Alan Barnett is 43 years old and has accumulated $78,000 in his self-directed defined contribution pension plan.Each year he contributes $1,500 to the plan, and his employer contributes an equal amount.Alan thinks he will retire at age 60 and figures he will live to age 83.The plan allows for two types of investments.One offers a 4% risk-free real rate of return.The other offers an expected return of 10% and has a standard deviation of 34%.Alan now has 40% of his money in the risk-free investment and 60% in the risky investment.He plans to continue saving at the same rate and keep the same proportions invested in each of the investments.His salary will grow at the same rate as inflation. How much can Alan expect to have in his risky account at retirement?
A)$158,982
B)$309,529
C)$543,781
D)$224,651

Question

Quizplus · Accepted Answer

Using the formula for expected return for a portfolio, we have:

E(r) = w₁E(r₁) + w₂E(r₂)

where:
w₁ = proportional weight invested in risk-free asset (0.4)
w₂ = proportional weight invested in risky asset (0.6)
E(r₁) = risk-free rate of return (4%)
E(r₂) = expected return of risky asset (10%)

E(r) = 0.4(4%) + 0.6(10%) = 7.6%

So Alan can expect a portfolio return of 7.6% per year.

Using the formula for future value of a series of payments, we have:

FV = PMT[((1 + r)ᴺ - 1) / r] + PV(1 + r)ᴺ

where:
PMT = annual contribution made by Alan and his employer ($3,000)
r = rate of return on the portfolio (7.6%)
N = number of years until retirement (17)
PV = present value of portfolio ($78,000)

To calculate the future value of the risky account, we need to subtract the portion of the portfolio that is invested in the risk-free asset:

FV = 0.6[PMT[((1 + r)ᴺ - 1) / r] + PV(1 + r)ᴺ] = 0.6[$43,529.17 + $155,117.24] = $118,870.57

Therefore, Alan can expect to have $118,870.57 in his risky account at retirement. Adding back the portion invested in the risk-free asset, his total expected retirement savings would be:

$118,870.57 / 0.6 + $78,000 = $309,529.

Alan Barnett Is 43 Years Old and Has Accumulated $78,000