Present Value: Calculations and
Interpretation
Classes 3 & 4:
March 5 and 7 (LA) and
March 1 and 6 (OCC)
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
From last classes . . .
 What
should be the goal of financial
managers?
 What do we need to know to pursue goal?
 How can we assess progress towards that
goal?
 What is a firm’s market value? Market cap?
How do we compute them?
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Overview: Classes 3 to 6
 Discounted
present value: basic tool given
projections of cash flows and discount rate
–
–
–
–
–
Present value and wealth creation
One and multi-period cash flows
Patterns in cash flows = formulas
Applications to valuation: bonds
Application to valuation: stocks
 To
(Class 3 & 4)
(Class 5 & 6)
be addressed later: projecting cash flows,
choosing a discount rate
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Determinants of Value
 Cash,
Time, Risk determine value
 Present value analysis deals with the effect
of time or timing on value
 Cash flow estimation is the subject of the
next part of the course (classes 5 to 8)
 Risk is incorporated in the discount rate
that we discuss in Part 3 of the course
 In discussing present value analysis now,
we assume that cash flows and discount
rates are given
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Emphasis on Present Values
 Chapter
4 raises a number of topics relevant to the
calculation of present values:
–
–
–
–
–
Simple versus compound interest
Compounding interval
Continuous compounding
Future values
Calculation of number of periods of cash flows to
achieve a given present or future value
 We
will not emphasize these issues, we
concentrate on basic present value calculations
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Present Value of Cash Flows
 Calculation
of present values is key
technique to assign values
 Present value calculations are applications
or simplications of two basic formulas:
CFt
PV of single cash flow = (1  r ) t
PV of multiple cash flows =
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
tT

