Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
What is the Time Value of Money (TVM)?
• Because money earns money, the value of a dollar today
is greater than the value of a dollar in the future.
• Investors demand that not only do they get a return of the
money invested, but that they get a
return as well.
• Sound financial decisions depend on an
understanding of the basic mathematics
of compound interest or return.
• To make financial decisions, compare
the value of two investments at the same
point in time.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
TVM is Essential to Understanding:
• The effect of time on the profitability of an investment.
• How the projected value of an investment’s future
returns affects the price that should be paid for it.
• How to compute the value of an investment’s future
return.
• How to determine appropriate financial goals for future
needs such as retirement planning, education funding
and insurance, allowing for the effects of inflation and
taxes.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
How Does Time Value of Money Analysis Work?
• Three basic underlying rate of return principles that
should govern every investment:
– Timing,
– Quality, and
– Quantity.
• In addition, the individual or
family will consider personal
risk preferences and financial
resources.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Timing
• Money now is better than money later.
• Since money now can be invested to earn
a return, the investor will have more money
later.
• Example:
– Timing affects investment in the treatment of depreciation in the
tax law.
– Since accelerated depreciation reduces taxes faster than straight
depreciation, thus putting more money in the hands of the
investor sooner, the better of two otherwise financially equal
investments will be the one that qualifies for accelerated
depreciation.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Quality
• Quality is another way of describing lower risk.
• An investment that has a lower chance of losing
money is a higher quality investment.
• If two investments have an equal
investment potential, but one is of
higher quality, it is the better
investment.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Quantity
• Higher rate of return is better, all other things being
equal.
• Choosing an investment
becomes like shopping:
– Weigh the relative merits of
two investments on the basis
of all three factors.
– Using the client’s risk preferences, make the choice
offsetting higher risk against higher return.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
When Do Financial Planners Use Time Value of
Money Analysis?
• When weighing potential investments against the client’s
risk preferences and timing needs.
• When planning for future financial needs such as:
– Education
– Retirement
– Estate planning
• While taking into consideration future income sources
such as:
–
–
–
–
Investments
Insurance
Social security
Retirement benefits
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Advantages Using TVM Analysis
• Permits a quantitative comparison of alternative
investments that have different rates of return and
investment maturities.
• To determine whether a particular
investment is affordable.
• Identifies situations in which current
savings and investment will not be
enough to fund future needs such as
retirement, and can be used to
determine how much additional
savings is needed.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Computing the Future Value of a Lump Sum
Problem
• If I have a lump sum of $______ today, how do I
calculate the value of that lump sum _____ years in
the future assuming I earn ___% on my investment?
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Choice of Tools to Solve the Problem
• Use the Future Value of a Lump Sum Table in
Appendix D.
• Use a scientific calculator and the formula.
• Use a financial calculator.
• Use a computer spreadsheet program such as Excel.
People who are preparing for the certification examinations need to practice
using the financial calculator, since it is the only method that makes sense to
use on the examination. Using the PC and spreadsheet software is often the
best solution in the office, but skill with the financial calculator is useful in outside
appointments.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Notation Used in the TVM Formulas
• PV = Present Value
• PMT = Payment (as in a loan or annuity calculation.)
• FV = Future Value
• N or n = number of periods
• I, I/Y, or I/P = interest rate or yield
• Begin or End – denotes whether the payment is
made at the beginning or end of the time period.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Future Value of a Lump Sum (FVLS)
• Start with a lump sum: That is the Present Value or PV.
• If i = the interest or return earned per period, then the
Future Value (FV) at the end of 1 period is
FV= PV (1+i)
– after two periods it is
FV = PV (1+i)(1+i) or FV = PV (1+i)2
– After three periods it is
FV = PV(1+i)(1+i)(1+i) = PV (1+i)3
– Therefore, after n periods it is
FV = (1+i)n.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Example: FV of $10,000 @ 10% for 5 Years
Beginning
Balance
Interest @
10%
Ending
Balance
Year 1
$10,000
$1,000
$11,000
Year 2
$11,000
$1,100
$12,100
Year 3
$12,100
$1,210
$13,310
Year 4
$13,310
$1,331
$14,641
Year 5
$14,641
$1,464
$16,105
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Using a Table
• Use Appendix D, Future Value of a Lump Sum Table.
• This table reflects the amount $1 will be worth in a given number of
years at various interest rates.
• Multiply your Present Value (PV) by the number in the table
representing the number of years and the interest rate to get the
future value.
• Drawbacks of the Table:
– May have to interpolate if your interest rate falls between those in the
table.
– A comprehensive table is a thick book.
– Rounding error can be significant.
– You cannot take the Table to the CFP® Certification Exam.
