BUS 365: Investments
Solution to Practice Problems
Time Value of Money
1) You recently inherited $50,000 and plan to invest that money at 10% interest. You also plan
to save $5,000 per year for each of the next five years. You then plan to spend two years
traveling before settling down. During those two years, you will not be able to save money.
Rather, you will spend $10,000 out of your savings each year. How much money will you have
in seven years? For simplicity, assume that the $5,000 in annual savings occurs at the end of
each year (the first one occurs one year from today). Also, assume that the $10,000 withdrawals
occur at the end of the year (the first one occurs six years from today).
Answer: V7 = $50,000 × 1.17 + $5,000 × FVIFA10%,5 × 1.12 - $10,000 × FVIFA10%,2
= $113,371.71
2) As an investment advisor, you are developing an investment plan for a new client.
Information on the client follows.
- The client plans to retire in 30 years.
- The client desires an annual retirement income that will provide purchasing power (measured in
today’s dollars) of $60,000.
- The client has saved $40,000 to date.
- The client’s salary will likely grow at 5% per year.
- The client’s daughter plans to go to college in three years and the client wishes to pick up the
expected cost of $10,000 per year for four years.
- The client is conservative and currently has a portfolio with a beta of 0.8. You believe a beta of
0.8 is appropriate given the client’s level of risk aversion. Once the client retires, however, you
believe a beta of 0.2 will be more appropriate for the client’s portfolio.
Other information follows.
- The yield on long-term Treasury securities is 4.0%.
- The expected return on the Russell 3000 index is 9.0%.
- Inflation is expected to be 2% per year indefinitely.
- To be conservative, your policy is to develop a plan based on a client life expectancy of 50
years following retirement.
Develop a reasonable retirement plan for the client. Specifically, describe a well-reasoned
savings plan that will meet the client’s goals. In doing so, you should assume that the client’s
savings will increase at the rate of his/her salary growth.
Answer:
Expected return, working years = 4.0%+0.8×(9.0%-4.0%) = 8%
Expected return, retirement years = 4.0%+0.2×(9.0%-4.0%) = 5%
Expected withdrawal, 1st year of retirement = $60,000×1.0231 = $110,855.33
Account balance needed at retirement = $110,855.33×PVIFGA5%,2%50 = $2,827,858.44
Value of savings = $40,000×1.0830 - $10,000×FVIFA8%,4×1.0823 + C×FVIFGA8%,5%,30
Then,
$2,827,858.44 = $40,000×1.0830 - $10,000×FVIFA8%,4×1.0823 + C×FVIFGA8%,5%,30
and C = $14,057.10
3) You recently inherited $100,000. You plan to save $8,000 during the next year (year 1) and
increase that amount by 5% per year until you retire in 35 years. During that period, you plan to
plan to invest entirely in stocks that are expected to earn 11% per year. During retirement, you
plan to invest in safer investments that earn 6% per year. The expected inflation rate is 2.5%.
a) If you plan as if you will live forever and you wish to maintain constant annual purchasing
power during retirement, how much can you withdraw each year during retirement?
Answer:
Value of savings = V35 = $100,0001.1135 + $8000FVIFGA11%,5% = $8,265,330
Value of withdrawals = V35 = C/(0.06-0.025)  $8,265,330
Setting these equal to each other and solving for C gives C = $289,287.
So, we can withdraw $289,287 36 years from now and then increase that amount by 2.5%
each year forever.
b) What is the purchasing power of those annual withdrawals in today’s dollars?
Answer:
Purchasing power in today’s dollars = $289,287/1.02536 = $118,924
4) You plan to save $7,000 each of the next 40 years, and invest that money in an account that
pays 9% annual interest. In addition, you plan to pay for your child’s college education
beginning in 20 years. You expect that education to cost $30,000 per year for four years. To pay
for the education, you will simply withdrawal money from your investment account. In addition,
you currently have an outstanding loan with a balance of $15,000 and an annual interest rate of
9%. You plan to pay off that loan over the next few years. A timeline depicting this situation
follows.
Date
Deposits
Withdrawals
Loan Balance
0
1-19
$7,000
20-23
$7,000
$30,000
24-40
$7,000
$15,000
a) How much money will you have just after you make your last deposit forty years from today?
Answer:
Value of deposits in 40 years = $7,000FVIFA9%,40 = $2,365,177
Value (i.e., opportunity cost) of debt in 40 years $15,0001.0940 = $471,141
Value of college withdrawals in 23 years = $30,000FVIFA9%,4 = $137,194
Value of college withdrawals in 40 years = 1.0917 = $137,1941.0917 = $593,725
Account Balance in 40 years = $2,365,177-$471,141-$593,725 = $1,300,311
b) [2 pts] How much money will you have 5 years later (year 45) if you make no additional
deposits or withdrawals?