CFt
t  1 (1  r )
t
Calculation of Present Values
C1
$1
PV1 

 $.9091
1  r 1.10
C2
$1
PV2 

 $.8264
2
2
(1  r )
1.10
C3
$1
PV3 

 $.7513
3
3
(1  r )
1.10
Cn
PVn 
(1  r ) n
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Examples / Applications
 U.
S. Treasury strip prices are examples of
market determined discount factors for
default-risk free cash flows
 The structure of present value tables like
those in the text (A.1 and A.2) are very
straightforward
 Time in discounting in in terms of periods,
usually one year, but often shorter intervals
 Compounding interval will affect present
or future values
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Present Value Calculations
 Present
values can be calculated using
present value tables and paper, calculators
and paper, routines programmed into
calculators, and spreadsheets
 All correct methods produce the same
answers
 There is often more than one way to
calculate the answers using formulas or
individual cash flows but, if correct, they
are all mathematically equivalent
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Example of Three Approaches
 Present
value of $1000 received at the end of
each year for five years discounted at 10%
 Three (at least) ways produce same answer:
PV  $3,791  $1,000  (.9091  .8264  .7513  .6830  .6209)
(Using Appendix Table A.1)
PV  $3,791  $1,000  ( 3.7908)
(Using Appendix Table A.2)
1
1
1
PV  $3,791  $1,000   
   $1,000  10  .6209  10
5
 .1 1.1 .1 
(Using Perpetuity formula and Appendix Table A.1 discussed later)
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Characteristics of Present Value
 Present
value calculations are non-linear in
the discount rate and growth rates, means
changes in present values are not
proportional to changes in the discount rate
 Changes in timing or patterns of growth
must always be calculated, relying on
intuition is dangerous
 Terminology may be confusing: discount
rate, discount factor, interest rate, cost of
capital, opportunity cost, and yield all can
mean the same thing in a calculation
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Example of Dangers
 Change
discount rate in previous example to 20%
from 10%, PV becomes $2,991, reduced to 78.9%
of $3,791 at 10%, not half.
 Change times to $1,000 for ten years at 10%, PV
becomes $6,146, not double.
 Delay first cash flow by one year, PV reduced by
about 10%, or if by three years, PV reduced by
about 25%, difference between delay of one or
three years is not three times greater.
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Meaning of Present Value and
Equality of Present Values
 Present
Value of $1,000 for five years at 10
percent (Table A.2)
 $3,790.80 is equivalent to $1,000 at the end
of every year for five years at 10 percent
 Future value of $3,790.80 at end of five
years is $3,790.80x(1.10)5=$6,105.12
 This is also future value of $1,000 for five
years at 10 percent (see Table A.4)
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Equivalence of Present Value
to Annual Cash Flows
Item
Previous Year
Interest
Accumulation
Cash Ouflow
BeginYear
Beginning of Period
Yr 0
Yr 1
Yr 2
Yr 3
Yr 4
Yr 5
3791.00 3170.10 2487.11 1735.82
909.40
379.10
317.01 248.71
173.58
90.94
4170.10 3487.11 2735.82 1909.40 1000.34
-1000.00 -1000.00 -1000.00 -1000.00 -1000.00
3791.00 3170.10 2487.11 1735.82
909.40
0.34
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Example of Future Value
Item
Previous Year
Interest
Accumulation
Cash Inflow
BeginYear
Beginning of Period
Yr 0
Yr 1
Yr 2
Yr 3
0.00 1000.00 2100.00
0.00 100.00 210.00
0.00 1100.00 2310.00
1000.00 1000.00 1000.00
0.00 1000.00 2100.00 3310.00
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Yr 4
3310.00
331.00
3641.00
1000.00
4641.00
Yr 5
4641.00
464.10
5105.10
1000.00
6105.10
Summary of PV/FV Examples
 Present
value is the amount that can
replicate cash flows if discount rate is the
future interest rate
 Maximizing present values also maximizes
future values if interest rates do not change
(in this case, they are equivalent)
 Present values and future values of different
patterns of cash flows will differ from
calculations using constant discount rate if
interest-rates vary through time
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Net Present Value
 Net
present value (NPV) is the difference between
the present value of the future cash flows and the
cost of acquiring the cash flows
 In most examples, costs are immediate and are not
discounted, while cash flows are in the future and
must be discounted
 More generally, costs and benefits may both be
discounted if some costs occur in the future
 Net present value is a measure of how much more
something is worth than it costs, or a wealth
increase, as we discuss and illustrate later
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Positive Net Present Values
 A positive
net present value means that
future cash flows represent earnings higher
than the discount rate
 Net present value represents the excess
returns (returns above the discount or
opportunity rate) represented by the future
cash flows
 Net present values represent value added
relative to the opportunity rate
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Seek Simplifying Patterns in
Cash Flows for Short-cuts
 Can
always evaluate individual annual cash
flows but this is cumbersome
 Simplest pattern is constant cash flow each
year --

Cash flow
 First
formula to memorize is
C
PV 
r
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
time
Useful Present Value Formulas
 Perpetuity:
C
PV 
r
 Growing
C
Perpetuity: PV 
rg
 Annuity:
1

