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University of Provence
Earle Traynham
Professor and Dean
College of Business Administration
University of North Florida
Jacksonville, Florida USA
February 2002
University of Provence
I.
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First Principles of Valuation
Future Value of Compounding
Present Value and Discounting
PV and FV of Multiple Cash Flows
Valuing Level Cash Flows: Annuities and
Perpetuities
University of Provence
II. Valuing Stocks and Bonds
• Bonds and Bond Valuation
• Common Stock Valuation
• Net Present Value and Other Investment
Criteria
• Opportunity Cost
University of Provence
III. Capital Budgeting Cash – The
Majestic Mulch and Compost
Company
IV. Acquisitions and Divestitures
• Future Value of Compounding
Investing in a single period
• FV = P(1+r) 1
Where P =
Principal invested, and r = the interest
rate on the investment
• What is value of $500 invested for 1
year at 10%
FV = $500(1+.10)1 = $500(1.1)1 =
$500(1.10) = $550
• Future Value and Compounding
– Investing for More Than One Period
FV = P(1+r)t
Where t = the
number of periods in the future
What is the FV of $500 invested for
2 years at 10%
FV = $500(1+.10)2 = $500(1.21)
= $605
• Present Value and Discounting
• PV = FV/(1+r)t, Where r is the
discount rate for t periods in the future
• PV for Single Period of Time
PV = $70,000/(1+0.12)1 = $62,500
PV for Multiple Periods of Time
PV = $100,000/(1+.075)8 =
$100,000/1.7835) = $56,070
• The Rule of 72
Approximate time (in years) required
to double an investment
FV/PV = 2.0, take 72/r
Example
How long will it take a $10,000
investment to reach $20,000 at 8%?
72/8 = 9 years (actual = 9.006 years)
The Present and Future Value of
Multiple Cash Flows
• Future Value – compound the accumulated
value period by period, or calculate FV of
each cash flow and sum them
• Example: assume you deposit $2000 today,
$1000 in one year, and $3000 in 2 years; all
@ 8%. Calculate FV
The Future Value of Multiple
Cash Flows
• Calculate FV of each cash flow and sum
• FV = ($2000)(1.08)3 + ($1000)(1.08)2 +
($3000)(1.08)1 = $2519.42 + $1166.40 +
$3240 = $6925.82
FV - compound the accumulated
value period by period
Time period 0: $2000
Time period 1: ($2000)(1.08) + $1000
= $3160
Time period 2: ($3160)(1.08) +$3000
= $6412.80
Time period 3: $6925.82
The Present Value of Multiple
Cash Flows
• Discount each amount to time period 0, and
sum them
• Discount back one period at a time,
summing as you go
• Example: What is present value of $1000
per year (at end of year) for 5 years, at 6%?
PV – Discount each amount to
time period 0, and sum
PV = ($1000)/(1.06)1 +
($1000)/(1.06)2 + ($1000)(1.06)3
+($1000)(1.06)4 + ($1000)(1.06)5 =
$943.40 + $890 + $839.62 + $792.09
+ $747.26 = $4212.37
PV – Discount back one period at at
time, summing as you go
End of year 5: $1000
End of year 4: ($1000)/(1.06) + ($1000) =
$1943.40
End of year 3 = ($1943.40)/(1.06) +
($1000) = $2833.40
End of year 2 = ($2833.40)/(1.06) +
($1000) = $3673.01
End of year 1 = ($3673.01)/(1.06) +
($1000) = $4465.11
Year 0 = ($4465.01)(1.06) = $4212.36
Example – you need $1200 one year
from now, $1500 after two years, and
$2000 after 3 years. How much will you
have to deposit today @ 8%
PV = ($1200)/(1.08)1 +
($1500)/(1.08)2 + ($2000)/(1.08)3
= $1111.11 + $1286.01 +
$1587.66 = $3984.78
Annuities – a series of constant cash
flows that occur at regular intervals for a
fixed number of periods
• Present Value of an annuity
1
1(1+r)t
APV = C x
r
Future Value of an annuity
(1 + r)t - 1
• AFV = C x
r
Present Value of a Perpetuity
• PV = C/r
• APV, as t goes to infinity =
C x (1-0) = C/r
r
Bonds and Bond Valuation
• Bond features
– Terms – refers to the number of years to
maturity
– Face value – the principle payment at maturity
date
– Coupon interest – specified rate of interest
based on face value.
Bond Values and Yield
• Market Value of Bond = PV of all coupon
payments plus principle repayment
discounted at opportunity cost for similar
bonds
• Example - $1000 bond, with 9% coupon
rate, with interest payable each October 1
and April 1, to be issued 4/1/2002 and a
maturity date of 4/1/2022
Bond Valuation
• If market rate of interest = 9%, then bond
market value = $1000
• If market rate of interest = 12%, then:
PV = $45 + $45 + $45 +…+$45 +1000
(1+.06) (1+.06)2 (1.06)3
(1.06)40 (1.06)40
This may be solved using the short-cut
equation for the present value of an
annuity
Bond – Discounted Yield to
Maturity
• The discounted rate of return (yield) is that
rate that equates the PV of the expected
bond cash flows to the current market price
(P0)
• In this case, you know the bond’s features
and you know its current market price. You
want to know its effective yield
Bond Yield to Maturity
• Example: A 9%, 10 year bond with face
value of $1000 currently sells for $920.
