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Future value
Present value
Annuities
Rates of return
Amortization
2-1

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Objective of the firm
Business forms
Agency conflicts
Capital budgeting decision and capital structure decision
2-2

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Time value of money concepts

 present value (PV) discount rate/interest rate (r)
Formulae for calculating PV of

 perpetuity annuity
Interest compounding
How to use a financial calculator
2-3



Which would you rather receive?

$1000 today

$1040 in one year

Both payments have no risk, that is, there is 100% probability that you will be paid
2-4

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
Why is it hard to compare ?


$1000 today
$1040 in one year
This is not an “apples to apples” comparison. They have different units
$1000 today is different from $1000 in one year
Why?

A cash flow is time-dated money
2-5

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To have an “apple to apple” comparison, we convert future payments to the present values or convert present payments to the future values
This is like converting money in Canadian $ to money in US $.
2-6

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
Finding the present value of some future cash flows is called
discounting.
Finding the future value of some current cash flows is called
compounding.
2-7

0
What is the future value (FV) of an initial $100 after 3 years, if i = 10%?


Finding the FV of a cash flow or series of cash flows is called compounding.
FV can be solved by using the arithmetic,
financial calculator, and spreadsheet methods.
1 2 3
10%
100 FV = ?
2-8

Solving for FV:
The arithmetic method




After 1 year:

FV
1
= c ( 1 + i ) = $100 (1.10)
= $110.00
After 2 years:

FV
2
= c (1+i)(1+i) = $100 (1.10)
=$121.00
2
After 3 years:

FV
3
= c ( 1 + i )
=$133.10
3 = $100 (1.10) 3
After n years (general case):

FV n
= C ( 1 + i ) n
2-9

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2 nd , “FORMAT”, set “DEC=9”, ENTER
2 nd , “FORMAT”, move “ ↓ ” several times, make sure you see “AOS”, not “Chn”.
2 nd , “P/Y”, set to “P/Y=1”
2 nd , “BGN”, set to “END”
P/Y=periods per year,
END=cashflow happens end of periods
2-10

Solving for FV:
The calculator method


Solves the general FV equation.
Requires 4 inputs into calculator, and it will solve for the fifth.
INPUTS
OUTPUT
3
N
10
I/YR
-100
PV
0
PMT FV
133.10
2-11

What is the present value (PV) of
$100 received in 3 years, if i = 10%?


Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding).
The PV shows the value of cash flows in terms of today’s worth.
1 2 3 0
PV = ?
10%
100
2-12

Solving for PV:
The arithmetic method
 i: interest rate, or discount rate

PV = C / ( 1 + i ) n

PV = C / ( 1 + i ) 3
= $100 / ( 1.10 ) 3
= $75.13
2-13

Solving for PV:
The calculator method

Exactly like solving for FV, except we have different input information and are solving for a different variable.
INPUTS
OUTPUT
3
N
10
I/YR PV
-75.13
0
PMT
100
FV
2-14

Solving for N:
If your investment earns interest of 20% per year, how long before your investments double?
INPUTS
OUTPUT
N
3.8
20
I/YR
-1
PV
0
PMT
2
FV
2-15

Solving for i:
What interest rate would cause $100 to grow to $125.97 in 3 years?
INPUTS
OUTPUT
3
N I/YR
8
-100
PV
0
PMT
125.97
FV
2-16

Now let’s study some interesting patterns of cash flows…
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Annuity
Perpetuity
2-17

ordinary annuity and annuity due
Ordinary Annuity
0 1 i%
PMT
Annuity Due
0 i%
1
2
PMT
2
3
PMT
3
PMT PMT PMT
2-18




Here C is each cash payment n is number of payments
If you’d like to know how to get the formula below (not required), see me after class.
PV

C

 i
1

(1

1
) n


2-19

Solving for FV:
3-year ordinary annuity of $100 at 10%

$100 payments occur at the end of each period. Note that PV is set to 0 when you try to get FV.
INPUTS
OUTPUT
3
N
10
I/YR
0
PV
-100
PMT FV
331
2-20

Solving for PV:
3-year ordinary annuity of $100 at 10%

$100 payments still occur at the end of each period. FV is now set to 0.
INPUTS
OUTPUT
3
N
10
I/YR PV
-248.69
100
PMT
0
FV
2-21

 you win the $1million dollar lottery! but wait, you will actually get paid $50,000 per year for the next 20 years if the discount rate is a constant 7% and the first payment will be in one year, how much have you actually won?
2-22

Solving for FV:
3-year annuity due of $100 at 10%



$100 payments occur at the beginning of each period.
FVA due
= FVA ord
(1+i) = $331(1.10) = $364.10.
Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity:
BEGIN
INPUTS
3 10 0 -100
N I/YR PV PMT FV
OUTPUT
364.10
2-23

Solving for PV:
3-year annuity due of $100 at 10%



$100 payments occur at the beginning of each period.
PVA due
= PVA ord
(1+I) = $248.69(1.10) = $273.55.
Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity:
BEGIN
INPUTS
3 10 100 0
N I/YR PV PMT FV
OUTPUT
-273.55
2-24

What is the present value of a 5-year
$100 ordinary annuity at 10%?

Be sure your financial calculator is set back to END mode and solve for PV:


N = 5, I/YR = 10, PMT = 100, FV = 0.
PV = $379.08
2-25

What if it were a 10-year annuity? A
25-year annuity? A perpetuity?