Answer:
Value in 45 years = $1,300,3111.095 = $2,000,690
c) As an intelligent and informed financial planner, you have been asked to evaluate the
assumptions and analysis above. What specific flaws do you see (if any)?
Answer: The analysis fails to incorporate potential increases in salary, which would allow
you to save more and more over time. It also fails to incorporate inflation, which can
dramatically affect the purchasing power of the retirement savings.
5) You plan to work for 35 years and then retire. You currently have nothing saved toward
retirement, but you plan to save 10% of your salary each year for the next thirty years. Your
current salary is $60,000 per year and you expect that salary to grow at about 6% per year.
Inflation is expected to be 2% per year indefinitely. All of your savings will be invested in a
portfolio of stocks that is expected to pay an 11% annual return. At the time you retire, you plan
to move your money into a safer portfolio that is expected to pay 7% per year. As an eternal
optimist, you expect to live forever. Assume that you want your purchasing power to be the same
each year during retirement and that you want to spend the maximum amount possible under the
assumptions above.
a) How much money can you withdraw during the first year of retirement (in 36 years)?
Answer:
Vsavings in 30 years = $6,000FVIFGA11%,6%,30 = $2,057,857
Vsavings in 35 years = Vsavings in 30 years1.115 = $3,467,608
Vsavings in 35 years = $3,467,608  Vretirement withdrawals = C/(0.07-0.02)
Solving gives C = $173,380
b) What is the purchasing power of a given annual retirement withdrawal in today's dollars?
Answer:
Purchasing power today = $173,380/1.0236 = $84,995
6) You have discovered a magical elixir that will allow you to live forever. You subsequently
sold the rights to that elixir for $15,000,000 and have invested that money at 10% interest. The
expected inflation rate is 2.5% forever. Finally, you wish to pay for your daughter’s wedding
which you estimate will occur in 25 years. Measured in today’s dollars, you plan to spend
$500,000 on the wedding. Assuming that you plan to never work again and that you wish to have
the same purchasing power each year (excluding the wedding cost) during your retirement, what
is the maximum amount you can withdraw during your first year of retirement? For simplicity,
assume that the first withdrawal will occur in one year, the second in two years, etc.
Answer: We expect to spend $500,000×1.025^25 = $926,972 on the wedding. Using the
concept of indifference, we know that spending $926,972 in 25 years is equivalent to
spending $926,972/1.125 = $85,556 today. Said differently, if we set aside $85,556 today, it
will have grown to $926,972 in 25 years. So, we effectively have $15,000,000-$85,556 =
$14,914,444 available today for our retirement planning. We know that the present value of
a growing perpetuity is V0 = C1/(R-g), which gives us C1 = V0×(R-g). It follows that we can
withdraw $14,914,444×(0.1-0.025) = $1,118,583 during our first year of retirement.
7) You plan to save $8,000 each of the next 35 years, and invest that money in an account that
pays 9% annual interest. In addition, you plan to pay for your child’s college education
beginning in 20 years. You expect that education to cost $30,000 per year for four years. To pay
for the education, you will simply withdrawal money from your investment account. In addition,
a long-lost relative recently died, leaving you $50,000. A timeline depicting this situation
follows.
Date
0
1-19
20-23
23-35
Deposits
$50,000
$8,000
$8,000
$8,000
Withdrawals
$30,000
How much money will you have just after you make your last deposit 35 years from today?
Answer: Using the concept of indifference, we can simply calculate the value of each set of
cash flows at date 35 and then add them together.
Value at date 35 of $50,000 inheritance = $50,000×1.0935 = $1,020,698
Value at date of 35 of $8,000 annuity = $8,000×FVIFA9%,35 = $1,725,686
Value at date 23 of $30,000 college payments = $30,000×FVIFA9%,4 = $137,194
Value at date 35 of $30,000 college payments = $137,194×1.0912 = $385,880
Total at date 35 = $1,020,698 + $1,725,686 - $385,880 = $2,360,504
8) You plan to save $8,000 each of the next 40 years, and invest that money in an account that
pays 8% annual interest. In addition, you plan to pay for your child’s college education
beginning in fifteen years. You expect that education to cost $20,000 per year for four years. To
pay for the education, you will simply withdrawal money from your investment account. In
addition, a long-lost relative recently died, leaving you $50,000. A timeline depicting this
situation follows.