What is the effective yield?
• P0 = $920 = 90 + 90 + … + 90 + $1000
(1+r) (1+r)2
(1+r)10 (1+r)10
Solve for r, using the present value of an annuity equation
Bond Prices
• Bonds will sell at Face Value when the
market rate of interest for similar bonds is
equal to the coupon rate of interest on the
bond
• Bonds will sell at a Discount when similar
bonds have higher yields
• Bonds will sell at a Premium when similar
bonds have lower yields
The Interest Rate Risk of Bonds
• Bond prices vary inversely with market
interest rates. Bond price falls as market
interest rates rise, and vice-versa.
• Example:
12-year bond with 10%
coupon rate, purchased at face value of
$1000. Two years later, the market rate for
10 year bonds is 14%. What is market price
of bond?
Bond Price and Market Interest
10
P0 = ∑$100/(1+0.14)n + $1000/(1+.14)10
n=1
P0 = $100 x [1 – (1/1.14)10]
+ $1000
.14
(1.14)10
P0 = $545.27 + $236.63 = $781.90
Total Bond Value
• Total Bond Value = Annuity Present Value
of Coupons + Present Value of Face Value
• TBV = C x [1-1/(1+YTM)t]/YTM +
F x 1/(1+YTM)t , where YTM = yield to
maturity on bonds in this risk class
Common Stock Valuation
• Same principles as Present Value of Bonds,
with qualifications
• Uncertainty of future cash flows in the
forms of dividends and share price
• Difficulty in determining appropriate
discount rate
Zero Growth Case
PVperpetuity = D/r , where D is
constant dividend, and r is your
opportunity cost or discount rate
Constant Growth Case
P0 = D0(1+g) = D1
r-g
r-g
where g is the constant growth rate
and r is your opportunity cost or
discount rate
This equation works as long as r>g.
Non-Constant Growth Case
Where dividend growth rate changes
during the period of evaluation
Each growth period must be calculated
separately, i.e., becomes a series of
“Constant Growth Case” Calculations
Investment Criteria
• Net Present Value – the difference, in
present value, between amount invested and
the sum of the future cash flows resulting
from the investment
• Net Present Value Rule – an investment
opportunity is worthwhile (economically) if
the NPV is positive, at the required rate of
return
Investment Criteria - continued
• Payback Period – the length of time until the
accumulated investment cash flows (nondiscounted) equal the original investment, i.e.,
how long to get your money back
• Payback Period Rule – accept an investment if
it pays back original investment within
acceptable length of time
• Shortcomings – timing of cash flows is
ignored; cash flows after payback ignored; no
objective period for choosing cut-off period
Investment Criteria - continued
• The Average Accounting Return – average net
income attributed to an investment divided by the
average book value of the assets
• AAR Rule – an investment is acceptable if the
AAR exceeds a specified target level
• Deficiencies – ignores time value of money;
accounting income not necessarily related to cash
flow; accounting return may not be related to
market rates and is arbitrary
Opportunity Cost
• Required Rate of Return – the rate an
investor can earn elsewhere in the financial
markets on investments of similar risk
• The higher the risk the higher the required
return
• Two types of risk – systematic and nonsystematic
Opportunity Cost - continued
• Firms usually rely on both debt and equity
sources of funds
• RD refers to the cost (interest rate) of debt
• RE refers to the cost of equity – the rate of
return required by investors to let you use
their money
Opportunity Cost - continued
• Estimating the Cost of Capital – requires
assessing the cost of equity
• RE = Rf + ß(Rm – Rf) , where Rf = risk-free
rate; ß = measure of sytematic risk of
particular investment; Rm = average market
rate of interest
Opportunity Cost - continued
• Weighted Average Cost of Capital
WACC = REXE + RDXD(1-t) , where
XE and XD are respective proportions of
equity and debt; and t = tax rate on
corporation
Capital Budgeting – important
terms
• Incremental Cash Flows – only incremental cash
flows are relevant. Sunk costs are irrelevant to
the decision. Opportunity costs are any cash
flow that is lost or forgone, and represent an
incremental cash flow. Incremental Net working
capital represent an incremental cash flow.
• Net Working Capital (NWC) = Cash + Inventory
+ (Accounts Receivable – Accounts Payable)
• Financing Costs – are separate from the
investment decision and are not part of the
project’s cash flows
Majestic Mulch and Compost
Company
• Selling Price: $120/unit for first 3 years, then
$110/unit thereafter
• Starting NWC: $20,000 plus 15% of sales
• Variable Costs: $60/unit
• Fixed Costs: $25,000/year
• Capital Equipment Costs: $800,000 initial
• Depreciation Rate: MACRS; 7 year property
• Equipment Salvage Value: 20% or $160,000 in
year 8
• Do we do it?
Majestic Mulch and Compost
Company
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Net Present Value: $65,488
Internal Rate of Return: 17.24%
Payback: 4.08 years
Average Accounting Return: 11.0%
Acquisitions and Divestitures
• Far less than half of acquisitions create
value for the acquirers
• Successful acquisitions require excellence
in three areas – strategic fit, acquisition
economies, successful strategy for
integration
Premium Recapture
Characteristics
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Undermanagement
Synergy
Restructuring
Financing and Tax Considerations
Undervalued Assets
Divestitures
• Unprofitable
• Strategic Misfit
• Need the cash