Basics of valuation
Value = Sum of discounted cash flows
Future cash flows have lower value. Discount rate R
= time value of money
Present value of a stream of cash flows:
PV0 = Σt CFt/(1+R)t
Perpetuity: Ct = C, t = 1, 2, 3, …
PV0 = C/R
Growing perpetuity (with const rate g): Ct = (1+g)t-1C
PV0 = C/(R - g)
Stocks with dividends growing with const rate g
PV0 = Div1/(R-g)
Annuity: Ct = C, t = 1,…,T
PV0 = (C/R) [1 – 1/(1+R)T ]
Growing annuity (with const rate g)
PV0 = (C/(R-g)) [1 – (1+g)T/(1+R)T]
Bond with coupon C and face value F (at T)
P0 = (C/R) [1 – 1/(1+R)T] + FT/(1+R)T
We will use DCF to evaluate projects (together
with other methods), mainly NPV = Σt CFt/(1+R)t
(where CF can be outflows too)
Two big issues:


How to compute cash flows?
What is the proper discount rate?
Investment decision rules
General criteria for investment analysis:
It should focus on cash flows rather than
accounting earnings
It should place higher weight on earlier cash
flows
It should penalize the expected cash flows
from riskier projects more heavily
For the time being we will abstract from the issues
of how to account for risk and uncertainty –
just assume a given discount rate
Discounted cash flow techniques


Net Present Value (NPV) criterion
Internal Rate of Return (IRR) criterion
Nondiscounted cash flow techniques


Payback Period (can be discounted)
Accounting Rate of Return
How to deal with:



Mutually exclusive projects
Capital rationing
Projects with unequal lives
NPV Rule
T
CFt
NPV  
 I0
t
t 1 (1  R )
Under no resource constraints, no mutual exclusive
projects accept the project if NPV > 0 and reject if NPV <
0.
When two projects are mutually exclusive and both have
NPV > 0, accept the project with the higher NPV.
Under resource constraints choose the combination of
projects such that NPV is max, s.t. to the constraints.
R is the opportunity cost of capital (or required return)
Value additivity property of NPV rule
Definition: Projects A and B are independent if they don’t
affect each other’s cash flows
Definition (Value Additivity): We will say that an
investment rule satisfies value additivity if the following
holds:
if C is independent of A and B, then
A is preferred to B  A + C is preferred to B + C
If X and Y are independent NPV(X+Y) = NPV(X)+NPV(Y)
– hence, NPV rule satisfies value additivity
Calculating A Project’s NPVAn Example
Quickie Enterprise’s Microprocessor Plant
Cash Flows in $ million
Year Cash Flow PVIF@12%
Present Value
0
-$400
1.0000
-$400.00
1
100
0.8929
89.29
2
110
0.7972
87.69
3
120
0.7118
85.41
4
130
0.6355
82.62
5
140
0.5674
79.44
NPV = $24.45
Internal Rate of Return (IRR) Rule
T
IRR solves
CFt
 I0  0

t
t 1 (1  IRR )
Accept the project if IRR > R, otherwise reject it

R is the opportunity cost of capital (or required return)
Among two mutually exclusive project choose the one
with the higher IRR
Isn’t it equivalent to the NPV criterion? For a decision
whether to accept or not a single project whose NPV is
monotonically decreasing with the discount rate – yes.
But in general – NO!
Problems with IRR:

Problems with ranking mutually exclusive
projects (can be: NPV(A) > NPV(B), but
IRR(A) < IRR(B))
scale effect
timing effect




Does not satisfy value additivity principle (can
be: IRR(A) > IRR(B), but IRR(A+C) <
IRR(B+C)
Multiple IRR when some CF are negative
NPV can be a positively sloping function of r.
Then IRR is nonsense.
Sometimes no IRR exists
Example of IRR problems
Year
Project 1 Project 2 Project 3 PV factor 1+3
@ 10%
2+3
0
-100
-100
-100
1.000
-200
-200
1
0
225
450
0.909
450
675
2
550
0
0
0.826
550
0
Project
NPV @ 10%
IRR
1
354.55
134.5%
2
104.55
125.0%
3
309.09
350.0%
1+3
663.64
212.9%
2+3
413.58
237.5%
• NPV(1) > NPV(3),
but IRR(1) < IRR(3)
• IRR(1) > IRR(2),
but IRR(1+3) < IRR(2+3)
Incremental IRR rule
When comparing two mutually exclusive projects A
and B one can use incremental IRR rule:
CF ( A)t  CF ( B )t
 ( I A0  I B 0 )  0

t
(1  IRR )
t 1
T
IRR(A-B) solves
Accept the project if IRR(A-B) > R, otherwise reject it

R is the opportunity cost of capital
This approach solves the problem of ranking mutually
exclusive projects if NPV(A-B)(r) is downward sloping.
Otherwise, usual IRR problems.
Is IRR ever helpful?
Useful to measure sensitivity of NPV to
estimation error in the cost of capital
A scale-independent measure of efficiency
(useful to compare businesses of different scale,
for comparison – no need to know the cost of
capital)
Aggregates the info about an investment into
one number
But anyway, if NPV is properly used then NPV is
the best:
maxm NPV(m), s.t. I  Ī, mM,
where M is the set of all possible combinations
of projects, Ī – resource constraint
Payback period rule
How long does it take for a project to recover
or “pay back” its initial investment?
PB
 CF  I
t 1
t
0
0
If recovery time < threshold – accept,
otherwise – reject.
DPB
CFt
Discounted Payback Period:
 (1  R)
t 1
t
 I0  0
Disadvantages of DPP:


Ignores the cash flows after the payback
period (what if they are negative?)
Arbitrary standard for setting the period
Advantages of DPP:



Simple
Measure of project liquidity
Measure (rough) of project risk
Average Accounting Return rule
AAR = Average Net Income / Average
Investment (book value)


Simple BUT ignores time value of money and is
based on accounting income rather than cash flow.
Moreover, what is the target rate?
Project selection with resource
constraints
Resource: capital, premises, people, time,
etc…
The straightest way is:
maxm NPV(m), s.t. I  Ī, mM,
where M is the set of all possible
combinations of projects, Ī – resource
constraint
But can be too complicated, hence…
Profitability Index
Profitability Index:
PI = PV of cash flows / Resource consumed
Variations:


PI = PV of cash flows subsequent to initial investment / I0
PI = NPV / Resource
Rule: rank the projects by the value of PI, then
select projects starting from the highest PI until the
resource is consumed
Doing so you will approximately maximize NPV
under the resource constraint if all projects are
independent
Example: PI with a human resource
constraint
Example (cont-d)
Shortcoming
You will never meet the resource
constraint precisely  can happen that the
selected combination does not maximize
NPV