Download The Forms of Business Organization

Introduction to Present
Value
Text: Chapter 4
How much are you willing to pay for
a project with payoff of $40,000 in a
year?
Present Value
 If the project is riskless, then use return of government bond as a
guide, say 8%, how much are you willing to pay?
$40,000/1.08 = 37,037
Discount rate
[ $40,000 * (1/1.08) = $40,000 * 0.926]
 Present value of this project is $37,037
 Future value of this project is $40,000
Discount
factors
Present Value
PV = C1 * [1 / (1+r)]
= C1 * discount factor
where r is discount rate, opportunity cost of capital
[1/1+r] is discount factor
Future Value
Future Value = PV * (1+r)
The First Rule in Finance
One dollar today is worth more
than one dollar tomorrow
Present Value and Future Value
Present Value
 What if the project is risky?
 Suppose
the return is as risky as the stock index,
which offers 12% in the long run, what is the PV
of $40,000 next year?
$40,000 / (1.12) = $35714 < $37037
(8% discount rate)
The Second Rule in Finance
A safe dollar is worth more
than a risky one!
Introduction to Net Present Value
 If the riskless project is sold at $37,000, is this a
good deal?
Net Present Value = C0 + C1* (1/1+r)
NPV = -37000 + 37037 = 37
Accept any project with positive net present value!
NPV and Cost of Capital
 In the previous example, the rate of return
[40,000 / 37,000] - 1 = 8.1% > 8% (discount rate)
NPV > 0 <==> rate of return > opportunity cost of
capital
Opportunity Cost of Capital
Example
You may invest $100,000 today. Depending on the state of
the economy, you may get one of three possible payoffs:
Economy
Payoff
Slump
Normal
Boom
$80,000 110,000 140,000
80,000  110,000  140,000
Expected payoff  C1 
 $110,000
3
Opportunity Cost of Capital
Example - continued
 The stock is expected to have a 15% rate of return.
 Discounting the expected payoff at the expected return
leads to the PV of the project
110,000
PV 
 $95,650
1.15
Initial Investment: $100,000
Opportunity Cost of Capital
Example - continued
Notice that you come to the same conclusion if you
compare the expected project return with the cost of
capital.
expected profit 110,000  100,000
Expected return 

 .10 or 10%
investment
100,000
But ….Cost of capital: 15%
A simple question …..
 Should DF2 always be less than DF1?
 [ie, 1/(1+r2)2 > 1/(1+r1) ]?
 If not, say DF2 = .8, and DF1=.7,
then we can
1. Borrow $.8 and return $1 in period 2
2. Lend $.7 and get $1 back at period 1, then hold it to period 2

We end with $.1 without investing anything!

But if everybody does this, then
r2  (DF2 ) and r1  (DF1 )
until no arbitrage conditions exists any more
No money machine exists for a lasting time.
The Calculation of
Present Value
Present Values
 PVs can be added together to evaluate multiple cash
flows.
PV 
C1
(1 r )
 (1r )2 ....
C2
1
Present Value of Multiple Cash Flow
$200
$100
Present Value
Year
0
Year 0
100/1.07
= $93.46
200/1.0772
= $172.42
Total
= $265.88
PV 
100
(1.07)1
1
2
 (1200
2  265.88
077)
Short Cuts
 Sometimes there are shortcuts that make it very easy to calculate
the present value of an asset that pays off in different periods.
These tolls allow us to cut through the calculations quickly.
Perpetuity
 A Perpetuity is a constant stream of CF without end.
PVt = Ct+1 / r
0
1
2
3
…forever...
|---------|--------|---------|--------(r = 10%)
$100 $100 $100
...forever…
PV0 = $100 / 0.1 = $1000
Growing Perpetuity
 a stream of cash flows that grows at a constant rate forever.
 Simplification:
PVt = Ct+1 / (r - g)
0
1
2
3 …forever...
|---------|---------|---------|--------(r = 10%)
$100
$102 $104.04 … (g = 2%)
PV0 = $100 / (0.10 - 0.02) = $1250
Annuity
Annuity - An asset that pays a fixed sum each year for a
specified number of years.
0
1
2
3
|---------|---------|---------| (r = 10%)
$100
$100 $100
Short Cuts
Annuity - An asset that pays a fixed sum each year for a
specified number of years.
Asset
Perpetuity (first
payment in year 1)
Perpetuity (first payment
in year t + 1)
Annuity from year
1 to year t
Year of Payment
1
2…..t
t+1
Present Value
C
r
1
C 
 
t
 r  (1  r )
 C   C  1 

    
t 
 r   r  (1  r ) 
Short Cuts
Annuity - An asset that pays a fixed sum each year for a specified
number of years.
1
1 
PV of annuity  C   
t
 r r 1  r  
Annuity Short Cut
Example
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 opportunity cost of
capital is 0.5% per month, what is the cost of the lease?
Annuity Short Cut
Example - continued
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 opportunity cost of capital
is 0.5% per month, what is the cost of the lease?
 1

1
Lease Cost  300  

48 
 .005 .0051  .005 
Cost  $12,774.10
Compound Interest
i
ii
Periods Interest
per
per
year
period
iii
APR
(i x ii)
iv
Value
after
one year
v
Annually
compounded
interest rate
1
6%
6%
1.06
6.000%
2
3
6
1.032
= 1.0609
6.090
4
1.5
6
1.0154 = 1.06136
6.136
12
.5
6
1.00512 = 1.06168
6.168
52
.1154
6
1.00115452 = 1.06180
6.180
365
.0164
6
1.000164365 = 1.06183
6.183
18
16
14
12
10
8
6
4
2
0
10% Simple
Number of Years
30
27
24
21
18
15
12
9
6
10% Compound
3
0
FV of $1
Compound Interest
Effective Annual Interest Rates
 Compounding periods (m)

How often is interest computed
 Stated (nominal) Annual Percentage Rate (r)

What the bank usually quotes
 Effective Annual Interest Rate (EAR):

EAR = (1 + r / m) m - 1

As m approaches infinity, (1 + r / m) m -----> er

EAR of continuous compounding = er - 1
Inflation
Inflation - Rate at which prices as a whole are increasing.
Nominal Interest Rate - Rate at which money invested grows.
Real Interest Rate - Rate at which the purchasing power of an
investment increases.
Inflation
Example
If the interest rate on one year govt. bonds is 5.9%
and the inflation rate is 3.3%, what is the real
interest rate?
1 + real interest rate =
1+.059
1+.033
Savings
1 + real interest rate =
1.025
Bond
real interest rate
=
.025 or 2.5%
Approximation =.059-.033 =.026 or 2.6%