Professor Thomas Chemmanur
MF 807
TOPIC 1 – FINANCIAL THEORY AND CORPORATE FINANCE
Invest Money
Firm
Projects
Financial Instruments
(Capital Structure Choice)
Cash
Cash flows
Cash from Securities
Financial Market
Investors
Funds
(Investors’ Portfolio Choice)
Figure 1: Capital Budgeting
Forms of organization:
1. Sole Proprietorship
2. Partnership
Private
3. Corporation
Public
Differences along several dimensions:
1. Limitation of liability
2. Taxes
3. Regulation, especially with respect to disclosure
4. Ease of raising external financing
5. Transfer of ownership (ease of transfer)
Objective of the firm manager:
•
Profit maximization is not an appropriate objective: Profits in what period?
•
How much risk should the firm take to maximize profits?
•
Correct answer in a market where frictions are small => Shareholder Wealth
Maximization
•
The above is equivalent to maximizing the Net Present Value of the Firm’s
projects (If no restrictions on capital, undertake all positive NPV projects).
•
“Fisher Separation Theorem”
Net Present Value (NPV)
NPV = PV of Benefits (Cash-out flows) from a project
PV of Investment in the project
If NPV > 0, the project, if implemented, will increase shareholder wealth (Firm Value)
Review of Present Value Calculations
1. F = P(1 + r ) n
Future value of an amount P collecting interest at a rate r for n periods
FVIFn ,r ≡ Future Value Interest Factor = (1 + r ) n
Example:
Future value of $1000 after 6 years, r=10% (Annual Compounding) =
1000[1 + 0.1]6 = 1772
Alternatively:
= 1000[ FVIF6,10% ] =1000[1.7716] ≈ 1772
2. Present value of a single amount, F payable n years from now: P =
F
(1 + r ) n
⎡ 1 ⎤
Present value interest factor PVIFn , r is tabulated ⎢
.
n ⎥
⎣ (1 + r ) ⎦
Example:
Present value of $1000 receivable after 10 years, if the interest rate is 12% is:
P=
1000
= $321
(1 + 0.12)10
Alternatively:
P = 1000* PVIF10 yrs ,12% = 1000*0.3220 = $322
3. Future value of an annuity of $A at the end of each year:
t=0
t=1
t=2
A
Period 1
t=3
A
Period 2
A
Period 3
For a three year annuity,
F = A + A(1 + r ) + A(1 + r ) 2
For an “n” year annuity,
F = A + A(1 + r ) + ... + A(1 + r ) n −1
Using the sum of a geometric series formula,
⎡ (1 + r ) n − 1 ⎤
F = A⎢
⎥
r
⎣
⎦
The term in square brackets is tabulated ⇒ FVIFn ,r .
Example:
If a person saves $1000 per year for 10 years, and deposits the money at the end of each
year in a bank which pays 8%interest, he will have, after 10 years
⎡ (1 + 0.08)10 − 1 ⎤
F = 1000 ⎢
⎥ = 14, 487
0.08
⎣
⎦
n=10 years; r=0.08; A=$1000
Alternatively,
FVIFA10 yrs ,8% = 14.487
F = 1000*14.487 = 14, 487
4. Present value of an annuity
P=
A
A
A
+
+ ... +
for a three year annuity
2
(1 + r ) (1 + r )
(1 + r )3
P=
A
A
A
+
+ ... +
for an n year annuity
2
(1 + r ) (1 + r )
(1 + r ) n
Using the sum of a geometric series formula,
1 ⎤
⎡
⎢1 − (1 + r ) n ⎥
P = A⎢
⎥
r
⎢
⎥
⎢⎣
⎥⎦
The term in square brackets is tabulated ⇒ FVIFn ,r .
⎡1 ⎤
If n → ∞, P = A ⎢ ⎥ , where an infinite annuity is called a “Perpetuity.”
⎣r ⎦
Example:
Find the present value of the following ordinary annuity:
$200 per year for 10 years at 10%
A = $200
n=10 years; r=10%
P = 200* PVIFA10,10%
1
⎡
⎢1 − 1.110
= 200 ⎢
⎢ 0.1
⎣
⎤
⎥
⎥ = 200*6.144 = $1228.92
⎥
⎦
Example:
The present value of $200 per year forever with r=10% annually is:
P=
200
= $2000.00 (A=200; r=0.1)
0.1
5.Compounding and Discounting more than once a year
If compounding is done “m” times a year, at a nominal interest rate, R per year, for T
years:
n= number of periods = mT
r= interest rate (effective) per period = R/m
⎡ R⎤
Future value, F = P ⎢1 + ⎥
⎣ m⎦
mT
Present value of an amount F received T years from now,
P=
F
⎡ R⎤
⎢⎣1 + m ⎥⎦
mT
Example:
How much does a deposit of $5000 grow at the end of 6 years if nominal rate of interest
is 12% and compounding frequently is 4 times per year?
R=0.12; r =
0.12
= 0.03
4
n = 6* 4 = 24 , number of periods
F = 5000[1 + 0.03]24
6. Continuous compounding and discounting
We know that,
⎡ R⎤
F = P ⎢1 + ⎥
⎣ m⎦
mT
m
⎡ R⎤R
When m → ∞, ⎢1 + ⎥ = 2.718
⎣ m⎦
RT
Thus, F = Pe as m → ∞ .
(Future value under continuous compounding)
Conversely, the present value of an amount F received T years from now an annual
interest rate (nominal) R, if discounting is done continuously is:
P=
F
= Fe − RT
RT
e
Example:
Under continuous compounding $5000 grows to (R=12%), after 6 years:
F = Pe RT = 5000e0.12(6)
Example:
If someone offers to “double” your money in 6 years, what is the interest rate you are
getting (under annual compounding)?
$1(1 + r )6 = $2
1/ 6
⎛2⎞
(1 + r ) = ⎜ ⎟
⎝1⎠
= 1.122
r = 1.122 − 1 = 0.122 = 12.2%
7. Present value of a growing annuity:
⎡ ⎛ 1 + g ⎞n ⎤
A ⎢1 − ⎜
⎟ ⎥
⎢⎣ ⎝ 1 + r ⎠ ⎥⎦
P=
(r − g )
g = annual growth rate
A is the first payment, second payment is A(1+g), third is (1 + g ) 2 … when g = 0, we get:
⎡ ⎛ 1 ⎞n ⎤
A ⎢1 − ⎜
⎟ ⎥
⎢⎣ ⎝ 1 + r ⎠ ⎥⎦
P=
which is the standard PV of an annuity formula.
r