Date
Deposits
Withdrawals
0
$50,000
1-14
$8,000
15-18
$8,000
$20,000
19-40
$8,000
How much money will you have just after you make your last deposit forty years from today?
Answer:
Value of deposits in 40 years = $8,000×FVIFA8%,40 = $2,072,452
Value of inheritance in 40 years = $50,000×1.0840 = $1,086,226
Value of college expenses in 18 years = $20,000×FVIFA8%,4 = $90,122
Value of college expenses in 40 years = V18×1.0822 = ×1.0822 = $489,953
Amount saved after 40 years = $2,072,452 + $1,086,226 - $489,953 = $2,668,725
9) You plan to work for 30 years and then retire. You currently have $25,000 saved toward
retirement and you plan to save 10% of your salary each year for the next thirty years. Your
current salary is $55,000 salary year and you expect that salary to grow at about 5% per year.
Inflation is expected to be 2% per year indefinitely. All of your savings (including the $25,000
saved and the money taken out of your savings) will be invested in a portfolio of stocks that is
expected to pay a 10% annual return. At the time you retire, you plan to move your money into a
safer portfolio that is expected to pay 8% per year. As an eternal optimist, you expect to live
forever. Assume that you want your purchasing power to be the same each year during
retirement and that you want to spend the maximum amount possible under the assumptions
above.
a) How much money can you withdraw during the first year of retirement (in 31 years)?
Answer:
Savings in 30 years
Value of $25,000 already saved = $25,000×1.130 = $436,235
Value of additional savings = $55,000×10%×FVIFGA10%,5%,30 = $1,444,021
Total savings = $436,235+$1,444,021 = $1,880,256
Retirement Withdrawals
Value of Withdrawals in 30 years = $C×PVIFGA8%,2%,∞ = $C/(0.08-0.02)
where C is the first retirement withdrawal
Total Savings = Total Withdrawals, so
$1,880,256 = $C/0.06
Solving gives C = $112,815
b) What is the purchasing power of that withdrawal in today’s dollars?
Answer:
Purchasing power = $112,815/1.0231 = $61,061
10) A university decides to offer students two different tuition options. In the first, a new
freshman can pay a one-time fixed payment of $100,000. In the second, the student would make
four annual payments of $29,000 each. The university would guarantee the student that there
would be no tuition increase during the four years. Assume that the first of the $29,000 payments
would be due at the same time that the $100,000 would be due. If the appropriate discount rate is
6%, which option should students prefer?
Answer: There are several ways to compute the value of the four-payment option. First,
V-1 = $29,000PVIFA6%,4 = $100,488
V0 = V-11.06 = $106,517.
Second,
V0 = $29,000 + $29,000PVIFA6%,3 = $106,517
In either case, the single fixed payment option is better.
11) You plan to save $10,000 next year. In each subsequent year, you plan to save 5% more than
the prior year. You plan to save for a total of 35 years. In addition, you currently have total
savings of $25,000. Those savings and all subsequent deposits will be invested in an account that
pays 10% annual interest. Upon retiring, however, you plan to move that money into a safer
account that earns 7% annual interest. A timeline depicting this situation follows.
Date
Deposits
Current Savings
0
1
$10,000
2
$10,0001.05
...
...
35
$10,0001.0534
$25,000
a) How much money will you have just after you make your last deposit 35 years from today?
Answer:
Value of current savings in 35 years = $25,0001.135 = $702,561
Value of deposits in 35 years = $10,000FVIFGA10%,5%,35 = $4,517,284
Total amount saved = $702,561+$4,517,284 = $5,219,845
b) Suppose that inflation is expected to be 2.5% per year indefinitely. Suppose further that you
want each of your annual retirement withdrawals to have the same purchasing power. If you plan
to live forever, what is the largest amount you can withdrawal during your first year of
retirement?
Answer:
Let W be the amount of the first withdrawal. We know that
WPVIFGA7%,2.5%, = $5,219,845
Solving for W gives W = $234,893
c) What is the purchasing power of an annual retirement withdrawal, measured in today's
dollars?
Answer: Purchasing power = $234,893/1.02536 = $96,563
12) You decide to wait for five years before saving money toward retirement. You then wish to
save a certain amount each year for 30 years, at which time you hope to have saved $1,500,000.
If you invest in an account that earns 8% annual interest, how much must you invest each year to
meet your goal? A timeline depicting this situation follows.
Answer: Let C be the amount we save each year. We know that
CFVIFA8%,30 = $1,500,000
Solving for C gives C = $13,241