• The advantage of the table is that it is simple to use.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Using Excel Software
• In Microsoft Excel 2003, move the cursor to the cell where you want
to have your answer.
• Click on the fx just above the worksheet.
• The Insert Function window will appear. Type FV in the search box
and press Go. Click OK.
• The Function Arguments window will appear. Fill in the numbers for
each argument. For example, if your interest rate is 6%, enter .06.
Note that as you move the cursor to each argument that the
software puts an explanation of the argument below. When you have
finished entering the arguments, click OK.
• Your answer will appear in the cell.
• If you have a different version of Excel, or use Lotus, the process
will be similar, but you may need to consult Help in that version of
the software you are using.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Advantages and Disadvantages to Using
Spreadsheet Software
• Advantages:
– Simple to use: Easy to do “What if” calculations.
– Very accurate.
– Can check the numbers in the arguments to make sure the
calculation was done correctly.
• Disadvantages:
– Must have a PC and the software to use it.
– Even a small laptop is sometimes unwieldy to take to a client
meeting.
– You cannot use a computer on the CFP® certification
examination.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Using a Financial Calculator
• Refer to your calculator manual for
detailed instructions.
• Clear all entries from the calculator.
Put it in TVM mode (if needed).
• Enter the quantity for the lump sum, then
press PV.
• Enter the number of periods (n) and press n.
• Enter the interest rate or return rate and press I/Y or I/P
(depending on make of calculator).
• On HP calculators, press FV. On TI calculators, press
compute then FV.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Computing the Present Value of a Future Lump
Sum
• PROBLEM: If I will have a lump sum of $______ in
___ years, how do I calculate the present value of my
investment, assuming it will earn interest at the rate of
___? In other words, what is the equivalent today of
$______ payable as a lump sum ___ years in the
future?
• You can use Appendix A – Present Value of a Lump
Sum Table, or
• Use the PV function in Excel, or
• Use a Financial Calculator.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Development of the Formula for Present Value
of a Future Lump Sum
• Given the formula for computing the Future Value of
a Lump sum, we can derive the formula for the
Present Value:
FV  PV (1  i ) n
• Solve for PV:
FV
PV 
(1  i ) n
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Tips on Solving TVM Statement Problems
• Write down the five variables: FV, PV, PMT, N, and I. As the
problem is read, fill in the quantities next to the appropriate
variable.
• If the return is expressed as an annual return, but it is
compounded, divide the annual return and multiply the
number of years by the appropriate number (4 for quarterly,
12 for monthly) to solve the problem.
• If doing a lump sum problem on a calculator, the PMT is 0. It
is recommended putting in the 0 to be safe, since prior
problem’s quantities may be saved in the calculator.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Setting Up a Calculator
• Many calculators come with the number of payments per
year set up as 12. If a person was doing only one kind of
problem that always had 12 payments per year, that would be
fine. However, for financial planning, where calculations are
varied, it may be advisable to set up a calculator for P/Y = 1
as a default. It can always be changed where needed.
• Also set up a calculator for as many decimal places as
desired. Rounding can always be done at the end.
• A calculator manual will have directions on how to do both of
these setup functions so that they become the default.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Sign and Flow on the Financial Calculator
• Remember that a minus sign designates an
outflow and a positive number represents an
inflow. If a Future Value is an inflow (positive)
then the Present Value must be an outflow
(negative). When investing, an outlay (PV) is the
negative number. This inflow and outflow
convention is also used in the Excel software.
• The flow of PMT is also signed. If receiving
money, it is plus: If paying money, it is minus.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problems Using Future Value of a Series of
Payments
• If, beginning today, I invest $______ a year for ___ years, how do I
calculate what the value of that series of investments would be ___
years from now assuming I earn a compounded interest rate of
___% on my investments?
• This type of problem requires the calculation of the future value of a
regular series of payments.
• Future Value of Annuity Due (FVAD): Where each payment is made
at the beginning of a compounding period (for example, at the
beginning of each year), the process is known as an “annuity due” or
an “annuity in advance.”
• Future Value of Ordinary Annuity (FVOA): If the first payment in the
series of investments is not made until one year from now, the
process is known as an “ordinary annuity” or an “annuity in arrears.”
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Future Value of an Annuity
• This calculation only works when the payments are made every
period and are for the same amount. Otherwise, a more complex
method must be used.
FVAD
• On a financial calculator and in Excel, choose “Begin.” In Excel,
“Begin” is coded as a 1 in the function argument TYPE.
• To use the tables, use Appendix E, Future Value of an Annuity Due
Table.
FVOA
• On a financial calculator and in Excel, choose “End.” In Excel, “End”
is coded as a 0 in the function argument TYPE.
• To use the tables, use Appendix F, Future Value of an Ordinary Due
Table.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Future Value of an Annuity (cont)
The formula for the FVAD is:
 (1  i) n  1
FV  PMT (1  i) 

