Chapter 3
Understanding The Time Value of
Money
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Time Value of Money
A dollar received
today is worth more
than a dollar
received in the
future.
 The sooner your
money can earn
interest, the faster
the interest can earn
interest.

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Interest and Compound
Interest
 Interest
-- is the return you receive for
investing your money.
 Compound interest -- is the interest that
your investment earns on the interest
that your investment previously earned.
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Future Value Equation

FVn = PV(1 + i)n
–
–
–

FV = the future value of the investment at the end
of n year
i = the annual interest (or discount) rate
PV = the present value, in today’s dollars, of a
sum of money
This equation is used to determine the value
of an investment at some point in the future.
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Compounding Period
 Definition
-- is the frequency that
interest is applied to the investment
 Examples -- daily, monthly, or annually
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Reinvesting -- How to Earn
Interest on Interest
 Future-value
interest factor (FVIFi,n) is a
value used as a multiplier to calculate
an amount’s future value, and
substitutes for the (1 + i)n part of the
equation.
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The Future Value of a
Wedding
In 1998 the average
wedding cost $19,104.
Assuming 4% inflation,
what will it cost in 2028?
FVn
FVn
FV30
FV30
FV30
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= PV (FVIFi,n)
= PV (1 + i)n
= PV (1 + 0.04)30
= $19,104 (3.243)
= $61,954.27
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The Rule of 72
 Estimates
how many years an
investment will take to double in value
 Number of years to double =
72 / annual compound growth rate
 Example -- 72 / 8 = 9 therefore, it will
take 9 years for an investment to
double in value if it earns 8% annually
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Compound Interest With
Nonannual Periods
The length of the
compounding period
and the effective
annual interest rate
are inversely related;
therefore, the shorter
the compounding
period, the quicker the
investment grows.
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Compound Interest With
Nonannual Periods (cont’d)
 Effective
annual interest rate =
amount of annual interest earned
amount of money invested
 Examples
-- daily, weekly, monthly, and
semi-annually
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The Time Value of a Financial
Calculator

The TI BAII Plus financial calculator keys
–
–
–
–
–
–
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N = stores the total number of payments
I/Y = stores the interest or discount rate
PV = stores the present value
FV = stores the future value
PMT = stores the dollar amount of each annuity
payment
CPT = is the compute key
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The Time Value of a Financial
Calculator (cont’d)
 Step
1 -- input the values of the known
variables.
 Step 2 -- calculate the value of the
remaining unknown variable.
 Note: be sure to set your calculator to
“end of year” and “one payment per
year” modes unless otherwise directed.
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Tables Versus Calculator

REMEMBER -- The
tables have a
discrepancy due to
rounding error;
therefore, the
calculator is more
accurate.
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Compounding and the Power
of Time
 In
the long run, money saved now is
much more valuable than money saved
later.
 Don’t ignore the bottom line, but also
consider the average annual return.
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The Power of Time in
Compounding Over 35 Years

$200,000
$150,000
,4
98
$1
46
,2
12
$1
$100,000
22

$50,000

$0
Selma
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Patty
Selma contributed
$2,000 per year in
years 1 – 10, or 10
years.
Patty contributed
$2,000 per year in
years 11 – 35, or
25 years.
Both earned 8%
average annual
return.
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The Importance of the Interest
Rate in Compounding
 From
1926-1998 the compound growth
rate of stocks was approximately
11.2%, whereas long-term corporate
bonds only returned 5.8%.
 The “Daily Double” -- states that you
are earning a 100% return compounded
on a daily basis.
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Present Value
 Is
also know as the discount rate, or the
interest rate used to bring future dollars
back to the present.
 Present-value interest factor (PVIFi,n) is
a value used to calculate the present
value of a given amount.
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Present Value Equation

