J. K. Dietrich - GSBA 548 – MBA.PM Spring 2007 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

Present Value: Calculations and Interpretation

Classes 3 & 4:

March 5 and 7 (LA) and

March 1 and 6 (OCC)


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?


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 be addressed later: projecting cash flows, choosing a discount rate

(Class 3 & 4)

(Class 5 & 6)


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


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


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: PV of single cash flow =

PV of multiple cash flows =

tt

)r1(

CF

Tt

1tt

t

)r1(

CF


Calculation of Present Values

9091$.10.1

1$

r1

CPV 1

1

8264$.10.1

1$

)r1(

CPV

222

2

7513$.10.1

1$

)r1(

CPV

333

3

nn

n)r1(

CPV


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


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


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:

)6209.6830.7513.8264.9091(.000,1$791,3$PV

)7908.3(000,1$791,3$PV

106209.10000,1$1.

1

1.1

1

1.

1000,1$791,3$PV

5

(Using Appendix Table A.1)

(Using Appendix Table A.2)

(Using Perpetuity formula and Appendix Table A.1 discussed later)


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


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.


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)


Equivalence of Present Valueto Annual Cash Flows

Beginning of PeriodItem Yr 0 Yr 1 Yr 2 Yr 3 Yr 4 Yr 5

Previous Year 3791.00 3170.10 2487.11 1735.82 909.40Interest 379.10 317.01 248.71 173.58 90.94Accumulation 4170.10 3487.11 2735.82 1909.40 1000.34Cash Ouflow -1000.00 -1000.00 -1000.00 -1000.00 -1000.00BeginYear 3791.00 3170.10 2487.11 1735.82 909.40 0.34


Example of Future Value

Beginning of PeriodItem Yr 0 Yr 1 Yr 2 Yr 3 Yr 4 Yr 5

Previous Year 0.00 1000.00 2100.00 3310.00 4641.00Interest 0.00 100.00 210.00 331.00 464.10Accumulation 0.00 1100.00 2310.00 3641.00 5105.10Cash Inflow 1000.00 1000.00 1000.00 1000.00 1000.00BeginYear 0.00 1000.00 2100.00 3310.00 4641.00 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


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


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


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 --

First formula to memorize is

PVCr

time

Cash flow


Useful Present Value Formulas

Perpetuity:

Growing Perpetuity:

Annuity:

Growing Annuity:

PVCr

PVC

r g

PV Cr r r T

1 1

1( )

PV Cr g r g

xgr

T

1 1 11


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


Graphical Representations

Perpetuity:

Growing Perpetuity: Time

Cas

h Fl

ow

0

0Time

Cas

h Fl

ow


Graphical Representations

Annuity:

Growing Annuity:0

0

Cas

h Fl

ow

TimeT

Cas

h Fl

ow

TimeT


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


Graphical Presentation of Four Present Value Formulas

time0

A B

DCCash Flow

E

T


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


Example: $1 growing at 10% Discounted at 20%

50

A =$ 2.99 B = $ 2.01

D = 1.23$ 1

E = $ 3.23

C = $ .54

PV = $ 10.00


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


Use of Present Value Formulas Familiarity with PV formulas important For example, what is future value of constant

annual cash flow? Using annuity

obtaining (see. p. 840)

Relations between present value formulas are really simple

FV r rT [( ) ] /1 1

FV Cr r r

rT

T

1 1

11

( )( )


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


Present Value and Wealth Wealth = Present value of consumption Wealth = Present value of cash income Wealth = Change in value of consumption

= Change in present value of cash income Wealth => Increase in utility from

consumption Wealth = Net present value Net present value > 0 => Wealth increased


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


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


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


Present Value Analysis: Review

Objectives Vocabulary Problem Assignments Relation to syllabus and requirements


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


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


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 Present Value: Calculations and Interpretation Classes 3 & 4: March 5 and 7 (LA) and March 1 and 6 (OCC)

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