i


The formula for the FVOA is:
 (1  i) n  1
FV  PMT 

i


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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Setting up a Problem to solve Using a
Calculator or Software
• Write down your five variables, the compounding period
and whether it is begin or end, thus:
• FV =
• PMT =
• PV =
• I/P =
• N=
• Begin/End =
Compounding:
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Setting Up the Problem for Your Calculator or
Spreadsheet
• Fill in the quantities from your problem. As an example,
consider “What is the future value of monthly payments
of $200 for 10 years, at 6% interest, compounded
monthly, that start today?”
• FV = ?
• PMT = 200 (Press the +/- key to make it negative, since
it is an outflow.)
• PV = 0
• I/P = 6% per year/12 months per year = .5 per period
• N = 10 years x 12 payments per year = 120 periods
• Begin/End = Begin
Compounding: Monthly
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Solving the Problem
• Enter the quantities into your calculator that you
recorded in the worksheet.
• Follow your calculator instructions to compute the
answer, OR
• Use the quantities that you recorded to fill in the function
arguments on your spreadsheet software.
Note: If you are studying for the CFP® certification examination, practicing by working
many problems each week will help you become skillful at using your calculator and
remove one source of anxiety on the examination. Some successful students make
up problems, work them on the calculator and then check their answer using the
Excel software, giving themselves double practice.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Computing the Present Value of a Regular
Series of Receipts
• This is the Present Value of an Annuity.
• If the payments start immediately and continue at the
beginning of each period, it is the Present Value of an
Annuity Due (PVAD).
• If the first payment is at the end of the first period and
every period thereafter, then it is the Present Value of an
Ordinary Annuity (PVOA).
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Computing the Present Value of a Regular
Series of Receipts
• PROBLEM: If, beginning one year from today, I
receive $______ a year for ___ years, how do I
calculate the present value of that series of
payments, assuming a ___% discount rate?
• SOLUTION:
– Use Appendix C, Present Value of an Ordinary Annuity
Table, or
– The PV function in Excel or
– A financial calculator.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Equation for the Present Value of an Ordinary
Annuity (PVOA)

1 
PVOA  PMT (1  i ) 1 
n 
(1  i ) 

An ordinary scientific calculator can be used to solve a TVM
problem using this formula, or any of the previous formulas.
Choose End for a financial calculator, or insert a 0 into or leave
blank the TYPE function argument in Excel.
Or use Appendix C, Present Value of an Ordinary Annuity Table.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Formula for the Present Value of an Annuity Due
(PVAD)

