

question 8
by the time tom retires, the balance in his 401k plan is $1.5 million. he can choose to use the money to buy a fixed annuity of 30 years (toms expected life expectancy) with a guaranteed annual interest rate of 5%. on the other hand, tom can leave the money in a stock index fund, which is expected (though not guaranteed) to earn an annual return of 8%. the advantage of the fixed annuity is its safety and certainty. on the other hand, the higher expected return of the stock fund means a higher expected annual withdrawal. tom is risk - averse and decides to buy the fixed annuity. how much more could tom expect to receive per year if he decides to leave the money in the stock index fund (in other words, what price does tom pay for the safety of the fixed annuity)?
a tom could expect to receive about 60% more than the annuity payment.
b tom could expect to receive about 36% more than the annuity payment.
c tom could expect to receive about 25% more than the annuity payment.
d tom could expect to receive about 3% more than the annuity payment.
e none of the choices.
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Explanation:

Step1: Recall annuity formula

The present - value of an annuity formula is $PV = A\times\frac{1-(1 + r)^{-n}}{r}$, where $PV$ is the present value, $A$ is the annual payment, $r$ is the interest rate per period, and $n$ is the number of periods. For the fixed annuity, $PV_1=1500000$, $r_1 = 0.05$, and $n = 30$. We can solve for $A_1$:
\[A_1=\frac{PV_1\times r_1}{1-(1 + r_1)^{-n}}=\frac{1500000\times0.05}{1-(1 + 0.05)^{-30}}\]
\[1-(1 + 0.05)^{-30}=1-\frac{1}{(1.05)^{30}}\approx1 - 0.231377 = 0.768623\]
\[A_1=\frac{1500000\times0.05}{0.768623}=\frac{75000}{0.768623}\approx97570.24\]

Step2: Calculate for stock - index fund

For the stock - index fund, $PV_2 = 1500000$, $r_2=0.08$, and $n = 30$. We solve for $A_2$:
\[A_2=\frac{PV_2\times r_2}{1-(1 + r_2)^{-n}}=\frac{1500000\times0.08}{1-(1 + 0.08)^{-30}}\]
\[1-(1 + 0.08)^{-30}=1-\frac{1}{(1.08)^{30}}\approx1 - 0.099377 = 0.900623\]
\[A_2=\frac{1500000\times0.08}{0.900623}=\frac{120000}{0.900623}\approx133241.38\]

Step3: Find the percentage difference

The percentage difference $\text{Diff}=\frac{A_2 - A_1}{A_1}\times100\%$
\[\text{Diff}=\frac{133241.38 - 97570.24}{97570.24}\times100\%=\frac{35671.14}{97570.24}\times100\%\approx36.66\%\approx36\%\]

Answer:

B. Tom could expect to receive about 36% more than the annuity payment.