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Time Value of Money
Many financial decisions require comparisons of cash payments at
different dates
Example: 2 investments that require an initial investment of $100
Timing
After 1
After 2
After 3
After 4
year
years
years
years
Inv 1
$30
$30
$30
$30
Inv 2
$20
$20
$40
$60
If you should choose one of them, which would you choose?
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Compounding



Future Value: amount to which an investment will
grow after earning interest
Compounding: the process of accumulating interest
in an investment over time to earn more interest
Compound interest: Interest earned on both the
initial principal and the reinvested interest from prior
periods
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Future Value
FV of $100 in 2 years if k=10%
Time
0
1
2
principal
$100
$100
$110
Interest
$0
$10
$11
So $100 today  $121 in 2 years
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simple and compounded interest?
What is the difference between simple and
compounded interest?


Compound interest assumes accumulated
interest is reinvested (therefore, interest
earns interest).
Simple interest assumes interest is not
reinvested. Interest is earned each period on
the original principal only.
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Present Value and Discounting
Present Value: value today of a future cash flow
 PV is simply the reverse of future value
 PV works backward through time, while future value
goes forward through time
Discounting: finding present value of some future
amount
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Example
3 different ways to find future value of a single cash flow
Find FV of $100 in 2 years @ 10%
FV2= 100*(1+10%)2
= 100 FVIF2,10%
100 PV 10
i
2
n
FV
formula
table
financial calculator
in general FVn= PV (1+i)n
PV, FV formulas are based on this equation
4 variables given any 3, you can calculate the 4th
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solving for n
in how many years will $100 grow to $121 @ i= 10%
Formula way:
100*(1+10%)n =121
(1+10%)n =1.21
n ln(1+10%)=ln 1.21
Table way:
100 FVIFn,10% =121
FVIFn,10% =1.21
Refer to FVIF Table. Look down the 10% column to find 1.21.
Financial calculator way:
100 PV 10
i
121
FV
n
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Solving for i
At what rate of return will $100 grow to $121 in 2 years
Formula way:
100*(1+i)2 =121
(1+i)2 = 1.21
1+i = (1.21)1/2 =1.10
i = 0.10 = 10%
Table way:
100 FVIF2,i =121
FVIF2.i =1.21
Refer to FVIF Table. Look across 2 period row to find 1.21.
Financial calculator way:
100 PV 2
n
121
FV
i
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Present and future value of multiple cash flows
Calculate PV(FV) of each cash flow and add them up: e.g. i=10%
PV = 100/(1+10%)+ 300/(1+10%)2 + 400/(1+10%)3
formula way
PV = 100 PVIF1,10% + 300 PVIF2,10% + 400 PVIF3,10%
table way
10
i
0
CFi 100 CFi 300 CFi
financial calculator
400
CFi
NPV
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Valuing Level Cash Flows: Annuities and
Perpetuities
We often deal with situations where cash flows are same
throughout the problem. For example, a car loan, rent
payment etc.
An annuity is a level stream of cash flows for a fixed period of
time. Cash flow must be the same in each period.


Ordinary annuity: Payments are at the end of period
Annuity due: Payments are at the beginning of period
Unless stated otherwise, assume you deal with ordinary annuity
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Future value of an annuity
FVA3
= A (1+i)2 + A (1+i) + A
= A {(1+i)2 + (1+i) + 1}
= A FVIFA3,i%
formula way
A
financial calculator way
PMT
r
i
3
n
FV
table way
again given any 3, we can solve for the 4th
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Present value of an annuity
PVA3 = A/(1+i)3 + A/(1+i)2 + A/(1+i)
formula way
= A {1/(1+i)3 + 1/(1+i)2 + 1/(1+i)}
= A PVIFA3,i%
table way
A
PMT
r
i
3
n
PV
financial calculator way
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deriving the PVIFA3,i% formula
use sum of infinite geometric series formula:
asa one can show that
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deriving the PVIFA3,i% formula
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Perpetuities
A special case of an annuity is when the cash flows continue forever.
The most common application of perpetuities in finance is preferred stock

Preferred stock offers a fixed cash dividend every period (usually every
quarter) forever.

The dividend never increases in value, so it’s similar to a bond with a
fixed interest payment.
Present value of a perpetuity
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Comparing Interest Rates
How do you compare interest rates?
Rates can be quoted monthly, annually or something in between,
and it quickly becomes confusing to try and determine the “real”
interest rate.
Stated Rate ( also called APR, Quoted Rate, Nominal Rate): rate
before considering any compounding effects
e.g. 10% APR quarterly compounding
Periodic Rate: APR/(# of times compounding occurs in a year)
It is the effective or “real” rate. It considers the compounding
effects.
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Effective Annual Rate
Effective Annual Rate (EAR)
Rate on an annual basis that reflects all compounding
effects
EAR= (1+APR/n)n – 1
You can compare different interest rate quotations
by using EAR
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Note: in TVM problems
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
Timing of cash flows tells you what the
period is
Find and use the periodic rate that is
consistent with the period definition
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Loan Amortization: There are many different
kinds of loans available

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Pure discount loan
With such a loan, the borrower receives money today
and repays a single lump sum at some time in the
future.
Interest-only loans
This kind of loan repayment plan calls for the
borrower to pay interest each period and repay
the entire principal at some point in the future.
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different types of loans

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Amortized loans
With a pure discount or interest-only loan, the
principal is paid all in once. An alternative is an
amortized loan where the lender may require the
borrower to repay parts of the loan amount over
time. The process of paying off a loan by making
regular principal reductions is called amortizing the
loan.
Partially amortizing loan
Similar to amortized loan except the borrower makes
a single, much larger final payment called a “balloon”
to pay off the loan.
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Example
You get a $10,000 car loan. It is a five year amortized loan with
annual installments. 12% is the interest rate charged by the bank.
Develop the amortization schedule.
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