1 
PVAD  PMT (1  i ) 1 
(1  i )
n 
(1  i ) 

An ordinary scientific calculator can be used to solve a TVM
problem using this formula, or any of the previous formulas.
Choose Begin for a financial calculator, or insert a 1 into the TYPE
function argument in Excel.
Or use Appendix B, Present Value of an Annuity Due Table.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Practical Examples, Problem 1:
• PROBLEM: Rich Stevens, age 53, has just inherited
$100,000 which he would like to use as part of his
retirement nest egg. Rich would like to know just how
much the $100,000 will be worth in 12 years, when
he will reach age 65, assuming the funds can be
invested for the entire period at a 12% annual rate.
He would also like to know what the future value of
the $100,000 would be in only 7 years, when he
reaches age 60, in case he decides to retire early.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 1
• Using the mini-worksheet below, write down the five
variables, the compounding period and whether it is
begin or end, thus:
• FV =
• PMT =
• PV =
• I/P =
• N=
• Begin/End =
Compounding:
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 1 Worksheet
•
•
•
•
•
•
FV = ?
PMT = 0
PV =100,000
I/P = 12
N = 12, then redo for 7 years
Begin/End = N/A* Compounding: Annual

Begin or End only matters when there is a series of payments. When it is a lump
sum calculation, Begin or End is irrelevant.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 1 (continued)
• Since this is the Future Value of a Lump Sum, Appendix D,
Future Value of a Lump Sum Table, could be used to solve
the problem.
• The FV function in Excel could be used, filling in the
function arguments with the numbers from the worksheet.
• An ordinary calculator could be used with the FV formula.
• The variables can be entered into a financial calculator.
• Whatever method is used, the answers are approximately
$389,600 in 12 years or $221,070 in 7 years.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 2
• PROBLEM: Now that Rich knows how much the $100,000
inheritance will be worth in both cases, he would like to know how
much he could withdraw from the fund in equal installments at the
end of each year from the year he retires until he reaches age 70½,
the year he must start withdrawing funds from his individual
retirement account (IRA). Rich assumes the funds will continue to
earn at a 12% annual rate.
• In other words, Rich would like to know the annual year-end
payment from (1) a 6-year annuity (from age 65 to the year he will
be 70½), earning 12% annually on a principal sum of $389,600, and
(2) an 11-year annuity (from age 60 to the year he will be 70½),
earning 12% annually on a principal sum of $221,070.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 2 Worksheet
•
•
•
•
•
•
Scenario A
(Retire at 65)
FV = 0
PMT = ?
PV = $389,600
I/P = 12
N=6
Begin/End = End
Scenario B
(Retire at 60)
FV = 0
PMT = ?
PV = $221,070
I/P = 12
N = 11
Compounding: Annual
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 2 (cont)
• Any of the four methods can be used.
– The Table that should be used is Appendix C.
Pmt equals PV divided by PVOA factor.
• Whichever method used should yield answers of about
$94,761 per year at age 65, and $37,232 at age 60.
• Note that using the financial calculator, since it saves the
variables unless actively cleared, all that is needed to
compute Scenario B is to enter new quantities for PV
and N, and then compute PMT. Since everything else is
the same, all the variables do not need to be re-entered.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 3
• PROBLEM: Rich has determined that he will need
$60,000 per year from the inheritance fund to handle
his living needs until he reaches age 70½. Assuming
the fund will continue to earn 12% annually, at what
age can Rich afford to retire? (Rich has already
decided not to touch his IRA funds until the latest
possible date, believing he can cover his living costs
with the inheritance until that time. He is even willing
to adjust his retirement date by a year or so if need
be.)
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 3 (cont)
• Based upon earlier calculations, the answer will be
greater than age 60 and less than age 65. It is important
to determine if an answer is reasonable. Figuring out an
approximation ahead of solving the problem can help
avoid errors.
• In this problem, N is being solved, but the starting PV is
not known, since it will vary according to the age of
retirement. Using a financial calculator or a spreadsheet,
the problem is still not difficult, since one variable can be
changed at a time to check various retirement scenarios.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 3 (cont)
• First, compute the value of the inheritance at each age
from 61 to 64.
– If there are more than 4 scenarios to check, start at the middle of
the range, compute it, then determine whether to go higher or
lower, thus halving the remaining work to be done with each trial.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 3, Worksheet 1
•
•
•
•
•
•
•
FV = ?
PMT = 0
PV =100,000
I/P = 12
N = 11, 10, 9, or 8
Begin/End = N/A Compounding: Annual