PV = FVn (PVIFi,n)
–
–
–

PV = the present value, in today’s dollars, of a
sum of money
FVn = the future value of the investment at the end
of n years
PVIFi,n = the present value interest factor
This equation is used to determine today’s
value of some future sum of money.
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Calculating Present Value for
the “Prodigal Son”
If promised $500,000 in 40
years, assuming 6% interest,
what is the value today?
PV = FVn (PVIFi,n)
PV = $500,000 (PVIF6%, 40 yr)
PV = $500,000 (.097)
PV = $48,500
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Annuities
 Definition
-- a series of equal dollar
payments coming at the end of a
certain time period for a specified
number of time periods.
 Examples -- life insurance benefits,
lottery payments, retirement payments.
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Compound Annuities
Definition -- depositing an equal sum of
money at the end of each time period for a
certain number of periods and allowing the
money to grow
 Example -- saving $50 a month to buy a new
stereo two years in the future

–
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By allowing the money to gain interest and
compound interest, the first $50, at the end of two
years is worth $50 (1 + 0.08)2 = $58.32
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Future Value of an Annuity
Equation
 FVn
–
–
–
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= PMT (FVIFAi,n)
FVn = the future value, in today’s dollars, of
a sum of money
PMT = the payment made at the end of
each time period
FVIFAi,n = the future-value interest factor
for an annuity
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Future Value of an Annuity
Equation (cont’d)
 This
equation is used to determine the
future value of a stream of payments
invested in the present, such as the
value of your 401(k) contributions.
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Calculating the Future Value
of an Annuity: An IRA
Assuming $2000 annual
contributions with 9% return, how
much will an IRA be worth in 30
years?
FVn
FV30
FV30
FV30
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= PMT (FVIFA i, n)
= $2000 (FVIFA 9%,30 yr)
= $2000 (136.305)
= $272,610
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Present Value of an Annuity
Equation
 PVn
–
–
–
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= PMT (PVIFAi,n)
PVn = the present value, in today’s dollars,
of a sum of money
PMT = the payment to be made at the end
of each time period
PVIFAi,n = the present-value interest factor
for an annuity
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Present Value of an Annuity
Equation (cont’d)
 This
equation is used to determine the
present value of a future stream of
payments, such as your pension fund or
insurance benefits.
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Calculating Present Value of
an Annuity: Now or Wait?
What is the present value of
the 25 annual payments of
$50,000 offered to the soonto-be ex-wife, assuming a 5%
discount rate?
PV = PMT (PVIFA i,n)
PV = $50,000 (PVIFA 5%, 25)
PV = $50,000 (14.094)
PV = $704,700
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Amortized Loans
Definition -- loans that are repaid in equal
periodic installments
 With an amortized loan the interest payment
declines as your outstanding principal
declines; therefore, with each payment you
will be paying an increasing amount towards
the principal of the loan.
 Examples -- car loans or home mortgages

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Buying a Car With Four Easy
Annual Installments
What are the annual payments to
repay $6,000 at 15% interest?
PV
= PMT(PVIFA i%,n yr)
$6,000 = PMT (PVIFA 15%, 4 yr)
$6,000 = PMT (2.855)
$2,101.58 = PMT
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Perpetuities
– an annuity that lasts forever
 PV = PP / i
 Definition
–
–
–
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PV = the present value of the perpetuity
PP = the annual dollar amount provided by
the perpetuity
i = the annual interest (or discount) rate
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Summary
value – the value, in the future,
of a current investment
 Rule of 72 – estimates how long your
investment will take to double at a given
rate of return
 Present value – today’s value of an
investment received in the future
 Future
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Summary (cont’d)
– a periodic series of equal
payments for a specific length of time
 Future value of an annuity – the value,
in the future, of a current stream of
investments
 Present value of an annuity – today’s
value of a stream of investments
received in the future
 Annuity
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Summary (cont’d)
loans – loans paid in equal
periodic installments for a specific
length of time
 Perpetuities – annuities that continue
forever
 Amortized
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