Present Value: Calculations and Interpretation

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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?
Present Value: Calculations and Interpretation

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Present value calculations and interpretationSlide 1

Present Value: Calculations and Interpretation

Classes 3 & 4:

March 5 and 7 (LA) and

March 1 and 6 (OCC)

From last classesSlide 2

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 6Slide 3

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 valueSlide 4

Determinants of Value

  • Cash, Time, Riskdetermine 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 ratethat 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 valuesSlide 5

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 flowsSlide 6

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 =

Calculation of present valuesSlide 7

Calculation of Present Values

Examples applicationsSlide 8

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 calculationsSlide 9

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 approachesSlide 10

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:

(Using Appendix Table A.1)

(Using Appendix Table A.2)

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

Characteristics of present valueSlide 11

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 dangersSlide 12

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 valuesSlide 13

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 value to annual cash flowsSlide 14

Equivalence of Present Valueto Annual Cash Flows

Example of future valueSlide 15

Example of Future Value

Summary of pv fv examplesSlide 16

Summary of PV/FV Examples

  • Present value is the amount that can replicatecash 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 valueSlide 17

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 valuesSlide 18

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 cutsSlide 19

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

Cash flow


Useful present value formulasSlide 20

Useful Present Value Formulas

  • Perpetuity:

  • Growing Perpetuity:

  • Annuity:

  • Growing Annuity:

Simple patterns in cash flowsSlide 21

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 representationsSlide 22

Graphical Representations

  • Perpetuity:

  • Growing Perpetuity:

Cash Flow



Cash Flow



Graphical representations1Slide 23

Graphical Representations

  • Annuity:

  • Growing Annuity:

Cash Flow




Cash Flow




Sources of present valuesSlide 24

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 formulasSlide 25

Graphical Presentation of Four Present Value Formulas











Graphical representation of the four important formulasSlide 26

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 20Slide 27

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

PV = $ 10.00

E = $ 3.23

D = 1.23

C = $ .54

$ 1

A =$ 2.99

B = $ 2.01



Present value and net pv npvSlide 28

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 formulasSlide 29

Use of Present Value Formulas

  • Familiarity with PV formulas important

  • For example, what is future value of constant annual cash flow? Using annuityobtaining (see. p. 840)

  • Relations between present value formulas are really simple

Using pv formulas to find ratesSlide 30

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 wealthSlide 31

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

Present value and mva eva iSlide 32

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 iiSlide 33

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 summarySlide 34

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 reviewSlide 35

Present Value Analysis: Review

  • Objectives

  • Vocabulary

  • Problem Assignments

  • Relation to syllabus and requirements

Basic steps to valuation in financeSlide 36

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 financeSlide 37

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 classesSlide 38

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

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