Recomputed for each N, changing only that variable, the value of
the inheritance at each year is:
– Age 61 – $247,596
Age 62 – $277,307
Age 63 – $310,584
Age 64 – $347,855
• Now it is simply a matter of determining which one of these lump
sums will yield an income of at least $60,000 per year until Rich
reaches the year in which he turns 70½.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 3, Worksheet 2
•
•
•
•
•
•
FV = 0
PMT = ?
PV = $247,596; $277,307; $310,584; or $347,855
I/P = 12
N = 9, 8, 7, or 6
Begin/End = End Compounding: Annual
Here, once the financial calculator is set up for the
first scenario, only the following need to be changed:
the PV and the N to solve each scenario, and
payments need to be set for End.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 3, Results
# of Years
to Age 70½
Annual
Income from
Inheritance
Retire
at Age
Lump Sum
Accumulated
61
$247,596
10
$43,821
62
$277,307
9
$52,045
63
$310,584
8
$62,522
64
$347,855
7
$76,221
(Rounded up to age 71)
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 3, Discussion of Results
• Thus, using scenario analysis and TVM calculations, it
has been determined that Rich should wait until age 63
to retire to meet his stated goals.
• The best tool for this problem would be a spreadsheet,
since the formula could be set up once and then simply
copied to cells to compute the numbers for every year.
• Note, however, that this analysis assumed that there
was no inflation, and that $60,000 per year would buy
the same goods and services then as now. Class
discussion: Will the extra $2,522 offset inflation?
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 4
• PROBLEM: Rich has decided that he wants to retire
at age 60. He would like to know how much of his
other funds need be set aside with his $100,000
inheritance in order to reach his goal of a $60,000
annuity from age 60 until the year he reaches age
70½. Rich assumes the funds can continue to earn at
a 12% annual rate.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 4, Worksheet 1
• The first step is to determine how much he needs to have at
age 60 to give him $60,000 per year until age 70½.
• FV = 0
• PMT = 60,000
• PV = ?
• I/P = 12
• N = 11
• Begin/End = End
Compounding: Annual
• The result is $356,262.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Problem 4, Worksheet 2
•
•
•
•
•
•
FV = $356,262
PMT = 0
PV = ?
I/P = 12
N=7
Begin/End = N/A
Compounding: Annual
Rich needs a $161,137 lump sum. Since he already
has $100,000 from his inheritance, he needs an
additional $61,137 to accumulate the needed
amount by age 60.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Including Taxes, Inflation, and Growth in TVM
Analysis
• Taxes and inflation reduce return on investment.
• Accounting for taxes and inflation is difficult since the
rates can be different at different times and in different
circumstances.
• However, to design reasonable strategies for personal
financial planning, it is necessary to make the best
possible approximation of what the effect of taxes and
inflation will be.
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50
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Taxes and Investment Return
• If t is the marginal tax percentage,
then:
– After-tax return = pre-tax return (1-t)
The result can be used as the Interest
Rate in any of the formulas where income is taxed annually.
• When taxes are deferred, use the before-tax return, then
apply the tax rate at the end of the deferral period.
• When there is a combination of ordinary income and
capital gains, the calculation becomes complex. Usually
the best tool will be to break down the return into its
component parts and use spreadsheet software.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Inflation/Growth
•
•
•
Usually, when calculating long-term needs, an adjustment to return is
needed to account for loss of purchasing power due to inflation, or to
account for systematic increments or decrements in payments or cash
flow over time.
The previous formulas can be used, with slight modification, if the interest
rate is adjusted for inflation or growth. Simple subtraction does not give an
accurate answer. Instead, use the formula: where ρ is the interest rate
discounted for inflation, r is the nominal interest rate, and g (or i) is the
growth (or inflation) rate. Use ρ for the interest rate in calculations.
The following two formulas are mathematically equivalent; either can be
used.
r  g 


1

g


Copyright 2009, The National Underwriter Company
1 r 

1

1  g 
52
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Inflation Adjusted Rate of Return and Annuities
• Some annuities (and the benefits on some
Long Term Care policies) have a built-in
growth rate meant to adjust for inflation.
• The following slides give the formulas
adjusting each of the previous TVM formulas
for an increasing payment with a growth rate
g.
• The growth adjusted rate of return is
expressed as ρ.
Copyright 2009, The National Underwriter Company
53
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Calculating ρ (growth adjusted rate)
rg
1 g
where r is investment return
and g is growth.