1
PV  C 
T
r
r (1  r ) 

T
 1
1  1  g 

x
 
 Growing Annuity: PV  C

 r  g r  g 1  r  
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Simple Patterns in Cash Flows
 Perpetuity
= Preferred dividend
 Growing perpetuity = Approximate cash
flows from new products or stock earnings
 Annuity = Retirement fund or car or
mortgage loan payments
 Growing annuity = Approximate cash flows
from investment with limited life or lifetime
earnings
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Graphical Representations
Cash Flow
 Perpetuity:
0
Perpetuity:
Time
Cash Flow
 Growing
0
Time
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Graphical Representations
Cash Flow
 Annuity:
0
T
Time
Cash Flow
 Growing Annuity:
0
T
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Time
Sources of Present Values
 Present
value of $1 perpetuity at 20% is $5
 Present value of $1 annuity for five years at 20%
is $2.99
 Therefore, present values of $1 from years six to
infinity at 20% is $5 minus $2.99 = $2.01 (less
than half of $5)
 Present value of perpetuity growing at 10%
starting at $1 and at 20% is $10
 Growing over infinite life is valued at $10 minus
$5 or $5
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Graphical Presentation of Four
Present Value Formulas
E
C
Cash
Flow
A
0
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
D
B
T
time
Graphical representation of the
four important formulas
 Areas
in graph represent parts of future cash
flows - Perpetuity = A+B
 Growing Perpetuity = A+B+C+D+E
 Annuity = A
 Growing Annuity = A+C
 You can solve for value added by a piece of
cash flows, for example cash flows after T,
by subtracting A from A+B
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Example: $1 growing at 10%
Discounted at 20%
PV = $ 10.00
E = $ 3.23
D = 1.23
C = $ .54
$1
A =$ 2.99
0
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
B = $ 2.01
5
Present Value and Net PV (NPV)
 Present
values are calculations assuming
expected cash flows and required discount
rates
 Each may differ for different analysts
– Knowledge and skill about future cash flows
– Assessment of risk and alternative investments
 Net
present value = Present value - cost
 Contrast present value with intrinsic value,
market value, under-valued and over-valued
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Use of Present Value Formulas
 Familiarity
with PV formulas important
 For example, what is future value of constant
annual cash flow? Using annuity
1
1

T
FV  C 
(
1

r
)
T
r
r
(
1

r
)


obtaining (see. p. 840)
FV  [(1  r ) T  1] / r
 Relations
between present value formulas are
really simple
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Using PV Formulas to Find Rates
 You
can solve for r given PV, in simplest
case of perpetuity r = C / PV
 With a value for g and PV in growth
formula, find r also easy and common in
stock analysis (we will use later)
 With annuities and other formulas you can
also solve for r although the equations are
non-linear requiring searches
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Present Value and Wealth
 Wealth
= Present value of consumption
 Wealth = Present value of cash income
 DWealth = Change in value of consumption
= Change in present value of cash income
 DWealth => Increase in utility from
consumption
 DWealth = Net present value
 Net present value > 0 => Wealth increased
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Present Value and MVA/EVA (I)
 Market
value added is how much more
assets are worth than they cost
 MVA is in part the present value of returns
above the opportunity rate on investments
thus represents management’s ability to find
investments better than alternatives
 EVA represents the returns above the
opportunity rate and is a measure of
management’s superior investment strategy
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Present Value and MVA/EVA (II)
 Market
values represent present value of
expected future cash flows
 If market value is above acquisition cost
(MVA), management is expect to produce
cash flows are above opportunity rate levels
 Excess returns (EVA) can be from existing
investments and future growth
opportunities or growth options
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Present Value Summary
 Present
values represent cash amounts that can
reproduce a pattern of cash flows in the future
given the discount rate
 Two equal present values can represent different
patterns of future cash flows
 Future values and present values are equivalent
measures of value given the discount rate
 Net present values are measures of the increase in
wealth representing increased utility from
increases in present and future consumption
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Present Value Analysis: Review
 Objectives
 Vocabulary
 Problem Assignments
 Relation
to syllabus and requirements
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Basic Steps to
Valuation in Finance
 Estimate
cash flows (CASH, TIME)
– Easy or hard depending on asset
– Look for patterns in cash flows
 Choose
a discount rate (TIME, RISK)
– Risk adjusted
– Opportunity cost
 Calculate
present value and net present
value
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
Valuation in Finance
 Applies
to all investment opportunities,
including
–
–
–
–
–
–
investments in fixed plant and equipment
starting a new business
selling a line of business (spin-off)
buying an existing business
values of bonds and stocks
real estate investments
 Used
by financial managers, stock and bond
analysts, real estate investors
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007
For Next Classes
 Read
Chapter 5, 14 and 20
 Do problems as assigned
 Download or call or write for annual report,
10K, and proxy statement, and any other
disclosures, for the group project firm
 Bring Value Line Investment Survey and
Standard and Poor’s reports for the
company to class
 Look for analysts’ reports and press
coverage of the group firm
J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007