University of Provence

Earle Traynham

Professor and Dean

College of Business Administration

University of North Florida

Jacksonville, Florida USAFebruary 2002

University of Provence

I. 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 CompoundingInvesting 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 TimePV = $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: $1000End of year 4: ($1000)/(1.06) + ($1000) =

$1943.40End of year 3 = ($1943.40)/(1.06) +

($1000) = $2833.40End of year 2 = ($2833.40)/(1.06) +

($1000) = $3673.01End 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 (non-discounted) 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 non-systematic

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

• 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

• Undermanagement

• Synergy

• Restructuring

• Financing and Tax Considerations

• Undervalued Assets

Divestitures

• Unprofitable

• Strategic Misfit

• Need the cash