Question

1. You are now 50 years old and plan to retire at age 65. You currently have a stock portfolio worth $150,000, a 401(k) retirement plan worth $250,000, and a money market account worth $50,000. Your stock portfolio is expected to provide annual returns of 12 percent, your 401(k) investment will earn 9.5 percent annually, and the money market account earns 5.25 percent, compounded monthly. (12 points)

a. If you do not save another penny, what will be the total value of your investments when you retire at age 65? (3 points)

b. Assume you plan to invest $12,000 every year in your 401(k) plan for the next 15 years (starting one year from now). How much will your investments be worth when you retire at 65? (3 points)

c. Assume that you expect to live 25 years after you retire (until age 90). Today, at age 65, you take all of your investments and place them in an account that pays 8 percent (use the scenario from part b in which you continue saving). If you start withdrawing funds starting at age 66, how much can you withdraw every year (e.g., and ordinary annuity) and leave nothing in your account after a 25th and final withdrawal at age 90? (3 points)

d. You want your current investments, which are described in the problem statement, to support a perpetuity that starts a year from now How much can you withdraw each year without touching your principal? (3 points)

Answer 1

Answer:

a. Stock Portfolio

Current value of stock portfolio = $150,000

Expected return on portfolio = i = 12%

Time to retirement = n = 15 years

Expected value of portfolio at age 65 = FV _Stock

FV _{Stock =} PV (1 +i)¹⁵= $150,000* (1.12)¹⁵= $821,034.86

410(k) Investment

Current value of 410(k) portfolio = $250,000

Expected return on portfolio = i= 9.5%

Time to retirement = n = 15 years

Expected value of portfolio at age 65 = FV_401k

FV_401k = PV (1+i )¹⁵= $250,000 (1.095)¹⁵= $975,330.48

Money market account

Current value of savings = $50,000

Expected return on portfolio = i = 5.25%

Time to retirement = n = 15 years

Frequency of compounding = m = 12

Expected value of portfolio at age 65 = FV_MMA

FV_MMA=PV*(1+i/m)^m*n

=$50000*(1+0.0525/12)^12*15

=50000*2.1941=$109706.14

Total value of all three investments = $821,034.86 + $975,330.48 + $109,706.14=

$1,906,071.48

Answer:b Planned annual investment in 401k plan = $12,000

Future value of annuity = FVA

FVA_n=PMT(((1+i)ⁿ-1)/i))

=$12000*(1.095)¹⁵-1/0.095)

=$12000*30.5402

=$366482.77

Answer:c

Amount available at retirement = PVA = $2,272,554.25

Length of annuity = n = 25

Expected return on investment = i = 8%

Annuity amount expected = PMT

Using the PVA equation:

PVA_n=PMT*[[1-(1/(1+i)ⁿ]/i]

PMT=$2,272,554.25/[[1-(1/(1.08)²⁵]/0.08]

[note: Add the totals from part a for the stock plan, the money market, and then adding the fv of the annuity in part b to the total from the 401k in part a. So 821,034+109,706.14 + 336,482.77+975,330.48= 2272554.25]

=$2,272,554.25/10.6748

=$212889.63

Each payment received for the next 25 years will be $212,889.63

Answer:d Type of payment = Perpetuity

Present value of perpetuity = PVA = $2,272,554.25

Expected return on investment = i = 8%

PV of perpetuity=PMT/i

$2,272,554.25=PMT/0.08

$181,804.34

You could receive an annual payment of $181,804.34 forever.