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

**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?

**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

**4. **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

**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

**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 =

**7. **Calculation of Present Values

**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

**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

**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:

**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

**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.

**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)

**14. **Equivalence of Present Valueto Annual Cash Flows

**15. **Example of Future Value

**16. **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

**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

**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

**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

**20. **Useful Present Value Formulas Perpetuity:
Growing Perpetuity:
Annuity:
Growing Annuity:

**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

**22. **Graphical Representations Perpetuity:
Growing Perpetuity:

**23. **Graphical Representations Annuity:
Growing Annuity:

**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

**25. **Graphical Presentation of Four Present Value Formulas

**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

**27. **Example: $1 growing at 10% Discounted at 20%

**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

**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

**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

**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

**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

**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

**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

**35. **Present Value Analysis: Review Objectives
Vocabulary
Problem Assignments
Relation to syllabus and requirements

**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

**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

**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