Time Value of Money (CH 4)
Future value
Present value
Annuities
Interest rates

TIP
If you do not understand
something,
ask me!
2-11
Last week




Objective of the firm
Business forms
Agency conflicts
Capital budgeting decision and capital
structure decision
2-2
The plan of the lecture

Time value of money concepts



Formulae for calculating PV of




present value (PV)
discount rate/interest rate (r)
perpetuity
annuity
Interest compounding
How to use a financial calculator
2-3
Financial choices

Which would you rather receive today?




TRL 1,000,000,000 ( one billion Turkish
lira )
USD 652.72 ( U.S. dollars )
Both payments are absolutely
guaranteed.
What do we do?
2-4
Financial choices


We need to compare “apples to apples”
- this means we need to get the
TRL:USD exchange rate
From www.bloomberg.com we can see:


USD 1 = TRL 1,637,600
Therefore TRL 1bn = USD 610.64
2-5
Financial choices with time

Which would you rather receive?



$1000 today
$1200 in one year
Both payments have no risk, that is,

there is 100% probability that you will be
paid
2-6
Financial choices with time

Why is it hard to compare ?





$1000 today
$1200 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


It has a money unit such as USD or TRL
It has a date indicating when to receive money
2-7
Present value




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 TRL to
money in USD
2-8
Some terms


Finding the present value of some
future cash flows is called
discounting.
Finding the future value of some
current cash flows is called
compounding.
2-9
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.
0
1
2
3
10%
100
FV = ?
2-10
Solving for FV:
The arithmetic method




After 1 year:
 FV1 = c ( 1 + i ) = $100 (1.10)
= $110.00
After 2 years:
2
 FV2 = c (1+i)(1+i) = $100 (1.10)
=$121.00
After 3 years:
3
3
 FV3 = c ( 1 + i ) = $100 (1.10)
=$133.10
After n years (general case):
n
 FVn = C ( 1 + i )
2-11
Set up the Texas instrument






2nd, “FORMAT”, set “DEC=9”, ENTER
2nd, “FORMAT”, move “↓” several times,
make sure you see “AOS”, not “Chn”.
2nd, “P/Y”, set to “P/Y=1”
2nd, “BGN”, set to “END”
P/Y=periods per year,
END=cashflow happens end of periods
2-12
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
10
-100
0
N
I/YR
PV
PMT
FV
133.10
2-13
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.
0
1
2
3
10%
PV = ?
100
2-14
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
2-15
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
10
N
I/YR
PV
0
100
PMT
FV
-75.13
2-16
Solving for N:
If your investment earns interest of 20% per
year, how long before your investments double?
INPUTS
N
OUTPUT
20
-1
0
2
I/YR
PV
PMT
FV
3.8
2-17
Solving for i:
What interest rate would cause $100 to
grow to $125.97 in 3 years?
INPUTS
3
N
OUTPUT
I/YR
-100
0
125.97
PV
PMT
FV
8
2-18
Now let’s study some interesting
patterns of cash flows…


Perpetuity
Annuity
2-19
ordinary annuity and annuity due
Ordinary Annuity
0
i%
1
2
3
PMT
PMT
PMT
1
2
3
PMT
PMT
Annuity Due
0
i%
PMT
2-20
Value an ordinary annuity



Here C is each cash payment
n is number of payments
If you’d like to know how to get the
formula below, see me after class.
1
1 
PV  C  
n 
 i (1  i ) i 
2-21
Example

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
Using the formula
 1

1
PV  50,000 * 


 0.07 1.07 20 * 0.07 
 $529,700 .71
2-23
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
10
0
-100
N
I/YR
PV
PMT
FV
331
2-24
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
10
N
I/YR
PV
100
0
PMT
FV
-248.69
2-25
Solving for FV:
3-year annuity due of $100 at 10%



$100 payments occur at the beginning of
each period.
FVAdue= FVAord(1+i) = $331(1.10) = $364.10.
Alternatively, set calculator to “BEGIN” mode
and solve for the FV of the annuity:
BEGIN
INPUTS
OUTPUT
3
10
0
-100
N
I/YR
PV
PMT
FV
364.10
2-26
Solving for PV:
3-year annuity due of $100 at 10%



$100 payments occur at the beginning of each
period.
PVAdue= PVAord(1+I) = $248.69(1.10) = $273.55.
Alternatively, set calculator to “BEGIN” mode and
solve for the PV of the annuity:
BEGIN
INPUTS
OUTPUT
3
10
N
I/YR
PV
100
0
PMT
FV
-273.55
2-27
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
2-28
What if it were a 10-year annuity? A
25-year annuity? A perpetuity?

