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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?

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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?

  • 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, 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 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 =

Calculation of Present Values

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:

(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

Example of Future Value

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 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

Cash flow


Useful Present Value Formulas

  • Perpetuity:

  • Growing Perpetuity:

  • Annuity:

  • Growing Annuity:

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:

Cash Flow



Cash Flow



Graphical Representations

  • Annuity:

  • Growing Annuity:

Cash Flow




Cash Flow




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











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%

PV = $ 10.00

E = $ 3.23

D = 1.23

C = $ .54

$ 1

A =$ 2.99

B = $ 2.01



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 annuityobtaining (see. p. 840)

  • Relations between present value formulas are really simple

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

  • 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 (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

  • Login