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10-year annuity

N = 10, I/YR = 10, PMT = 100, FV = 0; solve for PV = $614.46.
25-year annuity

N = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = $907.70.
Perpetuity (N=infinite)

PV = PMT / i = $100/0.1 = $1,000.
2-26

What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?
PV
1

$100

$100

$100

$100
1 2 3
(1.09) (1.09) (1.09) (1.09)
4

$323.97
$297.22
$323.97
$100 $100 $100 $100
0
PV
0

$ 323
1
.
97
1 .
09

$ 297 .
22
2 3 4 5
2-27

What is the PV of this uneven cash flow stream?
0
10%
1
100
90.91
247.93
225.39
-34.15
530.08 = PV
2
300
3
300
4
-50
2-28

Solving for PV:
Uneven cash flow stream

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Input cash flows in the calculator’s “CF” register:

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CF
0
CF
1
CF
2
CF
3
CF
4
= 0
= 100
= 300
= 300
= -50
Enter I/YR = 10, press NPV button to get
NPV = $530.09. (Here NPV = PV.)
2-29



To clear historical data:
CF
,
2 nd
,
CE/C


To get PV:
CF
, ↓,100 ,
Enter , ↓,↓ ,300 , Enter, ↓,2,
Enter , ↓, 50, +/-, Enter , ↓, NPV
,10,
Enter
,
↓,CPT

“NPV=530.0867427”
2-30

A 20-year-old student wants to start saving for retirement. She plans to save $3 a day. Every day, she puts $3 in her drawer. At the end of the year, she invests the accumulated savings
($1,095=$3*365) in an online stock account.
The stock account has an expected annual return of 12%.
How much money will she have when she is 65 years old?
2-31


If she begins saving today, and sticks to her plan, she will have $1,487,261.89 when she is 65.
INPUTS
OUTPUT
45
N
12
I/YR
0
PV
-1095
PMT FV
1,487,262
2-32

Solving for FV:
Savings problem, if you wait until you are
40 years old to start


If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146,000.59 at age 65. This is $1.3 million less than if starting at age 20.
Lesson: It pays to start saving early.
INPUTS
OUTPUT
25
N
12
I/YR
0
PV
-1095
PMT FV
146,001
2-33

Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated i% constant?

0
LARGER, as the more frequently compounding occurs, interest is earned on interest more often.
1 2 3
10%
0
0
100
Annually: FV
3
= $100(1.10) 3 = $133.10
5%
1
1
2 3
2
4 5
133.10
3
6
100
Semiannually: FV
6
= $100(1.05) 6 = $134.01
134.01
2-34

What is the FV of $100 after 3 years under
10% semiannual compounding? Quarterly compounding?
FV
3S
 
0.10
)
2
FV
3S
 6
$100 (1.05)

$134.01
FV
3Q
 12
$100 (1.025)

$134.49
2-35



1. Nominal rate (i
NOM
) – also called the APR, quoted rate, or stated rate . An annual rate that ignores compounding effects. Periods must also be given, e.g. 8% Quarterly.
2. Periodic rate (i
PER
) – amount of interest charged each period, e.g. monthly or quarterly.
 i
PER
= i
NOM
/ m, where m is the number of compounding periods per year. e.g., m = 12 for monthly compounding.
2-36

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3. Effective (or equivalent) annual rate
( EAR, also called EFF, APY ) : the annual rate of interest actually being earned, taking into account compounding.
If the interest rate is compounded
interest rate is m times in a year, the effective annual



1
 i nom m


 m

1
2-37


EAR= ( 1 + 0.10 / 2 ) 2 – 1 = 10.25%

An investor would be indifferent between an investment offering a
10.25% annual return, and one offering a 10% return compounded semiannually.
2-38

Texas Instruments BAII Plus keys:
[↓][NOM=] 10 [ ENTER]
[↓] [EFF=] [CPT] 10.25
description:
[2nd] [ICONV] Opens interest rate conversion menu
[↑] [C/Y=] 2
[ENTER] Sets 2 payments per year
Sets 10 APR .
2-39

Why is it important to consider effective rates of return?



An investment with monthly payments is different from one with quarterly payments.
Must use EAR for comparisons.
If i
NOM
=10%, then EAR for different compounding frequency:
Annual
Quarterly
Monthly
Daily
10.00%
10.38%
10.47%
10.52%
2-40

If interest is compounded more than once a year

EAR (EFF, APY) will be greater than the nominal rate (APR).
2-41

2-42

2-43

0
What’s the FV of a 3-year $100 annuity, if the quoted interest rate is
10%, compounded semiannually?
1
1
2 3
2
4 5
3
6
5%
100 100 100


Payments occur annually, but compounding occurs every 6 months.
Cannot use normal annuity valuation techniques.
2-44

0
5%
1
1
2 3
2
4 5
3
6
100 100 100
110.25
121.55
331.80
FV
3
FV
3
= $100(1.05) 4 + $100(1.05) 2 + $100
= $331.80
2-45



Find the EAR and treat as an annuity.
EAR = ( 1 + 0.10 / 2 ) 2 – 1 = 10.25%.
INPUTS
OUTPUT
3
N
10.25
I/YR
0
PV
-100
PMT FV
331.80
2-46

 i
PER is often useful if cash flows occur several times in a year.
2-47