10-year annuity


25-year annuity


N = 10, I/YR = 10, PMT = 100, FV = 0;
solve for PV = $614.46.
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.
2-29
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%?
PV1 
$297.22
0
$323.97
1
$100
$100
$100
$100



 $323.97
1
2
3
4
(1.09) (1.09) (1.09) (1.09)
$100
2
$327 .97
PV 
 $297 .22
0
1.09
$100
3
$100
4
$100
5
2-30
What is the PV of this uneven
cash flow stream?
0
1
2
3
4
100
300
300
-50
10%
90.91
247.93
225.39
-34.15
530.08 = PV
2-31
Solving for PV:
Uneven cash flow stream

Input cash flows in the calculator’s “CF”
register:






CF0
CF1
CF2
CF3
CF4
=
=
=
=
=
0
100
300
300
-50
Enter I/YR = 10, press NPV button to get
NPV = $530.09. (Here NPV = PV.)
2-32
Detailed steps (Texas
Instrument calculator)
To clear historical data:
 CF, 2nd ,CE/C
 To get PV:
 CF , ↓,100 , Enter , ↓,↓ ,300 , Enter, ↓,2,
Enter, ↓, 50, +/-,Enter, ↓,NPV,10,Enter,
↓,CPT
 “NPV=530.0867427”

2-33
The Power of Compound
Interest
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?
2-34
Solving for FV:
Savings problem

If she begins saving today, and sticks to
her plan, she will have $1,487,261.89
when she is 65.
INPUTS
OUTPUT
45
12
0
-1095
N
I/YR
PV
PMT
FV
1,487,262
2-35
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
12
0
-1095
N
I/YR
PV
PMT
FV
146,001
2-36
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.
10%
1
2
3
100
133.10
Annually: FV3 = $100(1.10)3 = $133.10
0
0
100
5%
1
1
2
3
2
4
5
Semiannually: FV6 = $100(1.05)6 = $134.01
3
6
134.01
2-37
What is the FV of $100 after 3 years under
10% semiannual compounding? Quarterly
compounding?
FV3S
FV3S
0.10 23
 $100 ( 1 
)
2
 $100 (1.05) 6  $134.01
FV3Q  $100 (1.025)12  $134.49
2-38
Classifications of interest rates

1. Nominal rate (iNOM) – 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 (iPER) – amount of interest
charged each period, e.g. monthly or quarterly.

iPER = iNOM / m, where m is the number of
compounding periods per year. e.g., m = 12 for
monthly compounding.
2-39
Classifications of interest rates


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 m
times in a year, the effective annual
interest rate is
m
 inom 
1 
 1
m 

2-40
Example, EAR for 10%
semiannual investment


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.
2-41
EAR on a Financial Calculator
Texas Instruments BAII Plus
keys:
description:
[2nd] [ICONV]
Opens interest rate conversion menu
Sets 2 payments per year
[↑] [C/Y=] 2 [ENTER]
Sets 10 APR.
[↓][NOM=] 10 [ENTER]
[↓] [EFF=] [CPT]
10.25
2-42
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 iNOM=10%, then EAR for different
compounding frequency:
Annual
Quarterly
Monthly
Daily
10.00%
10.38%
10.47%
10.52%
2-43
If interest is compounded more
than once a year

EAR (EFF, APY) will be greater than
the nominal rate (APR).
2-44
2-45
2-46
What’s the FV of a 3-year $100
annuity, if the quoted interest rate is
10%, compounded semiannually?
0
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.
2-47
Method 1:
Compound each cash flow
0
1
1
2
3
2
4
5
3
6
5%
100
100
100
110.25
121.55
331.80
FV3 = $100(1.05)4 + $100(1.05)2 + $100
FV3 = $331.80
2-48
Method 2:
Financial calculator


Find the EAR and treat as an annuity.
EAR = ( 1 + 0.10 / 2 )2 – 1 = 10.25%.
INPUTS
OUTPUT
3
10.25
0
-100
N
I/YR
PV
PMT
FV
331.80
2-49
When is periodic rate used?

iPER is often useful if cash flows occur
several times in a year.
2-50
Exercise
You agree to lease a car for 4 years at
$300 per month. You are not required to
pay any money up front or at the end of
your agreement. If your discount rate is
0.5% per month, what is the cost of the
lease?
2-51
Solution
 1

1
Lease Cost  300  

48 
 .005 .0051  .005 
Cost  $12,774.10
2-52