Example: The investment return is 12% and payments
are growing by 4% per year. 12% - 4% is 8%, and 1 + 4%
is 104%, so the growth adjusted rate is 7.69%.
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54
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Present Value of an Ordinary Annuity, Adjusted
for Growth (PVOAg)
PMT
PV 
1 g
1  (1   )  n 

 if   0



PMT (n)
PV 
if   0
1 g
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55
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Present Value of an Annuity Due, Adjusted for
Growth (PVADg)
1  (1   )  n 
PV  PMT 
 (1   ), if   0



PV  n( PMT ) , if   0
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56
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Future Value of an Ordinary Annuity, Adjusted
for Growth (FVOAg)
n

(
1


)
1
n 1
FV  (1  g ) 
 , if   0



FV  PMT ( n)(1  g ) n 1 , if   0
Copyright 2009, The National Underwriter Company
57
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Future Value of an Annuity Due, Adjusted for
Growth (FVADg)
n

(
1


)
 1
n 1
FV  PMT (1  g ) 
 (1  r ), if   0



FV  PMT ( n)1  g 
n 1
(1  r ) , if   0
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58
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Net Present Value and Internal Rate Of Return
• Not all investments result in regular payments. For
example, an investment in a business project typically
involves negative cash flow in the beginning, then
(hopefully) results in positive cash flow later.
• Making an investing decision among several projects,
each with differing cash flows, can be difficult.
• The use of Net Present Value (NPV) and Internal Rate
of Return (IRR) analysis allows comparison on the
same basis and aids decision.
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59
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Methods for Comparing Alternative Investments
• Net present value
• Internal rate of return
• Adjusted rate of return
• Pay back period
• Cash on cash
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60
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Net Present Value
• “Net” present value is the difference between
– (1) the present value of all future benefits to be realized from
an investment and
– (2) the present value of all capital contributions into the
investment.
• A negative net present value should result in an
almost automatic rejection of the investment.
• A positive net present value indicates that the
investment is worth further consideration.
Copyright 2009, The National Underwriter Company
61
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Evaluating Net Present Value
• A discount rate of return on investment must be
assumed in computing NPV.
• What is usually used as the discount rate is the
minimum acceptable rate of return.
• A negative NPV will indicate that the investment does
not meet the investor’s minimum.
• In comparing investments using NPV, the risk of the two
investments must be equal for the comparison to be
valid.
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62
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Example
Assume a beginning of the year investment opportunity requiring a lump sum
outlay of $10,000, which is currently invested in a money market fund at 6%
annual net after taxes. The investment proposal projects the following after-tax
cash flows at the end of each year. Assume 6% is the minimum required rate.
Year
1
2
3
4
5
Cash Flow
$2,000
1,500
750
500
10,000
Total 14,500
Computing the PV of each cash flow and adding them, the PV is $11,721, so
the NPV is $1,721, and the proposed investment deserves consideration.
But if the minimum required rate is 15%, the PV of the cash flow is only
$8,624, the NPV is -$1,376, and the proposed investment should be rejected.
Copyright 2009, The National Underwriter Company
63
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
How to Compute Net Present Value
(Lump Sum Investment, Single Future Receipt)
• Given: At beginning of Year 1 invest $10,000, receive
$15,000 at end of year 5. The investor’s discount rate
is 6%.
• PV of $15,000 at 6% for 5 years is $11,210.
• $11,210 - $10,000 = $1,210
– Therefore, the investment should be considered.
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64
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
How to Compute Net Present Value
(Lump Sum Investment, Multiple Future Receipt)
• Given: $10,000 outlay at beginning of Year 1. Future
Receipts are:
Year
1
2
3
4
5
Total Receipts
Amount
Received
$2,000
1,500
750
500
$10,000
$14,750
PV
@ 6%
$1,887
1,335
630
396
$7,473
$11,721
• Net Present Value is positive $1,721, so investment
should be considered.
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65
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
How to Compute Net Present Value
(Multiple Investments, Multiple Future Receipts
• Given: Extend last problem so that outlays at beginning
of Year 1 of $5,000 and Year 2 of $5,000. Receipts are
still the same:
Year
1
2
3
4
5
Total Receipts
Amount
Received
$2,000
1,500
750
500
$10,000
$14,750
PV
@ 6%
$1,887
1,335
630
396
$7,473
$11,721
