MGMT 221: Managerial Finance
Solution to Time Value of Money Practice Problems
1. If the appropriate discount rate is 12% per year, what is the present value of the following
expected cash flows?
Year
0
3
4
5
Cash Flow
-$8,000
$3,000
$4,000
$2,500
Answer: PV = -$8,000 + $3,000/1.123+ $4,000/1.124+ $2,500/1.125 = -$1,904
2. An elderly neighbor asks you for investment advice. She has found an annuity that currently
sells for $80,000. The annuity pays $800 per month (starting in one month) for the next ten
years. After a bit of research, you determine that annuities of similar risk earn annual percentage
rates around 8.5% [Note: the monthly rate would be 8.5%/12]. She would like to know whether
or not the annuity is a good deal. Is it? Why or why not?
Answer: At 8.5% APR, the bond should pay 8.5%/12 = 0.70833% per month. The present
value of the annuity is then PV = $800  PVIFA0.70833%,120 = $64,524. Since your neighbor
has been asked to pay more than that, the investment is not a good deal.
3. After carefully considering your future, you have decided that you would like to retire on your
55th birthday. Since you will be earning lots of money using your recently obtained financial
knowledge, you decide that you can put off saving any money until your 46th birthday. At that
time, you will simply make annual deposits into an account that earns 5% annually. The last
deposit will occur on your 55th birthday (there are 10 total deposits). Suppose that you wish to
withdraw $100,000 each year beginning on your 56tth birthday and lasting for 20 years. How
much must you deposit annually in order to meet your goal? Assume that after making the last
withdrawal, your account will have a balance of zero.
Answer: It is best to divide problems like this into two parts. First, calculate how much
money you will need in your account when you retire (age 55). You would like to withdraw
$100,000 per year for twenty years, so you will need PV = $100,000  PVIFA5%,20 =
$1,246,221 in your account at that time. Second, calculate how much you will need to
deposit in order to end up with that balance. You will be making ten deposits with the last
one occurring on your 55th birthday, so $1,246,221 = C  FVIFA5%,10. Solving for C gives C
= $99,080.
4. A client hopes to retire in 20 years with a retirement fund of $1,500,000. He plans to finance
his retirement by investing a certain amount each year. His child will be entering college in 12
years. Expected annual college costs are $25,000 per year (for four years). The expected
inflation rate is zero indefinitely. Draw a timeline depicting the cash flow situation.
Answer:
Date:
1 – 11
12 – 15
16 – 20
CF:
C
C - $25,000
C
Assuming that the client invests solely in the stock market (which returns 13% per year on
average), how much must the client invest per year in order to meet his retirement goal?
Answer: Value of savings in 20 years: C  FVIFA13%,20
Value of college costs in 15 years: $25,000  FVIFA13%,4 = $121,245
Value of college costs in 20 years: $121,2451.135 = $223,386
So, C  FVIFA13%,20 -$223,386 = $1,500,000
Solving gives C = $21,290
5. Suppose that the appropriate discount rate is 10% per year. What is the value today of the
following cash flows?
Date (in years)
Cash Flow
1-5
$1000
6-9
$0
10-12
$2000
13-15
$1500
To be clear, there are five $1000 cash flows, three $2000 cash flows, and three $1500 cash
flows.
Answer:
V = $1000PVIFA10%,5 + $2000PVIFA10%,3/1.19 + $1500PVIFA10%,3/1.112 = $7088.70
6. You plan to retire in 30 years. You are considering two different retirement plans. In the first,
you invest $10,000 today. In the second, you plan to live a life of luxury for the next ten years
and then invest $5,000 per year until retirement. The first investment occurs 11 years from today
and you investment a total of $100,000. You anticipate an annual return of 13.5%. Which plan
will give you a better retirement income? Justify your answer. BRIEFLY comment on the
implication (of your finding) for retirement planning.
Answer:
Value of first scenario = $10,0001.13530 = $446,556
Value of second scenario = $5,000FVIFA13.5%,20 = $429,143
Surprisingly, the scenario allows you to have more money available for retirement. The
example is a dramatic illustration of the importance of investing over a long period of time.
How much you invest appears to be less important than how long you invest it.
7. Your task is to compare and contrast a retirement plan under two different
assumptions. In the first, you will ignore the impact of inflation (i.e., treat the problem as
if there is no expected inflation). In the second, you will incorporate the impact of
inflation. Please list any assumptions you make in addition to those listed below.
As a financial planner, you recently discussed retirement planning with a client. She
revealed the following.
- She is 25 years old.
- She believes she can save $6,000 this year.
- She is slightly risk averse, but is willing to invest solely in the stock market until
retirement.
- Your sense is that upon retirement, she will want to invest approximately half her
money in a bond fund and half in a stock fund.
Market expectations are as follows.
- Expected inflation is 3% per year indefinitely.
- Expected return on a well-diversified stock fund is 13% per year indefinitely.
- Expected return on a well-diversified bond fund is 6% per year indefinitely.
You choose to use the following assumptions in the analysis.
- The woman will live forever (i.e., since you don’t know her life expectancy, you plan to
be conservative and assume and infinite life).
- The woman will retire in 35 years.
(a) Develop a reasonable investment strategy for the woman and describe her expected
retirement income ignoring the impact of inflation (i.e., assume that investment and
retirement cash flows do not grow).
Answer: With no inflation, it is reasonable to assume that the woman saves $6,000
each year for the next 35 years. At that time, she expects to have $6,000 
FVIFA13%,35 = $3,280,085. If she then invests at 9.5% (half bonds, half stocks), she
can withdrawal $C per year forever, where C satisfies $3,280,085 = $C / 0.095. This
gives C = $311,608.
(b) Describe the expected retirement income for the woman under the more realistic
assumption that inflation affects purchasing power.
Answer: Conservatively, we may assume that the woman increases her savings by
3% each year. In 35 years, she expects to have $6,000  FVIFGA13%,3%,35 =
$4,155,279. If she then invests at 9.5% (half bonds, half stocks), she can withdrawal
$C in year 36, where C satisfies $4,155,279 = $C / (0.095-0.03). This gives C =
$270,093. She can then increase the amount she withdrawals by 3% each year
forever. In today’s dollars, C=$93,190 (a nice retirement income).
(c) BRIEFLY comment on the importance (or lack thereof) of addressing inflation in a
retirement analysis.
Answer: It is important to consider inflation because it impacts purchasing power.
If we ignore it in retirement planning, we generally find that retirees end up with
less purchasing power than they expected.