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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)

- 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?

- 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)

- 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

- 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

- 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 =

- 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 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

- 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)

- 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

- 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.

- 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)

- 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 (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

- 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

- 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

time

- Perpetuity:
- Growing Perpetuity:
- Annuity:
- Growing Annuity:

- 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

- Perpetuity:
- Growing Perpetuity:

Cash Flow

0

Time

Cash Flow

0

Time

- Annuity:
- Growing Annuity:

Cash Flow

0

T

Time

Cash Flow

0

T

Time

- 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

E

D

C

Cash

Flow

A

B

T

time

0

- 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

PV = $ 10.00

E = $ 3.23

D = 1.23

C = $ .54

$ 1

A =$ 2.99

B = $ 2.01

0

5

- 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

- 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

- 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

- 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

- 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

- 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 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

- Objectives
- Vocabulary
- Problem Assignments
- Relation to syllabus and requirements

- 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

- 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

- 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