• PV of Outlays is $9,717. PV of Inflows is $11,721. NPV
is $2,004.
Copyright 2009, The National Underwriter Company
66
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Internal Rate of Return
• The Internal Rate of Return is the rate at which the NPV
of the inflows and the NPV of the outflows is equal.
• Determines what percentage rate of return cash inflows
will provide based on a known investment (cash outflow)
and estimated cash inflows.
• This is still a TVM calculation. Unlike the NPV
calculation, the discount rate is the variable that is being
sought, rather than the present value.
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67
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
How to Compute Internal Rate of Return
• Solving manually for the Internal Rate of Return is a
trial and error process. One computes the NPV using
the best guess of the IRR. If NPV is positive, then a
higher rate is tried. If NPV is negative, then a lower
rate is tried. Continue this process until the NPV is
roughly equal to $0; the result is the IRR.
• The simplest way to calculate is to use a financial
calculator or spreadsheet software, both of which do
the same iterative process, but much faster than one
can do manually.
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Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Shortcomings of the IRR Method
• Assumption that the cash flows are consumed and not
reinvested.
• Also cannot assume that the cash flows are reinvested
at the same rate.
• Despite these shortcomings, this is one of the most
widely used tools for evaluating investments.
Copyright 2009, The National Underwriter Company
69
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Consider 5 Possible Investments
Year
0
1
2
3
IRR
•
Project Cash Flows
A
B
C
D
E
($1,000) ($1,000) ($1,000) ($1,000) ($1,000)
100
50
(200)
200
600
100
50
(200)
200
600
1,100
1,215
1,793
869
(55)
10%
10%
10%
10%
10%
Each project has a 10% IRR. Yet each has different
unrecovered cash flows at a given point in time. The
interest rate at which the recovered cash flows could be
invested could make a great difference to the investor.
Copyright 2009, The National Underwriter Company
70
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Other Weaknesses of the IRR Method
• The investment with the highest IRR is not necessarily the
“best” investment among a mutually exclusive set.
• The unmodified IRR method does not consider realistic
reinvestment rates for positive cash flows or realistic
borrowing rates for negative cash flows over the holding
period.
• An investment project may have multiple IRRs.
• Solving for the IRR often requires a series of iterative
calculations to successively home in on the IRR.
• However, financial calculators and computer software
programs, have built-in functions that are adequate in most
cases.
Copyright 2009, The National Underwriter Company
71
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Modified Internal Rate of Return Methods
Determining the appropriate Rate of Return:
– Using a “safe” rate of return such as Treasury Bills.
– Using an interest rate available to the investor on another
investment.
– Using a rate at which money could be borrowed.
– The circumstances for each investment scenario need to be
analyzed to select the right rate to use.
Copyright 2009, The National Underwriter Company
72
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Pay Back Period
• Based on the concept that the sooner the original
investment is recovered, the better is the investment
proposal.
• However, this method may cause an investor to reject
a project with a much higher NPV that requires longer
to recover the original investment.
Copyright 2009, The National Underwriter Company
73
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Cash on Cash
• This method ignores time value of money and just
examines how much cash the investor recovers
annually.
• Can cause the investor to reject a project with a higher
NPV than one that returns money sooner.
Copyright 2009, The National Underwriter Company
74
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Risk, Probabilities, And Modeling
• In evaluating investments, there are numerous risks to
evaluate that are beyond the scope of this book. (See
Tools & Techniques of Investment Planning.)
• In addition, the financial planner must consider noninvestment risks, such as risk of dying and disability.
• Time Value of Money calculations are used in techniques
designed to evaluate and assess risks in investments.
• Monte Carlo simulation might attempt to take into effect
some or all these factors.
Copyright 2009, The National Underwriter Company
75
Time Value of Money and
Quantitative Analysis
Chapter 20
Tools & Techniques of
Financial Planning
Monte-Carlo Simulation
• Monte-Carlo simulation is the process of
assessing the likelihood of an expected
outcome.
• A computer program is used to randomly
choose returns for each period. The
program is run many times to achieve a
distribution of ending values. This process
can help access the probability of
achieving certain results in the future.
Copyright 2009, The National Underwriter Company
76