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. Of the total amount of new funds that will be invested by Stephanie and by her employer on her behalf, how much will she put into the safe account each year; how much into 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 find out how much Stephanie will put into the safe account and how much into the risky account, we need to use the information given in the question.

First, we need to find out how much money Stephanie will have in her pension plan at age 67. We can use the future value formula:

FV = PV * (1 + r)^n

Where: 
PV = present value (the amount Stephanie has currently) 
r = real rate of return 
n = number of years

Using the risk-free rate, we get:

FV = $4,000 * (1 + 0.035)^(67-23) 
FV = $17,827.46

Using the expected return and standard deviation, we can calculate the expected value and variance of the risky investment:

Expected value = 0.10 
Variance = (0.23)^2

Now, we can use the expected value and variance to calculate the expected future value of the risky investment:

EFV = PV * (1 + expected value)^n

EFV = $4,000 * (1+ 0.10)^(67-23) 
EFV = $215,542.20

We know that Stephanie has 5% of her money in the safe account and 95% in the risky account, so we can set up two equations to find out how much she will put into each account each year:

0.05 * total amount = amount in safe account 
0.95 * total amount = amount in risky account

We can use the formula for the present value of an annuity to find the total amount:

PV annuity = (PMT/r) * (1 - (1+r)^(-n))

Where: 
PMT = payment amount 
r = real rate of return 
n = number of years

We know that Stephanie will contribute $2,000 per year and her employer will match that, so her PMT is $4,000. We also know that she will continue saving for 44 years (from age 23 to age 67).

Using the risk-free rate, we get:

PV annuity = ($4,000/0.035) * (1 - (1+0.035)^(-44)) 
PV annuity = $102,463.50

Using the expected return and standard deviation, we can calculate the expected value and variance of the risky investment:

Expected value = 0.10 
Variance = (0.23)^2

Now, we can use the expected value and variance to calculate the expected future value of the risky investment:

EFV = PV * (1 + expected value)^n

EFV = $4,000 * (1 + 0.10)^(44) 
EFV = $858,463.03

So the total amount of new funds that will be invested by Stephanie and by her employer on her behalf is:

Total amount = $102,463.50 + $102,463.50 + $429,231.51 + $429,231.51 
Total amount = $1,063,389.02

Now we can use the two equations from before to find out how much Stephanie will put into each account each year:

0.05 * $1,063,389.02 = $53,169.45 in safe account 
0.95 * $1,063,389.02 = $1,010,219.56 in risky account

So Stephanie will put $53,169.45 into the safe account each year and $1,010,219.56 into the risky account each year.

Therefore, the answer is C: $200 into the safe account each year and $3,800 into the risky account each year.

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