Stephanie Watson is 23 years old and has accumulated $4,000 in her self-directed defined contribution pension plan.Each year she contributes $2,000 to the plan, and her employer contributes an equal amount.Stephanie thinks she will retire at age 67 and figures she will live to age 81.The plan allows for two types of investments.One offers a 3.5% risk-free real rate of return.The other offers an expected return of 10% and has a standard deviation of 23%.Stephanie now has 5% of her money in the risk-free investment and 95% in the risky investment.She plans to continue saving at the same rate and keep the same proportions invested in each of the investments.Her salary will grow at the same rate as inflation. How much does Stephanie currently have in the safe account; how much in the risky account?
A)$3,800; $200
B)$2,000; $2,000
C)$200; $3,800
D)$2,500; $1,500

Question

Quizplus · Accepted Answer

To solve this problem, we can use the concept of future value of a lump sum. We know that Stephanie contributes $2,000 each year, and her employer matches the contribution. Therefore, her annual contribution is $4,000. We also know that she has already accumulated $4,000 in her account, so her current balance is $8,000 ($4,000 contribution + $4,000 accumulated).

Since Stephanie plans to keep the same proportions invested in each of the investments, she currently has 5% of $8,000, or $400, invested in the risk-free investment, and 95% of $8,000, or $7,600, invested in the risky investment.

To calculate the future value of Stephanie's account, we can use the following formula:

FV = PV x (1+r)^n

where FV is the future value, PV is the present value (or current balance), r is the rate of return, and n is the number of years.

For the risk-free investment, we know that the real rate of return is 3.5%, so we can use the following formula to calculate the nominal rate of return:

nominal rate of return = (1+real rate of return) x (1+inflation rate) - 1

Since Stephanie's salary will grow at the same rate as inflation, we can assume that her inflation rate is also 3.5%. Therefore, the nominal rate of return for the risk-free investment is:

nominal rate of return = (1+0.035) x (1+0.035) - 1 = 7.1225%

Using this rate, we can calculate the future value of Stephanie's $400 investment in the risk-free account:

FV risk-free = $400 x (1+0.071225)^44 = $2,001.70

For the risky investment, we can use the expected rate of return (10%) to calculate the future value:

FV risky = $7,600 x (1+0.10)^44 = $371,183.22

Therefore, Stephanie currently has $2,001.70 in the safe account and $371,183.22 in the risky account. The answer is C.

Stephanie Watson Is 23 Years Old and Has Accumulated $4,000